journal of mathematics teacher education_4

GUEST EDITORIAL

PETER SULLIVAN

ICME AND TEACHER EDUCATION

One of the pressing challenges for mathematics educators is to identify thelanguage, concepts, principles, and practices that can be the shared basisof professional dialog. International meetings such as the 9th InternationalCongress on Mathematics Education (ICME 9) allow educators to considercommonalities and differences in ways that are perhaps not possible withinindividual national contexts. For example, mathematics educators haveexpressed considerable interest in teaching approaches seen as typicalin particular countries. ICME 9 provided opportunities for educators todiscuss, first hand, Japanese approaches to teaching mathematics. In asimilar way, ICME 9 allowed consideration of similarities and differencesin teacher education practices.

Of course, in seeking commonalities in professional knowledge,educators of mathematics teachers have a substantial challenge. The chal-lenge of teacher education is compounded in that virtually all of theissues of interest to teachers of mathematics are also relevant to teachereducators. At ICME 9 we saw active discussions on curriculum processes,mathematical topics, mathematical thinking and creativity, perspectiveson what constitutes knowledge, the role of technology in learning anddoing mathematics, social and political factors, equity, and assessment oflearning. These topics are all of interest to teacher educators and teachersalike.

There are, nevertheless, topics of particular interest to teachereducators. There were discussions on the tension between affirmingand challenging existing beliefs and knowledge of prospective teachers,methods for stimulating reflection and self learning, and practices forevaluating student professional learning and fitness to teach.

Discussions on pre-service education highlighted differences inapproaches. For example, there are countries where there is active politicalinterference in the evaluation of the quality of programs and of individualgraduates, others where there is active community interest in the content,

Journal of Mathematics Teacher Education4: 1–2, 2001.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

2 GUEST EDITORIAL

countries where pre-service education is well funded, and still others wherethe community has considerable trust that educators can prepare teachersappropriately. In all cases, though, there was a sense of the multidimen-sionality and complexity of pre-service teacher education, and the needfor ongoing renewal of the content and processes of teacher education –indeed of the educators themselves.

Even though practices in in-service teacher education are similarlydiverse, there seemed to be common assumptions across various countries.There seemed to be agreement that in-service professional developmentshould focus on education rather than on training, and extend over multiplesessions rather than be limited to a single session. Further, the focus andemphasis should be negotiated rather than imposed, with opportunities forparticipants to actively contribute rather than passively receive. One keydifference among countries seemed to be the extent to which there is asystematic requirement for teachers to participate in formal professionaldevelopment. Such requirements seem to stimulate more innovation in theapproaches to professional development programs.

A clear difference between those interested in elementary education(ages 4–11) and those interested in secondary education (ages 11–18) wasthe focus of the latter group on the content and processes of mathematics.Discussions included the study of old topics in new ways, the considerationof new topics, the role of technology, the nature of mathematical creativityand problem solving, and appropriate assessment mechanisms.

As with JMTE, the ICME meetings serve to facilitate understanding ofpractices and trends across cultural and national boundaries, and awarenessof innovations and alternatives.

Peter SullivanEditorial Board,JMTE

JEPPE SKOTT

THE EMERGING PRACTICES OF A NOVICE TEACHER: THEROLES OF HIS SCHOOL MATHEMATICS IMAGES

ABSTRACT. The relationship between teachers’ beliefs about mathematics and itsteaching and learning, and their classroom practices, has been investigated in manystudies. The results of these studies are by no means unanimous, and the purpose ofthe present report is to contribute to a further understanding of the significance of oneteacher’s beliefs and images of school mathematics on his teaching practices. The articlepresents a novice teacher whose images of school mathematics were strongly influencedby the current reform of mathematics education. The study focuses on how the teachercoped with the complexities of the classroom. On the basis of the teacher’s role withinthe classroom interactions, the construct ofcritical incidents of practiceis developed, andsome of the results of previous research in the field are re-examined.

The relationship between teachers’ beliefs about mathematics and itsteaching and learning, and the practices in their classrooms, has beeninvestigated in many studies. Although a teacher’s system of beliefs is notthe only relevant factor and “inconsistencies” between beliefs and practicedo appear, results from some studies suggest that there is an influencefrom the former to the latter (e.g. Carpenter & Fennema, 1991; Cooney,Shealy & Arvold, 1998; Ernest, 1991; Raymond, 1997; Schoenfeld, 1992;Thompson, 1992; Tobin & Imwold, 1993). Ernest (1991) considered ateacher’s personal philosophy of mathematics as the basic school mathe-matics concept and claimed that the philosophy of mathematics underpinsbeliefs about the teaching and learning of mathematics. In turn, thesebeliefs are mediated by the social contexts of education and transformedinto enacted models of mathematics teaching and learning, which thentranslate into specific uses of mathematical texts. Ernest claimed that alack of consistency between beliefs and practice is primarily due to insti-tutional and contextual constraints. Schoenfeld (1992) argued not onlythat beliefs influence practice, but that the relationship is one of directcausality. He stated that “the teacher’s sense of the mathematical enter-prise determines the nature of the classroom environment that the teachercreates. That environment, in turn, shapes students’ beliefs about the natureof mathematics” (p. 359).


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Other researchers, however, have argued that beliefs are the resultof practice rather than a main influence on it. Ruthven (1987), forinstance, noted that teachers need to broaden their perspective of whatability and quality in mathematical learning is in order to overcome theiroverly simplistic and stereotypical explanations of student failure. Thatperspective, he claimed, is probably more easily changed by changingpractice first, as the teachers’ understanding of mathematics teachingand learning is primarily tacit and based on their classroom experi-ences. Guskey(1986), in a more general educational setting, questionedthe extent to which a change in teachers’ beliefs may lead to changes inteaching practice. Guskey argued that increased student learning was themost important determinant of how the teachers evaluate their practice,and that therefore little change in teachers’ beliefs should be expectedbefore such an increase has been documented: “Improvement (positivechange) in learning outcomes of students generally precedes and may bea prerequisite to significant change in the beliefs and attitudes of mostteachers” (p. 7).

Like Ruthven (1987), Cobb, Wood, and Yackel (1990) found thatknowledge of formal models ofreformisteducational practices does notnecessarily inform teachers’ interactions with students, whereas the prac-tices may influence their beliefs about the respective roles of teacher andstudents in the classroom. Teachers’ reflections on practice, then, mayturn the classroom into a learning environment for teachers as well asfor students. As Cobb & Yackel (1996) later put it: “Our observationsconsistently indicate that teachers capitalize on the learning opportunitiesthat arise for them as they begin to listen to their students’ explanations”(p. 466). However, Cobb et al. (1990) did not disregard the role beliefs mayplay in relation to practice, and they consistently pointed to the dialect-ical relationship between the two. Thompson (1992), in a meta- study ofteachers’ beliefs, summarized previous results by claiming that teacher’sbeliefs about mathematics are enacted fairly consistently, but that the rela-tionship between the teachers’ beliefs about teaching and learning andtheir practices is less clear. She claimed further that although the teachers’beliefs are fairly robust, they might develop if the teachers are involvedin a cyclic process of changing the classroom and reflecting on thosechanges. Summing up, she questioned some of the underlying assump-tions of the previous research in the field: “Belief systems are dynamic,permeable mental structures, susceptible to change in light of experience.[. . .] The relationship between beliefs and practice is a dialectic, not asimple cause-and-effect relationship” (p. 140).

EMERGING PRACTICES OF A NOVICE TEACHER 5

Whereas Thompson (1992) and Cobb et al. (1990) pointed to a two-way relationship between beliefs and practice, others have argued that weshould not expect to find any such relationship. Hoyles (1992) focused onthe mismatch, and even on the incompatibility between teachers’ beliefsand classroom practices. Like Ernest, she pointed to the social practicesof schooling and to general and specific conditions of the classroom asframing factors of classroom interaction. She maintained further that ateacher’s beliefs also depend on the context in which they are expressed.Consequently – in contrast to Ernest (1991) – she argued that ratherthan talking aboutbeliefs and beliefs-in-actionor about espoused andenacted beliefs where the latter is seen as a watered down or contextu-ally constrained version of the former, we should do away with the ideaof decontextualised beliefs altogether and introduce a notion ofsituatedbeliefs. That is, situations are co-producers of beliefs, and as situationsdiffer, so do beliefs. Lerman (1994), in a somewhat similar fashion,reasoned that human activity is contextualised, and that Thompson’s(1992) description of a dialectic relationship between beliefs and practicedecontextualises beliefs and overlooks important qualitative differencesbetween the two. Although the membrane separating beliefs and prac-tice may not be impermeable, any similarity between the two was tobe considered one of family resemblance only, not one that suggestedessentially similar entities.

Bauersfeld (1988) not only did not deal explicitly with teachers’ beliefs,but even may have questioned their relevance as objects of research. Hisresearch project, developed during the 70s and 80s, seemed to reflect amuch more general trend in the theory of mathematics education. Hesuggested that an initial focus on mathematical structures and studentlearning be replaced by one on the role of the teacher and the generalcontext of learning, which in turn was substituted by an interest in theteacher-student relationship and the social interaction in the classroom.Elaborating on this last approach he claimed:

Both teacher and students contribute to the classroom processes. It is a jointly emerging“reality” rather than a systematic proceeding produced or caused by independent subjects’actions. . . . Teacher and students jointly constitute the reality of the classroom (pp. 29–30).

Bauersfeld’s point was that in order to understand classroom interactionsone has to perceive them as such and not focus on the alternating actions ofteacher and students as cause and effect, respectively, but on the evolvingpatterns and the “intersubjective constitution of norms for action” (p. 31).In this interpretation Bauersfeld called for interactionism as a programmefor classroom research, and he did not necessarily disregard the specialinfluence of the teacher. However, this call did question the relevance of

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the teacher’s beliefs as objects of research and at least implicitly questionedtheir practical significance.

It seems, then, that previous studies in the field have pointed to almostall possible relationships between mathematics teachers’ beliefs and theirclassroom practices. In part, this may be due to terminological and concep-tual problems with the very notion of beliefs. This notion may be describedin negative terms asnot requiring standardised canons of evidence,notrequiring common agreement, andnot requiring internal consistency.When it comes to positive definitions of the concept and to implied pre-understandings, however, there seems to be little consensus. First, beliefsare used to describe mental phenomena residing at very different levelsof consciousness, in some cases referring to personal priorities that are,at least potentially, conscious and may be explicitly stated, whereas inothers they are regarded as implicit by definition (Pehkonen & Törner,1996). Second, in some studies beliefs were seen as stable over time andacross contexts, whereas in a number of other studies the main point wasto question this stability. The stance taken on the issue of the temporaland contextual robustness of beliefs strongly influences the interpretationsmade, but rarely itself becomes an explicit focal point in belief research inthe sense of being empirically substantiated.

Though important, these different understandings of the notion ofbeliefs do not seem to account for all the differences between the studiesas far as the relationship between teachers’ school mathematical prioritiesand their classroom practices is concerned. In other terms, the role andthe significance attributed to teachers’ beliefs – whether explicit or notand whether stable or not – for the classroom atmosphere and its learningpotentials do differ in these studies. The aim of this paper is to contribute toa further understanding of these roles. I do not, however, investigate uncon-scious beliefs. Rather, I discuss the character and role of teachers’ explicitpriorities in relation to school mathematics, priorities that they described inquestionnaires and research interviews. Teachers’ subjectively importanttales of mathematics teaching and learning contributed as pieces of anendless puzzle to an image of the more or less precise, and more and lessstable, ideas about the teaching and learning of mathematics in schools.I use the termschool mathematics images(SMIs) to describe teachers’idiosyncratic priorities in relation to mathematics, mathematics as a schoolsubject and the teaching and learning of mathematics in schools. Theseschool mathematics images, then, are assumed to be expressions of uniquepersonal interpretations of and priorities in relation to mathematics, mathe-matics as a school subject, and the teaching and learning of mathematics inschools. The question I address in this article is the one of a possible rela-


tionship between the SMIs and the emerging practices in a novice teacher’sclassroom: I discuss the relationship between the teacher’s explicit prior-ities and the classroom interactions. In the process of doing so, I also dealwith the other non-consensual issue in relation to the notion of beliefs, thatis, the one of their temporal and contextual stability.

THE STUDY

In order to address the questions of whether and how novice teachers’SMIs are related to the ways in which they deal with the complexities oftheir mathematics classrooms, I discuss the case of Christopher, a 28-yearold novice teacher. Christopher was one of 115 Danish student teacherswho specialised in mathematics and who completed a questionnaire onschool mathematics approximately two months before their graduationin 1997. The questionnaire dealt with aspects of the teacher students’SMIs by asking open and closed questions about their conceptions ofmathematics as a discipline, about the specific role of mathematics in aneducational context including its relation to critical or democratic compet-ence, and about its teaching and learning. On the basis of the responses tothe questionnaire, 11 novice teachers, who represented a variety of SMIs,were selected for interviews. All but one of these interviews were carriedout immediately upon the students’ graduation, the last one having beenconducted slightly over a year later. Each of the interviews lasted between45 and 75 minutes and thematically focussed on the same issues as thequestionnaire. A semi-structured approach was used in which the inter-viewee was invited to describe his or her priorities in relation to schoolmathematics and if possible to share significant educational experiences.Also, the interviewees were asked to comment on three sets of short writtenmaterials: (a) a transcript of a Grade 5 student who proudly presented afalse conjecture about the area and perimeter of rectangles; (b) a quotefrom an experienced teacher who talked about her view of the differentroles of investigative activities for high and low ability students; and (c)five 12–17 line transcripts of imaginary interviews with teachers in whichvarious SMIs were illustrated.

Among the 11 interviewees, 4 teachers were asked to be part of acontinued study that included classroom observations and further inter-views at some time within the first 18 months after their graduation.These four teachers were selected because they showed a strong senseof commitment, and though they had different emphases in their schoolmathematical priorities, they all presented SMIs inspired by current reformefforts in school mathematics (e.g. NCTM, 1989, 1998). For example,

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they all depicted learning as active knowledge construction supported anddirected by social interaction, and they saw teaching in terms of facilit-ating learning rather than merely as explicating mathematical concepts andprocedures.

I followed and videotaped each of the four teachers in their mathe-matics classes for 2 to 3 weeks. Immediately after each lesson we had aninformal discussion in which the teacher described his/her general impres-sions of how the lesson had worked and responded to a few questions orcomments based on my field notes. Later, a more comprehensive interviewwas conducted, in which the teacher commented on a number of clipsfrom the video recordings. Looking for both congruence and conflict, Ianalysed the videotapes and the transcripts from the perspective of theteacher’s SMIs. This means that I used what appeared to be the valuejudgements and educational priorities inherent in those SMIs as an inter-pretative device: How, for instance, do the classroom interactions look,when viewed from the perspective of the teacher’s proclaimed view of thestudents’ mathematical learning? The clips used in the final interview wereselected as crucial and exemplary incidents of the teacher’s interaction withthe students when viewed from this perspective, especially with regard tohow the teacher organised and orchestrated the classroom communicationand conceived his or her own role in it.

CHRISTOPHER’S SCHOOL MATHEMATICS IMAGES ANDCLASSROOM

School Mathematics Images

Christopher’s school mathematics images – as presented in the question-naire and in the first interview – showed a strong resemblance betweenhis school mathematical priorities andthe reform. He did not disregardthe mathematical contents in a more traditional sense of the term, buthe consistently favoured process aspects of mathematics, and he arguedthat students should experiment and investigate, hereby assuming moreresponsibility for their own learning. Describing his view of schoolmathematics he proclaimed:

I hope that I can make the students realise that it [mathematics] is a way to view to world, toapproach the world –. If we have a problem or a project, what kinds of investigations maybe included to shed light on it? Can we use mathematics or can’t we use mathematics?Then [in the latter case] we shall have to use something else. (Interview 1, immediatelyafter graduation)


Further, Christopher saw his own role in terms of initiating and supportinginvestigative activities and inspiring the independent work of the students,individually and in small groups. He specifically referred to experiencesfrom his own upper secondary education, when his teacher had spent mostof the time explicating mathematics:

One of the things I shall do much less, is what the teacher did in thegymnasium[Grades10–12]. He just stood there lecturing a lot. He was bright and competent and a nice guy, themathematics teacher,. . . but I don’t want to lecture so much at the board. I wantthem[thestudents] to do the work, to getthemstarted, to maketheminvestigate. Yes, that is what Iwant (Interview 1).

One of the characteristics of Christopher’s response to the ques-tionnaire was that he emphasised aspects of education that were notspecifically mathematical. Describing the competencies of good studentsof mathematics Christopher said that they know how to:

systematise, plan, delimit open problems, reflect on their own learning, has the ability to co-operate, independently find solution strategies and models, relate critically [to informationand problems] (Questionnaire).

Asked to describe what he considered to be the main goals of schoolmathematics, Christopher pointed to the students’ development of exactlythese seven competencies. In the first interview immediately after hisgraduation Christopher showed a similar interest in broad educationalaims. In general, Christopher presented a coherent set of SMIs that wasclearly influenced by the reform, but that was also based on broader,educational, non-mathematical priorities.

The Classroom

Upon graduation Christopher got a job as a teacher in a Danishfolkeskole,that is, a municipal school for grades 1 through 10. The school, Gron-negardsskolen, is located in a small municipality in the outskirts of Copen-hagen. Socially the neighbourhood is neither particularly well to do, nor hitby social problems to any great extent. The school itself may – like manyother Danish primary and lower secondary schools – be characterised bya lack of explicit educational profile. Traditionally Danish teachers havebeen allowed a considerable degree of freedom with regard to their choiceof teaching methods. Combined with loose descriptions of the mathem-atical contents in the national curriculum for these grades and with aweak tradition for collaboration among teachers in each subject, individualteachers are given and left with significant influence and responsibility intheir classrooms. This is also the case at Gronnegardsskolen.

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Christopher primarily taught mathematics and music, the subjects hehad specialised in at college. One of his classes was a Grade 6 in mathe-matics, and between them, he and the students had developed a nicerelationship of mutual acceptance and respect. The previous teacher hadbeen, according to Christopher, a very traditional teacher who had usedan old textbook from the 1970s. Christopher had the opportunity to intro-duce a new textbook, and he had spent the first months of his teachingcareer trying to make the students accept the approach of the new book, anapproach that he found to be consistent with his own priorities. He pointedout that the textbook made the educational decisions and the mathemat-ical priorities of his day-to-day teaching as far as the aims, the contentsand the tasks were concerned. He had neither the time nor the energy tomake these decisions, given his many other obligations as a novice teacher.Initially, Christopher found his lesson preparations overwhelming, espe-cially the amount of homework he felt obliged to mark. Later he becameless concerned with grading. He prepared his lessons mainly by goingover the textbook tasks, dealing with practical and organisational issues(for instance, finding the supplementary materials recommended by thetextbook and making sure that he knew how to use them in the specificcontext), and deciding if he would have to go over some of the contents atthe board, or if the students could read it on their own.

Although Christopher did not make curricular decisions with respect tothe mathematical content and tasks, he strongly influenced the classroomin other ways. He avoided lecturing from the board; he used short whole-class sessions only to initiate the work of individuals and groups ofstudents. When working with small groups or individuals he asked studentsto elaborate the often vague and imprecise questions they had, and heencouraged them to come up with preliminary answers and to engage infurther explorations.

Generally Christopher’s class was fairly noisy, during individual orsmall-group work as well as during the rare whole-class sessions. Thenoise did not appear to be the result of significant disciplinary problems.Rather, it was due partly to the level of physical activity and commu-nication among the students when they solved the tasks and partly toChristopher’s priority that the students should arrive at their own modusvivendi. When asked about the noise level Christopher referred to hisintention of “allowing children to be children, also in school.” Ratherthan regulating the students’ behaviour, he spent most of the time in classrushing around from one group of students to the next helping them withwhatever mathematical problems they had.


AN INSTRUCTIONAL SEQUENCE ON AREA

During my first visit to Christopher’s classroom we agreed that I wouldfollow his teaching of area, the next topic in the textbook. His generalapproach to mathematics teaching could be exemplified by the way heintroduced this topic. He initiated a brief class discussion (7–8 minutes)and asked the students if they could explain what area meant. One of thestudents came up with the suggestion that area was “what is inside the rect-angle.” Accepting the answer Christopher showed a transparency with unitsquares and asked if anybody remembered how they had previously usedunit squares. A brief discussion followed during which everybody agreedthat the transparency was a good tool if one wanted to find the size of smallfigures, but that it was not very appropriate if one wanted to find the size ofa new carpet. Next Christopher asked a student to read the introduction tothe chapter to the class. When the student had finished, a few difficult, butnon-mathematical terms were explained. Christopher identified 21 pagesfrom the students’ textbook and workbook that the students were to finishwithin the next two weeks as they worked on the problems in pairs. Thestudents complained loudly about the amount of work, but gradually theyfound their partners and got started.

The textbook and the workbook included stories of walls to paint andlawns to mow; the tasks included cutting parallelograms into pieces tomake rectangles and cutting pairs of congruent triangles to make paral-lelograms. Further, the students were to measure and find the area of theclassroom, the desk, the blackboard, and other items of their own choice.

In the second lesson on area two boys, Rune and Christian, called forhelp as they were working on a task on how to find the area of a polygonallawn. When Christopher joined them, the following exchange took place:

1 Rune: We are working on this one. It is pretty difficult.2 Christopher: How can you solve that?3 Christian: I thought, that maybe you could put this one up here –

[pointing to a small rectangular piece that he wants to move to theother side apparently to make a large rectangle]

4 Christopher: Yes [puzzled but supportive].5 Christian: Then you only have to measure [is interrupted]6 Christopher: I haven’t measured that actually, but I could easily –

[reaches out for Christian’s ruler, but puts it back down]. How canyou do that?

7 Christian: That is 4.5 and that is 5 [points to the relevant lengths on thedrawing].

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8 Christopher: This is 4.5? [Picks up the ruler anyway and checks theresults].

9 Christian: Yes.10 Christopher: Yes. And this one is – [measures the length of the other

side].11 Christian: 5. But it is the easiest.12 Christopher: OK. Couldn’t you do something else?13 Rune: You could cut it into pieces –

In the exchange that followed Christopher vehemently and successfullytried to involve Christian in the solution strategy that Rune had suggested, astrategy the two boys had already discussed before Christopher had joinedthem. However, Christopher never returned to Christian’s initial suggestionabout moving one part of the original figure in order to obtain a largerectangle. Christopher’s reaction to the situation when shown on the videowas as follows:

14 JS: How do you interpret your reaction to this? What happens here?15 Christopher: Well, the piece doesn’t fit. It is like – I know that Rune

is very good in maths, and that Christian is very weak. I don’t wantto reject some line of thought from Christian, because I think hehas developed somewhat. At least he has gained more confidence; heplunges into things, which I don’t think he did in the beginning. But,hmm, I am taking over at a certain point, when I am going to measurethe lengths, but then I say “How would you do it Christian?”

16 JS: Yes, you pick up his ruler and then put it back down again.17 Christopher: Well, it is like, sometimes you want to push things a bit,

and – buthe is the one who has to do it,he is the one who has to learn,not me. And it is quite funny, when I leave them I don’t even knowif they have found the right way to do it, or I’ve probably sensed thatthey have found a method that works, but I don’t say “Yes, that’s fine,”I go just off to somebody else who has called me.

18 Christopher: When I put it [the ruler] down I think: “Oh no, he is theone to do it, not me. He is the one to explain his suggestion.”

19 JS: Do you often find yourself in such a situation thinking “I wish Icould – but no I shouldn’t.”

20 Christopher: Yes, I think I do. For instance today. . . [goes on todescribe a situation from another lesson].

In the next lesson two other boys, Martin and Kaspar, tried to find the areaof rectangular lawns drawn with different scales. In the task in questionthe picture was 8 by 212 centimetres and the scale was 1:200. Martin andKaspar found it difficult to solve the task, and Christopher intervened:


21 Christopher: 1 centimetre is 2 metres: And there are 8 centimetres –You still don’t follow?

22 Kaspar: No.23 Christopher: We’ll start up here again – [points to the task above]

one centimetre equals 100 centimetres in the real world – [waits fora reaction from the students].

24 Martin: 1 centimetre is 1 metre in the real world.25 Christopher: Yes, you can put it like that, too. Why is that?26 Martin: Well, [inaudible] that was just the way we calculated it.27 Christopher: But that was because 1 metre is the same as 100 centi-

metres. Every time you have 1 centimetre then in reality it is 100centimetres. And 100 centimetres that is the same as –

28 Both students: 1 metre.29 Christopher:1 metre. That is why we say:1 centimetre on the drawing

is the same as1 metre in the real world. Now you look at the onebelow.

30 Kaspar: That one, then, is 2 metres.31 Christopher: Yes. That means that 1 centimetre is the same as 200

centimetres in the real world or 1 centimetre is the same as –32 Kaspar: 2 metres33 Christopher:2 metres. So that means – and how many centimetres

were there?34 Martin: 8 centimetres.35 Christopher: 8 centimetres. And how many metres is that in the real

world – [waits a second for an answer] We can just start here [pointsto the drawing in Martin’s book and moves his finger one centimetreat a time]. How many metres is this?

36 Martin: 2.37 Christopher: And this?38 Martin: 439 Christopher: Here?40 Martin: 641 Christopher: Here?42 Martin: 8 [Christopher speeds up the process and moves his finger one

centimetre at a time while Martin answers] 10, 12 –, eh 14, 16.43 Kaspar: That is just the two-times table.44 Christopher: So how many metres are there? When there are 8

centimetres, how many metres is that?45 Kaspar: 16.46 Christopher: 16. 212 centimetres, how many metres is that?47 Martin: That is then 4 –, 5 centimetres. 5 metres.

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48 Christopher: 5 metres. Do you follow? [to Kaspar]49 Kaspar: Yes.50 Christopher: So, now we have 16 and 5. What are we going to do with

those two numbers?51 Kaspar: Add them.52 Martin: Multiply.53 Christopher: Multiply.54 Kaspar: Oh well, yes.55 Martin: 16 by 5 [using the pocket calculator]. That is 80.56 Christopher: Now I want you to write it all down. . . [Christopher

continues by telling exactly what to write].

After Christopher had watched the video I asked him about his reaction tothe situation:

57 JS: There are two boys, Martin and Kaspar, who are doing the taskwith the lawns drawn in different scales. They obviously don’t find itvery easy. What is your reaction to a situation like this?

58 Christopher: I get excited, I can see that. I start talking quickly, butin fact what happens is what I want to happen –. Although I talk alot here, I find out that Kaspar doesn’t understand it at first. . . but hecomes to understand it. It takes some time, but they understand, when Ileave them. . . also that the lawns are drawn in different scales. I thinkI can tell that they know the difference between when it says 1:100,and when it says 1:200. Well, yes that there is a difference and whatcomes out of it. I think it was good.

59 JS: OK. I don’t know your students too well, but when I see this, I amin doubt how much Martin understands – [goes over the situation indetail]. He seems to be mystified, even when you count the centimetreswith him . . .

60 Christopher: OK, I show on the line, and when we have moved 1 centi-metre on the lawn, we have in reality gone 2 metres. . . But Martin,I think that somehow he is OK –, if we say that mathematically he islike this [holds out his arm horizontally in the air], he thinks that heis like this himself [lowers the arm], while I maybe put him up here[raises the arm higher than to the original position].

61 JS: So he underestimates himself, and you may overestimate him?62 Christopher: But I’d rather overestimate somebody, and then maybe

they’ll begin believing in themselves, pull themselves up by thebootstraps. But at least in this situation I think that he gets someunderstanding, but maybe that is because I expect him to. . . I don’tknow.

63 [. . .]


64 Christopher: Well, it may be that I say what they are to write in thebook, but they know that they have found out how many metres thesides are. And they also know that they have to multiply the two. So Ithink they have understood.

65 JS: But maybe they have – you play a very different role here than inthe other bits we have seen, don’t you?

66 Christopher: Yes, I say more [. . .] Very often when I help Kaspar, hegets stuck. It is not only in this situation, he keeps coming and askingquestions. And then I have to. . . even when I try to get him goingon his own and make them help each other it is the role I have to playsometimes. Even when I try to let go. I don’t know, but there is a wholespectre of different roles that you have to play.

In the rest of the article I discuss these two episodes in some detail. Istart by interpreting them (a) from the perspective of Christopher’s SMIsand (b) from broader aspects of his educational priorities. I then charac-terise the two episodes by focussing on their common features and claimthat they are both examples of what I termcritical incidents of practice.I discuss the short-term and expected long-term relationship between theSMIs and the teacher’s contribution to the classroom interaction. In theconcluding paragraph I relate this study to some of the previous findingsin the field as outlined in the introduction.

INTERPRETATION OF THE EPISODES

Christopher played very different roles in the two episodes. In the firstepisode a student, Christian, came up with a suggestion that Christopherhad not considered: to move a small rectangular piece in order to transformthe lawn into one large rectangle [paragraphs 3–4]. Christopher was aboutto check the suggestion by measuring the relevant sides with Christian’sruler. But he decided against it, leaving it to Christian to elaborate on hissuggestion [6]. Christian already knew that the sides were not the samelength, but he still thought that his idea would somehow work, possiblybecause he was mentally caught up in a previous activity of cutting updifferent figures. So far, Christopher hadsaid “yes” to the suggestion, butthen began toreact “no” [12]: He asked for other solutions. When Rune,the other boy, came up with a second idea, Christopher never returned toChristian’s suggestion. He left the two students at an early stage, when hebelieved that they were able to continue on their own.

The episode can be viewed as one that strongly reflects Christopher’sSMIs. He attempted to initiate and support the students’ learning by letting

16 JEPPE SKOTT

them come up with suggestions and by refraining from turning thosesuggestions down, if they proved to be wrong or insufficient. Christopheremphasised that Christian was the one to carry out the physical and mentalactivities related to the task, and he used a terminology very similar to theone used in the interview six months earlier [17; compare to Christopher’ssecond quote in the sectionSchool Mathematics Images].

In the second episode the students had problems with a task thatcontained a drawing in a scale of 1:200. Christopher referred to a previoustask, in which the drawing was in a scale of 1:100, a task they had solvedsuccessfully before [compare paragraphs 23–28]. Still, the students neededmore support and, over the next couple of minutes, Christopher narroweddown his questions to the extent that the task was depleted of any math-ematical challenge and was reduced to carrying out a simple prescribedmultiplication with a pocket calculator [50–56].

This second episode may be seen as one in which Christopher’steaching degenerated to formally fulfilling his part of the didacticalcontract (Brousseau, 1984), and in which he deserted or compromised hismathematical priorities in favour of others with a stronger product orient-ation. In this interpretation – which I will question shortly – Christopherseemed to replace his general reformist school mathematical priorities withothers that emphasised traditional values in mathematics education to amuch greater extent.

The two episodes, then, invite an interpretation of Christopher asinvolved in mutually conflicting modes of interaction with the students.However, this description of Christopher as inconsistent and using oscil-lating practices may be questioned. When confronted with the two epis-odes he was – maybe surprisingly, as he was often very critical of his ownteaching – quite satisfied with both of them. There were, though, threeimportant aspects of his approach to teaching that became apparent inthe episodes and in the interview that followed, aspects that were not inan immediate way related tomathematicsteaching and learning but thatnonetheless shed light on his reaction to the video-clips.

First, Christopher’s evaluation was not exclusively related to thestudents’ mathematical learning. In the questionnaire, in the first interview,and throughout our discussions about his teaching Christopher consistentlypointed to the need to think of the children in broader terms than thoserelated to mathematics. In general, and in relation to individual children,he showed a lot of sensitivity to many other aspects of his students’ lives,in particular to their development of self-confidence, both mathematic-ally and otherwise. In Christopher’s comments on the two episodes herepeatedly referred to these general educational priorities.


Second, Christopher’s final comments on the second episode indicatethat the classroom atmosphere compelled him to use fairly direct instruc-tional methods when he dealt with Kaspar. This points to a conflict inherentin his approach. Christopher encouraged and expected his students to inter-pret and solve the tasks on their own and to discuss solution strategies andresults among themselves. As part of that goal he provided them with verylittle support before they started working, and consequently, they were inalmost constant need of assistance during the group work. It put a lot ofpressure on Christopher because often many students required his assist-ance at the same time. From this point of view, Kaspar was one of the mostdemanding students in the class.

Third, Christopher’s constructivist views of learning should not,according to himself, determine his own role in his interaction with thestudents. In other words, he did not consider the link between the epistem-ological notion of constructivism and the more practical notion of teachingmethods as one of cause and effect. On the contrary, in his comments onthe second episode he explicitly pointed to the multiplicity of differentroles that he had to play.

Given these three points, Christopher’s positive evaluation of both epis-odes is less surprising. His interactions with the students were shaped orat least influenced by his impression that the mathematical and generalself-esteem of the students, one in each group, were generally weak andvulnerable. Consequently, in the first episode he avoided explicitly turningChristian’s suggestion down. In the second episode he made sure that thestudents came up with the correct answer, because he thought they neededjust that in order to boost or – less ambitiously, to avoid yet another blowto – their self-confidence. Also, in the second episode he was concernedwith the general management of the classroom: He made sure that Martinand Kaspar came up with a correct answer to the task and would need nofurther assistance for a while, because he needed to reduce the pressure onhim as many students simultaneously wanted his attention. Finally, he didnot consider the interaction in the last episode incompatible with his visionof teaching: It was clearly within the spectre of different roles he thoughthe had to play (cf. paragraph 66).

Christopher’s comments not only serve to make sense of his positiveevaluation of the episodes; they also point to the need to view classroominteractions from a broader perspective than that of school mathematics.The second episode should not be seen merely as one in which Chris-topher’s practice degenerated into a funnelling approach that deprived thestudents of the opportunity to learn. What appear to be oscillating prac-tices, then, should not be seen as a result of teacher inconsistencies. Rather,

18 JEPPE SKOTT

it should be seen as an attempt to forestall Kaspar’s next call for help whenmany other students also needed assistance, and as a situation in whichChristopher’s reformist intentions were embedded in or submerged by a setof different and at least equally legitimate educational priorities related tobuilding the students’ confidence. My point here is not that Christopher’sreaction in the last episode was the only possible one or the most sensibleconsidering his own mathematical and other educational aims. Nor is it thepoint that there is no way of reconciling his school mathematical prioritieswith the aim of building student confidence. Rather, this interpretationsuggests that there are multiple and sometimes conflicting educationalpriorities, and that in this case the priorities related to Christopher’s SMIswere dominated by others concerned with managing the classroom andwith broader educational issues. Consequently, the former lost some oftheir practical significance.

CRITICAL INCIDENTS OF PRACTICE

A common feature of the two episodes was that the significance of thepriorities reflected in the SMIs for Christopher’s interaction with thestudents was challenged by the emergence of motives unrelated to studentlearning. As a response to these multiple motives, viable alternatives to hisdominant teaching strategy evolved, alternatives that were not immediatelyrelated to his SMIs.

The two episodes share two further characteristics. First, the challengeto Christopher’s SMIs encompasses an implicit evaluation of their poten-tial to address all aspects of the classroom interaction or, in extreme cases,even of their potential to address the complexities of the classroom alto-gether. The two episodes are different examples of that challenge. Ashis comments about the first episode show, Christopher felt successfulabout his attempt to strike a balance between facilitating the students’mathematical learning and supporting their general self-confidence. Thefirst episode, for him, exemplified the compatibility of his SMIs andclassroom interaction, and most probably substantiated his SMIs. In thelatter episode, Christopher played a different role, and his satisfactionstemmed primarily from his impression that he managed to support thestudents’ self-confidence while at the same time managing the classroomin an acceptable manner. This second episode, then, not only challengesthe enactment of Christopher’s present SMIs in the sense that the objectsand motives of his activity – building student confidence and managing theclassroom – required him to downplay the role of his SMIs; the episodealso suggests a criticism of his SMIs. In particular, the parts of his SMIs


most closely related to his own role in the classroom were challenged inthe sense that their applicability seemed to be limited to co-operation withstudents who did not need the same amount and types of support as Kasparand Martin. The episode may, then, have contributed to the developmentof a more critical stance towards the SMIs. In summary, the two episodescarried an implicit criticism – one primarily positive the other primarilynegative – of the SMIs and their suitability as means to understand anddeal with classroom interactions.

Second, the two episodes are both instances of teacher decision makingin which the further development of the classroom interaction was stronglyinfluenced by the decision made. For instance, the episode with Christianand Rune would probably have evolved very differently and led to verydifferent learning experiences, if Christopher had asked the two studentsto cut out the polygonal lawn and cut off the small rectangle, as Christianhad proposed, in order to discuss when the procedure would work. Or itmight have evolved differently had he used Christian’s ruler to measure therelevant sides and told the students what they were to do with these lengthsin order to find the area. Similarly, in the second episode, Christopher couldhave contributed to a different type of learning experience about scales andarea and to a different conception of what counts as mathematics altogetherhad he chosen to ask the students to put the books aside for a while andto make two different drawings of the classroom. Christopher’s decisionmaking, then, was critical in the sense that the learning potential inherentin the episodes significantly depended on the outcome of that decision.

It follows that both episodes encompass instances of Christopher’sdecision making

• in which multiple and possibly conflicting motives of his activityevolved;

• that were critical to his SMIs; and• that were critical to the further development of the classroom interac-

tion and for the students’ learning opportunities.

I term such an instance of teacher decision making acritical incident ofpractice[CIP], and I use CIPs as an analytical focal point for two reasons.First, they provide a window on the role of teachers’ school mathematicalpriorities when these are challenged as informants of teaching practiceby the emergence of multiple motives of their activities. Second, CIPsmay prove significant for the long-term development of a teacher’s schoolmathematical priorities.

20 JEPPE SKOTT

The Role of the SMIs in the Short Term

I have argued that Christopher’s contribution to the interaction in the firstepisode strongly resembled his SMIs. Further, this episode indicates a rela-tionship between his SMIs and his reflection on the classroom interaction.Christopher – when he picked Christian’s ruler and put it back down,as he realised that he was about to take over [paragraph 6] – evaluatedhis own activity on the basis of his SMIs. In other words, he instantan-eously reflected on his own practice on the basis of his school mathematicsimages. Those images shaped and filtered the objects of his reflection, thatis, what he reflectedon, and simultaneously framed his interpretation ofwhat he saw. In one sense this observation is just a way to rephrase andexemplify the basic constructivist tenet that the individual’s sense makingis necessarily built on one’s preconceptions of the situation at hand. Butit is also a way of saying that the school mathematics priorities relatedto the SMIs were an important part of Christopher’s preconceptions as heengaged in a practice of mathematics teaching. The termreflection, inter-preted literally, i.e. in the sense of casting back or mirroring, means thatthe SMIs are what he reflected in: they were the mirrors of his reflectivepractice. In this sense the SMIs were important for the way the interactiondeveloped.

Christopher’s interaction with Kaspar and Martin, on the other hand,seemed to provide a counterexample to the claim that there was a positivecorrelation between his SMIs and the classroom practices. My pointabove, however, was that it was mainly because of two different typesof mediating factors, each of which introduced competing motives ofChristopher’s activity, that the last episode developed the way it did. Oneof these factors was his own general educational priorities that in thissituation were at odds with his SMIs; the other factor was the way Chris-topher’s organisation of the classroom required him to rush around fromone group of students to the next in order to assist them in solving thetasks.

The last of these issues, the dominance of his organisational approach,deserves further attention. With regard to classroom organisation, Chris-topher’s main concern was to avoid standing at the board. He wasconvinced that lecturing from the board deprived him of the opportunity tosupport the individual students’ learning and deprived the students of thechance to engage in types of activity that allowed them to construct theirmathematical knowledge and skills. His orientation reflects an individual-istic tendency in Christopher’s conception of mathematical learning. First,in accordance with basic constructivist tenets, he argued that the studentshave to construct their own concepts and skills. Second, he combined this


understanding of learning with the view that the individual student had tobe the basic organisational unit of the classroom. It is indeed a view of theindividual student pitted against reality. This emphasis on the individualbecame the dominant aspect of Christopher’s planning, and despite theapparently social elements in his classroom – the small group interactionand his own communication with the students – he consistently focusedon the individual learner. When asked, for example, why he sometimespreferred group work to individual student’s activity he gave only oneanswer, even though he was prompted for other possibilities: communic-ation supports student learning, because you explain things to yourselfwhile you explain it to others. When I first visited Christopher’s classroom,his rejection of class discussion even encompassed situations in whichstudents presented their work to each other. He only later decided to tryout such a strategy.

From the students’ point of view, Christopher’s approach turned theclassroom into an often seen combination of epistemological construct-ivism and ontological absolutism: Mathematics was a set of well definedrules and routines inherent in the textbook tasks, and the mathematicalprocesses of the classroom were exclusively a matter of gaining accessto this world of mathematical results. Christopher generally recognisedthe importance of student command over standard results. There wereobviously concepts and procedures that students were expected to master.However, the dominance of his organisational approach appeared incom-patible with other aspects of his SMIs. For instance, it conflicted with hisview of mathematics as a way of approaching problems and his intention ofunobtrusively supporting student learning, and it even provoked modes ofinteraction that were counter-productive in relation to these other aspects.In this latter sense the organisational issue, the most practice-dominatingaspect of his SMIs, also turned into one of the key CIP-producing elementsof his classroom.

In summary, in the short term the priorities inherent in Christopher’sSMIs played a part for his contribution to the classroom interactions andfor the ways he reflected on them. However, the influence of the SMIswas sometimes regulated or overshadowed by more general educationalpriorities such as building students’ confidence or by practical concernssuch as managing the classroom. Further, the most influential element ofhis SMIs was his conception of the organisational consequences of hisview of teaching and learning. This influence, by default, illustrated theweak position of his reformist images of mathematics and helped explainthe emergence of some of the critical incidents of practice in the class.

22 JEPPE SKOTT

The Role of the SMIs in the Long Term

To indicate a possible long-term relationship between Christopher’simages and practice I return to the second episode presented aboveand describe a way in which Christopher’s classroom actions tended toreplicate themselves. Struggling to find an appropriate response to thestudents’ problems, Christopher used a funnelling approach in order toensure that the students produced a correct answer. In fact, he ended upmoving his finger one centimetre at a time along the side of the rectangle,asking Martin to count in twos in order to find the corresponding distancein metres. Over the next half an hour Christopher used exactly the sameapproach with a number of other groups of students, and he did so withouthaving struggled to make them come up with suggestions of their ownwith less direct instruction. In these situations, specific features of Chris-topher’s modes of interaction with Martin and Kaspar were disengagedfrom the context in which they had developed. The original grounds forthose modes of interaction – the motives of building student confidenceand managing the classroom – no longer appeared to be necessary forChristopher to resort to this approach. The way he coped with the originalproblem became a prototypical action type that was applied without thesame constellation of objects that initially conditioned it.

There is some resemblance between Christopher’s initial routinisationof elements of his teaching practice and Leont’ev’s description of opera-tionalisation (Leont’ev, 1979). Leont’ev described how shifting gears in acar for the novice driver is an action with its own goal, and how it laterbecomes an unconscious operation carried out in order to drive the carfrom A to B. Similarly, carrying out a multiplication algorithm may bean action in its own right with the goal of mastering the procedure fora student in Grade 2, whereas later it may become an operation carriedout for other purposes like living up to the expectations of the teacheror solving a subjectively important problem. The structural similaritybetween these examples and Christopher’s repetitious use of the sameinstructional procedure is that in all three cases the process in questionwas being removed from the framework of the original activity in whichit served a specific and subjectively well-defined goal. Also it was nowcarried out more or less automatically, for other purposes and with noconscious reference to the initial situation. The difference between theexamples is that in the last two the operationalised actions were compatiblewith or even necessary for the realisation of the new goals, whereas inChristopher’s case they push aside the goals of his actions in the new situ-ation. In other words, the actions-turned-operations surreptitiously becamea dominant mode of interaction irrespectively of the their lack of coher-


ence with Christopher’s overall attempt to facilitate learning. One obviousreason for this was the apparent success of his approach: Martin andKaspar produced an acceptable answer, and Christopher managed to easethe psychological strain he was under in a noisy classroom in which severalstudents constantly called for his assistance. The action-turned-operation,then, carried with it and defined new objects and motives of Christopher’sactivity that were related to fulfilling his part of the didactical contract andto managing the classroom. In the process he substitutes these objects andmotives for facilitating mathematical learning.

It is not clear to what extent this process of initial routinisation of Chris-topher’s actions left his educational priorities and the SMIs untouched. Thequestion is whether the SMIs, that for Christopher appeared to reflect arather coherent system of priorities encompassing many aspects of mathe-matics teaching and learning, constituted a separate and fairly isolatedcluster that was beyond the influence of his classroom experiences. Themany situations in which Christopher’s SMIs were positively correlatedto the classroom practices confirmed his existing school mathematicalpriorities. It is less obvious what role the CIPs played, when the motiveof teaching mathematics was dominated by other concerns. It may be thatChristopher developed a repertoire of operations that broke with his SMIsin the same way as those stemming from his interaction with Kaspar andMartin, and that took on a life of their own in the sense that they wererepeatedly enacted in spite of a lack of compatibility with his generalintentions. If this is the case these operations might at some point expli-citly challenge the existing operations, and the relationship between theteacher’s classroom practices and his SMIs may be described as dialectic.Based on the above analogy of the SMIs as the mirror of the teacher’sreflective practice, this suggests not only that the teacher’s classroom activ-ities are reflected in this mirror, but also that in the long-term they shapeit. Another possibility is that reflections on the actions-turned-operationsestablish a separate set of priorities related to a separate domain of activ-ities, namely the domain of his immediate classroom practice. This wouldsuggest the development of multiple and mutually isolated sets of schoolmathematical priorities each related to their own domain of practice.

CONCLUSIONS AND IMPLICATIONS

It is apparent that the relationship between Christopher’s SMIs and hisclassroom practices was very different in different situations. FollowingErnest (1991) this may be taken to indicate that institutional or contex-tual constraints play different roles even within the same classroom.

24 JEPPE SKOTT

Another and more radical interpretation applies Hoyles’ (1992) conjec-ture, that beliefs are situated. Accordingly, no clear relationship shouldbe expected either between Christopher’s professed educational prioritiesand his teaching practice, or between his practices under different circum-stances. In this interpretation, then, the school mathematical priorities aresituated to the extent that they change with every group of students theteacher works with, even within the same class and on the same topic.

The episodes, however, lend themselves to yet another interpretation,according to which it is neither the institutional constraints nor Chris-topher’s priorities that change with the situation. He still thought – and heprobably did all along – that mathematical processes are as important as theproducts, and he still considered it a main task for himself to initiate andmonitor student activity in a rather unobtrusive manner in order to facilitatelearning. The point is that the second episode in which he apparently useda very different approach should not be seen as a situation that establishednew and contradictory priorities, but rather as one in which the energisingelement of Christopher’s activity was not mathematical learning. He was,so to speak, playing another game than that of teaching mathematics. Thishappened as the object and motive of his activity changed in the course ofthe interaction. To be more specific, when students’ mathematical learningwas Christopher’s primary interest, he struggled to establish one type ofinteraction characterised by support of their individual construction ofmathematical concepts and skills, and he tried to create a conception ofwhat counted as mathematics that included the process of developingindependent solution strategies to given tasks. On the other hand, whenhis activity was primarily directed at other and more general educationalgoals, e.g. building student confidence, his contribution to the interac-tion was dominated by these other goals, and the influence of his SMIswas challenged. Consequently, he engaged in much more direct modes ofinstruction than those immanent in his school mathematics images.

This interpretation questions and elaborates some of the previousresults on the roles of the teacher’s school mathematical priorities. First,the case of Christopher does not substantiate Hoyles’ (1992) contentionthat beliefs are situated, or at least it calls for a specification of what maybe meant bysituated. It is not the situation that produces a new set ofschool mathematical priorities. Situations that are outwardly different –for example, a research interview as opposed to a classroom setting – donot necessarily create or radically change a teacher’s beliefs. Rather, thesituations, that is, very specific situations each concerned with a specificinteraction between student(s) and teacher, create or co-produce competingobjects and motives of the teacher’s activity. It is these competing objects


and motives that form the basis of what apparently, but only apparently, isa new set of priorities produced under new and slightly different circum-stances. In other words, the contextual embeddedness of Christopher’sactivity did not necessarily lead to a similar contextualisation of his schoolmathematical priorities.

Although the relevance of the notion of situated beliefs is questioned asa means of understanding novice teachers’ day-to-day practice, the conceptis not necessarily irrelevant in general. It may be that in the longer term theroutinised elements of Christopher’s activity do not challenge his out-of-school SMIs, but develop into a separate cluster of priorities in isolationfrom them. In this case, the idea of a dialectic relationship between theteacher’s priorities and the classroom practices does not properly describethe situation at hand and may be limited to situations in which researchers,in-service teacher trainers, or colleagues provoke a reflective activity onthe part of the teacher(s) in question. Which of these two possibilitiesbetter describes the long-term relationship between images and practice isstill unsettled. A possible answer requires studies that follow the teachersfor much longer periods of time than was done in the present study.However the only general conclusion so faraboutbelief research is thatno general conclusions should be expectedin belief research. Teachersare different, and so are the contexts in which they work. Consequently,long-term studies should not be expected to present a complete pictureof the relationship between classroom practices and teacher’s SMIs. Butdetailed studies of the idiosyncrasies of different teachers’ approach inspecific classrooms may generate preliminary understandings or theoret-ical constructs that provide insight into the possible relationships betweenthe two.

The above interpretation of the interactions in Christopher’s classroomalso questions some of the previous results that point to a verydirect connection between teachers’ priorities in school mathematics andclassroom interactions. Insisting on a very direct relationship between thetwo, Schoenfeld (1992) apparently missed important social constituentsof the classroom environment, constituents that strongly influence andsometimes even dominate teaching practices as well as learning oppor-tunities. Some of those social constituents are included in Ernest’s (1991)conception of the relationship between beliefs and practice, as he pointedto school culture and other contextual constraints on the enactment ofthe teacher’s beliefs. Ernest, however, only includes those social aspectsthat form the macro structures of school mathematics. He did not discussthe role of the classroom atmosphere nor the interactions between theteacher and specific students or groups of students. Further, Ernest viewed

26 JEPPE SKOTT

a teacher’s beliefs about mathematics as the basic construct that determineor strongly influence the conceptions of teaching and learning and, inturn, the enacted beliefs. In the case of Christopher this is clearly notthe case. The most dominant construct related to his practice is his viewof learning and, more specifically, what he considers to be the organisa-tional consequences of that view. In other words, the degree to which otheraspects of Christopher’s SMIs influence the classroom is contingent ontheir compatibility with the dominant organisational approach.

Those micro-aspects of the social context of mathematics education thatErnest (1991) largely ignored and that in Christopher’s case proved to beessential to an understanding of the social determination of the mathe-matics classroom, are the core of Bauersfeld’s (1988) conception of theclassroom as ajointly emerging reality(cf. p. 5). Whether that constructallows for the influence of the SMIs found in the case of Christopher isopen for discussion. Bauersfeld called for a different focus in classroomresearch, and he was not primarily concerned with an understanding ofthe role of the teacher. Bauersfeld was, then, asking and answering aquestion different from the one of the significance of the teacher’s schoolmathematics priorities. There are, however, two conclusions from thepresent study that are related to Bauersfeld’s article. First, it is obviousthat although Christopher’s SMIs do play a part, their degree of influenceis not determined by him alone. It is the specific interaction between Chris-topher and his student(s), it is the jointly constituted classroom reality thatimposes limitations and provides opportunities for the enactment of hisschool mathematics priorities. This confirms Bauersfeld’s suggestion thatan interactionistic perspective on the specific situation must be included inorder to understand classroom practices. Second, Christopher did have abig influence, both when he decided to take a particular course of actionand when for different reasons he did not play a pro-active role. The pointhere is that in order to understand the developments of the classroom, thespecific role of the teacher in it must be taken into account. In relation toBauersfeld, then, the question he did not ask should still be addressed.

ACKNOWLEDGEMENTS

I would like to thank Paul Cobb (Vanderbilt University), Anna Sfard(University of Haifa), and Paola Valero and Lisser Rye Ejersbo (the RoyalDanish School of Educational Studies) for valuable comments on previousversions of this paper.


REFERENCES

Bauersfeld, H. (1988). Interaction, construction, and knowledge: Alternative perspectivesfor mathematics education. In D. Grouws, T. Cooney & D. Jones (Eds.),Perspectives onresearch on mathematics education(27–46). Reston, VA: National Council of Teachersof Mathematics.

Brousseau, G. (1984). The crucial role of the didactical contract in analysis and construc-tion of situations in teaching and learning mathematics. In H.-G. Steiner (Ed.),Theoryof mathematics education(Occasional paper 54, 110–119). Bielefeld, Germany: Institutfür Didaktik der Mathematik.

Carpenter, T.P. & Fennema, E. (1991). Research and cognitively guided instruction. InE. Fennema, T.P. Carpenter & S. Lamon (Eds.),Integrating research on teaching andlearning mathematics(1–16). Albany, NY: State University of New York Press.

Cobb, P., Wood, T. & Yackel, E. (1990). Classrooms as learning environments for teachersand researchers. In R.B. Davis, C.A. Maher & N. Noddings (Eds.),Journal for Researchin Mathematics Education: Constructivist views on the teaching and learning of mathe-matics (Monograph No. 4, 125–146). Reston, VA: National Council of Teachers ofMathematics.

Cobb P. & Yackel, E. (1996). Sociomathematical norms, argumentation, and autonomy inmathematics.Journal for Research in Mathematics Education, 27, 458–477.

Cooney, T.J., Shealy, B.E. & Arvold, B. (1998). Conceptualizing belief structures ofpreservice mathematics teachers.Journal for Research in Mathematics Education, 29,306–333.

Ernest, P. (1991).The philosophy of mathematics education. London: The Falmer Press.Guskey, T.R. (1986). Staff development and the process of teacher change.Educational

Researcher, 15(5), 5–12.Hoyles, C. (1992). Mathematics teaching and mathematics teachers: A meta-case study.

For the Learning of Mathematics, 12(3), 32–44.Leont’ev, A.N. (1979). The problem of activity in psychology. In J.V. Wertsch (Ed.),The

concept of activity in soviet psychology(37–71). New York: M. E. Sharpe, Inc.Lerman, S. (1994). Towards a unified space of theory-and-practice in mathematics

teaching: A research perspective. In L. Bazzini (Ed.),Theory and practice in mathe-matics education. Proceedings of the fifth international conference on systematic co-operation between theory and practice in mathematics education(133–142). Pavia, Italy:ISDAF.

National Council of Teachers of Mathematics (1989).Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1998).Standards 2000. Principles andstandards for school mathematics(Discussion draft). Reston, VA: Author.

Pehkonen, E. & Törner, G. (1996). Mathematical beliefs and different aspects of theirmeanings.Zentralblatt für Didaktik der Mathematik, 96(4), 101–108.

Raymond, A.M. (1997). Inconsistency between a beginning elementary school teacher’smathematics beliefs and teaching practice.Journal for Research in Mathematics Educa-tion, 28, 550–576.

Ruthven, K. (1987). Ability stereotyping in mathematics.Educational Studies in Mathe-matics, 18, 243–253.

Schoenfeld, A.H. (1992). Learning to think mathematically: Problem solving, metacog-nition, and sense making in mathematics. In D. Grouws (Ed.),Handbook of research

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on mathematics teaching and learning(334–370). New York: Macmillan PublishingCompany.

Thompson, A.G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research.In D. Grouws (Ed.),Handbook of research on mathematics teaching and learning(127–146). New York: Macmillan Publishing Company.

Tobin, K. & Imwold, D. (1993).The mediational role of constraints in the reform ofmathematics curricula. In J.A. Malone & P.C.S. Taylor (Eds.),Constructivist interpret-ations of teaching and learning mathematics. Proceedings of Topic Group 10 at theseventh International Congress on Mathematical Education(ICME-7, 15–34). Perth,Australia: National Key Centre for School Science and Mathematics, Curtin Universityof Technology.

Department of Mathematics, Jeppe SkottThe Danish University of Education,Emdrupvej 115 B,DK-2400 Copenhagen NV,DenmarkE-mail: [email protected]

MURRAY S. BRITT, KATHRYN C. IRWIN and GARTH RITCHIE

PROFESSIONAL CONVERSATIONS AND PROFESSIONALGROWTH

ABSTRACT. A professional development program for 18 teachers was conducted overa two-year period. The participating teachers taught in intermediate schools (studentsaged 11–13) and secondary schools. The teachers worked collaboratively to improve theirmathematics teaching, with encouragement to reflect on their practice but with minimalinstruction from the researchers. Results, as defined by change in teaching practices,beliefs, and reflections, and student achievement, indicated that the collaborative programwas particularly useful for experienced secondary school teachers but less useful forintermediate school teachers. We concluded that this type of professional developmentwas most useful for teachers who had sufficient knowledge of mathematics; theseteachers were able to focus on pedagogy and to draw connections between aspects of themathematics they taught, without recourse to a specialist’s advice.

The professional development program described here arose from twoissues. The first issue, addressed in different ways in other professionaldevelopment projects, was the need to enable teachers to make lastingchanges to their teaching, a process that would take considerable timeand professional involvement. The second issue addressed the differencebetween teaching in intermediate schools and in secondary schools. Webelieved that offering a program with teachers from both levels would givethe teachers the opportunity to learn aspects of pedagogy from one anotherand to better understand the transition students faced.

Our project required teachers to decide what changes they wanted tomake to their teaching, to evaluate these changes, and to discuss changeswith other teachers as well as with the researchers. The aspects of thisproject that we discuss here include changes in teachers’ beliefs andclassroom practices, the effect of teachers’ integrated or connected know-ledge of mathematics, and the role of professional conversation in thesechanges. Other aspects are discussed in Britt, Irwin, Ellis, and Ritchie(1993).


30 MURRAY S. BRITT ET AL.

A LOOK AT PREFESSIONAL DEVELOPMENT PROGRAMS

Teachers’ Beliefs and Classroom Practices

Professional development programs for teaching mathematics have helpedbring about change in teachers’ beliefs and classroom practices. Mostof these programs have been conducted with either elementary, middle(intermediate), or secondary school teachers, but seldom with teachersfrom more than one school level. Programs for elementary school teachersinclude those of Carpenter, Fennema, Peterson, and Franke (e.g., Franke,Carpenter, Fennema, Ansell & Behrend, 1998), Schifter and Simon (e.g.,Schifter, 1998), and of Cobb and colleagues (e.g., Cobb et al., 1991).Programs for middle school teachers include those of Snead (1998),Sowder, Philipp, Armstrong, and Schapelle (1998), and Swafford, Jones,Thornton, Stump, and Miller (1999). Programs for secondary schoolteachers include those of Jaworski (e.g., 1994) and Shulman and Grossman(1988, cited in Brown & Borko, 1992). Programs at elementary as wellas secondary levels have focussed on improving teaching by developingteachers’ knowledge of students’ mathematical concepts and by encour-aging teachers to reflect on the effects of different aspects of theirteaching.

Schifter and Simon’s (1992) program, at the elementary level, empha-sized that teachers learn or expand specific mathematical concepts andthen reflect on the processes of that learning. Like Schifter and Simon,Cobb et al. (1991) emphasized the importance that teachers negotiate theirown changes in classroom practice, with ongoing support from researchersand colleagues. Snead (1998) as well as Sowder, Philipp, Armstrong &Schapelle (1998) demonstrated the benefit of one- or two-year courses anddiscussions on content and pedagogical content knowledge on classroompractice. In Jaworski’s study (1994), teachers and researcher focused on thebalance needed between mathematical challenge, sensitivity to students,and management of learning.

In none of the programs did the researchers tell the teachers what to do;rather, they trusted the teachers to find their own ways of improving theteaching of mathematics. Several of the studies found that once teachersstarted to observe their students carefully and to reflect on the effect oftheir teaching, they moved from a focus on transmission of knowledgeto a focus on guiding students toward better understanding. Cobb et al.(1991) especially noted a change in a teacher’s beliefs as being crucial tothe process of pedagogical change, an aspect also discussed by CarpenterFennema, and Franke (1996), Schifter (1998), and Jaworski (1994).

PROFESSIONAL CONVERSATIONS 31

Teachers’ Understanding of Mathematics

Fennema & Franke (1992) reviewed the relationship between teachers’knowledge and their ability to teach. Their review established that the rela-tionship between the number of mathematics courses teachers have takenand teachers’ effectiveness was not a simple one. Rather, teachers’ effect-iveness was related to the integration of their mathematical knowledge,as demonstrated by the connections they saw between different areas ofmathematics. Teachers who saw connections and appreciated their import-ance could also see the connections their students were making. Studiesby Leinhardt (1989) and Shulman & Grossman (1988, cited in Brown &Borko, 1992) showed that teachers who saw connections were able to drawstudents’ attention to these connections. Leinhardt and Smith (1985) notedthat as teachers increase their conceptual knowledge of mathematics theybecome more fluid in using this knowledge as a basis for lessons that canlead to improvements in the students’ conceptual knowledge.

Askew and colleagues (Askew, Brown, Rhodes, Wiliam & Johnson,1997; Brown, Askew, Rhodes, Wiliam & Johnson, 1997) extended theanalysis of teachers’ connected knowledge of mathematics with conceptmaps that were drawn by teachers and then used as an index of suchconnections. They found three distinct patterns of teaching and beliefs thatthey termedconnectionist, discovery, andtransmission. Teachers classifiedas connectionist were likely to have classes that made greater gains thanwere teachers with the other two belief systems. As a consequence, Askewet al. (1997) suggested that teachers need to develop a fuller, deeper, andmore connected understanding of mathematical concepts in order to teachmore effectively. They believed that this could be done, in part, throughdiscussion of the beliefs and the respective success of teachers who demon-strate the three patterns of belief. However, the authors did not suggest thatthe transition would be easy.

PROFESSIONAL CONVERSATIONS AS A FOCUS FORPROFESSIONAL DEVELOPMENT

We define professional conversations as discussions among those whoshare a complex task or profession in order to improve their understandingof, and efficacy in what they do. An example of such conversations is foundin a teacher telling a group of colleagues about a particular lesson, followedby several other teachers reflecting on the lesson’s strengths or weaknessesand what they might have done in a similar situation. We decided to focusour professional development program on such professional conversations


among teachers because of the specific conditions of teacher developmentin New Zealand.

In New Zealand, mathematics teachers have few opportunities to parti-cipate in professional development programs that address their particularconcerns and needs. Although a reform-oriented curriculum is mandated(Ministry of Education, 1992), there are few courses offered that wouldhelp teachers cope with the new curriculum. What courses are availableare of short duration, varying between a half-day and six half-days over sixweeks. For the most part, teachers are left to their own resources. Althoughsome teachers read journal articles that provide ideas for reform-orientedteaching, few teachers put these ideas into practice simply because theyoften fail to see, without continued support, how to incorporate the ideasinto their classroom teaching situation. Because external in-service supportfor teachers occurs infrequently, we argued that teachers needed to lookfor alternative sources to use as sounding boards as they raise concernsand consider new ideas related to their classroom practices. We believedthat teachers and their teaching colleagues can act as that professionaldevelopment resource given the appropriate conditions.

The findings from a professional development study by Clarke (1997)supported such a collegial approach. In his study, he ranked 12 factorsteachers had identified, at different times and in different settings, asleading to their development. Of these, collegial factors were ranked inthree of the four top positions. Teachers claimed,

The support they received from each other, the opportunity to work together in plan-ning and debriefing each day and the opportunity to have a sounding board present(whether another teacher, a project staff member, or a researcher) all appeared to facilitateprofessional growth more than the four in-service sessions (p. 302).

Like the teachers studied by Clarke, we expected that the teachers inour project would find professional conversations useful for reflection onclassroom practices and on student learning. We believed that with thehelp of such reflection their teaching would improve. We expected that along time period would be needed for teachers to establish an appropriatefocus for their professional development, be willing to try new techniques,trust their colleagues, and risk potential failure or criticism from colleagueswho saw little justification for change. We intended to provide a forum inwhich they could engage in professional conversations with each other. Wewanted to know the effect of such a program on their beliefs and practicesabout mathematics teaching and whether change in beliefs and practicesmight result in improved understanding for their students.


METHOD

Overall, we used a qualitative research model in which the change ofeach teacher was evaluated individually. We collected information fromclassroom observations, group discussions, questionnaires of beliefs andpractices, interviews, and evaluations of student competence and attitudesat the start and the end of each year. We developed case studies andsummaries of the nature of change seen in each teacher.

Participants

Initially, 8 teachers from four intermediate schools and 10 teachers fromfour secondary schools participated in the study. The teachers wereselected after two discussion sessions with interested teachers, staff, andprincipals of 25 schools. In the sessions we emphasized that participantswould select their own area for development and discuss the develop-ment with other teachers. Criteria for selecting the teachers were: (a)expressed interest of the teachers and principals in a two-year professionalenhancement project in which the teachers initiated their own reforms,(b) the pairing of secondary with intermediate schools that served thesame students, and (c) an overall geographic distribution so that pairsof schools came from the north, south, east, and west of the Auckland.This geographic distribution was chosen because it roughly represented thecity’s distribution of demographic variables such as ethnicity and income.

The intermediate school teachers were familiar with procedurescommonly used in reform mathematics, such as group work and inclu-sion of open-ended problems, because they used these procedures in othersubjects. They knew their students well, as they taught them all day long,covering every subject in the curriculum. Most secondary school teacherstaught only mathematics; they were more familiar with the subject but lessfamiliar with the practices of reform mathematics. Teachers at both levelstaught from a common curriculum that covered all mathematical topics ateach level rather than separate topics in any particular year.

All 10 of the secondary teachers were in the project for the full twoyears. Three of the intermediate school teachers left during the period andwere replaced with other teachers so that each participant continued tohave a partner in the same school. Table I shows the teaching experienceand years of tertiary study in mathematics for the teachers who were withthe program for the full two years.


TABLE I

Characteristics of Teachers

Teacher School type and location Years of teaching Years of tertiary

before project study in

mathematics

Intermediate schools

Tracey North 3 0

Joseph South 3 0

Adrienne East 18 0

Ngaire East 25 0

Naomi West 9 0

Secondary schools

Colin North 9 3

Beryl North 24 3

Simon North 4 3

Gareth South 14 1

Alex South 2 2

Robyn East 28 3

Leslie East 11 4

Jocelyn East 4 1

Petra West 10 3

Grace West 17 3

Note: It is usual for intermediate school teachers to have no tertiary mathematics,as this was not required when these teachers were trained.

Program

For two school years, the teachers met with two of the researchers for half-day sessions at approximately monthly intervals. Between meetings, theresearchers visited classes and provided teachers with full reports of whatthey had observed. Using reflections on these observations, ideas fromfellow teachers, and ideas gained from other sources, teachers identifieda teaching strategy or approach that they wanted to try in their classrooms.They were encouraged to discuss the effect of their innovation before theytried something else. Changes tried included (a) focusing on the summa-tion of a lesson, (b) trying new ways of teaching particular mathematicaltopics, (c) trying to teach with less telling, and (d) having students writeproblems that would become the focus for a class lesson. For the purposeof the project, teachers limited their attention to a specific class level.Secondary teachers focused on Year 9 classes (the first year of secondary


school), and intermediate school teachers focused on one of the two classlevels in their schools.

Most group sessions started with a mathematical activity. The teachersworked on the activities in small groups. Teachers then discussed theirsolution processes and how a similar lesson might be used in their classes.Teachers also chose to meet within their schools or to exchange visits withthe school of a different level in their district. Over the two-year period,adjustments to the procedures were made, primarily at the suggestion ofthe teachers. One such change was to include two whole-day sessionsduring which teachers developed lessons that were in line with the spirit ofreform mathematics.

Sessions covered many topics, including topics suggested by teachers.Many teachers presented a lesson that they had taught, including a descrip-tion of the lesson, displays of students’ work, and videotape clips. Onesession covered the difficulties teachers faced in their schools, and anotherfocused on group work. At many sessions there were quick reports by themajority of teachers on what they were trying, often followed by furtherdiscussion in small groups of teachers who shared common goals. On oneoccasion, a videotaped interview with Kath Hart was shown and discussed,and on another occasion Elizabeth Fennema spoke with the group. Both ofthese educators emphasized the importance of listening to students.

Data Collection and Analysis

While the teachers experimented with their own practices, they and theresearchers collected data from several sources in order to evaluate theproject. Teachers’ practices and beliefs were assessed through observa-tions; transcribed audio- and videotapes of lessons; comments at groupmeetings and in interviews; responses to questionnaires; and entries inteachers’ journals. Full notes of two initial class observations were givento each teacher. This was followed by a summary of the observations,analyzed by factors such as whole-class versus group teaching, teachingstyle, use of different periods of the lesson, use of materials, participationof students, and use of students’ existing knowledge. For later observationsteachers told us what they wanted us to attend to. The effect of teachers’practice on their students’ mathematics and attitudes toward mathematicswas measured by comparing achievement tests and attitude questionnairesin the cohort taught the second year with that taught in the first year.A greater change in the second year would indicate improved teachingpractice. Table II summarizes the data collected.

This report draws primarily on three sets of data: (a) the teachers’responses to a questionnaire and to interviews administered by an external


TABLE II

Data Collected

Year Data collected

Year 0 1. Teachers’ and principals’ reasons for wanting to participate in the project.

Year 1 2. Demographic data for teachers and their schools; teachers’ responsesto three questionnaires: Objectives, Resources Used by Teachers andObstacles to Student Progress1 (February).

3. Assessment of student attitudes (questionnaire, Maths and Myself1) andassessment of students’ mathematical competence on one or more ChelseaDiagnostic Test2 chosen by teachers, followed by interviews of selectedstudents, at the start and end of the year (March, November).

4. Classroom observations of each teacher and tape-recording of lessons(throughout year).

5. Teachers’ responses to reports of March classroom observation andreflections on changes they wanted to make (April).

6. Reports of individual teachers’ proposed and effected changes in groupmeeting, in telephone interviews, and in transcribed interviews in schoolsat the end of the year (throughout year).

7. Journals kept by each teacher of their attempted changes, reactions, etc.(throughout year).

8. Notes taken at each of 8 half-day meetings of the contributions made byeach teacher (throughout year).

Year 2 9. Student assessment on diagnostic mathematics tests and questionnaires asin Year 1 (March, November).

10. Classroom observation of each teacher including at least one videotapedlesson and one lesson for which the audiotape was fully transcribed(throughout year).

11. A one-lesson innovation by each teacher (April).

12. Continuation of journals, collection of work samples, and recording ofthe contribution of each teacher at each of 10 group meetings (throughoutyear).

13. Questionnaire and interviews by an independent researcher (the thirdauthor), on teachers’ practices, beliefs, and perceived effect of the project(May).

14. Lessons developed by groups of teachers, trialed, and revised, in line withNew Zealand’s reform curriculum (July, August).

15. Readmission of questionnaires to teachers, same as at the beginning ofYear 1 (November).

Year 3 16. Teachers’ post-project evaluation of what the project had meant to them(September).

1Binns, Carpenter, Elliffe, Irving and McBride, 1987.2Hart, Brown, Kerslake, Küchemann and Ruddock, 1985.Note: The school year runs from February through early December.


TABLE III

Questionnaire Items

Items scored positively, as likely to beconsistent with reform mathematics

Items scored negatively, as not likely tobe consistent with reform mathematics

Teaching practices: Teaching practices:

Teacher-initiated investigations Teacher-led discussion

Self-initiated investigations Teacher demonstration

Exploring ideas Teacher explanation

Exploring misconceptions Written task sheets

Discussing as a class

Discussing in small groups

Writing maths in own words

Tackling new challenges

Beliefs about what teaching practiceaided learning:

Beliefs about what teaching practiceaided learning:

Working out practical problems Listening to teacher explaining

Talking and listening to other students Taking good revision notes

Asking questions Practicing exam and test questions

Tinkering around with equipment Doing written textbook exercises

Testing out their own ideas Watching a teacher work through a

Solving puzzles problem

Reflecting on their own ideas

Examining misconceptions

Drawing and justifying conclusions

researcher in Year 2 (#13 in Table II), (b) classroom observations (#4 and#10 in Table II), and (c) the teachers’ journals (#7 and #12 in Table II).

Questionnaire. The external researcher (the third author) based his ques-tionnaire and interviews on those used by Kirkwood, Bearlin & Hardy(1989) and by Bell and Pearson (1993). The questionnaire had beendeveloped for the Learning in Science Project in order to analyze thechanges that project teachers thought they had made in their teaching andbeliefs (see Table III). It consisted of 17 items about teaching practice and19 items about beliefs. Additional items, not included in the analysis, askedabout the use of outside speakers, videotapes, and other issues. For eachquestion, teachers ranked the frequency of a practice or agreement with abelief on a five-point scale, fromalways(1) to never(5). They also rated


the change in the frequency of a practice or belief, as compared to theprevious year, as 1 (more than last year), 0 (about the same as last year),or –1 (less than last year).

The external researcher weighted items in the two parts of the question-naire as positive or negative depending on whether the practices and beliefswere consistent with the reform curriculum. On items scored as negative,frequency ratings of high scores were converted to (positive) low scoresand vice versa. This allowed the development of scales for reform-orientedteaching practice and reform-oriented beliefs. Reform-oriented teachingpractice and belief scores for each teacher were obtained by summingthe converted teaching practice or belief ratings. Table III indicates whichitems were scored as in line with reform mathematics.

Interviews. The interview schedule consisted of 10 open-ended questionsand provided an opportunity for teachers to expand on issues raised inthe questionnaire. Teachers were asked what they saw as helping studentslearn mathematics, what they looked for in students’ behavior, what theyexpected their students to be able to do, what their most important tasksas a teacher were, and whether or not they integrated different topics ormathematics with other subjects. They were also asked if they had changedeither their methods or ideas about teaching as a result of the project. In theanalysis of the interview data we attended to whether or not the teachers’responses were consistent with or contrary to reform teaching.

QUESTIONNAIRE RESPONSES: TEACHER BELIEFS ANDTEACHING PRACTICES

Data from observations and from the questionnaire were generallyconsistent and revealed that all but one teacher believed that he or shehad made marked changes. This one teacher was an experienced interme-diate school teacher who already taught in a manner consistent with reformmathematics (see Britt et al., 1993). Students’ mathematical performancealso showed improvement between the first and second year of the project.However, because of missing information no conclusions can be drawn onthis aspect. We elaborate on this issue in the discussion.

The change scores for individual teachers on both scales of the ques-tionnaire indicated a significant move towards reform-oriented teachingpractices (t(15) = 4.84, p < 0.001), and reform-oriented beliefs (t(15) =5.2, p < 0.001; see Table IV). The data also suggested that the helpfulnessof the project in making these changes was related to the experience ofthe teachers and the level at which they taught. Table IV shows that the


TABLE IV

Means and (Standard Deviations) for Reform-Oriented Scale Measures

Reform-oriented scale measure Experienced Inexperienced t-value

teachers1 teachers

(n = 11) (n = 5)

Average teaching practice rating 36.89 (4.15) 33.80 (3.77) 1.42

Average teaching practice change rating 5.8 (4.02) 2.75 (4.41) 1.37

Average belief rating 60.15 (6.10) 51.68 (4.55) 2.72∗Average belief change rating 6.84 (4.75) 1.60 (2.51) 2.29∗

∗p < 0.051Experienced teachers were those with more than four years of teaching by the startof the project.

TABLE V

Means and (Standard Deviations) for Reform-Oriented Scale Measures

Reform-oriented scale measure Intermediate Secondary t-value

teachers teachers

(n = 7) (n = 9)

Average teaching practice rating 37.04 (4.78) 35.07 (3.70) 0.93

Average teaching practice change rating 2.03 (3.37) 7.04 (3.60) 2.83∗Average belief rating 59.02 (7.23) 56.32 (6.80) 0.77

Average belief change rating 3.32 (3.79) 6.66 (5.18) 1.43

∗p < 0.05

experienced teachers were significantly more likely than the inexperiencedteachers to affirm beliefs consistent with reform-oriented pedagogy.

Table V shows that the secondary teachers perceived that theyhad moved more towards teaching consistent with a student-centeredpedagogy. However, despite this greater perception of shift in teaching,the secondary teachers were not different from the intermediate schoolteachers in terms of how they perceived their own teaching.

When the individual items in the reform-oriented scale were analyzedwith nonparametric Mann-WhitneyU tests, few items showed significantdifferences between the experienced and the inexperienced teachers, butthere were significant differences on a number of items between the inter-mediate school and the secondary school teachers (see Table VI). Theintermediate school teachers were more inclined to use student investiga-tions and group work than the secondary teachers. On the other hand, inter-


TABLE VI

Means for Change in Behaviour Items

Teaching dimension Intermediate SecondaryU value

teachers teachers

(n = 7) (n = 9)

Teacher explanation –0.16 –0.66 13.5∗Teacher initiated investigations 0.17 0.67 13.5∗Tinkering to see how ideas fit together –0.14 0.67 9∗∗∗Students discussing in small groups 0.14 0.87 7.5∗∗∗Students writing mathematics in their own words 0.29 0.89 12.5∗∗

∗p < 0.10, ∗∗p < 0.05, ∗∗∗p < 0.01. A positive change indicates that the behaviourmentioned is more frequent than prior to the project. A negative change indicates thatthe behaviour mentioned is less frequent than prior to the project.

mediate school teachers were less inclined to think that they had changedin either teaching or beliefs. Again, this may have been because there wasmore room for such change for the secondary teachers than for the inter-mediate school teachers whose lessons had been more student-centeredall along. The change in teaching practices professed by the secondaryteachers was consistent with the findings from classroom observations,participation in group sessions, journal entries, and other indicators ofchange.

PROFILES OF THREE TEACHERS

Each teacher in the project represents an interesting and distinct story ofchange. The teachers whose stories we have summarized in this sectionillustrate different patterns of change. The first teacher can be seen astypical for the secondary school teachers, although she had less confidencethan many. The second teacher is representative of the intermediate schoolteachers. We included the third teacher’s story because we saw her storyas a cautionary note for programs similar to ours. These three teacherscan be seen, at the end of the project, as fitting Askew’s categories ofconnectionist, discovery, and transmission styles (see Askew et al., 1997).We discuss this feature in the final section of this article.

Robyn

Robyn was an experienced secondary school teacher although she had notalways taught mathematics. She said that she joined the project because


she “was not good at teaching mathematics” and “want[ed] to get insidethe kids’ heads.” She wanted help “in the race to meet the bottom line,”which meant that she wanted to help students pass the examinations. Inher school, the same tests were administered to all heterogeneous classesat the end of each unit. Year 9 students had only three one-hour mathem-atics classes per week, therefore Robyn and other teachers felt pressuredto cover the material the students would be tested on.

In one of the first lessons observed (March, first year) Robyn wasworking on “angle facts” or rules that could be used to deduce unknownangles (for example, opposite angles are congruent, the angles of a triangleadd up to 180◦). She showed five diagrams and asked students to writethe angle fact for each example. She then asked the students to checktheir answers by referring to their notes from a previous lesson. Thiswas followed by a discussion of a homework problem that required useof the angle rules. In one instance, when Robyn did not get the answershe wanted, she told the class, “These angles are on a straight line. Theyadd up to 180◦.” In some of the exercises following the whole-class work,students, working in small groups, matched diagrams to statements of rulesand identified what rule was violated in other examples.

At the end of this lesson, Robyn commented that the lesson had notworked. She was disappointed because for a similar class the previous year“It had worked like a charm.” She identified the problem as due to thestudents not working cooperatively, whereas the observer thought that thelack of understanding of the rules in question was an important factor.

By April of the first year, Robyn had focused on her goal of “giving kidsownership of what they were learning in the classroom.” Her comments inher journal centered on students’ understanding. A typical statement, aftera lesson on drawing different triangles with the same base and height, was“Found out that A., T., B., and J. (names of students) thought that baseis the same as height. How could I have avoided this? I think it meansthat the drawing activity was not for those students” (July, first year). Thisexample, like others, demonstrates that she saw students’ learning diffi-culties as a problem that she, the teacher, had to solve, and not just as anindicator of students’ ignorance.

In August of the first year, Robyn told other teachers about the conceptof teacher lust, as described by Maddern and Court (1989). Teacher lustrefers to the desire of a teacher who knows something to tell it to studentswho do not know. She felt that her teacher lust interfered with students’learning. This became a theme of the project for her and for some of theother teachers as well.


For the second year of the project Robyn further refined her goal of“verbalizing as a part of owning your own mathematics; having [students]participate in setting their own goals: exploring, predicting, confirming andsummarizing,” and keeping a learning log for herself to know if she waslistening to the students.

In one of the lessons observed in the second year (September) studentsinvestigated powers, as calculated on calculators. Although this lessonwas not part of their school syllabus, the class had become interested inpowers in the previous lesson. Robyn started by writing3

√125= 5 on the

board. She then asked, “Do you or your neighbor have the same words forthese symbols?” Answers, written on the board, were:

Five cubed = 125 Five to the power of three = 125

Five squared three = 125 The cube root of 125 is five

The third root of 125 = 5

Robyn noted the use of the termsquaredbut did not comment on it until asimilar error had been made later in the lesson. Another example she gaveto the class was?

√1024= 2. She highlighted one student’s response of “It

can’t be 44 = 1024 because 16 times 16 should end in a 6.” Later a studentasked, “Do you think 21 = 4? But then 22 = 4, and it can’t be both.” Afterthe students had explored further examples of powers and roots, the lessoncontinued with a clarification of square, square root, cube, cube root, andthe use of calculator keys.

At the end of the lesson Robyn praised the students for knowing partof Year 10 mathematics. She commented to the observer that, “[Earlier] Iwould have shut this discussion down because it was not in the syllabus,”but now she felt that it was appropriate to interact with the students and tochange her pace as she listened to what they had to say. During the latterportion of the lesson she was hoping that the students would come up witha way to talk about inverse relations, but they did not, and she held ontoher teacher lust and did not tell.

Robyn reported that her greatest achievement in the project had been toproceed through two units without her giving explanations of the mathem-atics. She realized that students actually discovered the relevant conceptsand did well on the school tests. She shared this success proudly with theother teachers, who had previously heard of many of her perceived failures.Robyn also asked her students to write about their overall experience withmathematics (November, second year). Two typical comments were:

My opinion of what maths is has changed. Working in groups and discussing is better thanhaving a teacher talk on and on while people in class switch off until no one is listening.


and

I feel differently because there is more commitment to take time to understand it.

At the beginning of the project, Robyn used the common technique oftelling students what she wanted them to know. As she refined her goalsfor the project she encouraged her students to use their own language todescribe their understanding. She used what they said to encourage themto make connections, for example, between aspects of powers and knownmultiplication facts. She also heard students’ misconceptions, and usedthese to help make correct connections. These changes in her practice, andher ability to reflect on them, were noted in the major changes she reportedin her beliefs on the questionnaire.

Tracey

Tracey was a mature woman who had trained for teaching when her threechildren were teenagers. She had studied mathematics through Year 12 atschool. Although she was only in her fourth year of teaching when theproject started, she was generally a more confident teacher than Robyn.She taught Year 7 students in an intermediate school, where her classes ofabout 35 students included several with special needs. She was accustomedto having students work in groups, especially when they researched topicsin social studies. She chose to enter the project because it was her planto concentrate on development in one curricular activity each year. Shewondered why she taught mathematics differently than other subjects, andshe felt less confident with the subject matter. Her goal was to focus onwhat was important for students to learn. In the second year she clarifiedthe goal to, “[for] pupils to be responsible for their own learning and raisingachievement in maths.” Her goal was accompanied by a list of seven waysin which she would foster this development and how she would monitorher success.

In February of the first year, Tracey was observed teaching statistics.She wrote “Statistics” on the board, asked what it meant, then wrotestudents’ answers on the board. Most answers were rather vague, such as“in maths,” “the census,” and “graphs.” She then asked why graphs wereused. Responses to this question included: “neater and easier,” “quickerto read,” and “one mark tells a lot.” Later she collected data on what thestudents had brought for lunch, using a method suggested by a student.Finally, Tracey held a class discussion on what to do next. The classconcluded with group work. Students wrote their ideas for the next stepinto their books. For homework, students were asked to collect data athome on people’s occupations. In our view, this lesson reflected back the


students’ limited ideas rather than advanced their understanding. It raisedthe broader question of how a teacher could move from valuing students’knowledge to advancing that knowledge.

Tracey was a reflective and helpful member of the group of projectteachers; she offered suggestions and appreciated the suggestions made byothers. She reported that she gained ideas from her visit to the secondaryschool and from her observation of her partner teacher in her own school.In July of the first year she described to the group an investigation onperimeter and area that had been suggested to her. Her comments to thestudents in that lesson led them to a next level of generalization. Ratherthan just praising their efforts as she had done previously she asked ques-tions such as, “What have you found out?” or, “Have you found a rule?”Two secondary school teachers commented that they had used a similarmethod with Year 9 and Year 13 students.

In a class observed late in the second year (November), Tracey’sstudents studied volume and area using a context that involved packingup class sets of textbooks. In groups, students had to decide what shape ofcarton would be best and how much floor space would be required to stackall cartons for the whole school. The students measured a single textbookand then predicted the best way to stack the books. As they worked, Traceytalked with pairs of students, encouraging them to focus on aspects of thetasks that she knew would lead to the mathematics she wanted them toconsider. For example she asked, “Is that all we need to know?” when a pairof students had found only two dimensions of a book. When she felt thatthe class had made sufficient progress, Tracey asked students to report onwhat they had found. Students subsequently recommended and explained arange of different stacking methods. Some students had used centimeters,and others had used square centimeters rather than cubic centimeters forvolume units of measurement. Tracey used the incorrect responses to evokea whole-class discussion on which units were correct and why they werecorrect. The discussion led students to refocus on the three-dimensionalaspect of volume and provided a wrap-up to the lesson.

This lesson provided an investigation with clear mathematical goalsin mind. Tracey posed a problem students needed to determine how tosolve. She picked up important mathematical concepts and initiated a classdiscussion in order to resolve confusions. The lesson contrasted markedlywith the earlier lesson described above.

In her final evaluation of the project, Tracey said that the major factorfor her was deciding what she wanted to change. This “came from listeningto other teachers in [the project].” The best aspect of the project “forme [was] meeting with other teachers, discussing ideas, gleaning ideas,


reinforcement, recognition that similar situations exist, encouragement.”She “got validation from the other teachers and heard what didn’t work.”She felt more confident because the project had given her “license toexperiment. Risks were taken because I felt confident.” She wrote that“she identified the maths more often,. . . used students’ misconceptionsto teach,. . . asked pupils to identify what had been learned/practicedin a lesson,” and “listened to pupils more as a means of evaluation”(Journal, November, second year). She knew that she was not an expertin mathematics, but she welcomed the help of others in becoming a bettermathematics teacher. The changes that she described in the questionnaireindicated that her beliefs had not changed (they were already aligned withreform mathematics) but she believed that her practice had changed.

Naomi

Naomi was also a mature woman, who began teaching in her 30s. She wasa more experienced intermediate school teacher than Tracey. She had takenmathematics at school through Year 11. At the end of the project she toldthe group that she had not volunteered to participate, as other teachers had,but had come because her principal had asked her to, a fact of which wewere not aware.

When first observed in March of the first year, Naomi ran a busybut controlled classroom of relatively low-achieving Year 7 students. Shetaught four groups, with students divided by competence. Each group hada different activity. She first worked with one group at the board askingthem to tell her the number that went in each box in several paired numbersentences such as, 143 + 52 = 195, so 195 –� = 143. Students chorusedtheir responses as soon as each pair of sentences was presented. Whilethey proceeded to write similar paired-number sentences for each other tocomplete, Naomi turned to a second group that was working on problem-solving task sheets. These students were confused about how they might goabout solving word problems that could have been solved by a trial-and-improvement strategy or by an algebraic method. Rather than providinghelp to the students, she told them, “Put on your thinking caps,” a sugges-tion that did not help them out of their impasse. She then returned to thefirst group and worked on place value: changing a number from expandedto compact form and vice versa. Students appeared to enjoy this activity;but when one student made an error of magnitude Naomi did not notice it,and praised the student. She did not work with the third or fourth group inthis period.

Her first recorded goal for the project (July, first year) was that herstudents would enjoy mathematics. In August she said that she would work


on “problem solving, reading, and using maths language.” In November ofthat year she said that she wanted to look at attitude and achievement inthe context of geometry.

Although Naomi said that she was interested in talking with other inter-mediate teachers, she rarely contributed to group sessions, even at specialsessions set up for intermediate teachers only. She was interested in listingthe impediments to change, citing the amount of paperwork that teacherswere expected to do, and that the fact that, “[The school schemes] come atus three-weekly. There is pressure to teach to a test.” Her view of herselfwas that of an explainer. “I’ve actually said to my kids if you ask mesomething I’ll explain it and explain it until you get it. I’ll keep workingon it until I figure out how to get it across” (Interview, May, secondyear).

Naomi’s journal included examples of students’ work or attitudestoward mathematics, but there was little reflection on her teaching. Herstatements and journal entries repeatedly reflected her lack of confidencein mathematics and the fact that the project was not meeting her needs.She did make some reflective statements at meetings, including, “I haven’tbeen as adventuresome as I should perhaps” (August, first year). Once,when talking to another teacher, she said, “Is there anyone at school whocould help me?” and then immediately followed this with, “But I amdoing too many things.” At the final session of the project she said thatshe had needed “a feedback loop to know whether she was successfulor not” (November, second year). Professional conversations and repeatedclassroom visits had provided this loop for other teachers, but Naomi hadnot participated in conversations enough to benefit from them as much asother teachers had. She repeatedly found reasons why her class could notbe visited.

In a class observed in September of the second year, she was helpingone of her groups of students to find the missing digit in problems like

4� 7

+ 2 8 6

7 0 3

She assisted three students by leading them step-by-step through theproblem. After assisting these students, Naomi explained the algorithmsherself.

We found it interesting that in this lesson, Naomi attempted to havestudents explain their own reasoning. It is also of interest that she had diffi-culty letting them explain. Rather, she used a sequence of leading questions


to lead them to answers. It was our impression that this approach amountedto another way of providing students with carefully sequenced explana-tions of the algorithms. As with the lesson described earlier, this lessonalso confirmed the difficulty she had in designing tasks that would developher students’ understanding. However, it did show that she had begunto experiment with having her students explain their own mathematicalreasoning, possibly a first step in her change.

Although we were concerned about Naomi’s engagement with theprocess of development, it was not until the third author interviewed herthat we became aware of the extent of her difficulties. She told him inMay of the second year, “When I went through teachers college, they wereright into new maths and I cried a year through college. A year of mathsat college put me right off maths and it’s taken me a long time to sortof say it’s okay, I can cope again.” Her responses to questionnaire itemsindicated that she thought of herself as having changed her beliefs, butnot her practices. She believed strongly that learning facts and proced-ures came first and that few of her students would benefit from teachingthat focused on conceptual knowledge. “My confidence as a teacher washeavily undermined due to the project. I was given the impression by thesecondary school teachers that we did no work and did nothing and wereparticularly looked down on” (November, second year).

DISCUSSION

The changes in reported practices and beliefs and the profiles presentedhere suggest that professional conversations can play an important role inprofessional development, although they are not sufficient for all teachers.In this project, they were particularly useful for secondary school teachers.

Changes in Teachers’ Beliefs and Classroom Practices

The value of observing students and reflecting on teaching practices wassimilar to that found in the projects of Carpenter, Fennema & Franke(1996), Schifter & Simon (1992), and Jaworski (1994). The teachersgained considerable insight into students’ thinking when interviewingthem after diagnostic testing. Some, like Tracey, followed the developmentof five students throughout the year to aid their understanding. This greaterknowledge of students appeared to play a major role in changes in beliefsand teaching practice. The finding that secondary school teachers changedtheir beliefs and practices more than did intermediate school teachers mayhave been because the intermediate school teachers started with a better


knowledge of their students, as well as having different teaching practicesand beliefs.

Schifter (1998) discussed the role of reflection in enabling teachers tounderstand students’ concepts and difficulties. In this project it becameevident that reflection was not an easy process. Some teachers did littlemore than describe what they did in lessons while others wrote abouttheir anxieties or commented on how they might meet the difficulties ofparticular students. It appeared that reflection was something that had to belearned. We did not focus on teaching this, although another such projectmight do so.

Teachers’ Understanding of Mathematics

Differences in the degree to which teachers benefited from the projectappeared to be related to their knowledge of mathematics and their capa-city to make connections within mathematics. Although we have no directmeasures of how connected teachers’ mathematical knowledge was, wecould observe this aspect in their attempts to solve problems presented tothe group and in their teaching. The two groups of teachers, who wereso different in their mathematical knowledge, were also different in theirability to make these connections. The studies summarized by Fennema& Franke (1992) did not include teachers who were as diverse as ours.Teachers who had difficulty making connections, and who lacked self-confidence in their mathematical ability, had difficulty helping studentsmake these connections. In Naomi’s case, this also led to difficulty inengaging in professional conversations.

Robyn was an example of a teacher who came to the project with a solidunderstanding of mathematics and its connections. With this knowledge,she was able to move from transmission of information to lessons in whichher students, under her guidance, discovered concepts and found importantconnections. By the end of the project she acted similar to the connectionistteachers described by Askew et al. (1997). Like other teachers with a goodgrasp of the subject, she was ready to try some of the pedagogical strategiesthat were used by the intermediate school teachers.

Both intermediate school teachers had less mathematical knowledgethan the secondary teachers, but their knowledge and their willingnessto extend that knowledge differed markedly. They also had less teachingexperience than Robyn, but we do not believe that this was a major factorfor their different levels of change. Tracey’s desire to fill the gaps in hermathematical knowledge was related to her goal of drawing out the math-ematics from students’ learning activities. At the end of the project, sheshowed evidence that she was able to lead her students to better concep-


tual understanding. However, her limited knowledge in some areas had theresult that at times conceptual knowledge was subordinated to proceduralknowledge. She had most in common with Askew’s (1997) teachers whoemphasized discovery. Her participation in the project led her to valueconnections. Naomi, in contrast, primarily taught by transmission, both atthe start and at the end of the project. From her example, we would suggestthat teachers with limited knowledge of mathematics need a different typeof professional development. The fact that she had not volunteered to parti-cipate was an additional indicator that this was not the type of program sheneeded.

These differences contributed to our assessment of the value ofcombining teachers from different levels. Whereas the secondary schoolteachers seemed to gain pedagogical knowledge from their associationwith the intermediate school teachers, the converse was not often true.The mathematical activities provided at group sessions had a divisiveeffect, in that secondary school teachers dominated discussion of themathematics, and further discouraged teachers like Naomi. Professionalconversations enabled teachers to gain insight into the mathematics of theirstudents and of effective teaching methods, but they were not particularlyuseful in extending conceptual or connected knowledge of mathematics.This finding is consistent with that of Shulman (1987), who worked withsecondary school teachers. He saw the focus of professional developmentto be on teachers’ pedagogical content knowledge, and this would be truefor the secondary school teachers in the project. Without prior knowledgeof mathematical concepts, such a focus appeared to have much less benefitfor other teachers.

Pairing intermediate and secondary schools provided initial advantagesto both parties, especially as the result of visits to one another. However,the organization of each school system made such cooperation difficult tomaintain. Some New Zealand schools are trying to overcome difficultiesin this transition by developing either Year 7–13 schools or Year 7–10schools.

The Issue of Control

When teachers base their teaching on what students know, they give awaysome of their control of what will happen in a lesson. Similarly, in givingcontrol to teachers to determine the nature of the changes they wanted tomake, we had difficulty keeping control of some of the measures of changewe thought important. One aspect of this problem was the collection ofdata on students’ knowledge, especially at the end of a year. This is a busytime for teachers, and although the diagnostic tests had been useful to the


teachers at the start of the year, they had little interest in them at the end ofthe year. Because of the time pressure, teachers like Robyn and Tracey didnot test at the end of the first year. Similarly, a teacher like Naomi, over halfof whose students failed to progress during the first year, was not eager todiscover this for a second time. We could request and remind teachers ofour needs, but, by giving them control of the changes they chose to make,we made it clear that their needs were more important than ours. Hencesome of the data we wanted was not collected. For those teachers who didcollect data on students’ achievement at the beginning and end of eachyear, a marked improvement was evident in progress made in the secondyear of the project in relation to the first year.

There were long-term benefits of this project, evident in teachers’comments at a meeting nine months after the project ended and in laterinformal contacts with researchers. For example, one teacher who didnot believe that she had changed much was later put in charge of themathematics program in her intermediate school. Given this responsibility,she revised her thoughts about the benefit of the project and used manyprocedures from the project in her school.

Professional Conversations and Professional Development

There were many components to this project. We focused on professionalconversations because these were what stood out to many of the teachersand to us as being the most important impetus for development, as wasfound by Clarke (1997). For many teachers, these professional conversa-tions provided an important vehicle for the examination of their beliefsand practices. They encouraged experiments with alternative classroompractices. This encouragement to experiment was crucial for the devel-opment of knowledge about students’ knowledge of mathematics andof beliefs about what practices would further this knowledge. We asresearchers played a role in promoting these professional conversations,but the conversations soon began to take place without our presence. Theconversations were a place where teachers felt safe to talk about what didnot go well, something that rarely was analyzed in their schools wheremost of the discussions among teachers related to organizational matters.

The secondary school teachers appeared to engage in particularlyuseful professional conversations, both in the whole group and in theirhome schools. They discussed what they wanted to get out of particularlessons and why these appeared to work or not to work. They reflectedon the opportunity to talk with individual students and discussed ways inwhich other teachers built this into their practice on a regular basis. Theydiscussed the difficulty of redesigning lessons to give greater emphasis


to students’ conceptual development, and balanced this difficulty with thebenefit of doing so. It was they who initiated the request to spend daystalking among themselves and jointly designing lessons. As two of thesecondary schools chose to send three teachers rather than two, there weremore people within those schools to share the professional conversations.

Although the intermediate school teachers appeared to change less,they also engaged in useful conversations. Despite the fact that onlyfive intermediate school teachers participated in the full project, eachteacher did have a partner from the same school throughout the program.Conversations often happened in the car on the way to and from meet-ings. They occurred when teachers showed the whole group videoclipsof their lessons. The depth of discussion observed for the secondaryschool teachers also occurred for the one pair of intermediate teachers whoparticipated in the entire project.

A question, especially for an education system that cannot afford long-term professional development programs, is how such conversations canbe encouraged and sustained. Our view is that time for professional devel-opment, when administrative issues cannot be discussed, needs to beset aside for teachers. These professional conversations need to involveteachers from more than one school, to encourage discussion of differ-ences. Initially they may need a moderator whose role is to encouragereflection and develop the trust that permits discussion of lessons thatdid not run well. A fellow teacher who is aware of initial difficulties inreflection could do this, if this teacher was willing to share some of his orher difficulties, both in reflection and in teaching.

We also believe that more could be done in pre-service teacher educa-tion to aid future development. A major focus for pre-service mathem-atics teacher-education programs might be to develop more connectionistbeliefs and classroom teaching practices. For example, rather than student-teachers examining strategies for implementing curricula, more attentionmight be given to understanding how the conceptual underpinnings of keyaspects of the curricula connect with each other. We believe that if thiswere done, then when these student teachers began teaching they wouldbe prepared to benefit from involvement in school-based professionaldevelopment where there were opportunities to engage in professionalconversations related to their beliefs about teaching as well as to theirclassroom teaching practices. Our findings suggest that such conversationscan result in teaching that encourages greater conceptual knowledge ofmathematics for their students.


ACKNOWLEDGMENT

We are grateful to the New Zealand Ministry of Education for a grant tosupport this research.

REFERENCES

Askew, M., Brown, M., Rhodes, V., Wiliam, D. & Johnson, D. (1997, July). Effectiveteachers of numeracy in UK primary schools: Teachers’ beliefs, practices and pupils’learning. In E. Pehkonen (Ed.),Proceedings of the 21st Conference of the InternationalGroup for the Psychology of Mathematics Education(Vol. 2, 25–32). Lahti, Finland:PME Program Committee.

Bell, B. & Pearson, J. (1993).The teacher development that occurred: A report ofthe learning in science project. Hamilton, NZ: Centre for Science and MathematicsEducation Research, University of Waikato.

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Britt, M.S., Irwin, K.C., Ellis, J. & Ritchie, G. (1993).Teachers raising achievementin mathematics: Final report to the Ministry of Education. Auckland, NZ: AucklandCollege of Education.

Brown, C.A. & Borko, H. (1992). Becoming a mathematics teacher. In D.A. Grouws (Ed.),Handbook of research in mathematics teaching and learning(209–239). New York:Macmillan.

Brown, M., Askew, M., Rhodes, V., Wiliam, D. & Johnson, D. (1997, July). Effectiveteachers of numeracy in UK primary schools: Teachers’ content knowledge and pupils’learning. In E. Pehkonen (Ed.),Proceedings of the 21st Conference of the InternationalGroup for the Psychology of Mathematics Education 2(121–128). Lahti, Finland: PMEProgram Committee.

Carpenter, T.P., Fennema, E. & Franke, M.L. (1996). Cognitively guided instruction:A knowledge base for reform in primary mathematics instruction.Elementary SchoolJournal, 97, 3–20.

Clarke, D.M. (1997). The changing role of the mathematics teacher.Journal for Researchin Mathematics Education, 28, 278–308.

Cobb, P., Wood, T., Yackel, E, Nicholls, J., Wheatley, G., Trigatti, B. & Perlwitz, M.(1991). Assessment of a problem-centered second-grade mathematics project.Journalfor Research in Mathematics Education, 22, 3–29.

Fennema, E. & Franke, M.L. (1992). Teachers’ knowledge and its impact. In D.A. Grouws(Ed.), Handbook of research in mathematics teaching and learning(147–164). NewYork: Macmillan.

Franke, M.L., Carpenter, T., Fennema, E., Ansell, E. & Behrend, J. (1998). Understandingteachers’ self-sustaining, generative change in the context of professional development.Teaching and Teacher Education, 14, 67–80.

Hart, K., Brown, M., Kerslake, D., Küchemann, D. & Ruddock, G. (1985).Chelseadiagnostic tests. NFER-Nelson: Windsor.


Jaworski, B. (1994).Investigating mathematics teaching: A constructivist enquiry.London: Falmer Press.

Kirkwood, V., Bearlin, M. & Hardy, T. (1989). New approaches to the in-service educationin science and technology in primary and early childhood teachers: (or Mum is not dumbafter all!) Research in Science Education, 19, 174–186.

Leinhardt, G. (1989). Math Lessons: A contrast of novice and expert competence.Journalfor Research in Mathematics Education, 20, 52–75.

Leinhardt, G. & Smith, D.A. (1985). Expertise in mathematics instruction: Subject matterknowledge.Journal of Educational Psychology, 77, 247–271.

Maddern, S. & Court, R. (1989).Improving mathematics practice and classroom teaching.Nottingham, UK: The Shell Centre for Mathematics Education.

Ministry of Education. (1992).Mathematics in the New Zealand Curriculum. Wellington,NZ: Learning Media.

Schifter, D. (1998). Learning mathematics for teaching: From a teachers’ seminar to theclassroom.Journal of Mathematics Teacher Education, 1, 55–87.

Schifter, D. & Simon, M. (1992). Assessing teachers’ development of a constructivist viewof mathematics learning.Teaching and Teacher Education, 8, 187–197.

Shulman, L.S. (1987). Knowing and teaching: Foundations of the new reform.HarvardEducational Review, 57, 1–22.

Snead, L.C. (1998). Professional development for middle school mathematics teachers tohelp them respond to NCTM standards.Journal of Teacher Education, 49, 287–295.

Sowder, J.T., Philipp, R.A., Armstrong, B.E. & Schapelle, B.P. (1998).Middle-gradeteachers’ mathematical knowledge and its relationship to instruction: A research mono-graph. SUNY Series, Reform in Mathematics Education, Ithaca NY: State University ofNew York Press.

Swafford, J.O., Jones, G.A., Thornton, C.A., Stump, S.L. & Miller, D.R. (1999). Theimpact on instructional practice of a teacher change model.Journal of Research andDevelopment in Education, 32, 69–82.

Centre for Mathematics Education, Murray S. BrittAuckland College of Education,Private Bag 92601, Symonds St.,Auckland, New Zealand,E-mail: [email protected]

School of Education, Kathryn C. IrwinUniversity of Auckland,Private Bag 92019,Auckland, New Zealand,E-mail: [email protected]

School of Education, Garth RitchieUniversity of Waikato,Private Bag 3105,Hamilton, New Zealand,E-mail: [email protected]

GEOFFREY B. SAXE, MARYL GEARHART and NA’ILAH SUAD NASIR

ENHANCING STUDENTS’ UNDERSTANDING OF MATHEMATICS:A STUDY OF THREE CONTRASTING APPROACHES TO

PROFESSIONAL SUPPORT1

ABSTRACT. This report provides evidence of the influence of professional developmentand curriculum on upper elementary students’ understandings of fractions. Threegroups of teachers and their students participated. Two groups implemented a fractionsunit that emphasized problem solving and conceptual understanding. The IntegratedMathematics Assessment (IMA) group participated in a program designed to enhanceteachers’ understandings of fractions, students’ thinking, and students’ motivation. TheCollegial Support (SUPP) group met regularly to discuss strategies for implementingthe curriculum. Teachers in the third group (TRAD) valued and used textbooks andreceived no professional development support. Contrasts of student adjusted posttestscores revealed group differences on two scales. On the conceptual scale, IMA classroomsachieved greater adjusted posttest scores than the other two groups, with no differencesbetween SUPP and TRAD groups. On the computation scale, contrasts revealed nodifferences between IMA and TRAD, although TRAD achieved greater adjusted scoresthan SUPP (p < 0.10). Our findings indicate that the benefits of reform curriculum forstudents may depend upon integrated professional development, one form exemplified bythe IMA program.

Reform documents in mathematics education call for a shift from instruc-tion that fosters the practice of procedures and memorization of definitionstoward instruction that emphasizes mathematical inquiry and conceptualunderstanding (California State Department of Education, 1992; NationalCouncil of Teachers of Mathematics [NCTM], 1989, 1991; NationalResearch Council, 1989, 1990). These recommendations require complexchanges in practice, and there is concern that the visions of reform arenot becoming classroom realities despite teachers’ efforts (Ball, 1990b;Cohen, 1990; Heaton, 1992). Research is needed to identify the conditionsthat support effective implementation of reform principles.

The present study was designed to provide bottom-line evidence ofthe influence of professional development programs on student learning.We chose to focus on student learning of fractions for several reasons.First, the domain of fractions is deeply related to other forms of importantrational number concepts, including rates, quotients, operators, measures,percents, and decimals, and therefore is a critical curriculum target for the


56 GEOFFREY B. SAXE ET AL.

upper elementary grades (Behr, Lesh, Post & Silver, 1983; Hart, 1988;Kieren, 1988; Lamon, 1993; Tournaire & Pulos, 1985). Second, fractionsis a domain that poses difficulties for students. Many upper elementarychildren do not understand what fraction symbols represent (Carpenter,Lindquist, et al., 1988; Hart, 1981; Hope & Owens, 1987; Kerslake,1986; Kieren, Nelson & Smith, 1983; Mack, 1990; Nik Pa, 1989; Peck &Jencks, 1981; Post, 1981; Post, Behr & Lesh, 1986; Silver & Carpenter,1990). Third, one plausible source of children’s difficulties with frac-tions may be teachers’ difficulties understanding fractions (Ball, 1990a;Post, Harel, Behr & Lesh, 1991) and the ways that children make senseof fractions in instructional interactions (Lehrer & Franke, 1992; Marks,1990).

Reform Curriculum and Professional Development: Resources forChange?

To date, the social science of professional development is immature(Loucks-Horsley, 1994; Stipek, Gearhart & Denham, 1997). Standards andframework documents can orient teachers to key ideas. Their purpose isto promote new policies, but they rarely provide teachers usable modelsof mathematical inquiry. Consequently, these documents may motivateteachers but not effect change (Cohen, 1990; Heaton, 1992; Prawat,1992; Putnam, 1992; Saxe, Gearhart, Franke, Howard & Crockett, 1999).The adoption of curriculum materials can support teachers’ efforts toimplement educational reforms. Although good curriculum materials canprovide rich tasks and activities that support students’ mathematical inves-tigations, such materials may not be sufficient to enable deep changes ininstructional practice. To guide students in conceptual thinking and theexploration of mathematical conjectures (Ball & Cohen, 1996), teachersmust transform the ways they use curriculum materials with their students.Professional development strategies are designed to support teachers’efforts to transform their practices, but, to date, we have little informationregarding the influence of the materials on student learning. Indeed, prin-ciples for the design of appropriate professional development programsto support effective implementation of reform curricula and pedagogy areonly just emerging.

To understand the conditions that support effective instructional prac-tices, we need to know what works. How do the learning gains of studentstaught with a new curriculum compare with the gains of students taughtwith existing textbooks? If teachers decide to implement a new mathe-matics curriculum, what kinds of professional support programs willenable greater gains in student learning? Do the answers to these ques-

ENHANCING STUDENTS’ UNDERSTANDING 57

tions differ when we examine particular components of student learning,for example, understanding of concepts versus facility with skills? Theseare critical questions for educational policy and practice that requireresearch-based answers.

Professional Development Programs and Student Achievement

The purpose of this study is to understand the ways that professional andcurricular supports for reform implementation may strengthen students’developing knowledge of fractions. In particular, we have focused onstudents’skills with fractions proceduresand understandings of frac-tions concepts. In prior research on children’s learning of mathematics,the distinction between understanding and procedural skill has taken theform of contrasts between procedural and conceptual knowledge (Greeno,Riley & Gelman, 1984; Hiebert & Lefevre, 1986; Silver, 1986), the syntaxand semantics of mathematics (Resnick, 1982), or skills and principles(Gelman & Gallistel, 1978). Each of these contrasts captures in some-what different ways a distinction between (a) knowledge of step-by-stepalgorithmic procedures and memorized facts that can be deployed to solvecomputational problems and (b) knowledge that is conceptual, and richwith understandings of connections between aspects of mathematics.

There exist two prior studies of the influence of professional devel-opment programs on children’s procedural skill and conceptual under-standing, both at the primary level. One is a study of the role of CognitivelyGuided Instruction (CGI), a program focused on enhancing teachers’knowledge of children’s strategies for solving addition and subtractionword problems (Carpenter, Fennema, Peterson & Loef, 1989). With greaterunderstanding of student mathematics, CGI researchers argued, teachersshould be empowered to structure classroom practices in relation to theirstudents’ thinking. The second is a study of the Problem-Centered Mathe-matics Project (Cobb, Wood et al., 1991; Cobb, Wood & Yackel, 1992).Focused on arithmetic and place value, this program is designed to supportteachers’ understanding of children’s mathematics as well as teachers’ ownknowledge of the relevant mathematics.

The findings of the CGI study were impressive: Children whoseteachers participated in the project were more advanced in knowledge ofboth basic arithmetical facts (simple sums) and problem solving strategiesthan a comparison sample of children participating in non-CGI classrooms.Further, as teachers gained expertise with CGI approaches to studentthinking, they created practices that led subsequent cohorts of students toshow even greater improvements than prior cohorts (Fennema et al., 1996).Cobb, Wood, et al. (1991) reported similar findings; students in the project


TABLE I

Groups Participating in the IMA Comparative Study

Group name N Curriculum Staff development

Integrating Mathematics Assessment 9 Reform Knowledge and Assessment

Support 8 Reform Collegial support only

Traditional 6 Traditional None

classrooms surpassed those in comparison classrooms on both conceptualand procedural items.

There is a third study of the influence of professional developmenton elementary children’s learning, but in this study the researchers werelimited to students’ performance on standardized tests, often considered ameasure of basic skills. The Educational Leaders in Mathematics Project(ELM) was designed to enhance teachers’ knowledge of mathematics andknowledge about students’ conceptual struggles with mathematics (Simon& Schifter, 1991). Simon & Schifter (1993) found that the participatingupper elementary students did not lag behind norms on standardized testsdespite the teachers’ focus on conceptually oriented instruction.

Our research builds upon the strategies employed in the prior studies intwo ways. First, we examined the influence of professional developmentprograms on upper elementary level students’ rational number under-standings, distinguishing between (a) achievements involving memorizedfacts and procedures and (b) conceptual knowledge and problem solving.Second, we compared the influence of contrasting professional develop-ment programs on student learning. One program was designed to enhanceteachers’ subject matter knowledge and knowledge of children’s mathe-matics and motivation, and the other was designed to provide opportunitiesfor collegial interaction among teachers; teachers in both programs imple-mented the same curriculum. This design enabled us to determine howstudent learning gains were related to these two contrasting professionaldevelopment programs.

AN INTEGRATED STUDY OF PROFESSIONALDEVELOPMENT AND STUDENT LEARNING OF FRACTIONS

In the research reported here, students in 23 classrooms completed a grouptest of fractions knowledge at the beginning and end of a unit on fractions.The teachers in all classrooms were volunteers, willing to let us document


their practices and to budget time for participation in the project. In twogroups of classrooms, teachers used a reform curriculum unit; in the thirdgroup, teachers used existing, more traditional texts and methods (Table I).

All teachers in the two reform groups usedSeeing Fractions(Corwin,Russell & Tierney, 1990), a unit promoted by the State of California;teachers were selected for participation only if they had previouslyreceived training in the unit and had taught it at least once. These teacherswere provided one of two contrasting programs of professional develop-ment – Integrating Mathematics Assessment (IMA) or Collegial Support(SUPP), both described below. Teachers in the traditional group (TRAD)were chosen for their expressed commitment to textbooks that emphasizedtraditional skills. Our comparative design enabled us to investigate howteachers’ choices of curriculum and their opportunities for professionalsupport may lead to different patterns of student learning. We describe thethree study groups in further detail.

The IMA Professional Development Program

Guided by findings from prior research on classroom practice and studentlearning, we designed the IMA program to address four areas of need:(a) teachers’ understanding of the mathematics that they teach (Ball,1990a; Fennema & Franke, 1992; Post et al., 1991; Shulman, 1987;Thompson, 1992); (b) teachers’ understanding of children’s mathematics(Carpenter, Fennema, Peterson & Carey, 1988; Cobb, Yackel & Wood,1991; Fennema & Franke, 1992; Peterson, Fennema, Carpenter & Loef,1989); (c) teachers’ understanding of children’s achievement motivationsin mathematics (Stipek, Salmon et al., 1998); and (d) the opportunityfor teachers to work with other professionals concerned with effectiveimplementation of reform (Little, 1993; Maher, 1988; Richardson, 1990;Schifter & Simon, 1992; Sparks & Loucks-Horsley, 1989). We integratedthe IMA program with a specific curriculum, viewing curriculum as thecommon ground for productive consideration of mathematics and of theways that children understand and learn. We chose two curriculum units ofimportance to many teachers in California, one on fractions (Seeing Frac-tions) and a second on measurement and scale (My Travels with Gulliver,Kleiman & Bjork, 1991).

The IMA program was organized as a repeating set of activities asdepicted in Figure 1: Teachers’ Mathematics, Children’s Mathematics,Children’s Motivation, and Integrated Assessment. Each set of activitiesfocused on one key lesson in one of the curriculum units; the lessons weselected focused on core concepts in area and linear models of fractionsor in measurement and scale. We began with a 5-day summer institute,


Figure 1. Activities in the IMA professional development program.

followed by 13 meetings – a meeting held approximately every 2 weeksduring the year (12 evening meetings and one full Saturday meeting).Because this report is concerned with student outcomes only in the domainof fractions, we will not provide further information on the measurementand scale components.

Teachers’ Mathematicssupported teachers’ construction of sophis-ticated understandings of fractions, measurement, and scale. Because


teachers’ knowledge of mathematics should be deeper than the contentof the curriculum they are teaching, these activities were more complexinvestigations than those inSeeing Fractions. Linked to big ideas regardingfractions concepts and strategies for solvingSeeing Fractionsproblems,each activity provided teachers opportunities to participate as learners inpractices reflective of documents like theStandards(NCTM, 1989). Atany given session, teachers might work independently to solve an open-ended problem, and then analyze differences among their methods in smallgroups; or teachers might work collaboratively in pairs or small groups,and then consider how their separate contributions to problem solvingbenefited their learning and the quality of their solutions. For example, inone activity, teachers were asked to play the role of a pizza store managerand propose a strategy for distributing leftover pizza to the homelesseach evening; teachers worked in pairs to partition sets of partially eatenpizzas (sets of fractional parts of units such as 3/4 of a circle or 2/3 of arectangle) into fair shares. This activity was a more challenging versionof the lessons for elementary students, where students partition a wholenumber of cookies (circles) or brownies (squares) into fair shares. Afterthe activity, the facilitator engaged the teachers in reflection on part-wholerelations, and relationships among different representations of fractions.At the conclusion, teachers were invited to step back into their roles asteachers and to reflect on practices they had just participated in as learners.

The Children’s Mathematicscomponent was designed to enhanceteachers’ knowledge of children’s mathematical thinking and to fosterteachers’ interest in the assessment of that thinking. Each activity waslinked to a Teachers’ Mathematics activity (and thus to aSeeing Fractionslesson). We presented samples of students’ written work or videotape snip-pets of children solving fractions, measurement, and scale problems (cf.Cobb, Wood et al., 1991); these resources were drawn either from pilotclassrooms or from individual interviews with children. In the sessions, weengaged teachers in quests to understand children’s efforts to solve mathe-matical problems; explain concepts and strategies; and use, interpret, andrelate different mathematical representations. We shared with teachers thegeneral pattern of children’s developing understandings of fractions. Ourprogram contrasts with the groundbreaking Cognitively Guided Instruc-tion program (Carpenter et al., 1989) in that the complexity of the domainof fractions and the diversity amongSeeing Fractionsproblems requiredthat we identify developmental issues, rather than a sequenced model ofdevelopment. Thus one theme addressed the ways that children use theirunderstandings of whole numbers and correspondence relations to solvefractions problems. For example, a child may divide a quantity represented


as a circle or square into four unequal parts and give each of four people“one of these” by creating a piece-to-person one-to-one correspondence.Another theme emphasized the challenges children face in their efforts tocoordinate the meanings of diverse forms of representation. For example, achild who is asked to partition a set of 12 cookies into fair shares for eightpeople might produce one solution with a diagram (“one whole cookie andone of these” [a half]) and another solution with numbers (“8 into 12 is1 R4”); each representation affords the child certain interpretations andconstrains others.

The Children’s Motivationcomponent provided teachers with back-ground on children’s orientations to learning in classroom settings. Moti-vational orientations discussed included (a) beliefs about ability (e.g., asstable and uncontrollable vs. flexible and influenced by effort), (b) percep-tions of competence and self-efficacy in mathematics, (c) goals (e.g., todevelop understanding vs. to perform), (d) perceptions of the usefulnessof mathematics outside of the classroom, (e) interest in and enjoymentof mathematics activities, and (f) emotions associated with mathematics(e.g., shame, fear, anxiety, pride). The primary emphasis was supportingteachers’ assessment of student motivation, with secondary emphasis ondesigning strategies for addressing the motivational problems identified.

The goal of theIntegrated Assessmentcomponent was to enhanceteachers’ competence with assessment that builds upon students’ thinking.We focused on a range of practices: Whole class discussions (e.g., how tointerpret and address “wrong” answers); observation, inquiry, and guid-ance during student activities (e.g., how to focus observation on a keydevelopmental issue); assessment of students’ written work (e.g., samplerubrics); peer problem-posing and peer assessment; and portfolio assess-ment. Teachers analyzed these practices, role played, piloted assessmenttools, and shared assessments of their own design.

Support Program

The Support Program (SUPP), like the IMA program, provided teachers anopportunity to participate with a community of practitioners implementingthe two targeted curriculum units. It was an approach to professional devel-opment that was promoted in the Greater Los Angeles area at the time ofour study; the goal was to provide teachers opportunities to reflect on theirpractices with teachers engaged in similar efforts (Little, 1993; Loucks-Horsley, 1994; Maher, 1988; Richardson, 1990; Schifter & Simon, 1992).Unlike IMA, the Support program offered no focused help with subjectmatter, children’s mathematics, or reform-minded approaches to instruc-tion. Although many issues that were the target of our IMA intervention


were brought up in the Support group, none of these issues became a focusof sustained inquiry and discussion.

Support teachers met nine times during the year; they began theirwork on each curriculum unit with a full day session and continued withseveral monthly evening meetings. Topics were suggested by the teachers,and the facilitator focused these topics around the same key lessons inSeeing Fractionsand My Travels with Gulliverthat were a focus ofIMA sessions. Beyond that role, the facilitator supported the teachers’agendas by helping everyone stay on topic and by sending remindersabout the new topic before the next meetings. In some support meetings,teachers discussed particular practices: instructional methods appropriatefor specific lessons; the role of manipulatives; assessment methods suchas portfolios and open-ended tasks; and homework. At other meetings,teachers raised issues about the curriculum units, for example, concernsthat there were no correct answers to many problems, conflicts betweenthe curriculum and what was tested in the teachers’ school districts, andconcerns about the reduction of attention to skills. Each month, teachersbrought relevant curriculum materials and students’ work to share. Some-times teachers shared approaches that they felt were successful, andcolleagues considered whether those methods were applicable in theirown contexts. At other times, teachers shared methods that were notsuccessful, or they showed dilemmas that they were experiencing, andsolicited guidance.

Traditional Classrooms

The TRAD teachers were committed to use of textbooks. They did notparticipate in a professional development program. Like the IMA andSupport teachers, TRAD teachers were experienced and were volunteersin the project. They were willing to have their practices documented andto budget time for project participation.

Assessing Students’ Knowledge of Fractions

Our assessments of student achievement in the domain of fractions weredesigned to measure students’ performance on items requiring computa-tional skills and items involving conceptual understanding. We recognizedthat the distinction between computation and conceptual understandingis somewhat problematic, despite the utility of the distinction in under-standing student achievement. Indeed, a child might solve what weregarded as a computation task with conceptual understanding, or mightsolve what we classify as a conceptually oriented item using a memorizedsolution. However, the items that we constructed provided a heuristic-


ally useful way to measure students’ skills with fractions and problemsolving with fractions. The computational items could be readily solvedwith routine algorithmic procedures or commonly memorized facts. Theconceptually oriented items could not readily be solved by such proceduresand generally required insight into mathematical relations involving frac-tions. We validated the distinction between the two groups of items withour content analysis as well as confirmatory factor analytic techniques.

METHOD

Study Design and Teacher Characteristics

Volunteers were solicited through mailings to upper elementary teacherswithin a 40-mile radius of UCLA. Two letters were distributed. One letterrequested applications from teachers who had experience withSeeingFractionsandMy Travels with Gulliver; a second letter requested appli-cations from teachers committed to teaching with traditional textbooks.Both letters informed recipients that the study would contribute insightsregarding the role of curriculum in children’s understandings of frac-tions, measurement, and scale; they were also informed that the studyrequired a commitment of one school year. Applicants were asked tocomplete a pre-screening questionnaire regarding: (a) curriculum (use ofSeeing Fractions, My Travels with Gulliver, and textbooks); (b) yearsof teaching experience; (c) degrees and certificates; (d) participation inprofessional development workshops in mathematics education; (e) gradelevel(s) taught and currently teaching; (f) student characteristics at theirschool; and (g) availability for participation in professional development.Teachers who responded were interviewed to confirm and clarify theirresponses.

From the respondent pool we selected teachers who (a) were willing tocommit to participation in the project for the year (for example, budgettime, allow their practices to be documented) and (b) had a history ofusing traditional texts or the two state adopted reform units. We assignedto the TRAD group teachers who had used and planned to continue to usetraditional texts; these teachers were chosen for their expressed commit-ment to textbooks emphasizing fraction skills. None of the TRAD teachershad been trained in or taught either of the two reform replacement units.We used a stratified random assignment procedure to assign the IMA andSUPP teachers. The sample of volunteers who met our curriculum criteriavaried on characteristics that were plausibly related to instruction, forexample, prior participation in professional development linked to reform


TABLE II

Years of Teaching Experience

Years

Mean Range Participants

IMA 16.7 1–26 9

SUPP 13.4 3–22 8

TRAD 20.7 4–34 6

and number of years teaching. A simple random assignment procedurewas inappropriate because, with a small sample, the groups might beunbalanced with respect to these characteristics. We describe the groupcharacteristics of IMA, SUPP, and TRAD teachers.

Years of experience. Mean and range of number of years of teaching expe-rience and number of participants for the three groups are summarized inTable II.

Experience with the reform units. Almost every teacher in the IMA andSUPP group had been (a) trained in both the fractions and the measure-ment/scale unit, and (b) had previously taught each unit. There were threeexceptions: One of the IMA teachers was not trained in the fractions unitalthough she had taught it, and two of the IMA teachers had not taught themeasurement/scale unit although they had participated in training.

Additional professional development. IMA and SUPP teachers werematched for the extent of their participation in recent mathematics reformworkshops: We created a scale from 0 to 2 for “additional participationin professional development activities” by assigning one point for trainingin any other “reform” curriculum unit and one point for any other profes-sional development in mathematics education; the mean for IMA teacherswas 1.3 (range 0–2) and for SUPP teachers 1.1 (range 0–2). The mean forTRAD teachers was 0.6 (range 0–1); at the time of our study, there existedfew professional development opportunities for teachers committed to askills approach to mathematics teaching.

Student Participants

Table III shows background data for each study group: median gradelevel and indicators for English fluency and knowledge of fractions.


TABLE III

Classroom and Student Characteristics

Classroom type

Classroom / student characteristics IMA SUPP TRAD

Median grade level 5 4/5 5

Proportion of classrooms containing some 0.67 0.63 0.71

students not fluent in English

Proportion of classrooms containing more 0.11 0.63 0.29

than 25% of students not fluent in English

The measure of English fluency was the proportion of students in eachclassroom who were rated3 or 4 on a four-level rating of fluency andcapacity to participate in English-only instruction; our ratings were derivedfrom the school’s categorical assignment as well as teachers’ judgments.Ethnicity of children varied in the study. In the entire sample, 64% wereLatino, 14% were White, 8% were African American, and 7% were Asian.Because there was uneven distribution of English fluency across groups,we adjusted for language background statistically in our analyses.

Assessment Instrument

To document children’s fractions understandings, we developed a paper-and-pencil test that contained both computation and more conceptuallyoriented items. Resources for the construction of items included texts inuse in our area as well as more reform-oriented curricula. For the compu-tation items, we selected recurring problem types from standard texts,pilot tested these items for clarity and difficulty, and winnowed our poolto a manageable number given the time constraints of our assessment.We applied a similar procedure to generate conceptually-oriented items.Project staff members administered the paper-and-pencil test to students inall participating classrooms both before and after the intervention. Whenappropriate, students used a Spanish translation of the test. The duration ofthe test was about 40 minutes.

Based upon an item analysis of the fractions test, we created twosubscales, one that contained the computation items and the otherthat contained conceptually oriented items. These items are listed inAppendix A.

We used a confirmatory factor analysis to evaluate the appropriatenessof distinguishing the conceptually-oriented and computational items as


two separate scales. We entered all item types in a three-factor model.We interpreted the first factor as a measure of general fractions know-ledge, the second as a measure of computation skills, and the third as ameasure of conceptual understanding. The general fractions knowledgescale included all computation and conceptual items. All items are repro-duced in Appendix A. Cronbach’s alpha indicated internal consistency foreach scale: For the conceptual scale, the indices were 0.73 (pretest) and0.83 (posttest); for the computation scale, the indices were 0.86 (pretest)and 0.87 (posttest).

The confirmatory factor analysis showed strong support for the scales.For the posttest, the confirmatory factor analysis resulted in a chi-square(df = 11) of 17.254,p = 0.10058; all fit indices were high (Bentler-BonettNormed fit index = 0.984, Bentler-Bonett Nonnormed fit index = 0.985,Comparative fit index (CFI) = 0.994). When the model was applied topretest data, the confirmatory factor analysis resulted in a chi-square (df= 11) of 19.1,p = 0.059. Again, all fit indices were high (Bentler-BonettNormed fit index = 0.981, Bentler-Bonett Nonnormed fit index = 0.979,Comparative fit index (CFI) = 0.992). Analyses of the pretest data indi-cated a linear dependency, a problem that probably resulted from a heavilyskewed distribution towards the floor, given lack of prior instruction infractions. Overall, our analyses indicated that the two sets of items wereindexing independent areas of competence in children.

Data Analysis

To document children’s learning as indexed by the computation andconceptual scales, we conducted two types of analyses. First, we analyzedpre- to posttest gain in achievement for all classrooms. Because all chil-dren were receiving instruction in fractions, we expected to find evidenceof achievement from pre- to posttest across classrooms. Second, weexamined students’ posttest scores associated with teachers’ professionaldevelopment GROUP. In this analysis, we contrasted IMA, SUPP, andTRAD classrooms. Here our focus was whether teachers’ group affilia-tion was associated with differences in student posttest achievement whencontrolling for pretest scores and language background.

Our next step was to analyze whether there were differences in studentachievement as a function of professional development GROUP. We hadtwo options. One was to use student level data: We could analyze students’scores as a function of the professional development group of their teachers(IMA, Support, Traditional), using posttest performance on the concep-tual and computation scales as dependent variables and pretest scoresand language background as covariates. Such an approach has various


merits, one of which is that it provides considerable statistical power.However, because subsets of children were in the same classrooms andthus instructed by the same teachers, the student achievement outcomeswithin classrooms could be expected to be correlated, violating a coreassumption of ANOVA designs.

We chose a different approach that provided less statistical power butwas more appropriate, given the properties of our data. We aggregatedstudent scores by classrooms, taking mean classroom scores on the pre-and posttest conceptual and computation scales as our dependent vari-ables. We then used an ANCOVA procedure with classroom mean posttestscores (conceptual and computational) as dependent variables, GROUP(IMA, SUPP, and TRAD) as the independent variable, and classroom meanpretest score and English Language Fluency scores as covariates.

RESULTS

Change of Performance Within Groups

To provide preliminary evidence of student gain from pre- to posttestperformance in each teacher group, we calculated mean gain scores forboth the computation and conceptual scales for each classroom. Everyclassroom, regardless of group, showed gains on both the conceptual andcomputation scales. For the conceptual scale, classroom gains varied frommeans of 0.39 to 4.56 (mean = 2.53, s.d. = 1.26). For the procedural scale,classroom gains varied from means of 0.25 to 7.55 (mean = 4.22, s.d. =1.92).

Change of Performance Across Groups

As noted in our data analysis section, we used ANCOVAs to determinewhether posttest scores of classrooms differed as a function of professionaldevelopment group, adjusting for pretest scores and language background.

The ANCOVA on theconceptualscale revealed a main effect forGROUP (F(2,18) = 7.21,p < 0.005). The overall means and standarddeviations of groups for IMA, SUPP, and TRAD were 6.17 (0.89), 4.73(1.0), and 4.10 (0.68), respectively. Tukey-HSD post hocs (p < 0.05) onadjusted scores revealed that the IMA classroom means were greater thanboth the SUPP and the TRAD classroom means.

The ANCOVA on the computation scale did not reveal an effect forGROUP at conventional levels of significance (p < 0.05), although therewas a trend (F(2,18) = 2.82,p = 0.086. Although this difference did notachieve the 0.05 alpha level, it is nonetheless worthy of note in light of the


Figure 2. Adjusted posttest means on the conceptual scale for IMA, SUPP, and TRAD(classroom level data).

relatively small sample (reduced statistical power) and the added supportthat SUPP group received. The overall means (and standard deviations)of groups on the computation scale for IMA, SUPP, and TRAD, were7.32 (1.78), 6.01 (1.40), and 8.36 (1.92), respectively. Tukey-HSD posthoc comparisons (p < 0.05) on the adjusted scores revealed a significantdifference between TRAD and SUPP groups: TRAD classrooms achievedgreater scores than the SUPP classrooms on the computation scale.2

Figure 2 contains boxplots for the conceptual scale scores for each levelof GROUP. The plot shows that only a small portion of the distribution ofthe IMA classroom means for the conceptual scale overlapped with theTRAD classrooms. Further, one IMA classroom was an outlier, achievingconsiderably lower scores than all of the others (technically, between 1.5and 3 box-lengths from the lower edge of the box). The performances ofthe SUPP classrooms appeared more similar to the TRAD classroom thanto the IMA classrooms; we detected no differences between the SUPP andthe TRAD classrooms on the conceptual scale.

The boxplots for the computation scale by GROUP are containedin Figure 3. The comparison of the computation scale plots with theconceptual scale plots reveals a markedly different pattern of achievement.In particular, there is substantial overlap between the IMA and TRADclassrooms on the computation scale.

We summarize the results for student achievement by focusing onthe role of professional development support in the implementation of


Figure 3. Adjusted posttest means on the computation scale for IMA, SUPP, and TRAD(classroom level data).

reform curriculum. The IMA program was associated with greater studentachievement on the conceptual items: Achievement on the conceptual scalein IMA classrooms was greater than in SUPP and TRAD classrooms. Atthe same time, the IMA program was associated with student achievementon the computation items that did not differ from TRAD classrooms;TRAD achievement on the computation scale were greater than SUPP.These findings point to the advantages of the reform curriculum whensupported by the IMA professional development program. They also pointto the problems with reform curriculum when such curricula are notaccompanied by focused supports for teachers’ subject matter knowledge,knowledge of children’s mathematics, and the implementation of reform-oriented pedagogical practices. We discuss these results in the followingsection.

DISCUSSION

This study addresses a critical issue in mathematics education reform –the effects of reform curriculum and support for its implementation onstudent achievement. To address this issue, we selected the domain offractions in the upper elementary grades, and designed a study that allowedus to compare the effects of instruction with more traditional curriculumto instruction with reform-oriented curriculum under two conditions ofprofessional development support. Our findings revealed that, although


most participating classrooms showed increases on both the conceptualand computation scales from pre- to posttest, the patterns of gains instudent learning differed among our three participating study groups.

Importance of Professional Development Program

It was the premise of our study that effective implementation of reformcurriculum requires integrated and ongoing professional development. Theliterature on professional development that existed at the outset of ourstudy raised concerns about the training model that was the only optionavailable to teachers usingSeeing Fractions. The California State Depart-ment of Education encouraged districts and county offices to provide one-or two-day training to all interested teachers, and most of the IMA andall of the Support teachers had completed one of these sessions prior toour study. We felt that such brief exposure to the curriculum activitieswould not provide enough guidance with the mathematics, enough insightinto the ways that children interpret the mathematics, or enough practicewith new pedagogies to significantly impact student achievement. Thusboth the Integrating Mathematics Assessment and the Collegial Supportprofessional development program were designed to provide teachers theopportunity to deepen their expertise with the curriculum, not to train them.With experience in having taughtSeeing Fractionsat least once before, theteachers viewed the IMA and Support programs as the next step.

The Support and IMA programs reflect professional developmentstrategies that have been promoted in theory and in practice. Engenderedby notions of professional communities of practice, the Support programprovided teachers a context for reflective discussions with colleagues, aplace where they could collectively identify problems in teaching and sharesuccessful strategies. IMA teachers participated in activities designed toenhance their knowledge of the curriculum, the mathematics underlyingthe curriculum, and the ways that children understand and solve tasks in thedomains targeted by the curriculum, an approach similar to the CognitivelyGuided Instruction program (Carpenter et al., 1989), the Problem-CenteredMathematics Project (Cobb, Wood et al., 1991), and the EducationalLeaders in Mathematics Project (Simon & Schifter, 1993).

The three programs just cited have documented positive effects onstudent achievement. Our study extends findings from this body ofresearch. Building on the prior studies, our design included two controlgroups of teachers, one provided no professional support and the other aless intensive (but valued) professional support program. In addition, weadministered an assessment that included both conceptual and computationitems to students in the upper grades, thus adding to Schifter & Simon’s


(1992) upper elementary findings based on standardized test scores. Thusour study represents an important next step in the design of research on theroles of professional support and curriculum in student learning.

Our findings revealed both the problems and prospects for ongoingeducational reforms and the key role of professional support in thesereforms. Contrasts between IMA and Support classrooms showed greatergains for IMA classrooms on the conceptual scale, but no differencesbetween these groups on the computation scale. We attribute the strongerconceptual performance of the IMA students to the greater opportunitiesthe IMA program provided teachers to enhance their understanding ofmathematics, the ways that children make sense of mathematics, and thepedagogies that integrate assessment of student thinking with instruction.Support teachers touched on some of the same issues in their meetings, butthese issues never became the focus of sustained discussion and explora-tion. In a companion paper (Gearhart et al., 1999), we provide evidencethat the differences in professional support were realized in teachers’classrooms. Using observational data, we found evidence that IMAclassrooms showed significantly greater scores than Support classroomson a measure of opportunity for students to engage in conceptual analysisof fractions problem solving.

Contrasts with Traditional Classrooms

Our findings contrasting classrooms implementing reform curricula withclassrooms using traditional texts provide a window into the potentialpromise and problems with ongoing reforms. We found that studentachievement did not necessarily benefit from use of reform curriculum.Indeed, when we contrasted student achievement on computation items inthe Traditional vs. Support classrooms, we found that use of the reformcurriculum materials in the Support classrooms was associated with lessgain in skill with computation. Traditional and Support classrooms didnot differ in achievement on the conceptual items; both groups achievedless than IMA students, a pattern that suggests that reform curriculummay not necessarily advantage children’s problem solving compared withtext-based instructional practice.

The contrasts between Traditional and IMA groups lead us to be morehopeful regarding the potential of reforms to enhance student learningunder certain conditions of support for the professional development ofteachers. The use of reform curriculum when implemented with focusedsupport for teachers’ knowledge (1) may lead to gains in student concep-tual understanding greater than traditional practices, and (2) may not leadto performance decrements in computational skills.


CONCLUDING REMARK

We recognize that the IMA program was complex, providing support ina number of key areas, including teachers’ understanding of the subjectmatter, children’s conceptual struggles with the subject matter, children’smotivations, and integrated assessment. Our study cannot identify the rolesof any one of these components or their interaction in student learning.Further, our sample of classrooms was small, necessarily so, given limitedresources. Future research of broader scope is needed to identify charac-teristics of professional programs that enable teachers to implement newpractices in ways that enhance student learning.

ACKNOWLEDGEMENT

Francine Alexander, Tom Bennett, Randy Fall, Elana Joram, David Niemi,Steve Rhine, Michael Seltzer, and Tine Sloan aided in the project.

APPENDIX A

COMPUTATION ITEMS

V3: Procedural: Computation (adding and subtracting values) [3, 4, 5, 6,7, 8, 9, 21a]

3) 35 4) 2

10 5) 13 6) 75

8

+ 15 + 2

5 + 12 + 41

2

7) 710 8) 5

6 9) 23

– 110 – 1

3 – 12

21) John ran25 of a mile on Thursday and35 of a mile on Friday. How far did he run

altogether on the two days?

V4: Procedural: Fraction equivalencies [14a, 14b, 14c, 14d]

14) Write one fraction that is the same as each fraction below,

for example:12 = 24

a. 26 = b. 1

5 = c. 1216 = d. 7

6 =


V5: Procedural: Computation with values in pie [20]

20) Circle a, b, c, or d below to show what part of this circle is gray:

a. 12 + 1

3 b. 36 + 1

6 c. 1+ 13 d. 4

V9: Procedural. Missing Value Equivalence Problems [18a, 18b, 18c 18d]

18) Fill in the missing numbers:

a. 15 = 10 b. 3

4 = 8 c. 212 = 2 d. 31

4 = 8

PROBLEM SOLVING ITEMS

V6: Conceptual: Constructing Fractions for Unequal Parts of Wholes [1e,1f, 1g]

1) For each picture below, write a fraction to show what part is gray:

e. f. g.

V7: Conceptual: Estimating Fractional Parts of Areas [2a, 2b]

2) Circle the fractions that show what part of each circle below is gray:

a) 14

35

910 b) 1

913

25


V8: Fair Share Problems [15a, 15b, 16a, 16b, 17a, 17b]3

15) a. Four people are going to share these two pizzas equally. Color inoneperson’spart.

b. Write a fraction that shows how muchoneperson gets .

16) a. Threepeople are going to share these pizzas equally. Color inoneperson’spart.

b. Write a fraction that shows how muchoneperson gets

17) a. Sixpeople are going to share these five chocolate bars equally. Color inoneperson’s part.

b. Write a fraction that shows how muchoneperson gets .


V10: Procedural/Conceptual: Graphical Depiction of ComputationalWord Problem [21b]

21) John ran25 of a mile on Thursday and35 of a mile on Friday. How far did he run

altogether on the two days?

b. Draw a picture to show your work

NOTES

1 The research reported in this article received support from the National Science Found-ation, grant No. MDR 9154512, and the first author received additional support from theSpencer Foundation, Grant No. M-89-0224. The findings and opinions expressed in thisreport do not reflect the position or policies of either organization.2 To corroborate the classroom level analyses, we repeated our analyses with student leveldata. The analyses yielded similar results. For the conceptual scale, the ANCOVA yieldeda significant main effect (F(2,513) = 27.00,p < 0.0001) with means of 6.14, 4.72, and4.00 for the IMA, SUPP and TRAD groups, respectively. For the computation scale, theANCOVA yielded a significant main effect (F(2,513) = 16.71,p < 0.0001) with means of7.26, 5.88, and 8.26, for the IMA, SUPP and TRAD groups respectively.

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Hope, J.A. & D.T. Owens (1987). An analysis of the difficulty of learning fractions.Focuson Learning Problems in Mathematics, 9, 25–40.

Kerslake, D. (1986).Fractions: Children’s strategies and errors, a report of the strategiesand errors in secondary mathematics project. Windsor, England: NFER-Nelson.

Kieren, T.E. (1988). Personal knowledge of rational numbers: Its intuitive and formaldevelopment. In J. Hiebert and M. Behr (Eds.),Number concepts and operations in themiddle grades(162–181). Reston VA: The National Council of Teachers of Mathematics.

Kieren, T.E., D, Nelson & G. Smith (1983). Graphical algorithms in partitioning tasks.Journal of Mathematical Behavior, 4, 25–36.

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Lehrer, R. & M.L. Franke (1992). Applying personal construct psychology to the study ofteachers’ knowledge of fractions.Journal for Research in Mathematics Education, 23,223–241.

Little, J.W. (1993). Teachers’ professional development in a climate of educational reform.Educational Evaluation and Policy Analysis, 15, 129–151.

Loucks-Horsley, S. (1994, November). Teacher change, staff development, and systemicchange: Reflections from the eye of a paradigm shift. Prepared for Reflecting on ourwork: NSF Teacher Enhancement in Mathematics K-6. Arlington, VA.

Mack, N.K. (1990). Learning fractions with understanding: building on informal know-ledge.Journal for Research in Mathematics Education, 21, 16–32.

Maher, C.A. (1988). The teacher as designer, implementer, and evaluator of children’smathematical learning environments.Journal of Mathematical Behavior, 6, 295–303.

Marks, R. (1990). Pedagogical content knowledge: From a mathematical case to a modifiedconception.Journal of Teacher Education, 41, 3–11.

National Council of Teachers of Mathematics (1989).Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1991).Professional standards for teachingmathematics. Reston, VA: Author.

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National Research Council (1990).Reshaping school mathematics: A framework forcurriculum. Washington, D.C.: National Academy Press.

Nik Pa, N.P. (1989). Research on children’s conceptions of fractions.Focus on LearningProblems in Mathematics, 11, 3–25.

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Peterson, P.L., E. Fennema, T.P. Carpenter & M. Loef (1989). Teachers’ pedagogicalcontent beliefs in mathematics.Cognition and Instruction, 6, 1–40.

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Post, T.R., M. Behr & R. Lesh (1986). Research-based observations about children’slearning of rational number concepts.Focus on Learning Problems in Mathematics, 8,39–48.

Post, T.R., G. Harel, M.J. Behr & R. Lesh (1991). Intermediate teachers’ knowledgeof rational number concepts. In E. Fennema, T.P. Carpenter & S.J. Lamon (Eds.),Integrating research on teaching and learning mathematics(177–198). Albany: StateUniversity of New York Press.

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Schifter, D. & M.A. Simon (1992). Assessing teachers’ development of a constructivistview of mathematics learning.Teaching & Teacher Education, 8, 187–197.

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Silver, E.A. (1986). Using conceptual and procedural knowledge: A focus on relationships.In J. Hiebert (Ed.),Conceptual and procedural knowledge: The case of mathematics(181–198). Hillsdale, NJ: Lawrence Erlbaum Associates.

Silver, E.A. & T.P. Carpenter (1990). In M.M. Lindquist (Ed.),Results from the fourthassessment of the National Assessment of Educational Progress(10–18). Reston, VA:National Council of Teachers of Mathematics.

Simon, M.A. & D. Schifter (1991). Towards a constructivist perspective: An interventionstudy of mathematics teacher development.Educational Studies in Mathematics, 22,309–331.

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Graduate School of Education, Geoffrey B. Saxe4315 Tolman Hall, Maryl GearhartUniversity of California, Berkeley,Berkeley, CA 94720-1670,E-mail: [email protected]

School of Education, Na’ilah Suad NasirStanford University,485 Lasuen Mall,Stanford, CA 94305,USAE-mail: [email protected]

TEACHER EDUCATION AROUND THE WORLD

PRIMARY MATHEMATICS TEACHER EDUCATION IN GREECE:REALITY AND VISION

DESPINA POTARI

ABSTRACT. This paper focuses on two main issues concerning the mathematics educa-tion of prospective primary school teachers in Greece: the integration of mathematics andpedagogy and the relation of theory to practice. In particular, specific teaching approachesare discussed concerning the problem of integration both in mathematics and in mathe-matics education courses. The problem of “theory – practice” is examined through ananalysis of the kind of teaching practice in which the prospective teachers are involved.Finally, the constraints that the mathematics educators face and the impact of their workon the professional life of the teachers in the future are discussed.

Since 1983, universities in Greece have been responsible for the educa-tion of prospective primary school teachers. This means that to becomea teacher in a primary school, one has to complete a four-year degreeprogram rather than pursue a two-year diploma. Apart from the longerduration of study, the nature of the courses in universities is scientificand theoretical rather than practical. This reform had a big educational,political, and social impact on teacher education (Stamelos, 1999).

Nine new departments in different areas of Greece have been estab-lished. The autonomy of these departments brought into effect a differentkind of mathematics education. The nature of this education depends onthe educational policy of the departments, on the educational and mathe-matical background of the mathematics educators, and on the mathematicseducators’ research interests and views about the needs of future teachersconcerning mathematics, its teaching, and its learning.

Questions about what prospective teachers’ mathematical and profes-sional development means, how it relates to actual teaching practice, andwhat kinds of support the teachers will need in order to continue to developin their professional life are still open (Civil, 1993; Cooney & Krainer,1996). The ways that we, as mathematics educators, approach these ques-tions shape our decisions and goals with respect to the mathematicseducation experience we offer to our students. Moreover, the commondilemma about whether we prepare teachers for the realities of the existing


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classroom or for the ideal school acquires a different dimension in theGreek educational context because the gap between the two poles is verywide. The problem becomes more apparent when our students becometeachers and experience a reality shock (Weinstein, 1988). Do they survivethis shock and, if so, how? Are the teachers’ experiences at the universityappropriate for the development of their own effective practices, or dothey copy existing practices? Do teachers see their own development as acontinuous and difficult process, or do they finally feel safe in the familiartraditional setting?

Moreover, the apprenticeship model of teacher development, whichappears to work to some extent in some countries, for example, Britain,is not effective here, as school teaching is very much constrained bytraditional approaches to exposition. This underscores the problem of howtheory and practice can interact in mathematics education programs in sucha way that prospective teachers are able to envisage alternatives to mathe-matics teaching which are closer to children’s learning needs. It is alsoimportant that teachers are actually able to implement such approacheswithin existing school practices.

Other important issues in relation to the mathematics education ofprospective teachers in Greece are their poor mathematical backgroundsand their negative attitudes towards mathematics (Troulis, 1995). Moststudent teachers will not have taken mathematics as an examination subjectin order to gain admission to a university. Some of them are surprisedwhen they realise that they have to take some mathematics courses at theuniversity. Common views that prevail among student teachers are that “itis enough to know the mathematics that is taught in primary school,” andthat “what is needed is to know only how to teach mathematics.” Theseviews also reflect those held by university teachers, which are often dual-istic: “what prospective teachers need to learn is mathematics,” or “whatthey need is to learn how to teach mathematics.” Furthermore, studentteachers’ view of mathematics is constrained by a rather instrumental,product-oriented school experience.

This creates a vicious circle, wherein prospective teachers carry theirpersonal theories about mathematics and mathematics teaching from theirown schooling, and the current school reality does not challenge theirprior experience. The fact that many of those teachers reproduce existingpractices is the main concern of Greek mathematics educators. I attemptto show how mathematics teacher educators confront the above problemwithin the constraints of their departmental environments and the broadereducational and social context. I discuss issues from a range of sources: myexperience as a mathematics educator; discussions I have had with some of

TEACHER EDUCATION AROUND THE WORLD 83

my colleagues from other departments; the content and philosophy of thecourses offered; my colleagues’ responses to a questionnaire; the teachingmaterials used in the courses (for example, student teachers’ notes, text-books); case studies of some student teachers who have graduated from mydepartment; and informal discussions with graduates from other educationdepartments. These different sources of data are intended to compensatefor the fact that I have not had the opportunity to have personal contactwith all my colleagues about this issue.

RELATING MATHEMATICS TO PEDAGOGY

The integration of mathematics and pedagogy is supported in the researchliterature and is viewed in different ways (Cooney, 1994; Even, Tirosh &Markovits, 1996). Formally, in all departments, separate courses in mathe-matics and mathematics education are offered. Mathematicians (who havea Ph.D. in mathematics) usually teach the mathematics courses whereasmathematics educators (who have a Ph.D. in mathematics education) teachthe mathematics education courses. Table I lists the number of mathem-aticians and mathematics educators who work in the nine departments andthe number of compulsory courses that are offered. For example, in Depart-ment 6, one mathematician teaches two compulsory courses in mathe-matics, and one mathematics educator teaches one compulsory coursein mathematics education. The course “Teaching methods in primaryschool” is a mathematics education course, whereas the course “Units fromschool mathematics curriculum” focuses on a mathematical analysis ofunits in the mathematics school curriculum. In Departments 1–3, mathe-matics educators teach both kinds of courses whereas in Department 5 amathematician teaches both courses. In the rest of the departments, bothmathematicians and mathematics educators work together but in differentdegrees of co-operation. In some cases, they discuss and plan their coursestogether, consult about teaching practices in schools, and establish co-operation in research matters concerning the mathematics education ofteachers. In other cases, this co-operation is not ideal because of differentattitudes and positions on both sides.

Below follows an indication of the different approaches towards inte-grating mathematics and pedagogy, both in mathematics and in mathe-matics education courses. The extent to which mathematics is related topedagogy will depend on the lecturer’s mathematical background and onhis or her commitment to taking into account student teachers’ difficulties,attitudes, and needs. Concerning mathematics courses, we can see differentexpressions of this relationship. There are examples in which mathematics

84 DESPINA POTARI

TABLE I

Number of Lecturers and Compulsory Courses in Mathematics and MathematicsEducation

Department Lecturers Compulsory courses

Mathematics Math. Education Mathematics Math. Education

1 1 1 1

2 1 1 1

3 1 1 2

4 1 (temporary) 1 3 2

5 1 1 1

6 1 1 2 1

7 1 1, 1 (temporary) 1a 1b

8 1 2 2 2

9 2 2 2 2

aUnits From the School Mathematics Curriculum.bTeaching Methods in Primary School Mathematics.

is taught in a formal way, yet its content does not bear any relationshipto what is taught in primary schools. The idea behind this approach isthat mathematics teachers need to encounter advanced mathematics, forexample, calculus, in order to acquire a broader mathematical experience.A first attempt towards considering student teachers’ pedagogical needscomes from a decision to provide a content that is relevant to what theteachers are going to teach. This leads to an emphasis on the structureof numbers, including elements from number theory, and to a consider-ation of geometrical problems. In this approach, we can distinguish twodifferent teaching goals. One is to cover certain gaps in the mathematicalknowledge of the student teachers and to emphasise problem solvingrelated to problems they are going to teach. The other is to encouragethem to face challenging problems and to develop a variety of strategiesfor solving problems. Discussion of the historical development of mathe-matical concepts could provide student teachers with an appreciation of thecreation and evolution of mathematical ideas. It could also provide themwith an understanding of the nature of children’s thinking and its develop-ment. Exploration of the connections of phylogenesis and ontogenesis canbe seen as a way of relating mathematics to pedagogy.

These attempts to relate mathematics and pedagogy are reminiscentof the first steps of mathematicians to involve themselves in researchon mathematics education (Kilpatrick, 1992). A deeper consideration


of pedagogy starts when the lecturers consciously attempt to adoptteaching approaches similar to those their student teachers would like toconsider as teachers. In this case, they act as models for their students.This is expressed by an emphasis on concept development and problemsolving. Here, the communication of student teachers’ ideas becomes moreimportant. It is believed that through this experience, student teachers willconsider a similar role in their teaching.

In some cases children’s thinking serves as a starting point for talkingabout a certain mathematical topic. For example, children’s approachesto multiplication serve as a starting point for talking about algorithms.Another example occurs when student teachers encounter operations inother number systems; they then start to appreciate the complexity of thedecimal system, and children’s difficulties with it becomes a matter ofdiscussion.

These types of approaches are more often met in the mathematicseducators’ courses. The following extract from a response of a mathe-matics educator indicates a more integrative relationship between mathe-matics and pedagogy.

When we teach mathematics we try to relate it to teaching situations. On the one hand, thisis achieved by an appropriate choice of mathematics content, content which is useful andcan be applied in school teaching. On the other hand, our teaching approach is crucial. Forexample, the way we teach mathematics needs to reflect the way we want future teachersto teach (discovery, constructivist teaching) which is not always possible.

The focus in the mathematics education courses varies from depart-ment to department. Some adopt a realistic approach, others a construc-tivist approach, and still others an epistemological one that is closer tothe French didactics of mathematics. The courses include issues aboutgeneral principles of teaching, children’s thinking on specific mathematicsconcepts, problem solving, the use of technology, and socio-cultural issues.In these courses, student teachers get involved with mathematics when theyattempt to analyse children’s thinking and when they design and tackleproblems.

RELATING THEORY TO PRACTICE

Teaching practice is a compulsory part of all the programs in educationdepartments in Greece. Papoulia-Tzelepi (1993) described a program inwhich the prospective teacher is viewed as a reflective practitioner. Overall,the teaching practice is school-based, and the student teachers observeregular teaching and exemplary teaching, plan their own teaching, and

86 DESPINA POTARI

teach in a variety of classrooms. They discuss this experience with amentor, their lecturers, and the classroom teacher; they complete obser-vation sheets, teaching plans, and evaluation reports of their lessons andprojects. The schools are often regular schools but, in some cases, are clin-ical schools, which are typically associated with a university. In reality, theclinical schools are not exceptional in terms of their teaching approaches.The achievement of a balance between theoretical and practical knowledgeis rather difficult for the student teachers in their attempts to realise theirrole as teachers. Most of them adopt the role of practitioner in which theyare driven by existing traditional school practices and by their personalexperience as pupils. For them, the theory is not relevant, as it doesnot offer them immediate solutions. Overall, teaching practice gives thestudent teachers a sense of the reality of school and existing constraints.However, most students do not view the classroom as an opportunity forexperimentation, reflection, and development.

With respect to mathematics teaching, differences in the ways in whichteaching practice is actually realised are evident. In most departments,teaching practice is connected to the mathematics education courses, andits actual implementation depends on the mathematics educators’ perspec-tives and on the constraints they face. In some cases we see mathematicseducators acting as technicians. Their role is to provide student teacherswith an immediate solution in situations in which they identify a specificproblem. For example, when the student teachers do not understand amathematics concept they plan to teach, the mathematics educators offerthem a clear explanation.

Another approach is based on providing student teachers with oppor-tunities to observe exemplary teaching of certain mathematics conceptsand to discuss this teaching. In most cases, the student teachers base theirlessons on the formal mathematics curriculum and use the children’s text-books as their main teaching resource. They do not use the situation toexperiment with alternative teaching approaches.

The model of the teacher as a researcher seems to offer the possib-ility for getting student teachers involved in an inquiry and reflectiveprocess (Potari, 1997). This model is realised through a variety of activitiessuch as reading and presenting research papers that address the learningand teaching of mathematics; an exploration of children’s ideas aboutcertain concepts; an analysis of student-teacher interactions; an epistem-ological analysis of textbooks; the planning and evaluation of teachinginterventions; and the writing of reports.


In some cases, these experiences remain only at the level of planning,and focus on “what is expected.” Questions asked by the mathematicseducators such as “What kind of difficulties do you expect the childrento have?” and “What kind of strategies do you expect the children todevelop?” prompt this hypothetical experience. In other cases, the studentteachers bring problems they experienced in their practice for discussionduring the mathematics education course. Reasons for not directly relatingthe actual classroom experience to theoretical considerations are the factthat some mathematics educators do not participate in the teaching prac-tice, the unwillingness of the schools to accept these kind of activities,and the large number of student teachers. Overall, the constraints that themathematics educators face make them pessimistic about the effectivenessof their attempts: “It is not easy to change student teachers’ attitudestowards mathematics and its teaching in schools. You can only expectthings to become a little better.”

REFLECTIONS

Offering a vision for the future is what most mathematics educators feelthey achieve for their student teachers. More specifically, they believethat their student teachers are challenged to explore alternatives to thetraditional expository approach in mathematics teaching. An approach thatplaces more emphasis on children’s learning is indicated in the followingextract.

I hope that they are unblocked from the logic of prescription concerning mathematicsteaching, and that they learn to recognize and understand better children’s difficulties inmathematics. They get a set of “tools” to intervene in learning that are not magical butprobable. (questionnaire response)

Whether this vision can inspire prospective teachers in their practiceand support them throughout their professional life as teachers is still anopen problem. Certainly, it is unrealistic to expect in a short time that thewhole philosophy and way of thinking both of the student teachers andof those involved in their education, either in schools or in the university,can be changed. Changing beliefs is a rather slow and difficult processin pre-service teacher education (Nesbitt & Bright, 1999). In Greece, thismeans that we, as mathematics educators, need to confront and overcomea range of constraints in an attempt to bring about change in mathematicsteaching in primary schools. We need to face constraints arising fromour work at the university as well as from extant school practices. Some

88 DESPINA POTARI

examples of these constraints are the large number of students; the smallnumber of mathematics educators and mathematicians in the educationdepartments; student teachers’ mathematical background; attitudes andbeliefs about mathematics teaching held by student teachers, universitylecturers and classroom teachers; and existing classroom realities, such asthe prevailing traditional approach towards teaching and the centralisedcurriculum. Building up communication among all those who are involvedin the mathematics education of prospective primary school teachers ispossibly a useful start towards facing these constraints. I hope that thispaper will contribute to this communication.

ACKNOWLEDGEMENTS

I would like to thank all my colleagues and the teachers who discussedwith me their experiences. I also thank Vassiliki Spiliotopoulou for herhelpful comments on an earlier draft.

REFERENCES

Civil, M. (1993). Prospective elementary teachers’ thinking about teaching mathematics.Journal of Mathematical Behavior, 12(1), 79–109.

Cooney, T.J. (1994). Research and teacher education: In search of common ground.Journalfor Research in Mathematics Education, 25, 608–636.

Cooney, T.J. & Krainer, K. (1996). Inservice mathematics teacher education: The import-ance of listening. In A.J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde(Eds.),International handbook of mathematics education(1155–1185). Dordrecht, TheNetherlands: Kluwer.

Even, R., Tirosh, D. & Markovits, Z. (1996). Teacher subject matter knowledge andpedagogical content knowledge: Research and development. In L. Puig & A. Gutierrez(Eds.),Proceedings of PME 20(119–134). Valencia, Spain: University of Valencia.

Kilpatrick, J. (1992). A history of research in mathematics education. In D.A. Grouws(Ed.),Handbook of research on mathematics teaching and learning(3–38). New York:Macmillan.

Nesbitt V.N. & Bright, G.W. (1999). Elementary preservice teachers’ changing beliefsand instructional use of mathematical thinking.Journal for Research in MathematicsEducation, 28, 89–110.

Papoulia-Tzelepi, P. (1993). Teaching practice curriculum in teacher education: Aproposed outline.European Journal of Teacher Education, 16(2), 147–162.

Potari, D. (1997). Developing preservice elementary teachers’ theories about mathematicsteaching and learning. Paper presented in the WG14 at ICME VIII, Sevilla.

Stamelos, G. (1999).The university education departments. Athens, GR: Gutenberg (inGreek).


Troulis, G.M. (1995).The relationship of the students of the education departments withmathematics. Athens, GR: Hellenic Grammata (in Greek).

Weinstein, C.S. (1998). Preservice teachers’ expectations about the first year of teaching.Teaching and Teacher Education, 4(1), 31–40.

University of Patras,Department of Education,261 10 Patras, Greece,E-mail: [email protected]

EDITORIALAPPRECIATING THE CHALLENGE IN

RECOGNIZING THE OBVIOUS

It seems rather trite to say that teaching is a complex phenomenon. Highet(1950) claimed that teaching is an art, not amenable to scientific inquiry.Although his notion of science was probably more deterministic thaninterpretive, his writing provided little comfort for those seeking supportfor their study of teaching. Begle (1968) would later conclude that therewere few nuggets to be mined from the body of research on teachingmathematics. As the field began to sense a methodological crisis, alterna-tive methodologies, often grounded in fields such as anthropology, gainedacceptance and even prominence. Concomitantly, or perhaps consequently,a different perspective about the value of research on teaching and teachereducation emerged. It was not that we had uncovered a phenomenon notpreviously seen but rather that we began to appreciate the value of viewingteaching from different perspectives.

RECOGNIZING THE OBVIOUS

Recent research on teaching mathematics does not deny the complexityof which Highet (1950) spoke but rather enables us to document thiscomplexity. Articles in recent issues ofJMTE have emphasized howdifferent dimensions of school culture facilitate or impede teachers’professional development, including the role that administrators play in thetransformation and transmission of ideas developed in inservice teachereducation programs. We have learned that teachers have the capacity toreconceptualize their mathematics and their methods of teaching providedthey are given the material support of reform curricula and human supportfrom colleagues. Nevertheless, change is an uncertain process as teachersare willing to reform certain aspects of their teaching but are reluctant tomove too far from those methods with which they are most familiar. Thatis, teachers have some sort of an intuitive compass that suggests how farthey are willing to move away from their core teaching methods, but notso far that they could not easily return to those methods. We have learned


92 EDITORIAL

about different ways to engage teachers in the reform of their practice andthe role that reflective thinking plays in dealing with the complexity ofenacting reform. We have also seen instances in which teachers interpretreform curricula or documents in a variety of ways, not always consistentwith the authors’ intentions.

I suspect all of these findings could be couched under the umbrellaof a practitioner’s wisdom. What moves us beyond the more common-sense notion of teaching is the insight gained fromthinking about what isoften obvious about teaching. Yes, administrators are an important partof school culture, and yes, there is variance among the interpretationsteachers place on reform, and yes, some teachers are more adventuresomein accepting reform than others. The issue is not just about documentingthe obvious but also about conceptualizing the contexts that give rise tothe obvious. The question then becomes one of how we can facilitateteachers’ ability to deal with that context should they decide to reform theirteaching.

APPRECIATING THE CHALLENGE

Mason (1998) argued that “to be a real teacher involves the refinement anddevelopment of a complex of awarenesses on three levels, and that this ismanifested in alterations to the structure of attention” (p. 243). AlthoughMason was referring to elevating students’ awareness of what they arelearning and educating teachers about the value of students having such anawareness, his basic argument speaks to teacher educators and researchersas well. Our awareness of the complexity of teaching can remind us ofthe need to engage multiple perspectives and methodologies in an effortto unveil what on the surface seems obvious but, in reality, is enormouslycomplex.

This complexity enriches our professional life. In some sense it freesus from the burden of expecting our results to be definitive and determin-istic. It enables us to consider a variety of alternatives, to engage in whatBauersfeld (1988) calledfundamental relativism. Our challenge is to seethe uncommon amidst the common. Each of our studies can help providea dot on our mosaic of understanding teaching but never the entire mosaic.Let us honor the artistry of which Highet spoke but, at the same time, main-tain our resolve to better understand this artistry in our efforts to facilitateteachers’ professional growth. Let us appreciate the distinction betweenSchön’s (1983) notion of technical knowledge and the professional artistrythat gives rise to that knowledge. It is a matter of accepting the challengeof understanding the obvious.

EDITORIAL 93

REFERENCES

Bauersfeld, H. (1988). Interaction, construction, and knowledge: Alternative perspectivesfor mathematics education. In D. A. Grouws & T. J. Cooney (Eds.),Effective mathe-matics teaching(27–41). Reston, VA: National Council of Teachers of Mathematics.

Begle, E. (1968). Curriculum research in mathematics. In H. Klausmeier & G. T. O’Hearn(Eds.), Research and development toward the improvement of education(44–48).Madison, WI: Dembar Educational Research Services.

Highet, G. (1950).The art of teaching. New York: Macmillan.Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and

structure of attention.Journal of Mathematics Teacher Education, 1, 243–267.Schön, D. A. (1983).The reflective practitioner: How professionals think in action. New

York: Basic Books.

Thomas J. CooneyEditor,JMTE

ROBYN ZEVENBERGEN

PEER ASSESSMENT OF STUDENT CONSTRUCTED POSTERS:ASSESSMENT ALTERNATIVES IN PRESERVICE

MATHEMATICS EDUCATION

ABSTRACT. In this article I discuss a four-year action research project that involvedthe development of effective assessment tools for preservice mathematics teachers. Thefocus of the article is on peer assessment in which students reviewed posters created bytheir peers. The article discusses the strategies that were used and the implications thatarose from the project. I argue that peer assessment is an effective tool for assessment inpreservice mathematics teacher education but must not be seen as an alternative to teacher-based assessment due to the variability in marks between and within student cohorts. Thevalue of peer assessment is its potential as a learning tool.

One of the concerns that has been expressed by preservice teachers istheir lack of hands-on experience with assessment, with respect to boththeir university level courses and the practical, school-based component.In attempting to address this concern, I instigated peer assessment ofposters constructed by students. This initiative has been a progressivelyevolving project over a period of four years with the explicit intention toprovide a learning experience that addresses the concerns of students andis sustainable within the contracting resources of higher education.

Background and Context

Over the past two decades, there has been an increasing recognition ofinnovative assessment practices. Yet, as Williams (1992) suggested, suchpractices are underrepresented in contemporary higher education despitetheir broader applications at all levels of education. He suggested thatthe lack of innovative and effective alternatives to the more traditionalmodes of assessment may be due, in part, to the changing nature ofhigher education where there have been massive increases in demandsand workloads. Further compounding the problems related to assess-ment is the tighter scrutiny by employer agencies who demand that skillsand knowledge should be transferable from the university to the work-place (Stefani, 1994). Larisey (1994) suggested that the clients of highereducation, that is adult learners, need to be given opportunities for self-directed learning and critical reflection in order to mirror the world of


96 ROBYN ZEVENBERGEN

learning beyond formal education. Such learning experiences are seento produce better learning outcomes for students and the wider society.Teacher education facilities are not exempt from this pressure for effi-ciency. Given the use of traditional forms of assessment and other practicesof the economically driven university, economically driven reforms inteacher education may be inadvertently reinforcing the legitimation ofmore traditional forms of pedagogy and assessment that have dominatedpast practices in mathematics. Hence, the call is for assessment practicesthat are economically and practically sustainable within the new lean insti-tutions but which expose preservice mathematics teachers to exemplarypractices that challenge widely-held, conservative beliefs and practices.

It is essential to develop sustainable practices that produce desiredlearning outcomes for preservice teachers. This article reports on a projectwhich had three very distinct goals:

• to develop skills in preservice teachers that will enable them toproduce stimulating resources for their students;

• to develop assessment skills in preservice teachers; and• to develop effective and innovative assessment tools for mathematics

preservice teacher education.

The peer assessment project was an integral part of the students’ studyof mathematics teaching and learning. To this end, the peer assessmentproject was developed and tested over a period of four years with studentsenrolled in their first mathematics education course. I anticipated thatthe peer assessment project would facilitate exposure to a broad array ofresources in a structured context.

Literature Review: Peer Assessment in Higher Education

One method of assessment that has been trialed effectively within thehigher education sector is peer assessment and self-assessment. Literatureon self-assessment and peer assessment encompasses a wide range ofdisciplines in a wide range of contexts. Assessment tasks included ratingof individual and group presentations, artwork, and posters. Such practicesseem to motivate deeper learning, but have not been widely implementedin higher education (Williams, 1992).

Group work is becoming more popular and is more widely imple-mented due to its sustainability within higher education and its replicationof competencies expected in the workplace. One limitation of group workis that it is often seen as unfair assessment for those in the group who dothe work and “carry” the others. In attempting to address this concern,Freeman (1995), in his work within a business course, found that the

PEER ASSESSMENT IN PRESERVICE MATHEMATICS EDUCATION 97

process of peer assessment reduced the chance for students to have a freeride because students must assess the contribution of each member to thegroup project. Conway, Kember, Sivan and Wu (1993) documented theneed for peer evaluation of participation in group work in an optometrycourse and concluded that the process of peer assessment of individualcontributions was well supported by students. Although peer assessmentallowed for evaluation of each student’s contribution, Conway et al. alsonoted that assessment of individual contributions was not unreserved. Theycommented that the subjectivity associated with mark allocations wasinfluenced by the students’ different standards of what numerical scoreshould constitute a pass. The issue of subjectivity and interpretation ofgrades remains an area of concern.

When peer assessment is undertaken together with other forms ofassessment, there is a concern that as novice assessors, students’ evalu-ations of their peers’ work may be considered unreliable. When experts areinvolved in the assessment process, the process is perceived to have someobjectivity. However, as Freeman (1990) noted, examinations assessed bymultiple staff assessors also produced unreliable results. That is, evenexperienced assessors leave room for variability. In contrast, Freeman(1992) noted cases in which students reliably marked essays (see also,Orpen, 1982). Furthermore, when staff marks were available as references,students’ assessments were reliable (Williams, 1992).

Other studies have undertaken correlations between staff members’and students’ marks, with varying results. Marks allocated by teams oftutors and teaching staff have been reported to be unreliable (Marshall &Powers, 1969; Beard & Hartley, cited in Freeman, 1995), particularly whenissues of subjectivity related to the assessment item, such as an art work,were involved. In contrast, Hughes and Large (1993) found little differ-ence between the staff scores and students’ scores on oral presentations ofpharmacology students.

Burke (1969) found that peer assessment was more reliable than self-assessment in the selection of high- and low-performing students. Freeman(1995) found similar results with his business students. Orsmond, Merryand Reiling (1996) also found that students were able to discriminatebetween good and poor work. However, the authors suggested that therewas a poor correlation between the students’ grades and those of staff.They concluded that the lack of experience with marking could be a factorin the low correlation and suggested that students be given an externalreference point for their marking.

Freeman (1995) found that at the extreme ends of the distributionof grades, students tended to mark good presentations down and poor


presentations up. His findings suggest that there is no strong support forgiving students the responsibility for marking. However, he also recom-mended that more training in assessment and a less detailed assessmentschedule (his had 22 items to mark) may produce more reliable results ingrading.

Some of the discrepancy in assessment may be due to the lack ofclear criteria. Williams (1992) supported the notion of clear guidelines formarking as it makes the task more objective for students and in so doingreduces their feelings of personalising the criticism. Orsmond, Merry andReiling (1996) extended this point and suggested that some of the discrep-ancy in marking may be due to the different interpretations of criteria, apoint iterated by Boud (1989).

Students’ reactions to peer assessment are not always positive, andhence, the adoption of peer assessment needs to be tempered with arecognition of its limitations. Peters (1996) found in his study of artstudents that 53% of the students did not support self-assessment while76% did not support peer assessment. Larisey (1994), in her study ofnursing students, found that students believed that it was the responsibilityof the faculty to evaluate what and how they learned. Williams (1992)reported that students felt that peer assessment was criticism of one’sfriends and colleagues. Although these authors contended that studentsmay be resistant to the process, they found that the learning outcomes forstudents were significant. In addressing the concerns raised by students,Larisey (1994) suggested that practices needed to be embedded withinthe structures of the institution so that such forms of assessment becomenormalised for students.

Despite student resistance to self-assessment and peer assessment,Peters (1996) recognised that forms of continuous assessment werepreferred over examinations and that when criteria were made explicit,students were more open to sharing responsibility for assessment. Becausethe future of students is closely linked to assessment outcomes, it isimportant that peer assessment involves validity checks (Freeman, 1995).This is particularly the case when subjective assessment is involvedbecause such assessment “always involves making fallible human judge-ments” (Ramsden, 1992, p. 186). Freeman (1995) raised concerns aboutthe certification process and the importance for peer assessment to reliablyreflect student learning. Because peer assessment may not be as reliable asthe academic marking, Freeman (1995) suggested that the peer-assessmentcomponent should be weighted lightly.

Despite the issues raised, peer assessment has considerable potentialwithin teacher education, and even more so within mathematics education.


As an assessment tool, correlation between staff and students appears tomeet with mixed success. As a learning tool, it appears that peer assess-ment compels students to look at the quality of the outputs rather than onlyat the inputs and efforts (Freeman, 1995). Boud (1990) suggested that thefocus on outputs fosters the growth of critical thinking skills and evalu-ation. By focussing on the criteria for assessment, students are compelledto be more analytical about the actual requirements of the tasks. However,there is little evidence in the literature of how the peer assessment processhas contributed to the learning outcomes for students.

In the sections that follow, I analyse the development and implementa-tion of a peer assessment project in mathematics education. Based onthe literature reviewed and the more philosophical components of peerassessment, I discuss recommendations and limitations of the approach.I propose that peer assessment has value within teacher education butmust be carefully developed and implemented for it to be authentic andto provide a valuable learning component in initial teacher preparation. Iargue that peer assessment should not be seen as an alternative, or shortcut, to teacher-based assessment. Rather, it can be a valuable supplementto traditional forms of assessment. Furthermore, I suggest that the valueof peer assessment may not lie in its use as an assessment tool but in itspotential as a learning tool.

THE PROJECT

The project reported here evolved over a period of four years. It is imple-mented in the second of a suite of three elementary mathematics educationcourses. The first course, Course id1, focuses on mathematics competency;the second and third courses, Course id2 and id3, are concerned withmathematics education. Course Id2 focuses on an understanding of thepedagogical aspects of teaching mathematics; Course Id3 focuses on anunderstanding of social, cultural and political issues in teaching mathe-matics. The project has adopted an action research methodology and hasevolved into the form reported in this paper.

An Approach to Peer Assessment

My approach to peer assessment consisted of having students constructa poster – a stand-alone resource containing a maximum of 25 words(excluding heading) which can be used in the classroom. Students couldselect from a series of poster topics that had been identified. Alternatively,a topic could be negotiated with the course instructor/tutor. In the early


stages of the project, topics were open to self selection. It was found,however, that the topics chosen often were not substantive and did notprovide a challenge to learning. The other major restriction related to open-ended task selection was due to the lack of experience of the studentsin mathematics education, resulting in a range of topics that was notcomprehensive. The current offerings of topics appear to work well. It isimportant, however, that students are able to select a topic that has personalrelevance to their understanding or lack of understanding of a mathematicseducation topic. The topics were selected on the basis of problems oftenposed to preservice teachers in their content classes. Topics included:

Area: The difference between area and perimeterMass: What is mass? (difference between mass and weight)Time: Using 24 hr. clocks – representing 24 hour timesTime: Using timetables – adding and subtracting timesSpace: The various types of angles (acute, obtuse, etc.)Space: Geometry on a sphereNumber: Subtraction with internal zerosNumber: Operations with fractionsNumber: Equivalence of fractionsRatio: What is the difference between ratio and factions?Data: Graphs – using appropriate graphs to represent data

The intentions of the posters were two-fold. First, the poster had todocument the learning of the student. The student had to display theselected concept in a concise, innovative, and user-friendly manner thatdemonstrated appropriate understanding of the selected concept, thusaddressing the concern that many of the mathematical concepts are poorlyunderstood by preservice teachers. Second, through the construction of theposter, students would develop skills, techniques, and knowledge aboutposter construction and effective communication. Similarly, the purposeof peer assessment was two-fold. First, students would practice assess-ment – something often lacking in teacher preparation courses. Second,students were exposed to a broader experience, and hence learning, incarrying out the intentions of the posters. Through their participation inpeer assessment, students were to gain substantially more knowledge dueto the imposed need to critically reflect and evaluate their peers’ posters.

During preparation activities, students were exposed to a range ofposters from previous years and engaged in discussions about effectiveand ineffective posters. An analysis of what worked and what did notwork provided a catalyst for identifying key factors in the construction ofeffective posters. Students discussed criteria for the posters and analysedthe criteria, in particular, what each of the criteria meant. According


to Freeman’s (1995) recommendations, that criteria should be kept to aminimum. Consequently, only three criteria were used – one for visualimpact, one for mathematical content, and the third for remaining withinthe 25 word limit. These criteria corresponded to the overall rationale forthe assessment approach.

Criteria for Constructing Posters

As result of earlier trials, a number of conditions are placed on the posters.The first was that the posters had to be confined to the size of DIN A2(about 16 1/2in. by 23 1/4 in.). In part, this was due to students’ tendency toconstruct posters that were large and cumbersome and difficult to display –their weight meant that they kept falling off walls. In many cases, studentsassumed that bigger was better and thus constructed posters that were largebut contained little content. Consequently, they would become distressedby their marks because they believed that their posters had been expensiveto construct, so therefore had to be good. Finally, knowing that a clusterof ten posters of the same size would be confined to an area, it was thenpossible to undertake the spacing of the posters.

Restrictions on durability were also imposed. As the posters weredisplayed in a common area – the hallways in the education building –they needed to sustain abuse from other students. In one year, a studenthad constructed an excellent bar graph with chocolate bars, all of whichwere removed and eaten by other students. When the marking occurred,the poster was not judged to be of high standard. Similarly, posters invest-igating probability with coins were often destructed. A common themefor the notion of durability was whether the poster might be laminated sothat it could not be vandalised. Students agreed that this aspect was alsoimportant in the classroom, where similar acts of vandalism are likely tooccur. In addition, students felt that if they put a lot of work into theirposters, they would like to preserve them for later lessons on the topics.

Marking of Posters

In line with Freeman’s (1995) comments that peer assessment that involvessubjectivity should be weighted lightly, the overall weighting of the assess-ment item was kept at 15% (or 15 marks). The artistic and mathematicalcomponents were weighted evenly (5 marks each) because students didnot feel comfortable with the mathematics content at this early stage oftheir career development. Because the restriction to stay within 25 wordsrequired considerable thinking, 2 marks were allocated to this componentof the poster. The remaining 3 marks were allocated for peer assessment.Peer assessment was also weighted lightly because students had little


experience with assessment at this point in their program. However, theallocation of some marks made students accountable. In previous years,when the peer assessment aspect of the project was under trial, no markshad been allocated for peer assessment with the intention that it was theintrinsic value of the learning experience that would provide the moti-vation to undertake the assessment. However, less than 50% of studentscompleted the task. This confirms Ramsden’s position (1992) that assess-ment becomes the curriculum for students. Feedback from this first cohortof students was highly favourable of the process, but students suggestedthat some marks be made available for the process. Their commentsconfirmed that students wanted to be rewarded for undertaking extrawork.

All posters were marked blind, that is, each poster was allocated anumber and all student identification were removed from sight. Eachstudent marked 10 of the 180 posters submitted by their peers. Posters to bemarked were assigned randomly according to the students’ number on theclass list to ensure that no student marked his or her own poster. Randomassignment also ensured that students would be likely to mark a range oftopics rather than be constrained to one topic. This ensured a wider rangeof experiences related to topics which they personally had not covered andso exposed them to further content. All posters also were graded by theinstructor. Students were given 2 weeks to complete their assessments. Inaddition to a quantitative evaluation, students had to provide qualitativefeedback by justifying the marks given and by offering constructive feed-back. Student results were compiled through the mean of the students’marks, then averaged with the instructor’s mark. All qualitative feedbackwas returned to the students.

In order to monitor student learning outcomes, a survey was distributedfollowing the peer assessment process to gauge students’ reactions to theproject, assess the effectiveness of the process, and develop more effectivemeans for implementing the project in the future. Focus groups at the endof each year evaluated the effectiveness of and reactions to the project.Some students in the groups had chosen to participate; others, to ensure arepresentative sample of students, had been asked to participate.

EFFECTIVENESS OF PEER ASSESSMENT

Although peer assessment may be a viable alternative to teacher-directedassessment within preservice teacher education, some form of modera-tion is needed to protect students, academic staff, and the employer asworkplaces increasingly become sites for accountability and litigation. In


part, this is due to the variability in the research outcomes as to the relia-bility of peer assessment. The results from this project indicate a degreeof reliability between student marks and staff marks, suggesting that peerassessment may be a viable alternative for assessment. This result supportsthe notion of peer assessment as a valid form of assessment.

Different from the approach in previous research, I adopted an explicitpedagogical tool in attempting to facilitate a stronger correlation inmarking between students and staff. I explicitly advised students of theassessment process and that the three allocated marks for peer assessmentwere for fair assessment. I defined the notion offairnessas the attempt tomark within the criteria outlined, with the expectation that there should bea fair degree of commonality amongst the marks. I also advised studentsthat marking which did not appear to be taken seriously would not receivethe three marks. That is, the three marks were to be awarded for seriouslyattempting to mark fairly, as opposed to completing the exercise. I recog-nised that this process was flawed methodologically and would be difficultto enforce. The practice arose from previous trials in which students didnot take the process seriously and awarded high/low grades (or in somecases the same grade) for each poster. After I introduced accountability,the distribution of marks was more consistent across the group. This resultindicates that students took seriously the process of peer assessment; thusa reliability check had been built into the peer assessment process.

There was an observable trend between high and low achievers and thescores they gave. That is, low achievers (as determined by their overallscore) allocated higher than average marks whereas high achievers (thosewho received higher marks) tended to mark less generously. The samplesize was too small to undertake a statistical analysis of this observation.

The qualitative comments showed similar trends. High-achievingstudents offered more insightful and critical comments whereas thelow-achieving students provided glib and often very generic comments.Furthermore, high-achieving students were more likely to provide morefeedback than their low-achieving peers. For example, the followingqualitative comments were offered on the poster dealing 24 hr clocks:

High achievers

• The pictures were great and relevant to the kids and curriculum forthat age group. The times were relevant to them too but they mightnot understand why you would use 12 hr or 24 hours. They can’t seeall the other times, so it would be hard for the kids to see what happenswith p.m. times [i.e., adding on the 12].

• Poster does not show the full range of the 24 hr clocks. Only 3 isolatedexamples. It is therefore a limited teaching aid.


Photo 1. TV with 24 hr clock.

• This was relevant to kids, however, I think it needed a little moreexplanation about how the p.m. times change for a stand alone item.

Low achievers

• Not thorough enough. Idea’s there, but not easy to follow, wouldconfuse kids

• Well presented, but not very mathematical• Great drawings. Kids would know the times


This outcome could be explained in terms of the students’ comprehen-sion or lack of comprehension of the task and/or criteria. Because thegroups were small, there was insufficient evidence to suggest statisticallysignificant or valid trends. However, this observation is worthy of furtherinvestigation as this may be one of the variables that contribute to thevariability in peer assessment scores.

PEER ASSESSMENT AS A LEARNING TOOL

In line with current conceptions of good practice in mathematics educa-tion, assessment is integral to the learning process. I anticipated that thepeer assessment process would contribute more to teacher education thantraditional forms of assessment.

Students’ Reactions

Focus-group discussions in previous years of the action-research cycleindicated that construction and assessment of the posters were novel exper-iences and enjoyed by the students. They felt that the process contributedto their practical knowledge; that is, they learned how to construct posterswhile at the time learning about one particular area of the mathematicscurriculum. Students indicated that they enjoyed looking at the postersdisplayed in the corridors, but did not look at them very critically. In one ofthe focus groups, students discussed the potential for learning more aboutposters and mathematics through a more systematic and critical exami-nation of the posters. This provided the catalyst for the introduction ofpeer assessment. I anticipated that learning could occur in three areas:(a) students could expand their mathematical knowledge; (b) they coulddevelop practical skills in constructing posters; and (c) they could havefirst-hand experience with practical experience of assessment. To evaluatethis third aspect of learning, students completed a brief survey in whichthey ranked their learning outcomes with respect to the three aspects (seeTable I).

The high mean scores indicate that students considered the processof peer assessment as valuable. To further assess the impact of peerassessment on learning outcomes, students were given the opportunity toelaborate on what they had learned through participation in peer assess-ment. The comments were sought through the open-ended question Whatdid you learn from this activity? As with any form of survey, the threeclosed questions served as a catalyst for the responses so that responsestended to cluster around these items. However, from the comments, it was


TABLE I

Peer assessment as a learning tool

Student response item Mean score

Peer assessment was useful for my knowledge about assessmenttechniques.

6.0

Peer assessment was useful for learning about other ways of makingposters.

6.6

Peer assessment was useful for learning about other aspects of mathe-matics education.

6.4

Note: Scores are on a Likert scale from 1 to 7: 1 – strongly disagree; and 7 – stronglyagree.

clear that students believed they had learned considerably from the processof peer assessment.

Learning about Assessment

As mentioned at the beginning of this paper, one of the key motivatorsfor this project was for preservice teachers to gain hands-on experiencewith assessment. Even when assessment and assessment alternatives arean integral component of the course, it is still difficult to organize situ-ations in which genuine assessment becomes an integral component ofthe learning experiences. Rather, assessment is often spoken about, butwith little chance to experience it. This project has enabled students toexperience genuine assessment first hand. In doing so, they came to realisethe pitfalls, difficulties, and ethical dilemmas of assessment.

The process of peer assessment gave students experiences in markingso that they became more aware of the complexity of assessment:

That marking is very difficult and time consuming. You have to look at a variety of aspectsduring marking which can make things difficult and challenging.

Need to consider the visual as well as the mathematical aspects.It is really difficult to mark and it gave me an idea of how mine will be marked.It highlighted the difficulty in having to assess material.I learned a lot of skills for marking.

They also reported that they learned how to go beyond their first impres-sions of the posters and look more deeply at both the content and the visualimpact. Students offered the following comments:

[I learned] how to critically and fairly evaluate others’ work.It’s good the students can assess the poster to get them used to making critical assessment.You must look below the surface.


Photo 2. Angles.

Numerous comments indicated that the immediate appeal of a postermight be seductive and trick the reader into assuming that the posterwas effective. However, as the last comment suggests, it is important tolook below the surface appearance. This was particularly the case if theposters were visually appealing but, upon closer examination were foundto have weaknesses. Comments indicated that students felt that the visualappeal seduced them into believing that the poster contained substantivecontent, yet upon the closer examination the first impressions were modi-fied so as to incorporate substantive content as well as the visual impact.The comments indicated the emergence of a more critical reader andassessor.

Because students had been involved in the construction of their ownposters and were personally aware of the decisions they needed to make,they were more critical of the decisions made by their peers. For example,students who had considered the use of colour and contrast to make theirposters more dynamic and appealing were critical of posters which had notincorporated these aspects.

Students offered the following feedback on the above poster


Great concept! Love the characters. Better use of colour – more contrast betweenbackground and actual angles would been great.

Dull background. Angle of noses don’t stand out enough for a clear explanation. But areally good concept.

Angles should have been more prominent/highlighted. If you used a fluoro cardboard andthen made the angles in black or another fluoro colour, they would have stood out moreand it would be clearer what the angles were.

Not clear maths content. If lines of angles were more clearly shown, poster would beexcellent.

As noted in the preceding overview of the literature on peer assessment,there is a strong recommendation that students are given clear criteria formarking, and that there is adequate discussion of these criteria. In currenteducational contexts, the use of criterion marking is becoming increas-ingly popular. Students, prior to this exercise, assumed that such criteriaare objective and are easily translated into effective marking. The processof peer assessment and the discussions in tutorials raised their awarenessof the subjective interpretations of criteria and subsequently, the problemswith criterion-based marking. Although the criteria for the evaluation ofartistic components of the poster were open to a far greater range of inter-pretations, there were problems even with criteria that seemed obviousto the students. The criterion of “no more than 25 words” seemed to bevery straightforward. The students realised that even such a black andwhite criterion is grey. Questions raised included whether the labels onan axis were to be included in the word count or whether numerals werewords. For example, “six million” are clearly two words, but how should“6,000,000” be counted? Similarly, one student constructed a poster whichhad minimal words, but a decorative border that contained words, symbols,and numerals related to fractions and ratios. Some students gave zeromarks as they saw the words in the border as violating the 25 word limitwhereas others gave the full two marks because they only counted thewords in the central poster.

In considering the issue of criterion-based marking, students began torecognise the subjectivity associated with marking – whether they realisedtheir own biases or the biases in their peers’ interpretations of the work.

Assessment is very subjective and personal.Understanding of how difficult [it] is can be to be fair with marking.It highlighted the difficulties of marking, of trying to maintain consistency and also of

“time consuminess.”There may be a fair amount of variation in criteria used by various students.It seems a bit unfair because some people are prejudiced when marking.

Such comments provided the basis for considerable in-class discussion asto the dangers and pitfalls in marking. This discussion provided a rich


context for debate and a valuable forum for considering their future workas teachers, particularly in a context where there is increasing account-ability demanded of teachers to provide documentation of students’learning.

The process of peer assessment also offered students the opportunity toexperience first-hand the problems associated with assessment. There werenumerous comments that indicated the students’ surprise at their peers’lack of compliance with the stated criteria, even when these criteria didnot involve substantive reading or comprehension.

I am amazed at how many people do not follow the criteria – how many posters were over25 words, or greater than the standard cardboard size (A2).

These observations and the ensuing discussions highlighted the problemsand issues associated with assessment and the subsequent marking ofstudents’ responses. The project also created opportunities for students todiscuss the process of assessment within their particular setting as theyrealized the breadth and quality of items submitted. For many students,this produced the realisation of where their work was located within thespectrum of their cohort. Perhaps as a consequence, there have been nostudent-initiated challenges to their marks – an increasingly importantissue within the current work environment.

Given Williams’ (1992) recommendation that criteria should be madeclear, this project has progressively involved students in developingcriteria. Prior to undertaking the construction and subsequent marking ofthe posters, students discussed the criteria and refined them in light ofthe comments made. Even when this was undertaken, the process of peerassessment provided a rich experience in which students were compelled toimplement and reflect on the criteria. Although William’s concerns aboutexplicitness of criteria had been addressed, it is clear from this project thatit is not always possible to achieve this explicitness, but this may not beas problematic as suggested by Williams. The richness of experiences incritically reflecting on the problems of interpretation of criteria provides avaluable learning experience for the students.

Learning about Posters

The process of undertaking focused observation of at least 10 postersdemanded that students consider the processes involved in the construc-tion of the posters and, in doing so, expanded their knowledge of effectiveposter construction. The examination of other posters compelled studentsto critically examine and appraise the work of others and consider theeffectiveness of strategies used in displays.


[Peer assessment provided me with] heaps of ideas for posters and resources. It is also goodto see the overall standard of my peers.

I could see what would/would not work.Very useful to see other people’s strategies of presentation and explanation.The opportunity to see others’ ideas and concepts afforded me the opportunity to add to

my resource ideas.Gave new ideas about posters – what kinds of things you can put on posters and what works

better.

The compunction to examine carefully the posters demanded that studentstake careful note of the techniques used by others and their perceivedeffectiveness.

Learning about Mathematics

The process also demanded that students examine the mathematics ofthe posters. In the construction of their own posters, students researcheda number of areas prior to settling on their displayed topic. Throughexamination of the posters, they were exposed to other aspects of themathematics curriculum that they might not have previously known.

I learned a lot about the maths concepts through constructing my own but also during theassessment of others.

Some basic concepts I had misconceptions about beforehand, now became clearer.Because we only picked one topic for our poster it also gave us a quick overview of other

topics.

Furthermore, comments indicated that the peer assessment processprovoked students into realizing that the same mathematical content canbe displayed effectively in a variety of ways.

There are different approaches that could be taken in teaching different maths content.There really are a lot of ways of doing the same topic.To give thought to what makes a good educational poster in regards to content and

presentation.

These comments suggest that the process of peer assessment expandedthe learning originally intended in the assessment item. The process of peerassessment exposed students to a variety of topics and hence

• broadened their awareness of those topics;• broadened their knowledge of poster construction technique; and• extended their knowledge of approaches to teaching the same content.

Despite my best efforts to implement peer assessment in accordance withsuggestions from the literature, there were still remnants of concernsechoed in previous projects. Some students felt that the onus and expertise


for marking was with the academic staff. Although such comments werefew, they are highlighted in the comment:

I believe it is better for the lecturer/teacher to mark assignments as they have the experienceupon which to base their judgements.

What seems to remain entrenched with some students is the perceptionthat marking can only be valid when undertaken by experts. In teachereducation, this can be a concern as preservice teachers assume responsi-bility for assessment in their near futures. In this project, the conceptualknowledge was not demanding for most students as they were required tocomplete a course in which they obtained an 80% rating on mathematicalcontent. As such, it could be safely assumed that they should have themathematical knowledge to be able to assess the content of the posters, andhaving constructed their own posters, would have a working knowledge ofposter construction.

CONCLUSION

From the results of this project, peer assessment offers potential as anassessment alternative within teacher education. There is a positive corre-lation between student marks and staff marks, particularly when studentsare provided appropriate training in marking. In our project, that prepar-ation included discussion of the set criteria, provision of examples, anddiscussion that focused on the key aspects of the criteria – in this case,the construction of the poster and the mathematical content. Students weregiven ample time to view and reflect on the posters. A low weighting wasgiven to the assessment item, and marks were allocated for peer assessmentthat made students accountable. Students were advised that moderation inmarking should occur. These factors appeared to enhance the effectivenessof peer assessment.

One of the issues identified in the literature, which was not addressedin this project, was the use of a referent mark. It was not implemented asit was raised in the focus groups as providing the potential for studentsto “copy what the lecturer wants.” In considering an alternative to theprovision of referents, some discussion of what constitutes certain marksmay be useful, as suggested by one of the students:

If we had some criteria to follow when assessing – knowing what was representational of5 marks was unknown.

Subsequent discussions with students revealed that they have little ideaof what constitutes a mark and hence some of the variability may lie in


the allocation of marks rather than in the clarity of the criteria. For somestudents, a 0 was seen to be representative of nothing being addressed,whereas for other students the mark was allocated for students who did notaddress the criteria for what was considered a pass standard.

It was expected that many students would reject peer assessment, as ithas the potential to jeopardise students’ futures if used unwisely or inef-fectively. In some cases it may even be open to abuse where it becomesa substitute for teacher-controlled assessment. The new market-placereforms that dictate the implementation of cost-effective practices mayfacilitate the co-opting of peer assessment as a perceived economically-sustainable assessment alternative. Nothing could be further from the truth.To co-opt a student comment, one only has “to scratch below the surface”to see that it will not be difficult for students to see that this will be a cost-cutting exercise. It is clear that moderation is needed so it is important tohave a referent from a staff member.

The project has provided evidence that peer assessment provides avaluable learning tool for students. By compelling students to undertakeconstructive criticism, they are provided a forum in which they mustcritically view and evaluate the item according the same criteria thatguided their own poster constructions. Through the construction of posters,students became more cognisant about assessment, poster construction,and other aspects of mathematics teaching.

Further areas for research should be explored. First, it may be usefulto follow through the trends noted whereby variability between and withinstudents and staff may be due to the achievement levels of the students.Second, although criteria should be discussed and made clear to students,this discussion may not be sufficient to address variability. Different inter-pretations of the criteria provided a basis for considerable discussion andhence were invaluable as learning tools. In contrast, more time may beusefully spent in discussing, and perhaps negotiating, the meaning of eachmark so that a clearer and consistent understanding is negotiated across thecohort.

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Assessment and Evaluation in Higher Education, 17(1), 45–58.

Faculty of EducationGriffith UniversityPMB 50 Gold Coast Mail CentreBundall, QLD 9726Australia

JANET BOWERS and HELEN M. DOERR

AN ANALYSIS OF PROSPECTIVE TEACHERS’ DUAL ROLESIN UNDERSTANDING THE MATHEMATICS OF CHANGE:

ELICITING GROWTH WITH TECHNOLOGY1

ABSTRACT. We analyze the interrelations between prospective and practicing teachers’learning of the mathematics of change and the development of their emerging under-standing of effective mathematics teaching. The participants in our study, who were allinterested in teaching secondary mathematics, were mathematics majors who had signifi-cant formal knowledge of the fundamental concepts of calculus prior to taking our courses,but who often experienced and expressed procedural orientations toward the teaching ofmathematics. To address this difficulty, we developed novel computer-based activities tochallenge the participants’ mathematical understandings and required them to use tech-nology during short teaching episodes they conducted with younger students. To analyzeour participants’ understandings, we developed a framework that juxtaposes the roles ofthe participants as students and teachers, and their understanding of mathematics andof pedagogical strategies. Our analysis of the participants’ views from these differentperspectives enabled us to see simultaneously the intertwined development of subjectmatter insights and specific views of teaching.

Students enrolled in mathematics education courses are simultaneouslylearners and teachers in transition. As learners, they are constructingnew ways of thinking about seemingly familiar mathematics and aboutnew ways that others might learn. As teachers in transition, they areanticipating how their experiences in learning mathematics will relate totheir future experiences as teachers in their own classrooms. There areseveral difficulties facing mathematics educators teaching such coursesfor preservice teachers. One obstacle relates to prospective teachers’ well-documented resiliency toward changing their views of effective pedagogy(cf., Cooney, Wilson, Albright & Chauvot, 1998; Hiebert, 1986; Lampert& Ball, 1998; Pajares & Bengston, 1995; Siebert, Lobato & Brown, 1998;Thompson, 1992). A second difficulty is that, as learners, prospectiveteachers are often content with what may be superficial understandingsof deep mathematical concepts. Once they have formalized a procedure,it is difficult to re-visit the underlying concept for deeper understanding(Hiebert & Carpenter, 1992; Lee & Wheeler, 1989; Skemp, 1978; Wilson& Goldenberg, 1998). As teacher educators, we would like pre-serviceteachers to realize that fragile mathematical understandings are inade-


116 JANET BOWERS AND HELEN M. DOERR

quate when teaching mathematics in ways that support more meaningfulunderstanding.

Our approach to challenging and perhaps changing prospectiveteachers’ understandings of mathematics and effective pedagogy wasto design novel computer-based activities that would elicit dissonancebetween what was expected and what occurred on the computer screen, andthen to discuss how this dissonance could promote reflection and learning.The goal of our study was to investigate two questions:

1) How do prospective teachers, acting in the role ofstudents, think aboutthe mathematics of change when using an exploratory microworld inthe context of their course work?

2) How do prospective teachers, acting in the role ofteachers, think aboutthe mathematics of change when using an exploratory microworldduring tutoring sessions with young children?

THEORETICAL FRAMEWORK

The theoretical framework that guides our work is based on a construc-tivist perspective in which learning is viewed as a process of experi-encing dissonance and working to resolve perturbations by building viableexplanations (von Glasersfeld, 1987, 1995). As Steffe and Thompson(2000) recently noted, Piaget contended that there are four factors thatcontribute to one’s cognitive development. These include social inter-action, maturation, physical experience, and self-regulation. “Individualsestablish equilibrium among personal schemes of action and anticipationas they interact in mutual adaptation – as constrained by local limitationsimposed by their abilities to accommodate those very schemes” (Steffe &Thompson, 2000, p. 193). The critical element of this general model is thatstudents are seen as cognizing individuals who are continually interactingwith each other and with their environment (which includes the computerand the accompanying activities) and adapting their own views throughprocesses of interactive accommodation. We take very seriously Steffe andThompson’s recommendation that “Researchers should not apply generalmodels like von Glasersfeld’s or Vygotsky’s directly to the practice ofmathematics education” (p. 204). In fact, we view the model as a generalway of looking at how the participants in our study accommodated theircurrent ways of knowing mathematics with the unanticipated outcomesthey experienced during some of their activities. To create the need for ourparticipants to adapt their mathematical understandings and their viewsof effective pedagogy, we designed activities that were grounded in the

CHALLENGING PROSPECTIVE TEACHERS’ VIEWS 117

well-established body of research regarding students’ conceptions of themathematics of change. The nexus of this research consists of studiesdescribing students’ understandings of motion and graphing (Bowers &Nickerson, in press; Cooney & Wilson, 1993; Kaput & Roschelle, 1997;Nemirovsky & Monk, 2000), of rate (Harel, Behr, Lesh & Post, 1994;Lobato & Thanheiser, 1999; Thompson & Thompson, 1996; Thompson,1994, 1996), and of calculus (Davis & Vinner, 1986; Kaput, 1994; Lauten,Graham & Ferrini-Mundy, 1994; Schoenfeld, Dubinsky & Gleason, 1997;Tall, 1992; Thompson, 1994; Williams, 1991).

One consistent finding in the research regarding the mathematics ofchange is the difficulty students have reading and interpreting graphs ofmotion. For example, several researchers have identified the tendency forstudents to interpret the graph of position versus time as a picture ofthe actual path of the motion (cf., Leinhardt, Zaslavsky & Stein, 1990).This tendency towardsiconic translationof the graph as a picture of thephysical event suggests that graphs of motion and the change in motionare difficult for students to construct and interpret. Likewise, studentsencounter difficulties in interpreting the global features of a graph, suchas change over time (Monk, 1992). Thus, we anticipated thatgraph aspathand point-wise versus over-time graphical interpretations might serveas potential sources of perturbation in the computer-based activities.

Other studies have revealed that students have particular difficultyunderstanding graphs of rates, since they do not have strong intuitionsabout rate prior to instruction (cf., Harel, Behr, Lesh & Post, 1994). Inthe case of the mathematics of change, one possible reason that rate ofchange (speed or velocity) is not as intuitive as position is that speed is anintensive quantity whereas position is an extensive (measurable) quantity(Schwartz, 1988). In exploring the distinction between intensive andextensive quantities, Lobato and Thanheiser (1999) found that students’everyday experiences with an intensive quantity like speed did not helpthem form meaningful ratios for measuring the speed of novel motionssuch as a mouse running along the floor. In fact, in some cases students’prior experiences may have promoted their tendencies to conflate thevarious quantities that could be measured in the situation. These find-ings guided our efforts to design tasks that would involve participantsin activities that demanded strong proportional reasoning about rates andtheir relation to accumulated position in order to interpret the relationshipsbetween the quantities in the computer-generated graphs.

Given that the mathematics of change is a central foundation forcalculus, we also drew on research on students’ difficulties in developingconceptual understandings of limits and functions. Although all partici-


pants had taken at least three undergraduate courses in calculus, we didnot assume that their understanding and experiences extended beyondsuperficial interpretations of differentiation and integration (Selden,Mason, & Selden, 1989; Tall, 1992; Thompson, 1994). Thompson (1994)explained that the danger of such fragile knowledge is that students whodevelop procedural understanding often think of algebraic expressions ascommands to do something, i.e., calculate, rather than as quantities thatare mathematical objects in and of themselves. For example, students mayview the expression 7x − x2 as a string of operations rather than as anentity in itself, namely a function ofx. One implication of this view is thatstudents come to view the integral of an algebraic expression as simplya formalized algorithm; they do not view the expression as a positionfunction that represents the accrual of distance resulting from travelingat a given velocity over time. We hypothesized that if such proceduralunderstanding was a prominent aspect of our own students’ understanding,then asking them to create position graphs over time based on velocitygraphs (rather than on their algebraic representations) would be a potentialsource for perturbation. Our hypothesis was that, for them, position graphswere the result of algebraic manipulations, not graphical interpretations.Likewise, being able to interpret families of functions and the effects ofparameters, such as adding the constantC when computing an integral,would be potentially challenging in that such interpretation depends onan understanding of the integral as a family of position functions alldetermined by the same velocity function.

In summary, the constructivist learning theory that guided our work wasbased on efforts to initiate perturbations in our participants’ views of themathematics of change and elicit shifts in their views of effective ways ofteaching these concepts. The way in which we initiated dissonance was tocreate novel, computer-based activities that might challenge our students’expectations. In creating these activities we drew on three main areas ofresearch: (a) students’ propensity to view a graph as a picture of the pathtraveled; (b) students’ difficulties in interpreting intensive quantities andtheir graphs; and (c) the difficulties inherent in superficial understandingsof calculus and the mathematics of change.

METHODS

Participants and Setting

The participants for this study were pre- and in-service secondary mathe-matics teachers enrolled in one of two technology-based mathematics


courses taught by each of the authors. The courses were taught at twodifferent universities on opposite coasts of the United States. The goalof both courses was to expand students’ ideas about the mathematics ofchange in conceptual ways by engaging the participants in three sharedinstructional sequences. The sites differed slightly in terms of courseemphasis and participants’ educational levels and experiences.

The course at Site A had 15 students, five of whom were in-serviceteachers enrolled in a master’s degree program. The remaining ten studentswere undergraduate mathematics majors, most of whom were seniors, whohad volunteered as teaching assistants in local schools, but had not yet beencertified to teach. The course at Site B had eleven students; ten were preser-vice master’s or doctoral students, and one was an in-service master’sstudent. All the participants at Site B had completed student teaching at thesecondary level, and most had taught introductory level courses as teachingassistants at the university level. All students enrolled in the courses hadcompleted all or most of the courses required for an undergraduate mathe-matics major. Each course met for 3 hours per week with some computerlab work completed in class and the remainder completed by the studentsas homework. All students enrolled in the two courses agreed to participatein the study.

Technology Environment

The shared activity sequences enacted at each site involved the use ofmotion detectors and the MathWorlds software environment, a simulationworld developed by a team of researchers at the University of Massachu-setts at Dartmouth (Kaput & Roschelle, 1997).2 This environment is adynamic microworld for exploring one-dimensional motion in which anycombination of three graphs (position vs. time, velocity vs. time, and accel-eration vs. time) can be linked to an animated simulation and to each other.Unlike most function graphing software that includes multiple, linkedrepresentations of the same data set, the central focus of the MathWorldssoftware is the exploration of the same phenomena that can be representedwith different data sets (i.e., position, velocity, and acceleration). In otherwords, unlike function graphing software that includes linked tables ofvalues, graphs, and algebraic equations to represent the same data set,the MathWorlds software includes an animation of a character or a setof characters moving in a horizontal direction that is directly linked to itsposition, velocity and acceleration graphs, which also are bi-directionallylinked to each other. This bi-directional link enables learners to change anyparameter of a character’s motion by manipulating any of the graphicalrepresentations.


Instructional Sequences

The three core instructional sequences that were shared between the twocourses were designed to engage participants in experiential and graph-ical ways of challenging their formal knowledge of the mathematics ofchange and to support the development of pedagogical content knowledgeregarding how younger learners might engage with these ideas. The firsttwo sequences, which involved investigations of relative and parabolicmotions, were specifically designed to provide opportunities to explore therichness of the Fundamental Theorem of Calculus by examining the rela-tionship between a velocity graph and its linked position graph. The thirdactivity sequence involved having the participants design, implement, andreflect on a MathWorlds-based lesson sequence to help younger studentsinterpret various concepts of the mathematics of change.

Sequence 1. The core idea of the first sequence was to create a situationin which the relative, one-dimensional motion of two characters could beinvestigated. In each task, one character travels at a constant rate whileanother travels at a linearly increasing or decreasing rate. A series of ques-tions focused on determining when and if the two characters will meet(e.g., whether or not a Cheetah would catch a Gazelle) under a variety ofconditions. The level of difficulty increased from simple chases in whichthe two characters started at the same time to more complex scenariosin which one character got a head-start in time (anx-axis translation) ora head-start in distance (ay-axis translation on the position graph only).When investigating overtake questions, the students were only given infor-mation about each animal’s velocity and were asked to solve the task bycreating a velocity graph and a linked simulation as shown in Figure 1.

The pivotal aspect of this sequence was that the position graph of eachanimal’s motion was never used. We attempted to perturb the students’thinking about their views of the relation between position and velocitygraphs by asking them to focus on how they could determine a finalposition by thinking primarily in terms of the graphical representation ofthe relative velocities. In this way, their prior knowledge of computingintegrals algorithmically would not support their efforts to interpret finalposition because they were not given any algebraic expressions. We insteadanticipated that they would reorganize their views of velocity graphs asshowing an accumulation of distance based on rate traveled up to thecurrent moment of time.

Sequence 2. The second investigation involved the use of a motion detectorto graph the position and velocity of a bouncing ball and then to simulate


Figure 1. Velocity graph (left) and Dots simulation (right) used with Cheetah and Gazelleactivities.

this experience in MathWorlds. We designed this activity to accentuatethe contrast between using the motion detector, which takes the motionof a ball as input and gives graphical data as theoutput, and the Math-Worlds software, which essentially inverts this process by taking idealizedgraphical data as theinput and giving an animation of a bouncing ballphenomena as output. As with the first sequence, we intended that apedagogical inversion, that is, a reversal of the traditional instructionalapproach, would encourage the participants to act in new ways withfamiliar mathematical objects and re-think their understanding of therelationship between rates and accumulations.

Sequence 3. The third shared activity sequence was designed to engagethe participants in their roles as teachers. Each participant was asked tocreate a three-lesson sequence that focused on the mathematics of changeand utilized the MathWorlds software. The participants were allowed tochoose the grade level (i.e., middle or high school), the content as long as itinvolved a concept within the mathematics of change, and the pedagogicalapproach, such as working with an individual student or a small group, orusing a single context or multiple contexts for developing concepts. Aftereach teaching session, the participants reflected on the lesson and modifiedsubsequent lessons. The intent of this activity was to have the participants


reflect on and consequently modify their view of their actions within themicroworld as they shifted from the role of student to the role of teacher.

Data and Analytic Method

The data consisted of copies of the participants’ written work on therelative motion assignment and the Bouncing Ball assignment; writtenreflections on their teaching; and the instructors’ daily teaching journals.Data analysis involved comparison of the data from the two coursesin three phases. In the first phase, each author identified the moststriking trends in her students’ mathematical and pedagogical thinking byanalyzing all student work and reviewing journal notes from each day’sinstruction. A trend was defined as an observed reorganization in thinkingor an “Aha” insight reported by a majority of students over the course ofan activity sequence. The second phase involved having each instructorcompare, contrast, and elaborate of each of the trends in order to differ-entiate constructs that could be linked, at least in part, to the students’participation in the common activities from those that were more likelyspecific to the norms and values developed at a particular site. The thirdphase of the analysis involved the documentation of the occurrences of thefinal list of the trends in the data from each course. Thus, all of the trendsreported in this paper were observed at both sites.

RESULTS

Our primary goal was to identify and understand the sources and typesof change that we observed in participants’ views of pedagogy and theirunderstandings of the mathematics of change. We found it useful tocategorize the different types of trends the participants reported as theyassumed the dual roles of student and teacher. In Figure 2, Cells I andIV refer to the more familiar paradigm in which mathematical knowingis examined from the perspective of participants as students (Cell I), andpedagogical knowing is examined from the perspective of participantsas teachers (Cell IV). Our hypothesis, however, was that the experiencesand insights listed in Cells II and III also contributed to the participants’overall changes in their mathematical and pedagogical knowledge. In thefollowing discussion of our results, we describe our findings in each ofthese four analytic categories in more detail.


Figure 2. Participants’ mathematical and pedagogical insights when acting with theMathworlds software.

Mathematical Insights from Participants as Students

In this section, we describe two mathematical insights that our participantsreported after experiencing perturbations in their work as mathematicsstudents using the MathWorlds software at each of the two sites.

Mathematical insight #1: A velocity graph determines a family of posi-tion graphs. One insight that all participants from both sites reported wasthat any given velocity graph determines a family of position graphs.The genesis of this insight occurred as follows. First, participants trans-lated the velocity graph vertically and noted that the linked position graphmoved accordingly. Next, they translated the position curve vertically, butnoted that the linked velocity curve did not change at all. Their efforts toresolve this perturbation ultimately led them to develop a deeper under-standing between the underlying quantities represented in velocity andposition graphs. They came to see that in varying the initial startingpoint by vertically translating the position graph, they were generating afamily of position graphs but were not changing the speed at which theanimal moved. At this point, many students reported an “Aha” insightregarding the meaning of the ubiquitous “+C” they had routinely beenadding when computing indefinite integrals in their prior calculus classes.One participant from Site A wrote:

First I made a velocity graph for the Gazelle, and then transferred this to a position graph.Then by chance I moved the final position of the Gazelle’s dot [in the simulation] and theGazelle’s line on the position graph shifted upward! “Great,” I thought, I messed everythingup, now I’ll have to redo the graphs – that is when it hit me. The velocity graph had notchanged. The velocity graph reflects the rates on the position graph – not the starting point.With a different starting point (simply shifted all up) all of the velocities remain the same.


Cool! I guess I knew that, but now I am aware of it and understand the workings behind it.. . . I did not ever realize (probably because it was never posed to me) that from the velocitygraph you could not draw the position graph, unless you were given the starting point!

This participant described her surprise when she realized that what sheexpected would happen, that is, that moving the character in the simula-tion would mess up the velocity graph, did not occur. She resolved herperturbation by forming what, to her, was a more viable interpretation ofthe velocity graph: “the velocity graph reflects the rates on the positiongraph – not the starting point.” This reconsideration of the velocity graphbrought forth another aspect of her previous formal knowledge, namely thevalue of they-intercept as a starting point. She also noted that she probablyalready was aware of this, but had not fully realized “the workings behindit.”

Mathematical insight #2: The difference between average and instantan-eous velocity. The Bouncing Ball activity required the participants to createa position graph given the velocity graph shown in Figure 3a. Like the firstsequence, we anticipated that this task would be difficult for students withfragile understandings of calculus who relied on rote methods of integra-tion and who had not formed an image of position in terms ofaccumulateddistanceaccrued by traveling at a linearly changing rate over time. Ourgoal was to challenge the participants’ current mathematical knowledge aswell as their views about the traditional ways the subject often is taught.As we anticipated, when first attempting this task with paper and pencil,over half of the participants at Site A attempted to solve the task by relyingon the formula of d = r * t. Thus, these participants created a table of valueswith three columns as shown in Figure 3b, and then calculated the positionat any given timex by multiplying the velocity at that point (as indicatedby they-coordinate of the velocity graph at timex) and the value of thetime at that point. One participant described his work by noting, “In orderto obtain the position graph, you must multiply the velocity by the time tohave the desired units of meters. The algorithm looks like this, (m/s) * (s)= m.” This student’s position vs. time graph is shown in Figure 3c.

When this participant compared the graph he had created on paper withthe position graph that he created in the MathWorld, he was surprisedto see that they did not match. This observation, which he brought upin class, led to a discussion of two critical mathematical concepts: theneed for the conventional definition of negative velocity and the differencebetween average and instantaneous speed. These discussions led the classto devise a more meaningful interpretation of the Mean Value Theorembased on a graphical interpretation of rate. The average rate was defined as


Figure 3. (a) Assignment asking students to create a graph that shows the ball’s positionat any time given the velocity graph (assumeP(0) = 0). (b) One student’s solution toassignment. (c) Student’s graph created by plotting time and product oft * v(t).


the constant rate at which another character would travel in order to coverthe same distance as the bouncing ball during the same given time interval.

Pedagogical Insights from Participants as Students

In this section, we continue to view the participants as students learningmathematics but now consider the ways in which their pedagogicalthinking was challenged as they worked with the microworld activities.

Pedagogical insight #1: The potential value of conceptual explanations. Atboth sites, the focus of most activities included an emphasis on explainingwhy a computer-generated graph appeared as it did. At Site A, this wasdiscussed in terms of a distinction between calculational and conceptualexplanations. Although the instructors had hoped that such a practicewould be helpful, some of the participants had difficulties understandingthe purpose and form that such explanations should take. Several of theparticipants questioned the value of this practice whereas others main-tained that conceptual explanations supported their own efforts to developimagery for motion and hence would support their future students’ effortsto reason conceptually as well. Although the class remained undecided,they did agree that teaching with technology involves rethinking the formatof activities and what counts as an acceptable explanation and solutionwithin any given classroom culture.

Pedagogical insight #2: The tradeoffs in having exploration before orafter symbolization. Participants discussed extensively whether mathema-tical formalisms should be introduced before, during, or after studentshave explored the mathematics that the symbols portend to signify (cf.,Doerr, 1997). Given that all of the participants in the study had alreadyencountered the formal symbols of calculus, and hence the formalisms ofcalculus preceded their explorations in graphically oriented microworld,one might expect that they would argue for the symbolize-then-exploreapproach. Indeed, some of the participants at each site maintained thatthis was a desirable pedagogical ordering given that it served as a sourceof perturbation and ultimate reorganization for them. On the other hand,others were eager to shift their pedagogical approach, based in part on theirenthusiasm for their own new-found conceptual insights and on the poten-tial for learners to meaningfully engage in conceptually oriented activities.The point here is not that the students should have come to an agreement,or even that there is one right answer to the question. Instead, the valueof these discussions was that the participants assumed the roles of bothteacher and student as they argued their points. Moreover, they realized


that they, as prospective teachers, do have choices in how they interact withtheir students and that these choices affect students’ views of mathematicsin general.

Mathematical Insights from Participants as Teachers

In this section, we shift from viewing our participants as students ofmathematics to viewing them as teachers who were tutoring and teachingyounger children in one-on-one or small-group settings with the Math-Worlds software. Given that this activity occurred toward the end ofthe semester at both course sites, we expected to see some evidence ofthe participants’ increased pedagogical content knowledge based on ourcurricular agendas as described earlier.

Across both sites, 38% of the participants chose to focus their lesson onthe relationship between the position and velocity graphs and, in particular,the concept relating area under the velocity graph accrued at each timexto the value of the object’s position at timex. This was not surprising,because the two primary activity sequences in the course (the Cheetahand Gazelle and the Bouncing Ball activities) focused on aspects of thisconcept. As a consequence of their experienced instruction, this may haveappeared to the participants as a natural starting point for their activitieswith younger learners. What was more surprising was that the remaining62% of the participants extended their own learning experiences withMathWorlds by designing learning activities for students that addressedother mathematical content (such asy-intercept and slope) in novel ways.

In the following two sections, we describe the mathematical andpedagogical insights that emerged as the participants transitioned fromtheir role as students to their role as teachers. First, we present two mathe-matical insights that emerged from the participants’ design and reflectionprocesses. Following that, we discuss two pedagogical insights that werereflected in the participants’ written descriptions of their work with theirstudents.

Mathematical insight #1: The importance of differentiating between localand global interpretations of graphs. One of the most widely reportedinsights by participants, most notably and explicitly at Site B, was thattheir students’ activity in the software environment provided occasions togauge whether the students were making local or global interpretations ofthe features of a graph. This focus was discussed at Site B after the partici-pants had read Monk’s (1192) distinction between local interpretations,which involve attention to specific point-wise features such as points ofintersection or relative extrema, and global interpretations, which involve


a focus on the behavior of the graph over its entire domain. For example,one participant reported that his students, who had investigated periodicgraphs, were easily able to interpret the features of both the position andthe velocity graphs as the character in his story walked back and forthbetween the garage and the end of the driveway. The participant describedhow his student, Jake, interpreted the situation:

When identifying the period of the function, Jake focused on the maximums and minimumsof the position graph. Jake has given meaning to those features as ones that will helphim find the period of the function. After the period was identified on the position graph,Jake would verify the period on the velocity graph. Jake has identified the maximums andminimums of the position graph as places where Toni [the character in the story] is at theend of the driveway or at the garage. When he is identifying the period of functions, he isimmediately drawn to these features even if the initial position is not a relative extrema ofthe graph.

Although this participant remained skeptical about Jake’s understandingof the relationship between the direction the character was moving andthe slope of the position graph, his final report indicated that he felt thathis pupil had developed a good understanding of the relationship betweenthe relative extrema (a local feature of the graph) and the period of thefunction as well as a solid interpretation of these features in terms of thecharacter’s actual motion. This distinction, which the participant madeon his own, is significant in that it reflected his shift away from a focuson correct or incorrect answer toward mathematical meaning in terms ofinterpretation. It also demonstrated strong pedagogical content knowledgein that he distinguished between his student’s formal understandings of themathematical concepts (such as slope as an interpretation of speed) and hisstudent’s interpretation of the various critical points on the graph.

Another participant, Romy, explicitly focused her lesson on herstudent’s interpretation of across-time or global features of a function.Romy wanted her students to predict the total distance a character wouldtravel given a velocity graph and to create a position graph that wouldmatch one character’s motion traveling at two different speeds. To Romy’ssurprise however, one of her students created a graph resembling thatshown in Figure 4, which is a non-standard representation that led Romyto re-think how graphs may be interpreted in unanticipated ways. Romyexpected her students to use piecewise constant line segments to representthe character’s changing velocity over time. This interpretation wouldhave involved a global interpretation of a position graph. In assessingher student’s work, Romy allowed for the possibility that the student mayhave misunderstood the question but she also felt confident that the studentunderstood the sequential nature of the motion. What is more significant isthat this participant appeared to be flexible enough in her own mathema-


Figure 4. An unexpected velocity graph generated by Romy’s student.

tical thinking to extend her mathematical interpretation of the graphicalrepresentation by seeing how the student could be reading along onlyone dimension of the graph and not making time an explicit part of therepresentation. By considering the student’s paper and pencil representa-tion as an alternative way of knowing, the participant gained insight intohow and why the student might choose an unconventional representationof sequential, one-dimensional motion.

Mathematical insight #2: The importance of appropriate contexts. Allparticipants created contexts that they thought would appeal to theirstudents’ interests, but several reported that this was more difficult thanthey had anticipated. For example, one participant from Site A explainedthat he began his sequence using the context of plant growth. Later herealized that this caused a problem when he wanted to include negativevelocity into the same context.

The context of the overtake race, where the one-dimensional motion ofone character overtakes another character given a range of initial startingconditions and varying velocities, as featured in the Cheetah and Gazellesequence, became a powerful metaphor for supporting the participants’development of short instructional sequences. All but one of the partici-pants at Site B and approximately half of the participants at Site A usedsome variation of a race in teaching their mathematical content.

For example, one participant who designed an activity to focus onRiemann Sums used the context of an overtake race to feature one dot’svelocity controlled by a half period of a sine curve. The task for his


students was to create another velocity graph using piecewise, horizontalline segments so that the second dot would end up in a tie with the firstdot. This participant drew on the students’ prior knowledge about the areaunder the velocity curve as determining position and created a mathe-matical environment that would allow his students to create a piecewiseapproximation for the area under a sine curve. Based on his two students’work, in which they generated their approximations by positioning thehorizontal line segments at the midpoint of the curve, this participantrefined his activity to include both upper and lower sums for the approxim-ation. What we find significant is that this refinement occurred as a result ofseeing his students interact with each other and with the learning task thathe designed. For this participant, the mathematical insight that he exper-ienced emerged as he tried to reconcile the contextual difference for hisstudents between approximating by positioning segments at the midpointof the curve on an interval and by sandwiching the curve between upperand lower segments.

Pedagogical Insights from Participants as Teachers

Pedagogical insight #1: The value of building on students’ incorrectexplanations. Although some of the participants such as Romy capital-ized on her student’s errors, other participants from both sites maintainedlimited views of what constituted a correct answer from a pedago-gical point of view. We see this as a stumbling block when it preventsparticipants from building on their students’ incorrect but reasonable andpotentially fruitful explanations.

One participant who missed an opportunity to capitalize on herstudent’s own mathematical explanations was Ellen, a participant from SiteA who was tutoring a talented high school sophomore who had never seena position or velocity graph. To begin Ellen asked him to draw a positiongraph in which a clown was traveling at a constant rate of 6 m/s for 3seconds. The student reasoned that the clown would be at 18 meters afterthree seconds. He then plotted the points (3,18) and (0,0) and then drew aline segment connecting these two points as shown in Figure 5a. Ellen thenasked him to draw a velocity vs. time graph showing the clown’s velocityat each second. In response, the student drew the graph shown in Figure 5b.Because this graph did not match Ellen’s expectations of what the studentwould draw, namely a velocity graph showing a constant speed of 6 m/sfor 3 seconds, she dismissed it as incorrect.

However, in her written reflection, Ellen realized that her student haddescribed a graph of rate byseeingvelocity in the ratios of the heightand length of each stair step. She noted that her student explained that


(a) (b)

Figure 5. (a) Student’s drawing of position graph. (b) Student’s drawing of velocity graph.

if the person was traveling for 6 meters per second, then, after 1 second,he would be at the point (1,6), which could be calculated as a speed of 6meters per second. Likewise, after two seconds, he would be at the point(2,12), which, the student explained, was 12 meters per two seconds, or6 meters per one second. For him, although they looked the same, graphs5a and 5b were entirely different, depending on how he interpreted them.After discussing this with her instructor, Ellen gained insight into thepotential value of building on students’ intuitions rather than assumingthat she could just erase what he was thinking and tell him how to createthe correct velocity graph.

A second example of this tension was evident in one participant’s reportfrom Site B. Linda described a student’s desire to work with the simulationto check his conjectures and contrasted that with her drive toward inter-preting it graphically and then with the formula for area. Even when theformalism followed the exploration in the lesson plan, Linda maintainedthat the formalism was her central goal for the lesson. She essentially sawthe lesson as “failing” (her words) because, in the end, the student didnot move successfully, in her opinion, to the formalism. This report, likeseveral others, indicated that the participants’ experiences as students andteachers highlighted the pedagogical dilemma of the relationship betweenthe explorations of ideas and their expression in the formalism of mathe-matical symbols as described in the earlier section on pedagogical insightsfrom participants as students.

Pedagogical insight #2: Influence of hidden supports and constraints oftechnology on students’ mathematical activities. As Greeno (1997) andCobb (1999) pointed out, the features of any computer software programprofoundly affect the nature of one’s activity with it. One implication ofthis claim is that teachers wishing to use technology in their classroomsneed to recognize the supports and constraints of the technology andto decide how to structure activities so that students act in ways that


are potentially productive. For example, one technological feature thatmay have constrained the types of mathematical activity in which thestudents were engaged was indicated in several participants’ descriptionsof their students’ dissonance with discontinuous velocity graphs. Julie,a participant from Site B, who worked with a group of four 11 year-oldstudents, reported the following dialogue about the velocity graph shownin Figure 6.

Figure 6. A discontinuous velocity graph.

A: But that’s impossible. Rumba [a clown in the Mathworlds software]has to pass through all the velocities between 5 m/s and 4 m/s beforeactually reaching 4 m/s.

T: Why do you say that?B: That means Rumba would have to be an alien!C: Yea, there has to be some number of seconds when Rumba is slowing

down before reaching the new slower velocity.

Julie interpreted the dialogue as follows:

The limited ability of MathWorlds to replicate real-world situations became apparent toeven the sixth graders with whom I worked. In fact, they saw what perhaps the creatorsof MathWorlds failed to see – the physical impossibility of instantaneously going fromwalking 5 meters/second to walking 4 meters/second.. . . Hence, in the eyes of thesestudents, discontinuous velocity graphs seemed as extraterrestrial as discontinuous positiongraphs.

Julie’s comment indicates that she understood that the use of discontinuousvelocity graphs was an artifact of the design of the software but saw this as


a problematic mismatch with the viable explanation the student had createdto explain his experienced reality. For Julie, the designers’ decision wasnot consistent with her intended pedagogy. She therefore interpreted theconstraint as presenting a serious pedagogical difficulty in that it pushedthe students to abandon their common sense about realistic situations.

The participants at Site A discussed the differences between the afford-ances of the MathWorlds software and those of the motion detector. In theirdiscussion, they noted that because the motion detector did reflect the real-world, it contained noise that distracted from the mathematical abstractionof the motion. Their work in MathWorlds following their work with themotion detector enabled them to differentiate the mathematical abstractionfrom the noise and to develop a deeper understanding of the way in whichthe limits of real-valued functions serve as a bridge between motion in thereal world and that of mathematical formalisms that model it.

DISCUSSION

The two research questions that we set out to investigate were: (a) Howdo participants, acting in the role ofstudent, think about the mathematicsof change when using an exploratory microworld, and (b) How do partici-pants, acting in the role ofteacher, plan, implement, and reflect on lessonsabout the mathematics of change when tutoring their students. In analyzingour results, we found it useful to coordinate the participants’ experiences inour classes with the perspectives through which they enacted those exper-iences. In so doing, we were looking for a way to view our courses fromstudents’ experiential perspectives, and perhaps gain more insight than justlooking at their pedagogical thinking as prospective teachers (Cell IV) ortheir mathematical knowledge as students (Cell I).

Our analysis revealed two critical perspectives that might have beenotherwise missed. First, the participants developed pedagogical insights asstudents of mathematics (Cell II), and second, the participants developedmathematical insights as teachers of mathematics (Cell III). Some of theparticipants’ most powerful pedagogical insights emerged as they wereassuming the role of mathematics students. For example, the debates overthe value of conceptual explanations and the benefits of exploring mathe-matical ideas (with the use of microworlds) before introducing formal-izations were rich because the participants argued from both perspectives.Although neither debate was fully resolved, the participants from both sitescame to value their own mathematical insights and appreciate these valuesas teachers more deeply than had they just been told of these pedagogicalstrategies in a methods class. In other words, we found that the students


who experienced the struggles from both sides came to develop an appre-ciation for the value of conceptual explanations and explorations withtechnology.

A second finding that confirmed our hypothesis regarding the value ofviewing participants in the dual roles was that some of the participants’mathematical insights developed as they created, taught, and reflected onmathematical lessons. For example, many participants realized the impor-tance and difficulty of choosing a rich context through which one couldexplain the mathematics underlying the relationship between velocitygraphs and their associated position graphs. Similarly, several participantsfrom each site found that they could learn new mathematics by listening totheir students’ interpretations.

The constructivist approach we assumed when planning and analyzingthis project focused on accounting for cognitive changes which were situ-ated in the context in which the individuals were acting. In following Steffeand Thompson’s (2000) premise that the source of perturbations is oftenthe social situation in which the student is acting, we are not claiming thatthe microworld alone caused any of these fruitful reorganizations. Instead,we claim that as the activities were realized in the social setting of each oftwo sites, the participants’ efforts to reconcile what they anticipated withwhat they found led to fruitful discussions. It was these discussions and theparticipants’ consequent reflections and abstractions that we believe led tochanges in their mathematical and pedagogical content knowledge as well.

CONCLUSIONS

We close with two conclusions that relate to our work as teacher educators.First, as noted above, we found that our use of computer-based activitysequences served as an effective means for eliciting perturbations amongprospective and practicing teachers. Our results indicate that many of theparticipants experienced “Aha” insights because they reorganized theirinitial understandings of the mathematical relations between position andvelocity, and, in so doing, gained a deeper understanding of the mathe-matical formalizations as well. We found that because the activities weresomewhat novel, they placed the participants in a learning situation inwhich familiar mathematics seemed unfamiliar but not threatening, thusevoking a dissonance that needed to be resolved.

A second implication for teacher educators is that our analytic frame-work enabled us to explicitly position our participants in the dual roles ofstudent and teacher while simultaneously considering their mathematicaland pedagogical knowledge. The value of this conceptualization can be


illustrated by considering a discussion that took place near the beginning ofthe semester at Site A. When the instructor asked the participants what thevalue of a conceptual explanation was, they seemed to almost uniformlyagree that good teaching involved “delivering clear explanations” to theirown students and hence the value of a conceptual explanation was that itenabled them to explain things more clearly. This view changed over thecourse of the semester, such that they came to value conceptual explana-tions not as tools for preaching but as tools for helping their own studentsexplain things for themselves. Had this not been an explicit discussionearly on, neither the participants nor the instructor at Site A would havebeen aware of the fact that they were talking past each other.

NOTES

1 The analysis reported in this paper was supported in part by the National Science Found-ation under grant No. REC-9619102. The opinions expressed do not necessarily reflect theviews of the Foundation.2 The version of MathWorlds software used in our classes was compatible with the Macin-tosh platform only. A Java version is now available at http://www.simcalc.umassd.edu/.

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Janet BowersCenter for Research in Mathematics and Science EducationCollege of SciencesSan Diego State UniversitySan Diego, CA 92120USAE-mail: [email protected]

Helen M. DoerrDepartment of Mathematics215 Carnegie LibrarySyracuse UniversitySyracuse, NY 13244-1150USAE-mail: [email protected]

DIANA F. STEELE

THE INTERFACING OF PRESERVICE AND INSERVICEEXPERIENCES OF REFORM-BASED TEACHING:

A LONGITUDINAL STUDY

ABSTRACT. This article contrasts four elementary teachers who were graduates of ateacher education program that incorporated a reform-based mathematics methods course.The report provides results from a four-year longitudinal study that extended from thetime that the participants were preservice teachers until the end of their second year ofteaching. The article provides background information of each teacher, vignettes from herteaching, excerpts from interviews, and an analysis of each teacher’s case. Results fromthe case studies indicate that two of the four teachers sustained their cognitively-basedconceptions about mathematics teaching and learning, and implemented these conceptionsinto practice. The analysis suggests that there were several factors that influenced theteachers’ conceptions and the choices they made in their teaching: personal commitment,professional strength, curriculum, planning, assessment, beliefs, knowledge, and supportfrom school administration. The article concludes with implications for teaching and ques-tions about the nature of what might be required in the beginning years of teaching if newteachers are expected to implement reform-based mathematics teaching practices.

Researchers in mathematics education suggest that teachers’ conceptionsabout what mathematics is and what it means to learn mathematicsdirectly impact how they teach mathematics (Ernest, 1989; Thompson,1992). According to Thompson, conceptions are belief systems that are“mental structure[s], encompassing beliefs, meanings, concepts, proposi-tions, rules, mental images, preferences, and the like” (p. 130). Theseconceptions do not need to be consciously held views, but rather couldbe implicitly held philosophies. Teachers’ conceptions are dependentupon their experiences as learners of mathematics (Ernest, 1989). Currentteachers, when students, often learned in teacher-centered classrooms bylistening to lectures, memorizing information, and practicing rote compu-tations. Ernest suggested that teachers who have learned mathematics inthese ways most likely conceive of mathematics in the instrumentalistview, a conception that mathematics is a collection of unrelated facts, rules,and skills.

In contrast, mathematics teachers, teacher educators, and researchersinvolved in the current reform movement in mathematics education recom-mend that students need to be actively involved in constructing theirown knowledge and developing mathematical concepts that require them


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to explore, explain, and justify solution strategies to mathematical tasks(NCTM, 2000). Teaching mathematics in this way coincides with theproblem-solving view of mathematics, a conception that mathematics iscontinually expanding through human inquiry (Ernest, 1989). For manyteachers, this reform-based approach to teaching mathematics requires achange in conceptions about mathematics and about what it means to learnand teach mathematics (Brown, Cooney & Jones, 1990).

The impact that the beginning years of teaching has on teachers’commitments to the conceptions they developed during teacher educa-tion has been the topic of research on teacher education. Some studiessuggest that many beginning teachers give up their new conceptions asthey struggle to survive and to fit into the institutional norms of tradi-tional educational practices (Wilcox, Lanier, Schram & Lappan, 1992).The excitement of becoming adult learners of mathematics during univer-sity methods courses is often diminished when beginning teachers confrontthe day-to-day problems of teaching their students. As a result of thecompeting issues of “disciplining their class, individualizing their instruc-tion, motivating their pupils, evaluating children’s work, dealing withpersonal problems of pupils, scheduling class work, relating to parents andcolleagues, discovering what materials are available, and finding time toplan lessons” (Goodman, 1987, p. 207) beginning teachers develop a “util-itarian perspective toward instruction in which teaching is separated fromits underlying educational or social dimensions” (Goodman, 1987, p. 208).Cooney and Shealy (1997) suggested that when beginning teachers entertheir first year of teaching and are confronted with uncertainties inherent inpractice they may take control of their classrooms by denying uncertaintiesor accusing reform-minded teacher educators of being unrealistic.

Lacey (1988) and Etheridge (1989) found several strategies that begin-ning teachers use to respond to administrative demands of schools. Thefirst strategy is called internalized adjustment. The teacher complies withenvironmental constraints, such as pressures from school administratorsand other teachers, and believes that the constraints are for the best. Thesecond strategy, strategic compliance, occurs when the teacher appearsto comply with the authority’s constraints, but retains private reserva-tions. The third strategy is strategic redefinition. Teachers who use thisstrategy respond to situations that conflict with their views by changingthe situation to match their beliefs.

Many studies have been designed to bring about changes in the concep-tions of preservice and inservice teachers (e.g., Cooney, Shealy & Arvold,1998; Raymond & Santos, 1995; Schifter & Fosnot, 1993; Steele &Widman, 1997; Wilcox et al., 1992). Few studies, however, have explored

INTERFACING OF PRESERVICE AND INSERVICE EXPERIENCE 141

whether perceived changes in conceptions of preservice teachers have beensustained over time, that is, through the first years of teaching. Such longi-tudinal studies can help teacher educators and researchers deepen theirunderstanding of how or whether teachers reconceptualize their teachingpractice to include their new conceptions.

In a prior study, I found that a group of preservice elementary schoolteachers had demonstrated evidence of changes in their conceptions aboutmathematics and about what it meant to teach and learn mathematics.These changes in conceptions had been precipitated by their experiencesin a reform-based mathematics methods course in which I, as instructor,elicited cognitive conflict with respect to their previously held conceptions(Steele & Widman, 1997).

PRIOR STUDY AND FRAMEWORK

I conducted this prior study at a large southern university that had a five-year teacher education program. In a report of this study, I describedchanges in conceptions that five preservice elementary school teachershad experienced as students in an elementary mathematics methodscourse (Steele & Widman, 1997). I taught the course with a reform-based approach, as outlined in theProfessional Standards for TeachingMathematics(NCTM, 1991), and used research fromCognitively GuidedInstruction (CGI; Fennema, Carpenter & Peterson, 1991) as a basis forteaching much of the content of the course. The guiding principle of CGI isthat instructional decisions should be based on teachers’ careful analysesof students’ knowledge. I used this approach to help preservice teachersunderstand children’s thinking, give preservice teachers an opportunity toplan how to use this knowledge in their classrooms, and give them timeto reflect on what happens when they use this knowledge (Fennema et al.,1991). I organized instruction to involve preservice teachers in activelyconstructing their own knowledge with understanding, as I wanted them todo with their own students.

At the end of the methods course the five preservice teachers espousedconceptions about mathematics and about mathematics teaching andlearning that had shifted toward a more cognitively-based perspective andproblem-solving conception of mathematics, as emphasized in reform-based mathematics education. Another researcher and I conducted inter-views before, during, and after the methods course with these fiverandomly selected participants. These interviews showed significantchanges in the images that the preservice teachers had of themselves asteachers of mathematics. They identified turning points they had experi-

142 DIANA F. STEELE

enced during the course that had led them to rethink mathematics teachingand learning. Eighteen months later, after their student teaching and internexperiences, we interviewed these preservice teachers again. They stillproclaimed cognitively-based conceptions about mathematics learning,and they still held images of themselves as using a reform-based approachto mathematics teaching.

In this follow-up study I focused on four of the original five preserviceteachers. In this study I investigated whether the preservice elementaryschool teachers sustained their new conceptions when they becameteachers, and if so, how they implemented these new conceptions intotheir own practices. The questions that guided the research were (a)Did preservice teachers sustain their new conceptions about mathematicsand mathematics teaching and learning and implement a reform-basedapproach to mathematics teaching when they became teachers? and (b)What factors influenced teachers’ new conceptions?

METHODS

I used qualitative research methods, in particular case studies, to explorehow or whether these elementary school teachers, who had clearly articu-lated a change in conceptions about mathematics and about mathematicsteaching and learning, sustained these conceptions (Glaser & Strauss,1975). Case studies as a research approach provided the richness and detailthat was needed for me to understand the process in which the teachersengaged as they implemented practices in or out of harmony with theirconceptions. Case studies provided the means for obtaining a descriptive,holistic view of these teachers’ conceptions in practice (Merriam, 1988;Stake, 1995).

Participants and Context

The four female elementary education graduates had been traditionallyaged college students. They were now teaching at different schools andcompleting their second year of teaching, four years after they had beenstudents in my methods course. They had graduated from a program thatemphasized a reflective approach to teaching; several professors from theuniversity had national reputations as experts in reflective teaching. Everyteacher education course was infused with the need for reflecting on prac-tice and reflecting in action for becoming a professional. The missionstatement of the department was centered around constructivist learningtheory. This teacher education program was also clinically intensive. By


their third year, preservice teachers were not only observing but alsoteaching. They performed what would be considered student teaching intheir fourth year. They began a full semester of internship as classroomteachers in their fifth year. For their final semester, preservice teachersconducted action research projects in the classrooms in which they hadbeen interns. If students scored a predetermined appropriate score on theGraduate Record Examination, they could receive a Masters Degree whengraduating from the university.

Data Collection

Qualitative data. In order to enhance the validity of the research findings(Lincoln & Guba, 1985; Merriam, 1988), I collected data in three ways. Igathered the majority of the data through formal and informal interviews.Each teacher participated in six formal interviews that I either videotapedor audiotaped and later transcribed verbatim. Each interview lasted aboutone hour. After each interview, I wrote a journal documenting my obser-vations from the interview and reflecting on future questions. Some of thequestions that I asked were:

• What is mathematics?• How do students learn mathematics?• Paint a picture of yourself in the classroom.• How do you choose which topics to teach and in what sequence?• How do you plan and prepare for a lesson?• Do you always complete the lesson you planned?• How do you assess students’ learning?• To what extent have you been able to implement the types of

classroom activities and tasks you believe most help students learnmathematics?

In addition, I conducted informal interviews with the principals and otherteachers in the schools. I asked questions that dealt with issues such assupport for teachers and their own views of teaching mathematics. Ingeneral, I wanted to understand the collegiality and learning environmentsof the schools.

Participant observation was the second form of data collection. I spenttwo days in each teacher’s classroom observing their teaching – mathe-matics and other subjects. In addition, in order to understand the schoolenvironment, I also observed the four participants interact with otherteachers at lunch time and in meetings. I audiotaped all observations inclassrooms and kept a journal of field notes and reflections. Finally, I

144 DIANA F. STEELE

collected documents that included teachers’ plans, worksheets, curriculumguidelines, and tests.

Quantitative data. I asked each teacher to complete a Mathematics BeliefsScales questionnaire (Fennema, Carpenter, & Peterson, 1987) from theCognitively Guided Instruction Project (Fennema et al., 1991). The partici-pants had completed this questionnaire at the beginning of the methodscourse, at the end of the course, and now after two years of teaching.The questionnaire is a Likert 5-point scale with possible responses rangingfrom “strongly agree” to “strongly disagree.” Items on the questionnairecan be divided into different constructs about teaching and learning mathe-matics that question beliefs along a continuum. I placed teacher responseson this continuum. At one end I classified the responses as more cogni-tively based, with students viewed as constructors of their own knowledgeand teachers as facilitators helping students to develop their own under-standings. At the other end of the continuum I classified the responses asless cognitively based, with the student viewed as the receiver of knowl-edge and the teacher as the provider of knowledge (Lubinski & Jaberg,1997). A lower score on the beliefs scales of one to five indicated a morecognitively-based perspective about learning and teaching.

Data Analysis

I systematically and rigorously organized, analyzed, classified, and consol-idated the data using the Developmental Research Cycle (Spradley, 1980)in order to determine patterns and cultural themes. This type of dataanalysis enabled me to make the transition from asking general ques-tions about the study to asking and answering more specific questionsthat followed directly from the data. The themes of personal commitment,professional strength, curriculum, planning, assessment, beliefs, knowl-edge, and support from school administration emerged as factors thatinfluenced and sustained conceptions. In the following section I use thesethemes to organize the discussion of the teaching cases after I presenteach case. Each case includes a description of the teacher, the context ofher teaching, a vignette from her mathematics lesson, and excerpts frominterviews.


THE CASE STUDIES

The Case of Mary

Mary’s change in the methods course. When Mary was a student in theelementary mathematics methods course, she was perhaps one of the mostresistant students in the entire class to reform-based ideas about learningmathematics. In the first interview before the methods course began, shehad declared,

As a teacher, I see myself at the chalkboard showing [students] how to work problems –the basic skills – addition, subtraction, multiplication, and all that. I think they need to getthe skills down – drill and practice, drill and practice. Whether they need to understand andall that, I don’t know. They should just be skillful in the four operations. If I do flash cards,I like to see them go like this [snaps her fingers – snap, snap, snap]. (Steele & Widman,1997, p. 188)

Mary’s attitude about how children learn mathematics changed after shehad completed an assessment interview of a gifted fifth-grade student’sunderstanding of multiplication and division. She had been sure that hewould understand the concepts. After her assessment, however, she wrotethat he was adept at using the algorithms but had not conceptually under-stood what he was doing either while multiplying or dividing. In fact, hewas well beyond the fifth-grade in the use of algorithms for multiplicationand division. This experience initiated a change in Mary’s thinking aboutteaching mathematics from a traditional drill-and-practice approach. Fromthat time until the end of the semester, Mary displayed a more open attitudeto what she was hearing in class. And, by the end of the semester, sheexpressed a new view of what it meant to teach and learn mathematics.

My teaching habits and philosophies have changed a lot during the course. I’m glad I hadthe opportunity to re-examine my mathematics beliefs. As a teacher, I have a vision of myrole as “tour guide.” My part would be to lead the students to a specific math concept Iwant them to obtain and let them explore and discover.

A snapshot from Mary’s teaching. At the time of the study, Mary wasa first-grade teacher who was excited about her teaching. She had beenteaching in the same public school for two years. Even though herelementary teaching specialization was special education, Mary taught allability levels. Her room was full of activities and projects on which herstudents were working. Their work was displayed on walls and windows,and hung from the ceiling. Shelves were filled with mathematics manipu-latives of all kinds. There were calculators for every student, and twocomputers at the side of the room. At the time I observed Mary’s classshe was using a unit called “The Sea” as the unifying theme for all the

146 DIANA F. STEELE

subject areas. On the days I observed, students were learning about linearmeasurement. They had read about different whales and were trying tovisualize these lengths. However, they were having difficulty visualizingthe length of the Blue Whale, which was 100 feet.

Mary said, “About how big is 100 feet? Anybody have an idea”? Severalstudents had different answers: “Bigger than this room,” “From here tohere (a student pointed from the bottom of the chalkboard to top of thechalkboard),” “From the parking lot fence to this wall,” “From here toNew Jersey,” “From Canada to Florida.” When it appeared that studentswere having difficulty visualizing 100 feet, Mary began a discussion aboutother whales with shorter lengths. They talked about the Killer Whale,which is 15 feet and went on to others. Mary got out different measuringtools, that is, tape measures, rulers, yardsticks, and asked students whichtools would be appropriate to measure the whales. By the end of the lesson,students were in the cafeteria measuring the lengths of different whales.Working in pairs, students rolled out adding machine tape and used tapemeasures and yardsticks to measure the lengths of different whales. Marysoon discovered that students were having difficulty knowing where tostart measuring when lengths were longer than the measuring tool eventhough they had learned earlier in the year how to use rulers. Studentsbegan disagreeing about how to start a new measurement space or how tokeep track of the number of times they used their measuring tool. Somestudents decided to draw a pencil line or put down a finger to keep theirplace. Some kept a tally of the number of times they used the measuringtool; some used calculators. They did not complete their measurement forthe different whales.

At the beginning of next day’s lesson Mary discussed with the studentssome of the issues that had been raised during the measurement activity,that is, which tools to use, how to use the tools, how to keep one’s place.Listening to the discussion, I heard her ask students about importantmeasurement foundations: Units of measure should be adapted to theobjects being measured; all the units must be identical when using repeatedunits to measure an item; and the object must be subdivided along its lengthwhile measuring. Students again measured the lengths of different whales,and by the end of this second lesson, most students appeared to have abetter understanding of how long 100 feet was, the question with whichthe measurement lesson had begun, and how it compared to their guesses.

Mary later discussed what the goal of her two lessons had been. Shesaid,


What I had hoped they would get was that they would be able to visualize or picture intheir heads how big 100 feet is or how big 15 feet is. They need to visualize lengths. Whatif a policeman says, “You are supposed to be four car lengths behind that car up there”?

It appeared that Mary had achieved her goal. Students estimated differentlengths and then measured to show how close their estimates had been tothe real measurement. Mary’s reference to the car lengths indicated thatshe wanted students to connect their learning to real life.

Being a reflective teacher. Mary showed important qualities of being areflective teacher when she talked about how she planned her lessons.For example, when Mary’s students couldn’t visualize 100 feet and thenhad difficulty measuring lengths of more than one foot, she planned activ-ities for the next day that would help them both understand the conceptof measuring and the procedure required to validate estimates. She saidthe first day’s lesson had been “okay, but I was not really satisfied.”During the lesson, she had informally assessed her students’ conceptualand procedural knowledge about measurement and planned the next day’slesson accordingly. During an interview Mary also articulated her changein perspective on teaching mathematics in general as she reflected,

I think I wasa paper-pencil kind of person. You know, okay, make sure you can do 5 + 5like this (snaps her fingers).. . . I still believe that. . . [and] before I would have started thatway – automatically, saying to all kids, “Okay, today we are going to learn that 2 + 2 = 4.”Everybody say it, “2 + 2 = 4.” Okay, let’s write it. I think that is how I would have beenbefore.. . . Now, I begin with problems they can act out and use their manipulatives.

Demonstrating flexibility in teaching. Mary’s statements about curriculumdemonstrated the flexibility she felt about planning the mathematicalcontent to teach. She discussed her use of the teacher’s manual. Initially,she said she did not use the manual at all, and then she amended this bysaying,

Well, I use it for scope and sequence, especially like last year because I didn’t know that Ineeded to teach them how to tell time. I didn’t know that I should teach them how to countmoney. Because I didn’t know first graders.. . . I went to the titles of all the chapters. I’dpick out the skills that I needed to teach while they were in first grade.

Even though Mary appeared to have flexibility in planning her lessonsin ways she thought helped her students understand, she did not always feelthis flexibility with assessment. She said that most of her assessment wasdone while observing students and asking them questions, but that someformal assessment had to take place. She needed paper-and-pencil gradesto show the parents. “They want to see papers come home.” So, in order toplease parents, every six-week grading period she had three or four papergrades that went home. “I have to give them tests. I need grades. It’s hard

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to give a grade [for] what we did yesterday and today. I could do a teacherjudgment grade, but how many parents are going to like all [grades] beingteacher judgment grades”? She also discussed how the achievement test,theCalifornia Test of Basic Skills(CTBS), affected her teaching:

I teach all the skills I am supposed to teach because [they are] going to be on the CTBS.Most of those things are things they do need to know – like telling time and countingmoney. I agree with that.. . . In addition to that, I use my unit – whatever we are studying.Like we are doing The Sea.

Mary said she wanted her students to be able to use mathematics in reallife and understand the concepts, but that they did need to know the mathe-matics on the CTBS “because it was important to the principal. But if theyknow my goal of getting the concepts, they’ll get his [the principal’s goalof good test scores on the CTBS]. It goes together.. . . Knowing [concepts]makes understanding so much easier.”

Internalizing a change in conceptions. It was evident that Mary had incor-porated her more cognitively-based cognitions about mathematics learning(conceived from the methods course) as I observed her build her lessonson her informal assessments of her students’ thinking.

Everybody is going to get it differently.. . . I don’t think that it is realistic to expect 22 kidsto get it the same way. For example, in learning how to add, some are going to count up.Or some are going to add on. Some are going to draw pictures.. . . I think Alice may get itbest by counting up or someone else might get it best because they know 6 + 6 is 12. Theyknow their doubles. I think that unfortunately some people think that [they] will teach itone way. [They will] teach how to subtract one way and that is it –this way. Then you havethose six or seven [students] who didn’t [understand, but the teachers] go on – next thing,next concept.

Mary had definitely internalized the cognitively-based strategies thatshe had learned in the methods course from the research on children’slearning, and she demonstrated her belief that students construct theirown understanding in individually unique ways. Her lessons demon-strated that she paid close attention to how her students were thinkingabout the concepts and procedures for measurement. During observationsMary took advantage of learning opportunities that presented themselveswhen students discussed their thinking. She was concerned that herstudents understood the underlying mathematical concepts she was tryingto teach. She encouraged students to explore mathematics and explaintheir thinking. Her sustained change in conceptions about teaching andlearning mathematics was again evident when she discussed what shewould recommend to a new teacher. She recommended that teachers getstudents actively involved in constructing their own knowledge and devel-oping mathematical concepts. She believed that teachers needed to know


as many different ways as possible that students might understand. WhenMary taught mathematics and talked about how she thought about hermathematics teaching, it became apparent that she tried to put into practiceher view of teacher “as tour guide.” Mary’s goal during observations hadbeen to guide students to understanding through appropriate choices fortasks and reflective questioning strategies.

The Case of Ann

Ann’s change in the methods course. From Ann’s first interview beforethe methods course began, she had definitely painted a picture of herselfstanding in front of class telling students how to solve problems. She said,

[In] math there are certain definite ways to do [it] – whether it is addition or subtraction, orany of thosecomputations or problem solving. And the child must learn to do itthat way.You shouldn’tshowthem all the different ways to do it because it might confuse them.Math is a separate thing. It’s not related to anything. (Steele & Widman, 1997, p. 188)

By the end of the semester in the methods course, however, she demon-strated more change in her thinking than any of the other students in theclass – not only from her contributions in class, but also in her office visitsduring which we discussed her new understanding. When asked at the endof the course what had begun the change, she replied,

The day we worked on the problem 1 3/4÷ 1/2 was the day that opened my eyes. I knewthe algorithm forward and backwards, but the algorithm didn’t mean anything to me. Ididn’t know what was really going on. I gained a tremendous amount of insight into aworld of math that I did not know existed. (Steele & Widman, 1997, p. 190)

Ann had never done well in past mathematics courses. By her ownadmission, she lacked mathematical knowledge, but during the methodscourse she had begun to understand many mathematical concepts andtheir connections to the procedures through problem-solving activitiesand through the use of manipulatives and diagrams. By the end of thecourse, she was still at the concrete stage when explaining her thinking.However, during the semester of the methods course she became tremend-ously excited about mathematics. She definitely was the student who wasmost enthusiastic about her new-found knowledge and understanding.

A snapshot from Ann’s teaching. Ann was a fourth-grade teacher in aprivate school. During her first year of teaching she had been a long-term special education substitute teacher (Special Education had beenher specialization when getting her elementary education degree). Ann isvery artistic, and she had many centers in her classroom that were decor-ated to encourage her students to use them. Students’ writing was found

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on several walls and bulletin boards. On the days I observed, she wasteaching decimal fractions. During her lessons, she was very energetic andconstantly moving among students seeking to motivate them with encour-aging words. She began her lessons with a review from the previous day.Ann drew a number line on the chalkboard and asked students to writedecimal numbers, between 1 and 10, on the number line. She found thatmany students had difficulty remembering from the previous lesson how toorder decimals, so she repeated much of what they had done the day before.While students marked points and wrote numbers on the number line, otherstudents called out the numbers (i.e., 3.4). Ann asked students to say thenumbers in two ways. For example, she asked, “What is that number”?Students answered, “Three decimal four.” Ann then asked, “How do yousay it in place value”? Students answered, “3 and 4 tenths.” This dialoguelasted about 10 minutes.

After this portion of the lesson, students worked in cooperative groupscoloring in base-ten grids for decimals. Ann said, “I want you to shadein the number with no wholes and two tenths [0.2].” Students held up ahundred grid with two tenths shaded and said the number, “Zero decimaltwo. Two tenths.” For the number 1.1, Ann asked them to say, “One wholeand one tenth.” After shading on grid paper, students then worked withbase-ten blocks in a similar way. The teacher called out numbers, andstudents held up the appropriate blocks, but did not discuss why they chosecertain blocks. After this portion of the lesson was over, students got outtheir textbooks and read orally over one page. Ann did not ask studentshow or why they arrived at their answers. This oral reading and answeringquestions lasted about 30 minutes. Afterwards Ann assigned a textbookpage on ordering decimals for homework, and students began it in class.They were allowed to work in pairs on the homework; students did not usethe base ten blocks or the graph paper.

Rigid sequence in planning mathematics lessons. When I asked Ann howshe planned this lesson or any mathematics lesson, she responded,

I always follow a certain sequence within math. Today, they had forgotten [what we didyesterday]. That’s why I kept going over it and over it. I always want the transitions inmath to be smooth and clear-cut. This leads to this, and this leads to this. Normally, whatI do is come straight from the book. I take the sequence of the book. I’ll read all of thesuggestions, their teaching tips, their introduction tips, their background tips. . . then I justthink of ways to do it.

Ann then talked about how she planned differently for mathematicsthan for any other subject. When she discussed this difference, it becameapparent that she was not as comfortable in teaching mathematics asteaching other subjects.


I plan for math a lot differently than I do my other subjects. For example, with Englishand reading, science, and even social studies, I can read the text and not read the teachingtips and the background.. . . I can make up events in my head. I’ll make up scenarios.. . .

We do a lot of situation things. Whereas, in math, I go to the book – like we did that page[today] – and we read through the book and through the exercises, and then we will pullsome of those exercises to paper or to manipulatives.

Recognition of change in thinking. Ann was very clear as to how herperspectives on teaching mathematics had changed since she had left theuniversity. She said,

I know I changed how I feel about math since school.. . . Before I guess I was thinkingeverything is going to be problem solving and critical thinking and applying – from theget go – all the way to the end of the lesson. I will use cooperative groups. I [will] usethe manipulatives.. . . Then I can move to the algorithm on the board. . . [b]ut now I feellike I should stay with the book. They should see the algorithm first and learn that processinstead of starting out with [manipulatives] and then going back.

Ann said that this change in thinking had begun when she had beenteaching special education as a long-term substitute. She had decidedat that school to move away from manipulatives for the whole lesson,at least not “bring them in at all in many of the lessons until laterafter [students] grasped the algorithm.” She said that it had taken toomuch time to begin with manipulatives and then move to procedures oralgorithms. In addition, some students had not learned the mathematics.And now, although it seemed that Ann did not really help her studentsmake connections between the more concrete (manipulatives) and themore abstract (algorithms), she was not ready to give up using the manipu-latives completely. Nevertheless, Ann had not asked her students to use themodels or grid paper when ordering the decimal fractions on the numberline.

Return to former conception of role of teacher. For all subjects exceptfor mathematics, Ann saw her role as one of making the subject matterrelevant and applicable to students’ lives. For instance, during observationsof her social studies class, she sought to help her students relate slavery inthe Civil War era to how there are still enslaved people today. However,when it came to teaching mathematics, Ann’s role was different. “[T]heyjust need to know how to do it,” and when she talked about why studentsneed to learn mathematics, she did not discuss the importance of makingit relevant. She stated,

They need to know it for school, and for tests, and for SATs [the achievement test in thisschool]. . . . For example, in the third grade they learn multiplication. When they comehere, they learn multiplication with two-digit numbers. They learn division with one-digitnumbers. In the fifth grade, they will learn division with two-digit numbers.

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Ann also found it difficult to incorporate mathematics with other subjects.

[E]very now and then I incorporate a math problem or situation in another area; it is notoften enough that they will relate it to other subject areas.. . . I have friendship stories inEnglish and reading and spelling. In social studies they are working on a long term projectthat has all these skills in it. When it is math, it is math. What are we working on today?Decimals. Here is how you do it. This is what it looks like. This is the processalways. . . . Itis very ABC 123. I never would have thought that is how I would feel the most comfortabledoing it.

Ann clearly realized that she taught mathematics differently fromother subjects. As she described her thoughts, her former beliefs aboutmathematics and how she should teach it came through. She said,

[I]n all other subjects, science and social studies included, either they are writing andtalking and discussing and making up their own questions. But [not] in math.. . . I ammore give.. . . It is a definite thing.. . . You are either right or wrong.

The Case of Dawn

Dawn’s change in the methods course. Dawn’s original definition ofmathematics, given before the methods course, was, “I think math is aright answer. And you have to come up with only one answer” (Steele &Widman, 1997, p. 188). At the end of the methods course Dawn stated herphilosophy about what she believed mathematics to be. “It’s not manipula-tion of numbers. It’s not just abstract rules anymore. It’s not just numbersbeing out here. It’s something more tangible. Now, I see there’s somethingconceptual behind the manipulations” (Steele & Widman, 1997, p. 188).Mathematics had been Dawn’s specialization when getting her elementaryeducation degree.

A snapshot from Dawn’s teaching. Dawn was teaching third grade in arural community. This was her first year in this school district whereshe moved to be closer to family. Dawn used scripted lessons to teachmathematics. These scripts were part of a textbook series noted for itsability to increase achievement test scores. This series had been pilotedin the school district and accepted as theonly method that elementaryteachers could use to teach mathematics. On the days I observed Dawn,she was not teaching any specific content, but was exposing studentsto several different mathematical skills. Dawn had established a regularroutine for her daily mathematics class and used the script all throughthe lesson routines. For the first portion of this lesson, students used acalendar and the days of the week to learn their multiplication tablesfor sevens. For example, Dawn asked, “How many days in one week?”Students answered, “Seven.” Dawn then asked, “How many days in three


weeks? Seven weeks?” Students answered and then recited their sevensboth forward and backward several times. Dawn moved on to the currentdate, May 12, which students used to make various number sentences (i.e.,17 – 5 = 12; 1÷ 12 = 1/12; 3× 4 = 12). Students then counted aloud bytwelves several times.

The next part of the lesson was about temperature. Students madepredictions about what they thought the outside temperature was and drewtheir predictions on their own paper thermometers. A student checked thereal outside thermometer to find the actual temperature, then recorded it onthe class adjustable thermometer at the mathematics bulletin board. Next,students found the pattern for the day (having to do with seven) which theydeveloped from a series of numbers the teacher had written on the bulletinboard: , , , 742, 749, 756. Dawn asked, “What’s the rule?” Severalstudents responded with the correct answer, “Add 7.” She did not ask themto explain.

The next part of the daily routine was a problem of the day. Today’sproblem was: “Eddie’s rectangular garden is 6 feet by 10 feet. Drawa picture of the garden. How many feet of fence will he need for thisgarden?” Dawn walked around to observe students’ answers, read theproblem aloud again, and asked, “How many equal sides on a rectangle?Tell me what you know about a rectangle.” A student answered, “One wayis longer than the other one.” Dawn asked,

Only one side longer? Does it matter if the line that goes this way [draws a vertical line]is longer or this way [draws a horizontal line]? What do I call this line [pointing to thevertical line.]? The vertical.

Dawn continued to draw the following figure:

I have a rectangle up here. Will you all agree with that? What about this line? Don’t I needto label this? Should I go ahead and label all these? A bunch of you added 10 + 6. Is thatall the way around?

Even though Dawn asked these questions of her students she answeredthem herself, and by the end of the discussion she had explained to studentswhy they needed to add both the 6 and the 10 twice in order to get theperimeter of the garden and find the amount of fence needed.

The next activity Dawn called “coin up.” She said, “Think of this inyour head. You have 8 quarters plus 2 dimes plus 6 nickels plus 2 pennies.”

154 DIANA F. STEELE

Luke said $2.52. Dawn asked, “Is there anybody who disagrees?” She thenworked the following out on the board instead of asking Luke to do so.She wrote: 8 quarters makes $2.00; 2 dimes makes $0.20; 6 nickels makes$0.30. Dawn then added these to get $2.50. She wrote, $2.50 + 2 morepennies = $2.52. After the “coin up” portion of the lesson, Dawn workeda few multiplication problems on the chalkboard (i.e., 28× 7) using thestandard algorithm. Students then spent about 30 minutes learning how towrite checks. They had to write out such amounts as 439 and 350 dollars,both in digits and in words on copies of checks. Dawn spelled out most ofthe numbers for them while they wrote. At the end of mathematics class,students completed a timed 45-second subtraction test that they exchangedand graded together.

Teacher as giver of information. During her lesson, Dawn kept studentson task and asked many questions. Except for the oral recitation whencounting by 7, 12, and so forth, she did most of the talking – using the scriptshe kept with her at all times. When asked what her role was in teachingmathematics, she discussed her role as one of facilitator. She responded,

I think that I should be a facilitator, like today, I. . . had some kids leading it. [One studenthad led the thermometer activity.] I did some where I was leading. I think that in the idealsituation if I had a lot of time, I would just like to let them figure out the patterns and figureout things – figure out how to do these things. Like, for example, if I had enough playmoney for every kid in the class, I would give them all some money and let them do theseaddition problems with money in the hundreds and figure it out themselves.

As Dawn discussed how she saw her teaching role, it appeared shebelieved what she was saying. During the observations, however, she hadnot been a facilitator nor had students done much of the talking. And, asshe continued to talk more about her mathematics class, she manifestedher real purpose of teaching mathematics. Her purpose was tocover thematerial. She declared,

I feel a lot of pressure to cover certain skills that are in the [text series]. The big thing atthis school is “this is going to be on the CTBS; this is going to be one the CTBS; this isgoing to be one the CTBS.” When they go to fourth grade, the fourth-grade teachers aresaying, “They have got to know this. It’s on the CTBS. They have to know it when theycome to us because we have got to teach them this and this.”

Dawn said that all the third-grade teachers in her school would probablybe on lesson 82 that day, but “for the most part” they would not spend thehour and a half that she usually spent. She said that she always completedthe scripted lessons “by the end of the week. . . like I had a section I didn’tdo today that I will either pick up tomorrow or. . . on Friday.”

During calendar time Dawn had asked students to solve the wordproblem about perimeter. She liked to have students solve word problems


that came from the text series and ones she created on her own. When shediscussed her use of word problems, she confirmed what I had noticed inthe observations – Dawn asked the questions, did most of the thinking forthe students, and then asked them to give only short numerical answers.She stated,

I keep the kids involved.. . .We do examples together, and then they practice. . . especiallywith word problems because a lot of them were scared to death of word problems at thebeginning of the year.. . . I’ll say “Think about if you, Joe, had 7 pencils. You are supposedto lend me 9. Can you give pencils? And he’ll say, No.” I’ll say, “You need to borrow fromyour neighbor, don’t you”? So let’s say the kid next to him has got 10 pencils. “You canborrow from him.. . .”

The textbook as lesson planner. When asked how she planned for thelessons, she said that she did not need to do much planning because theteacher’s manual told her what to do. She had to copy the worksheets andchange the bulletin board. If the manual suggested manipulatives, thenshe needed to get them out. Dawn said that she liked the way this textseries related mathematics to the real world. She thought students learnedmathematics best by “relating to their world. Like with the example of thethermometer.” She added that the students knew their sevens by the days inthe week, their fours by the quarters in one dollar, and their twelves by themonths in years. Dawn thought that she was connecting students’ mathe-matics learning to real life with these kinds of activities. She rushed inorder to finish the script in the teacher’s manual so that she would cover allskills in the lesson. The textbook series emphasized whole-class discussionduring most of the lesson; this was her predominant routine of instruction.She commented that students always took longer than the lesson in thescript said – especially the oral recitation of the multiplication tables. “Idid not do all the counting they [were] supposed to do today, and it stilltook a long time.”

Dawn’s planning was also influenced by other teachers. However, shesaid that she occasionally did not agree with how the other teacherschose to teach certain topics. For example, Dawn believed that studentsshould not use key words when solving word problems (this idea had beendiscussed at length in our methods course). Other teachers in her schoolthought using key words was important in understanding how to solveword problems. Dawn did not feel comfortable telling them her own viewof key words, but she felt strongly enough that she went ahead with herown view of using key words.

Constraints on teaching. Pressures from administration also affectedDawn’s teaching. During the previous year, Dawn had taught in another

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school district that had a more flexible curriculum. She said she had onlyused the text for a guide and created most of her own lessons. In her newschool, however, she did not feel that she could do this; she said, “I thinkthat it is expected that you are going to use a certain curriculum. It isreal [sic] political around here. People get moved all of a sudden if theyupset somebody.. . . Sometimes I might not say anything, but I will just doit.” The administration valued achievement test scores and expected thatteachers would teach the mathematics that students were expected to knowfor the CTBS. At her old school she said, “The CTBS was just somethingwe did.”

Dawn expressed her concerns about how she was teaching mathe-matics. “I don’t know if this is the right way to teach math.. . . I amputting a lot of blind faith” in the statistics quoted by those who had usedthis mathematics curriculum. Nevertheless, regardless of her uncertainly,Dawn definitely did not feel she had the freedom to teach mathematics anydifferently than the other teachers. I heard echoes of what she had learnedin the methods course, but the pressures of the school district administra-tion, teachers, and parents had convinced her she did not have the freedomto use what she had learned.

Return to former conception of teaching. Dawn cared about her students’learning, but in her lessons there was no depth of mathematics, eitherconceptual or procedural, and no connection to the real world except ina superficial way. All the activities were isolated and unconnected to eachother. She seemed to think that if she just covered the material, studentswould learn. She said that she knew she should emphasize such things asregrouping tens from the ones column, and referred to one mathematicsproblem during class, 28× 7. While Dawn worked the problem on thechalkboard she had said, “You carry the 5.” Later, during the interview, shesaid, “I just don’t have the time to always discuss the difference betweencarrying 5 and carrying 5 tens.” She did not ask students to explain theirthinking either with incorrect or correct answers. Dawn had returned to herformer definition of mathematics that “math is a right answer.”

The Case of Vanessa

Vanessa’s change during the methods course. From the beginning of themethods course Vanessa had demonstrated characteristics of being a goodmathematical thinker who needed to build confidence in her own thinking.Before the methods course began, when she had first discussed how shesaw herself as a teacher of mathematics, Vanessa described her image ofteaching:


Worksheets, workbooks, teacher-direction, things on the chalkboard, work to do yourself.The teacher shows you some examples of how to do the problems. Then you do it allyourself, usually at your own pace. By practicing computations you learn.. . . This wayyou can understand it. (Steele & Widman, 1997, p. 188)

This image reflected students passively learning rather than activelyconstructing knowledge. She saw mathematics as “all about numbers andmanipulation of numbers” (Steele & Widman, 1997, p. 188). She viewedthe teacher as an organizer and presenter of information. By the end of themethods course, she no longer saw her role as presenter of information tostudents. She said,

Overall, I think my role is to guide students so that they will value mathematics as theyapply math concepts to real-life situations. I want to be a facilitator of knowledge ratherthan a “body” delivering correct answers. By providing a problem-solving atmosphereand one that is secure enough to allow students to take risks, children will better learn toconstruct their knowledge.. . . My essential role as teacher will be to provide a learningenvironment that will spark thought, new ideas, and help students make connections.(Steele & Widman, 1997, p. 189)

A snapshot from Vanessa’s teaching. Vanessa was a third grade teacherwho had 33 students in her class. This was her second year teaching ina public school that was just two years old. The school was located on alarge well-landscaped campus. The lunchroom was spacious; there was acomputer lab; and all classrooms had several computers. The school wasovercrowded because many parents had taken their students out of olderschools in order for them to attend the newly completed, very attractiveschool. When I observed Vanessa, she was teaching division with remain-ders and introduced the concept with a children’s literature book entitledA Remainder of One(Pinczes, 1995). She liked to use reading as a bridgeto teaching all subjects and especially liked using literature for teachingmathematics. When Vanessa had been a student at the university, herelementary specialization had been reading. She commented,

I think using books makes [understanding] easier for [students]. I think a lot of people areafraid of math, afraid of getting it wrong. Literature helps them be a little more comfortable.It is also a form that they are used to learning.. . . We use it in science. We use it in socialstudies. We use it during reading time.. . . They get to see math going on in a differentsetting – not just paper-pencil. Even if the book is not about a realistic thing, they still getto see [math] a different way.

At the beginning of this lesson, all 33 students were seated on the floortogether in front of Vanessa as she readA Remainder of One. The bookwas about 25 hungry ants that were trying to line up in equal groups.Vanessa asked students to experiment with placing felt ants in groups ona feltboard. Each time before she turned the page, Vanessa asked them to

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predict the number of rows in which they thought the ants would be able tofit evenly. For the first prediction, several students chose two rows and triedthat number. On the feltboard one student placed the ants in vertical rows,one in horizontal columns. Both students found that the 25 ants wouldnot fit evenly in their chosen grouping. Students discussed the idea ofwhat a remainder means and then tried three rows, four rows, and finallydiscovered that the only way the ants would fit evenly were in rows of fivewith five ants in each row.

After students read and acted out the book, they worked at their deskson written division problems, such as 24÷ 3, and 8 divided into 40 with thelong division sign. They solved these problems and created word problemsto coincide with the numbers. During the seatwork, Vanessa walked aroundand asked students to explain how they were solving the problems. Shespent quite a bit of time with one student who was using tally marks todivide up numbers into group. Vanessa asked: “How are you grouping themarks”? “How are you deciding when to stop the tally marks”? Anotherstudent was writing 3× = 24 when solving the problem 24÷ 3. Vanessasaid to the class, “One of us is doing it this way (She showed the student’swork on the overhead, 3× = 24; 3× 8 = 24).” This student explainedhow he was making the problem 24÷ 3 into a multiplication problem.Vanessa made a suggestion to other students, “So maybe a couple moreof you could try that.” Some students used counters to solve the problems,and at least one student drew pictures.

Helping students make connections. After the lesson, Vanessa said that shewas pleased because students had begun to understand how to decide whenthey had equal numbers of ants in the rows, and some had begun to see thatmultiplication could be used to solve division problems. But she was alsodisappointed that no students had used their knowledge of multiplicationfacts to solve the problems with the ants while she read the book. All thestudents had needed to move around the felt ants to discover whether theycould be placed in equal rows. She said, “We talk a lot about if you knowyour multiplication facts, then division is simply a reverse of it. They cantell me that, but they don’t seem to understand it.”

On the following day all students used counters to model differentnumbers of ants and how they might fit in equal rows. Vanessa askedquestions like, “How many even rows of ants can you make with 36 ants?37 ants”? Students saw how they might have leftover counters in someproblems and learned how to write algorithms with the remainders in thequotient.


When teaching mathematics, Vanessa usually used manipulativesbefore doing the computations. She had tried both – using the manipu-latives and doing the computations first. “But with the paper-pencil first, Ifelt like I was backtracking a lot. There are some things that they can do[with] pencil-paper fine. I didn’t need to start with the manipulatives. I tryto do more starting with the hands on and moving on to the paper-pencil. Ifeel better about that.”

Flexibility and constraints in teaching. When Vanessa planned for herlessons, she said that she did not use the textbook much because the bookdid not help “students with understanding the concepts.” She said, “Thereare some neat ‘hands-on’ activities, but [they] seem separate from the restof the lesson. I think they need more hands-on activities along with theproblems before they start doing the pencil-paper.” Instead, Vanessa usedthe textbook more for knowing what topics to teach. She explained,

[When I am planning] first of all, I think of [resources] I have. Then I’ll look at the [textused in our methods course]. I look at that one a lot, especially for introductory lessons.I’ll then decide [on] what I know I have [and] what I can make.

Vanessa also discussed the effect of students taking the CTBS had onher planning and assessment. She said, “I felt like I had to stop before theCTBS and just teach what was on the test – I was behind because I justhate moving on and take a lot of time with each concept when [students]don’t understand.” However, it was different after students took the CTBS.Vanessa explained,

After the CTBS I feel much better. Now we can backtrack – like with division. We back-track and we start doing division from scratch and learning it the way I feel is the correctway of doing it – actually using the manipulatives and talking about how we use them.

In addition to the pressure of students’ scores on the CTBS, she alsodescribed how parental pressure sometimes caused her to move on whenshe was not comfortable. Many parents did not like for their students tospend three days in a row on the same topic. And parents also expectedworksheets to go home with children’s work. “When students are absent,parents don’t understand that I don’t necessarily have a worksheet to sendhome. If it is an activity that we do together, you can’t really send it home.”

Support from peers and administrators. Vanessa felt that she worked ina collaborative environment with other third-grade teachers. This supporthelped her deal with testing issues and parental pressures. Her principalwas supportive of alternative teaching methods and did not specify acertain curriculum that all third-grade teachers had to teach, nor did she

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dictate how they should teach. Vanessa said, “We have a lot of freedom tointerpret how we want to teach the lessons.”

Need for new mathematical experiences. Vanessa thought she now neededmore experiences with learning how to teach mathematics. She wanted tolearn more about helping students make the connections “from the handson to the paper-pencil.”

I don’t know how to make that connection all the time. It depends on what we are doing.Sometimes it is easier for me to see it. Sometimes it is like we are ending an activity andstarting a whole new one. I still have a hard time with that. I feel that there is somethingthat I am missing that is still not clicking for me. [I want to] help them see how things work. . . why it works . . . then to see how they might use this in real life.

Vanessa did not think that she was getting much help in improvingher mathematics teaching. The school had workshops in many areas otherthan mathematics (e.g., integrated thematic instruction, positive coaching,cooperative learning, and the seven intelligences). However, Vanessabelieved in what she was doing. She thought that teaching mathematicswith a reform-based approach “is worth the effort in the end.” She wasliving out the conception she had espoused at the end of the methods course– one of “spark[ing] thought, new ideas, and mak[ing] connections.”

ANALYSIS AND DISCUSSION

The teachers in this study made contrasting decisions about what to teachand how to teach their students. These choices were influenced by theinteraction of several factors: personal commitment, professional strength,curriculum, planning, assessment, beliefs, knowledge, and support fromschool administration. I will discuss these factors in the following sectionwhile answering the questions that guided the study: (a) Did preserviceteachers sustain their new conceptions about mathematics teaching andlearning and implement a reform-based approach to mathematics teachingwhen they became teachers? and (b) What factors influenced teachers’ newconceptions?

Personal Commitment and Professional Strength

Goodman (1987) cited personal commitment of teachers as the drivingfactor in the individual’s need to go beyond traditional approaches toteaching that they had experienced as learners. Goodman stated thatthe desire to make decisions about curriculum and instruction must beparticularly strong in beginning teachers in order for them not to be over-whelmed by meeting expectations of administrators. Without this level


of commitment, novice teachers cannot go far in becoming reflectiveteachers.

Mary had definitely sustained her conceptions about mathematics andmathematics teaching and learning, and demonstrated a personal commit-ment to her conceptions when she taught. The methods course had been thebeginning of a new way of thinking about mathematics teaching. She wasflexible about the direction her lessons went. She integrated mathematicalconcepts with other curriculum areas and created a rich learning environ-ment. As another confirmation of her sustained conceptions, Mary’s meanscore (see Table I) on the Mathematics Beliefs Scales showed that shehad become even more cognitively based in her conceptions about howchildren learn mathematics since the end of the methods course.

Mary also had a professional strength that helped her maintain herconceptions and align her practice with them. Kilgore and Ross (1993)found in their study that “there is no question that the professional strengthof the individual plays a role in the ability of the beginning teacher tosustain a reflective approach” (p. 35). Mary was a reflective teacher whohad learned to build her instruction upon assessment of her students’learning. She continued to reflect and ask questions about her teaching.

Ann had begun using a reform approach in teaching mathematics whenshe had been a substitute special education teacher, but she had not beenable to continue this approach. Ann had not had a positive first yearof teaching. Kilgore and Ross (1993) found that a negative first assign-ment can change a teacher’s commitment to student learning. Ann hadnot, however, allowed this negative experience to affect her less tradi-tional views of learning in other subjects. Ann’s answers to questions onthe Mathematics Beliefs Scales (see Table I) indicated that she still heldcognitively-based conceptions about mathematics learning – the concep-tions that I observed in her teaching practice ofother subject matter.However, in mathematics she taught “straight from the book” and in “thesequence of the book.” Ann did not feel comfortable in her own mathema-tical knowledge and was not willing to take risks in her teaching that wouldallow her to gain more knowledge. She used journal writing extensively inall subjects and talked about using it the following year in mathematics.Ann demonstrated a professional strength and commitment to her views ofteaching other subjects.

Dawn, ironically the teacher whose specialization had been mathe-matics, did not appear to sustain any of the conceptions she had espousedat the end of the methods class. She had said, “[Math] is. . . a way ofmaking sense around us.. . . It’s not just abstract rules. . . there’s somethingconceptual behind the manipulations.” This conception was definitely not

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TABLE I

Scores on beliefs questionnaire

Score before Score at end Score after two

course of course years of teaching

Mary 126 87 85

Ann 136 83 85

Dawn 113 84 114

Vanessa 112 73 71

Notes: Lower score signifies a more cognitively-based conceptions about teaching andlearning; Lowest possible score is 48; highest possible score is 240.

observed in her teaching, although she thought the curriculum was makinggreat real-world connections. She thought she was helping students makesense of the mathematics. She demonstrated in her teaching the belief thather role was one in which she needed to break up mathematical topicsinto small step-by-step pieces in order to provide sufficient practice for herstudents. She emphasized memory and repetition.

During interviews, Dawn did not express the desire to teach any differ-ently and planned to teach again the next year in this school district thatmandated a specific mathematics curriculum. Her acceptance of the scriptsfor teaching asthe true way seemed to cause her little worry about whatto teach. She just followed the script. Her mean score on the Mathe-matics Beliefs Scales (see Table I) indicated that she had not sustainedher cognitively-based conceptions. Her score was significantly higher (lesscognitively-based) than it had been at the end of the methods course. Shedid not demonstrate a commitment to the beliefs about teaching that shehad espoused at the end of the methods course.

Vanessa had sustained her cognitively based conceptions. Vanessa’smean score on the Mathematics Beliefs Scales (see Table I) was lowerby two points. Her scores, both at the end of the methods course and atthe end of two years teaching, were the lowest scores (more cognitively-based) of the four teachers. These conceptions were evident in her teachingpractice. Unlike teachers in Lortie’s (1975) seminal study, Vanessa hadbeen able to overcome the apprenticeship of observation of the many yearsshe had spent learning mathematics in traditional classrooms. She hadaccommodated these past negative experiences with her new reform-basedexperience and had constructed an evolving and dynamic approach toteaching mathematics. The knowledge about teaching and learning mathe-matics that she had constructed in the methods course had scaffolded herand given her some of the tools and confidence that enabled her to continue


to grow in her own mathematical knowledge. When she talked about whatshe wanted her students to know about mathematics, she demonstrated acommitment to her students’ learning and understanding.

Curriculum, Planning, and Assessment

How the teachers viewed and used the mathematics curriculum influencedtheir planning and assessment. Mary and Vanessa used their textbooks asa scope and sequence to help them know which concepts and skills toteach for their grade levels. They used multiple resources to plan theirlessons. Mary integrated mathematics along with all subjects with hercontent units. Vanessa connected mathematics and children’s literature. Incontrast, Ann isolated mathematics from all the other subjects. Finally, thecurriculum adopted by Dawn’s school district was “the curriculum” andhad to be taught directly from the teacher’s manual. When I examinedthe textbook series, it was clear that the authors had made no attempt tointegrate mathematics with other subject matter except in superficial ways.

The most significant aspect of Mary and Vanessa’s planning was thatthey both desired and made efforts to be curriculum decision-makers ratherthan managers of a prescribed mathematics program. Mary particularlyspent a substantial amount of time reflecting upon how her students reactedto a given lesson. Vanessa culled her lessons from several sources, alwayslooking for ways to help her students make mathematical connections.However, Ann went to the book and followed the book’s sequence. Dawn,too, was willing to allow the teacher’s manual to dictate what mathematicsto teach and when and how to teach it.

Regarding assessment, Mary and Vanessa were very comfortable usinginformal ways to assess students. They questioned their students andencouraged them to talk about how and why they solved problems. Theyhad difficulty finding enough written assessments to make parents happy.On the other hand, Ann and Dawn opted to use written assessments everyclass period. In Ann’s class, students always had time to do some of theirhomework that she would later grade. Dawn daily used timed quizzes.

Achievement tests also affected how the teachers planned their lessons.All of the principals emphasized the importance of high achievementtest scores. However, all of the teachers, except Dawn, were allowed thefreedom to prepare students in their own ways for the achievement tests.Mary was the only teacher who seemed to be comfortable with how shedealt with this preparation. She thought that by teaching the concepts alongwith the skills her students would not only understand what they needed toknow for the achievement tests, but would also meaningfully understandmathematics. Vanessa felt uncomfortable with moving on to new concepts

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before most students understood the present mathematics. The notion thatit was more important for them to understand in depth rather that breadth,however, became a dilemma when time drew closer for students to takeachievement tests. Therefore, before the tests she would stop how shetaught mathematics and teach the skills on the achievement test for a fewweeks. After the test was over, she backtracked to where she had stoppedbefore the drill and practice.

Both Mary and Vanessa wanted their students to develop a rela-tional understanding of mathematics that would enable them to make richconnections between conceptual and procedural knowledge. In contrast,Ann and Dawn taught mathematics in very procedural ways; even whenAnn used the manipulatives with her students, they learned a procedure ofusing the manipulatives rather than using the manipulatives as a meansto develop conceptual knowledge. When Ann asked them to name thedecimals with place value language, it was more like recitation eventhough it demonstrated that she did recognize the importance of studentsunderstanding place value.

Beliefs and Knowledge about Teaching and Learning Mathematics

Mary believed that everyone thinks about mathematics in his or herown individual and unique way, and that it is important for teachers toknow many different ways that students might solve problems. Mary wasknowledgeable about how children learned mathematics and comfortableasking open-ended questions. She demonstrated this knowledge during hermeasurement lessons when she sought for students to develop measure-ment sense. Additionally, in the interviews Mary discussed many waysstudents might think about addition. She was knowledgeable about usingmanipulatives to help students “see the concepts.” For her, it was importantthat teachers help students understand how mathematics relates to the realworld.

In contrast, Ann believed that it confused students if they heard morethan one way to solve a problem. She did not want her students to experi-ence cognitive conflict in learning mathematics. She wanted the transitionsbetween topics to be smooth. Ann believed that students learned mathe-matics in certain ways, and students should learn it in exactly thoseways. “Here is how you do it.. . . This is the process.. . . You areeither right or wrong.” Her goals for her students were that they knowthe mathematics that would be on the achievement tests and that theyunderstand the sequence of mathematics throughout their grades (i.e.,multiplication facts in the third grade, two-digit multiplication in the fourthgrade).


Ann’s failure to sustain her conceptions when teaching mathematicsmay be linked to her dependence on the textbook and its suggestions. Shewas unable to creatively incorporate and connect the mathematics to othersituations. She also believed that students needed to learn mathematicswith an underlying notion of sequencing that she based on efficiency –from procedural to conceptual knowledge. She took students from namingand writing numbers to modeling. She believed cooperative learningwas important, but she did not use it effectively in mathematics class.Her personal deficit in mathematical understanding prohibited her fromextending the given mathematics curriculum into something more relevantand personal for her students, as she strove to do in other subjects. Duringthe methods course she had just begun to understand mathematics forherself; perhaps it was too early in her own development for her to helpstudents understand. Despite her recognition that memorizing rules andformulas had not been an effective way for herself to learn mathematics,she returned to this approach as a teacher. She felt uncomfortable askingopen-ended questions that might lead her students into areas that sheherself did not understand.

Eisenhart et al. (1993) also found in their case study of an elementaryteacher, Ms. Daniels, that mathematical knowledge was a key weakness.Ma (1999) discussed shortcomings of elementary teachers’ conceptualmathematical knowledge. Ma wrote that these teachers were limited ingetting at conceptual knowledge when using manipulatives and often reliedtotally on computations and algorithms to teach mathematics. Etheridge(1989) found in her research that teachers who were less comfortable withand knowledgeable about their subjects were likely to teach skills ratherthan thinking.

Dawn, like Ann, also taught mathematics in very procedural wayseven though, unlike Ann, her specialization had been mathematics. Dawnbelieved that if she covered the content for the prescribed lesson andspent enough time on the mathematics lesson, students would learn. Shewanted students to relate mathematics to the real world. She felt if studentswereexposedto how mathematics could be related to the real world, thenthey could make the connections themselves. She thought students couldlearn if they heard her explainthe right way and then practice. Dawnhad been successful taking mathematics courses, and felt she understoodmathematics. Dawn was knowledgeable about mathematics. When shetoldstudents how to solve problems, she demonstrated this knowledge. Shedid not, however, demonstrate knowledge of how her students needed toconstruct the mathematical knowledge that she herself possessed.

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Vanessa believed, as did Mary, that it was important for students tomake connections. She wanted them to see “how it works, why it works. . .

then to see how they might use [it] in real life.” She believed that studentslearned by talking about and explaining their thinking. Vanessa said herrole was “to provide a learning environment that will spark thought, newideas, and help students make connections.” Vanessa’s teaching showedthat she wanted students to understand both conceptually and procedurally.During her teaching, she demonstrated that she possessed knowledgeabout how to facilitate her students’ making mathematical connections,even though she confronted dilemmas in how to do this. Her personalemphasis on the importance of making connections is the very nature ofthe mathematics of the reform-based classroom (Schifter, 1998).

Kilgore and Ross (1993) found that beginning teachers do recognizethe importance of helping students to think critically and to value their ownthinking. Some teachers also realize that it is equally important that theyvalue students’ ideas and knowledge. Kilgore and Ross noted that teacherswho value their students’ thinking often view themselves as learners. Inthis study, Mary and Vanessa discussed how they had learned from theirstudents and how important it was for teachers to understand how studentslearn. They both provided extensive evidence in their teaching practices toconfirm their views of learning as a key part of their roles as teachers. Marytalked about learning from her students, “[W]e know that kids can tell. . . better than [teachers] can sometimes. They have their own language.Hopefully, students’ contributing to the class get others thinking.” Vanessawanted more workshops about how to teach mathematics to help studentsmake connections. “I feel that there is something that I am missing.. . . It’sstill not clicking for me.”

Support from Community and School Administration

Some of Lacey’s (1988) and Etheridge’s (1989) strategies for respondingto school administrative demands were manifested from the teachers inthis study. Mary seemed to use Lacey and Etheridge’s second strategy ofstrategic compliance. She appeared to have complied with the constraintsof the school administration and pressures from parents while continuingto implement her own conception of teaching. Mary had purchased allthe manipulatives she used with her students, and she had not asked formoney to do this from the school system. She had also purchased enoughcalculators for cooperative groups, and she had begged for old computersfrom family and friends. Parents wanted worksheets to come home, soMary accommodated their wishes by sending home at least one work-sheet per week. Even though Mary’s principal emphasized achievement


test scores, she was able to maintain good scores while she taught herstudents to understand more than the mathematics on the tests. Mary wasable to negotiate her conceptions about teaching amidst the pressures fromschool administration and parents.

Ann’s response to administrative demands could not be categorizedwith one strategy from the Lacey and Etheridge model. Even though Annhad clearly preferred a reform-based approach to teaching mathematics atthe beginning of her teaching assignment, she had strategically set asidethis teaching approach (Lacey’s and Etheridge’s second strategy) whenshe confronted learning dilemmas of her students. There was no evidencethat she had set aside her conceptions because of lack of support fromschool administrators. In fact, there was evidence to confirm that admin-istrators valued her ideas of teaching. Ann felt that she would have beensupported for using creative methods of teaching as long as test scoresremained stable. She did not express any particular pressures from parents.Ann’s teaching definitely mirrored Lacy and Etheridge’s third strategy ofstrategic redefinition in teaching social studies and language arts. Admin-istrators encouraged her to share her ideas about teaching for these twosubject areas with other teachers.

Dawn is the only teacher who had internalized and complied with theconstraints of her school district and pressures from parents (Lacey’s andEtheridge’s first strategy). In Dawn’s school, the school administrationhad mandated a certain curriculum. Dawn felt that she should follow theexamples of the other teachers rather than implement any of her own ideas.The only time Dawn rebelled was in the use of key words. Parents expectedpapers to come home everyday, and stopped by school or telephoned tofind out why their students did not bring home the work they had donein class, so Dawn sent papers home every day. The district had adopteda teacher-proof mathematics curriculum that allowed for little flexibilityor adaptation by the teacher. Goodman (1987) stated that administratorswho expect teachers to use teacher-proof curriculum expect teachers tobe managers, rather than reflective practitioners who ask questions abouttheir teaching. During interviews Dawn spent a lot of time talking about“keeping her job.” She had observed that teachers who did not conform toadministrative expectations did not keep their teaching positions.

Institutional constraints did not seem to play a major part in Vanessa’steaching practices. Vanessa felt she worked in a supportive environment.Her team of teachers worked well together and held similar beliefs abouthow students should learn mathematics. The principal allowed alternativemethods of teaching. However, from the types of inservice experiencesthat the school administration provided for the teachers, there was very

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little emphasis placed upon teaching mathematics. The administration hadnumerous inservice training activities about latest trends in learning, butnone about teaching mathematics.

Depth of Cognitive Conflict

Perhaps individuals need a defining experience in order to sustain thecommitment to new-found conceptions. Looking back to these teachers’experiences in the methods course, Mary, Vanessa, and Ann had gonethrough major turning points in their conceptions about mathematics andabout mathematics teaching and learning. Mary had done well in mathe-matics courses, but learned through different assignments for the methodscourse that asking students to perform only procedures in mathematics wasnot getting at real understanding. She took this idea into her teaching andstill emphasized that everyone thought differently about solving problems.Vanessa had studied hard in mathematics, and had gotten good grades, buthad always lacked the confidence that she really understood mathematics.In the methods course, she began to see connections in mathematics thatshe had not seen before. This experience began a path that she was stillfollowing – to help her students see the connections. Ann also wentthrough profound changes in her mathematical understanding during thecourse. She had never understood mathematics, and in the methods courseshe had begun to fill in some gaps in her mathematical knowledge.

Dawn’s score on the beliefs questionnaire at the end of the courseshowed that she had accommodated cognitively-based conceptions intoher thinking about learning mathematics. However, Dawn did not have aparticular experience in the course that profoundly affected her thinking.It had seemed to be a gradual change. It was curious that Dawn’s scoreon the beliefs questionnaire was cognitively based at the beginning ofthe methods course and that her score at the end of the course showedless change than the other preservice teachers. Dawn thought she alreadyunderstood mathematics, and perhaps her past experiences of learningmathematics were in the end more influential in her approach to teaching.

IMPLICATIONS FOR INSTITUTINGREFORM-BASED TEACHING

Problems and Realities of Teaching

This study revealed some of the problems and realities of the work-place that interfere with teachers sustaining a change in conceptions thathave occurred when they are preservice teachers. Mary and Vanessa, who


were not pressured by realities of their working environments, remainedcommitted to their conceptions. In contrast, Ann and Dawn did not remaincommitted. Dawn consciously compromised her new conceptions whileimplementing a teacher-proof curriculum, and she almost unconditionallyaccepted institutional and societal pressures. Ann was almost paralyzedwhen confronted with planning how to teach mathematics.

Need for broad-based support. When preparing reform-oriented mathe-matics teachers, mathematics educators should help prepare the contextof teaching by working with schools in order to change the cultureof teaching mathematics in schools. School culture is a major problemwhen implementing reform-based mathematics instruction. Administratorsneed to support teachers’ enrollment in inservice experiences that involvemathematics. In addition, Nelson (1998) suggested that administratorsalso need to develop for themselves a new sense of what it means tounderstand mathematics, and then develop policies and procedures thatimplement these new understandings. In this context, administrators andmature teachers should be ready to support new teachers who want touse innovative approaches to teach mathematics. The National Councilof Teachers of Mathematics (1991) stated that we need new models forpreparing preservice teachers to enter the profession of teaching becausefew models currently support new teachers. “The constraints of the realworld of schools overwhelm the perceptions these new teachers hold aboutwhat mathematics teaching and learning could be [and]. . . the result is thatmany new teachers find it difficult to adapt what they have learned in theirteacher preparation programs” (NCTM, 1991, p. 5).

Need for reformed-based professional development and curriculum.Dawn’s use of the mandated curriculum identifies the need for reform-based curricula and workshops. For beginning teachers, textbooks are acritical tool in implementing reform-based ideas of mathematics teaching.Textbooks need to contain in-depth mathematical investigations and excel-lent instructional support. They need to help guide teachers in ways topromote students’ mathematical understanding. Dawn’s use of the skill-based text actually inhibited her ability to reflect on how her studentslearned mathematics. School districts should provide opportunities forteachers to increase their knowledge of teaching mathematics and provideteachers with the resources to help them implement this knowledge intotheir teaching practices. Vanessa’s school administration had not offeredany professional development in teaching mathematics. Perhaps if Ann’sschool had used reform-based materials she could have created meaningfulmathematics experiences for her students. She was unable to do it alone.

170 DIANA F. STEELE

Need for Sufficient Mathematical Content Knowledge

Content knowledge or the lack thereof has an effect on the teaching prac-tices of beginning teachers. Ann, in her preservice teacher education, hadtrouble with mathematics. Teacher educators must recognize and not betimid to act on evidence that some preservice teachers do not possess thecontent knowledge sufficient to teach. Teacher educators need to providemore assistance in these areas for preservice teachers. Disciplinary studyis important for all teachers, and sometimes teacher education programsneglect this area with the preparation of elementary teachers. In orderfor teachers to reflect on the mathematical learning of their students,they must understand the content they teach. In a sense, Ann was tryingto survive each day without teaching something she did not understand.Without content knowledge, teachers cannot make connections betweenmathematical concepts or to other parts of the curriculum.

Experiences During Teacher Education

Newly found conceptions often lead to uncertainty and conflict whenbeginning teachers are confronted with the real world of teaching. There-fore, teacher education programs should help preservice teachers identifyahead of time the possible sources and dilemmas that will create uncer-tainty and conflict when teaching mathematics with a reform-basedapproach. Reflective and rational decisions are key aspects of teaching. Butreflection in general ways on practice is not enough; preservice teachersmust also reflect on how to teach specific content. Wilcox et al. (1992)agreed with this implication. They suggested that reflecting on how indi-viduals learn mathematics and making necessary decisions about howto teach it requires helping preservice teachers develop the qualities andhabits of mind “to ask critical questions – about curriculum, instructionalpractices, educational policies, testing, their own learning and that ofothers, the contexts in which mathematics education takes place” (Wilcoxet al., 1992, p. 26). Mary and Vanessa’s ability to reflect deeply in theseareas helped them negotiate their conceptions of teaching and still play thegame of pleasing school administrators.

CLOSING REMARKS

Brown and Borko (1992) asserted that “to understand learning to teach,one must study how [belief] systems – and the relationships among them– develop and change with experience, as well as identify the factors thatinfluence this change process” (p. 211). The present study provides insight


into whether and how reform-based ideas learned in teacher education caninfluence preservice elementary teachers in long-lasting ways. Importantquestions for future research are: What factors continue to play a signifi-cant role in these teachers’ professional development? Will they improvetheir goals of mathematics teaching and learning with experience? Howcan teacher education programs influence the culture of teaching thatalready exists in schools? Should mathematics be taught by specialistsin elementary schools? Clearly, the answers to these types of questionsrequire longitudinal research studies.

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Brown, C. A. & Borko, H. (1992). Becoming a mathematics teacher. In D. A. Grouws(Ed.), Handbook of research on mathematics teaching and learning(209–239). NewYork: Macmillan.

Brown, S. I., Cooney, T. J. & Jones, D. (1990). Mathematics teacher education. In W.R. Houston (Ed.),Handbook of research on teacher education(636–656). New York:Macmillan.

Cooney, T. J. & Shealy, B. (1997). On understanding the structure of teachers’ beliefs andtheir relationship to change. In E. Fennema & B. S. Nelson (Eds.),Mathematics teachersin transition(87–109). Hillsdale, NJ: Lawrence Erlbaum.

Cooney, T. J., Shealy, B. E. & Arvold, B. (1998). Conceptualizing belief structuresof preservice secondary mathematics teachers.Journal for Research in MathematicsEducation, 29, 306–333.

Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D. & Agard, P. (1993). Concep-tual knowledge falls through the cracks: Complexities of learning to teach mathematicsfor understanding.Journal for Research in Mathematics Education, 24, 8–40.

Ernest, P. (1989). The knowledge, beliefs, and attitudes of the mathematics teacher: Amodel.Journal of Education for Teaching, 15, 13–33.

Etheridge, C. P. (1989). Acquiring the teacher culture: How beginners embrace prac-tice different from university teachings.International Journal of Qualitative Studies inEducation, 2, 299–313.

Fennema, E., Carpenter, T. P. & Peterson, P. L. (1987). Mathematics beliefs scales.Studies of the application of cognitive and instructional science to mathematics instruc-tion (National Science Foundation Grant No. MDR-8550236). Madison: University ofWisconsin-Madison.

Fennema, E., Carpenter, T. A. & Peterson, P. L. (1991).Learning mathematics with under-standing: Cognitively guided instruction. Madison: University of Wisconsin, WisconsinCenter for Education Research.

Glaser, B. & Strauss, A. (1975).Discovery of grounded theory. Chicago: Aldine.Goodman, J. (1987). Factors in becoming a proactive elementary school teacher: A

preliminary study of selected novices.Journal of Education for Teaching, 13, 207–229.Kilgore, K. & Ross, D. (1993). Following PROTEACH graduates: The fifth year of

practice.Journal of Teacher Education, 44, 279–287.Lacey, C. (1988). Professional socialization of teachers. In M. J. Dunkin (Ed.),The

international encyclopedia of teaching and teacher education. Oxford: Pergamon.

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Lincoln, Y. & Guba, E. (1985).Naturalistic inquiry. Beverly Hills, CA: Sage Publications.Lortie, D. C. (1975).School teacher. Chicago: University of Chicago Press.Lubinski, C. A. & Jaberg, P. A. (1997). Teacher change and mathematics K-4: Developing

a theoretical perspective. In E. Fennema & B. S. Nelson (Eds.),Mathematics teachers intransition(223–254). Mahwah, NJ: Lawrence Erlbaum.

Ma, Liping. (1999).Knowing and teaching elementary mathematics: Teachers’ under-standing of fundamental mathematics in China and the United States. Mahwah, NJ:Lawrence Erlbaum.

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Department of Mathematical SciencesNorthern Illinois UniversityWatson Hall 357IL 60115DeKalbUSAE-mail: [email protected]

GUEST EDITORIAL

As we are beginning the final editing process for the fourth issue ofVolume 4 of the Journal of Mathematics Teacher Education (JMTE), weare preparing the last issue of our tenure as Editor-in-Chief and ManagingEditor of the journal. Five years of sometimes hectic, sometimes slow,but always exciting work lay behind us. Little did I know in 1996 whata rewarding and fruitful journey I was about to join.

Of course, there was always the technical part of my involvement inthe enterprise: designing the data base; finalizing the layout for the Call ofPapers for the originally planned two-book series and later for the flyerthat announced the first issue of the journal; filing, sending, receiving;keeping track of manuscripts, authors, and reviewers. But from the begin-ning, the technical work was closely interwoven with the conceptual workon the manuscripts and thus with my professional growth as a mathe-matics educator. During my doctoral studies and research, my primaryfocus had been on young students’ mathematical thinking and the epistem-ological questions connected to that issue. Mathematics teacher education– although not absent from my professional education – had been ofperipheral interest.

With the arrival of the original 100 manuscripts in 1996, the teachereducation component of mathematics education became an integral partof my work and of my thinking. The articles on pre-service and practicingteachers’ mathematical content knowledge – the most frequently addressedissue – challenge me to reflect on my own practices and the decisionsI make about what to teach and how to teach it as I guide secondarymathematics students in their professional preparation. Which topics andconcepts from the wide array of secondary mathematics are essential andyet unfamiliar enough to challenge their attitudes toward and conceptionsof school mathematics? What mathematical concepts do I choose thatinvite preservice teachers to immerse in mathematical activities (Davis,1999) and – at the same time – leave room for reflection on those activities?What kinds of problems and contexts will entice them to step out of their

Journal of Mathematics Teacher Education 4: 173–175, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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traditional student roles and experience the college mathematics class as acommunity of learners?

The many contributions on professional development initiatives andprograms published in JMTE serve as invitations to look at my ownwork with practicing teachers from a broader perspective. For teachers tobe learners, the occasional summer workshop with its array of enticinginstructional activities does not suffice, as became abundantly clear inJMTE articles. Like preservice teachers, practicing teachers have tobecome doers of mathematics, they have to immerse themselves in relevantand challenging mathematics (Borasi, Fonci, Smith & Rose, 1999), andthey have to reflect on that mathematics from within and out-of theimmersion (Schifter, 1998). In addition, teachers have to transform theexperience of being-in and reflecting-on the doing of mathematics tothe practice of teaching mathematics. Again, they have to reflect on theactivity of teaching from within and out-of that activity. Ideally, thisreflection lays the ground for professional development, that is, teachersask the hard questions about their own practices (Jaworski, 1998). Ifteachers, such as Bujak and Keeney (Schifter, 1998) and Sarah (Edwards& Hensien, 1999), take ownership of their mathematics as well as oftheir questions about their practices of teaching mathematics they willalso own the process of change within their classrooms. Thus, teachersand teacher educators become partners in working towards the commongoal of constituting the mathematics classroom as a community oflearners.

Finally, I learned a lot about myself, about my professional weaknessesand strengths. I learned to trust myself and my judgement. My ability toattend to details and my need and sense for structure complemented theeditor’s experience and conceptual insights. I am grateful to Tom Cooneyfor giving me the opportunity to learn, for his willingness to teach, and,most of all, for his generosity in working with me as a colleague. I wishthe next editorial team will experience a working relationship that is asrewarding as the one I have enjoyed.

REFERENCES

Borasi, R., Fonzi, J., Smith, C.F. & Rose, B.J. (1999). Beginning the process of rethinkingmathematics instruction: A professional development program. Journal of MathematicsTeacher Education, 2, 49–78.

Davis, B. (1999). Basic irony: Examining the foundations of school mathematics withpreservice teachers. Journal of Mathematics Teacher Education, 2, 25–48.

GUEST EDITORIAL 175

Edwards, T.G. & Hensien, S.M. (1999). Changing instructional practice through actionresearch. Journal of Mathematics Teacher Education, 2, 187–206.

Jaworski, B. (1998). Mathematics teacher research: Process, practice and the developmentof teaching. Journal of Mathematics Teacher Education, 1, 3–31.

Heide G. WiegelManaging Editor, JMTE

MARIA L. BLANTON, SARAH B. BERENSON and KAREN S. NORWOOD

EXPLORING A PEDAGOGY FOR THE SUPERVISION OFPROSPECTIVE MATHEMATICS TEACHERS1

ABSTRACT. Our investigation explored a pedagogy for supervision through a casestudy of one prospective middle school mathematics teacher during her student teachingsemester. Classroom observations by the university supervisor, teaching episode interviewsbetween the supervisor and student teacher, and focused journal reflections by the studentteacher were coordinated to challenge the student teacher’s existing models of teaching.The emerging pedagogy of the teaching episodes, a central focus of this study, wascharacterized by (a) the use of open-ended questions that centered the student teacher inthe process of sense making; (b) a shift away from the supervisor’s direct, authoritativeevaluations of the student teacher’s practice; (c) a sustained focus throughout supervisionderived from the student teacher’s classroom experiences; and (d) an effort to maintainsensitivity to the student teacher’s zone of proximal development. We found our approachto be coincident with the notion of instructional conversation (IC) advanced by Gallimoreand Goldenberg (1992). The nature of the teaching episodes seemed to open the studentteacher’s zone of proximal development so that her practice of teaching could be mediatedwith the assistance of a more knowing other.

Few would seriously question the complexities of the student teachingpracticum. The practicum reflects the integration of sometimes dissonantagendas of teaching and learning that ultimately define a community intowhich the student teacher is acculturated. It demands that the studentteacher negotiate tensions imposed by the juxtaposition of school anduniversity cultures in the context of a practice still in its infancy. It is fromthe overabundance of pedagogical beliefs and practices constituting thesecommunities that the student teacher’s practice emerges.

Despite these challenges, the practicum still promises the optimalsetting in which knowledge of content and pedagogy coalesce in themaking of a teacher. This opportunity naturally invites questions aboutthe role of agencies associated with the practicum in effecting teacherchange. Of particular interest here is the place of university supervision.Specifically, does supervision act as teacher education, or does it insteadconfirm the student teacher’s pre-existing habits of teaching by focusing onperipheral issues of practice? More importantly, how can supervision func-tion as teacher education? Research on the supervision of student teachershas produced a continuum of responses to these questions. Whereas the


178 MARIA L. BLANTON ET AL.

more skeptical suggest that we abandon supervision altogether (Bowman,1979), others argue that we must fundamentally alter the way we superviseif we are to effect real change in the ways that student teachers teach (Ben-Peretz & Rumney, 1991; Borko & Mayfield, 1995; Feiman-Nemser &Buchmann, 1987; Frykholm, 1996; Richardson-Koehler, 1988; Zimpher,deVoss & Nott, 1980).

THE ROLE OF SUPERVISION:EDUCATION VERSUS EVALUATION

Historically, university supervision has tended toward more evaluativerather than educative interactions with student teachers. That is, tradi-tional supervision may be more closely described as an assessment of thestudent teacher’s existing habits of teaching, buried within an attentionto classroom bureaucracy, rather than prolonged interactions purposed tochallenge those existing habits. Quite possibly, this emphasis is a reflectionof the chronological placement of student teaching at the end of academicteacher preparation. It might also reflect the dilemma that supervisorsare sometimes inadequately prepared to seriously challenge a studentteacher’s ability to teach a particular subject. Furthermore, case loadsthat leave little time for one-on-one interaction between the supervisorand student teacher often relegate the supervisor to an evaluative role.However, Feiman-Nemser and Buchmann (1987) challenged us to recon-ceptualize the practicum, and hence supervision, as preparatory to futurelearning, that is, as educative rather than simply evaluative. Zeichner’s(1996) admonition that we “view the practicum as an important occasionfor teacher learning and not merely a time for the demonstration of thingspreviously learned” (p. 216) echoes the need for an educative approach tothe supervision of student teaching.

Research indicates that an educative approach is not currently assumedin all supervisory relationships. In an investigation of guided practiceinteractions between university faculty, cooperating teachers, and studentteachers, Ben-Peretz and Rumney (1991) pinpointed the lack of profes-sional reflection provided by support personnel. They found that theauthoritative demeanor adopted by supervisors was met with passivityfrom student teachers, with the result of little change in practice. Borkoand Mayfield (1995) found that supervisors focused on superficial aspectsof teaching, such as paperwork, lesson plans, and behavioral objectives,and avoided in-depth discussions about content and pedagogy, and thus,offered student teachers no specific directives on how to change theirpractice. Concluding that supervision seemed to exert little influence on

PEDAGOGY FOR SUPERVISION 179

student teachers’ development, they proposed that supervisors shouldactively participate in student teaching and “challenge student teachers’existing beliefs and practices and model pedagogical thinking and actions”(p. 52). Although these recommendations might be seen to conflict withthe physical parameters, such as time, that constrain supervision, a strictlyevaluative approach does not seem to engender substantive change inteaching. In short, active participation in student teaching will requiremore than peripheral commitments by the supervisor but could result in apracticum that functions as teacher education, not just teacher evaluation.

Why should we consider an approach to supervision that challengesstudent teachers’ models of teaching in the context of their practice? First,it is within the demands of the classroom that a student teacher’s inter-nalized models of teaching are most readily revealed (Feiman-Nemser,1983). Such models, a legacy of the “apprenticeship of observation”(Lortie, 1975) realized through one’s years of schooling, will persistthroughout the practicum if left unchallenged. Indeed, the assumption thatdesirable teaching habits necessarily derive from the activity of studentteaching is challenged by existing research. For instance, Feiman-Nemser(1983) cited studies in which successful student teaching was most oftenequated with the achievement of utilitarian goals affiliated with classroommanagement. This perspective on successful teaching could likely impedeany designs by teacher education programs to infuse theory into prac-tice. Feiman-Nemser (1983) and Feiman-Nemser and Buchmann (1987)also reported that student teachers tend to imitate the persona of theschool community into which they are acculturated. Such behavior, whichmight reflect the specific habits of the cooperating teacher or the moregeneral attributes of the school bureaucracy, could persist in the absenceof supervision that challenges student teachers’ models of teaching.

Taken together, these findings suggest that the way we conceptualizesupervision portends the nature of teacher development during the profes-sional semester. It is the supervisor who is most able to “provide supportand guidance for student teachers to integrate theoretical and research-based ideas from their university courses into their teaching” (Borko &Mayfield, 1995, p. 517). However, meaningful supervision rests on rein-terpreting the role of supervisor as teacher. Sporadic visits by a supervisorwhose primary function is to evaluate peripheral characteristics of teachingseem to be an ineffective route to changing practice (Borko & Mayfield,1995; Frykholm, 1996).

From this position, we explored the nature of educative supervisionduring one middle school mathematics student teacher’s practicum. Byeducative supervision, we mean supervision that prioritizes the devel-


opment of a student teacher’s practice through prolonged instructionalinteractions with and extensive classroom observations by the univer-sity supervisor. Additionally, we defined it to include but not be limitedto evaluations by the university supervisor. We conceptualized educativesupervision within the Vygotskian (1978/1934) tenet that the supervisor,as a more knowing other, can guide the student teacher’s development to agreater extent than the student teacher can alone. This notion, theorizedby Vygotsky as the zone of proximal development (ZPD), is unique inthat it “connects a general psychological perspective on [the individual’s]. . . development with a pedagogical perspective on instruction” (Hede-gaard, 1996, p. 171). As such, the ZPD supports the use of intentionalinstruction (not just the practice of evaluation) during supervision.

But what might this instruction resemble? Although student teaching isone of the most widely studied components of formal teacher preparation,the influence of supervision on teacher learning, and the educative formsit might take, is still unclear (Borko & Mayfield, 1995). In particular,understanding what educative supervision resembles within a frameworkthat reflects our current ways of knowing remains virtually unexplored. Assuch, our central question in this study was What does it mean to supervisefrom a theoretical orientation that situates the student teacher as an activeconstructor of his or her knowledge about teaching?

METHODS

Within the grounded-theory paradigm (Creswell, 1998), our goal was thedevelopment of a model or explanation inherent in the phenomenon beingstudied, i.e., educative supervision, that suggests how one might super-vise a student teacher if the purpose is to challenge his or her practice ofteaching. This focus necessarily guided our data collection and analysis.That is, our process involved repeated visits to the student teacher’s schoolinterspersed with ongoing, informal data analysis for the purpose of under-standing and describing the nature of educative supervision. Formal dataanalysis of descriptive categories about educative supervision emergingfrom the field began after data collection was complete.

In order to study the nature of educative supervision, the headauthor became the university supervisor of Mary Ann (pseudonym), aprospective middle school mathematics teacher. At the time of the study,the supervisor was a doctoral student in mathematics education and wasconducting research on various aspects of the development of a studentteacher’s practice during the practicum. This particular study grew outof that research (see also Blanton, Berenson & Norwood, 2001). The


dual role of supervisor/researcher was adopted because it offered aninside perspective to understanding supervision. That is, analogous to thegenre of research in which researchers become classroom mathematicsteachers (e.g., Ball, 1993; Lampert, 1992), the supervisor in this studybecame the teacher in a classroom in which learning to teach mathematicswas the content. Moreover, the supervisor’s advocacy of reform-basedmathematics teaching prompted an introspection about the practice of asupervisor that ultimately became the central question of this study.

Mary Ann, who was in her final year of a four-year teacher educationprogram, had successfully completed her academic studies and was eagerto begin the student teaching semester. Assigned to a seventh-grade mathe-matics classroom in an urban middle school, she was paired with a veteranteacher whose approach of sharing her own wisdom of practice withoutstifling Mary Ann’s ideas led to a positive, open relationship between them.In particular, the cooperating teacher worked to familiarize Mary Ann withthe rudiments of planning lessons by offering insights and techniques thatshe had found useful in her own instruction. She was enthusiastic aboutwhat she saw in Mary Ann’s practice and the direction in which it wasdeveloping. She was supportive of the university supervisor’s sustainedinteractions with Mary Ann, commenting at various points that she andMary Ann were both learning from each other.

Data Collection

During the student teaching practicum, Mary Ann met weekly withthe university supervisor for what we conceptualized as a constructivistteaching experiment (see e.g., Steffe, 1983, 1991). According to Steffe(1983), the teacher’s role in such an experiment is to challenge the modelof the student’s knowledge and examine how that model changes throughpurposeful intervention. In our case, the supervisor took on the teacher’srole, and the student teacher was seen as the student. Each prolongedconversation between supervisor and student teacher was considered tobe a teaching episode, that is, one session of the teaching experiment.

Each visit by the supervisor consisted of a three-hour sequence thatbegan with an observation of Mary Ann teaching a general mathematicsclass. Field notes taken during this observation focused on classroom inter-actions that reflected the nature of Mary Ann’s thinking about teachingmathematics. Immediately following the observation, Mary Ann collab-orated with the supervisor in a one-hour teaching episode to help makesense of these interactions. In particular, Mary Ann’s thinking aboutthe interactions, what these interactions suggested about how studentslearn mathematics, and how subsequent lessons might be modified, were


discussed. The visit concluded with a second classroom observation ofMary Ann teaching another general mathematics class. The second obser-vation provided the opportunity to document short-term changes in MaryAnn’s practice as she taught the same subject to a different class imme-diately after a teaching episode. The supervisor arranged each visit withMary Ann prior to the event.

In addition, Mary Ann was asked to keep a personal journal in whichshe reflected on what she had learned about her students, about mathe-matics, and about teaching mathematics subsequent to each visit by thesupervisor. Other written artifacts such as lesson plans, activity sheets,and quizzes were collected at each visit. At the conclusion of each visit,the supervisor audiotaped personal reflections about Mary Ann’s emergingpractice and how future visits could support her development. More gener-ally, by combining the supervisor/researcher roles, the researcher was ableto engage in ongoing informal reflections about what it meant to be aneducative supervisor. In all, there were eight supervisory visits followedby a separate exit interview. In addition, two clinical interviews with thecooperating teacher were conducted in order to explore the cooperatingteacher and student-teacher partnership. Each visit, documented throughfield notes and complete audio- and audiovisual recordings of the teachingepisodes and interviews, along with supporting written artifacts, providedthe data corpus for this investigation.

Data Analysis

From the teaching episodes with Mary Ann, complete transcripts of fourrepresentative episodes were selected for further analysis. The selectionswere made based on an earlier analysis of discourse in Mary Ann’sclassroom (see Blanton, Berenson & Norwood, 2001) from which fourvisits by the supervisor were selected as indicative of development inher practice. The teaching episodes analyzed here occurred during thosevisits. In particular, transcripts were coded by conversational subject witha speaker’s turn as the basic unit of analysis. Categories identified by thecoding process were based on what we perceived as the primary thrustof a speaker’s comment and included mathematics pedagogy, generalpedagogy, mathematical knowledge, knowledge of student understanding,classroom management, and teacher-student relationship. For example,Mary Ann’s description of the use of a balance scale to develop students’conceptual understanding of solving linear equations was coded as mathe-matics pedagogy because it demonstrated a principle in how she taughtmathematics. That is, the use of a concrete apparatus to contextualize the


abstraction of solving symbolic linear equations was important to her inthe development of students’ understanding. She explained:

It’s kind of hard for them to understand solving equations, especially negatives and posi-tives cancel each other out, so what we did is we used the balance [scale] with rainbowcubes and put [cubes] on each end. . . . They looked at it, they saw it, take it away, you knowif you took something from one side, then you had to take it away from the other side orthe scale was not balanced. So we worked for probably a day and a half on understandingwhy we do the things that we do. . . . I was always taught just move that around, subtract it,and I never understood why we do that, why does it work? So by seeing it on the balance,like, I took a Kleenex and covered up five cubes . . . so that was their unknown.

As another example, her comment “I’ve been trying really, really hard,you know, for [a student] to like me. I want . . . all of my kids to like me. Ofcourse they’re not going to like me sometimes when I have to disciplineand stuff.” was coded as teacher-student relationship because it addressedMary Ann’s attentiveness to and perception of her relationship with herstudents.

The teaching episodes were then quantified by a word count todetermine the emphasis given to each subject code and to establish theamount of conversational time used by the university supervisor and thestudent teacher. Transcripts from classroom observations and Mary Ann’sjournal reflections were used to corroborate changes in Mary Ann’s prac-tice as a result of the teaching episodes. One visit (hereafter referred toas the “problem-solving day”) was selected as an exemplar of the super-visory model of observation, teaching episode, observation presented here.Transcripts from the teaching episode on this visit were analyzed for char-acteristics of the supervisor’s pedagogy that promoted change in MaryAnn’s teaching. Data from the other teaching episodes were subsequentlyanalyzed to determine if these characteristics were representative of thosevisits as well. That is, once assertions had been generated about char-acteristics of the supervisor’s pedagogy from the problem-solving day,the remaining transcripts were analyzed in order to see if the identifiedcharacteristics were reflected in the other teaching episodes. In particular,each conversational turn was analyzed for the nature of the questions askedby the supervisor (e.g., open-ended), the nature of the supervisor’s voice(e.g., authoritative vs. facilitative), the supervisor’s adherence to a them-atic focus, and the supervisor’s sensitivity to Mary Ann’s development.Specific excerpts from transcripts on the problem-solving day and one ofthe other teaching episodes (the “pattern-finding day”) are included hereto substantiate our findings.


A PEDAGOGY FOR TEACHING EPISODES WITH MARY ANN

In this section, we describe the nature of the teaching episodes with MaryAnn and what the supervisor observed about her own practice in theseepisodes that seemed to promote change in Mary Ann’s practice.

Classroom Experiences as Entries to Teaching Episodes

Gallimore and Goldenberg (1992) maintained that “students must be‘drawn into’ conversations that create opportunities for teachers to assist,. . . including activating relevant prior knowledge” (p. 209). Indeed, ourimmediate challenge with Mary Ann was to bring her into conversationsthat would activate and build on her existing knowledge about teachingmathematics. An advantage of supervision is that it can use the context ofthe student teacher’s practice to scaffold his or her emerging ideas aboutteaching. For instance, the student teacher’s classroom experiences revealmuch about his or her relevant background knowledge, particularly inter-nalized models of teaching, which the supervisor can exploit to challengethe student teacher’s thinking.

We include the following excerpts from the problem-solving day toillustrate the use of an observed classroom experience to draw Mary Anninto conversation about, and subsequently analysis of, her practice. Webegin with an episode that occurred in Mary Ann’s classroom duringthe supervisor’s observation prior to the teaching episode. During thisobservation, Mary Ann began a lesson on “working backwards” as aproblem-solving technique by giving students the following problem tosolve during individual seatwork:

Problem 1: I’m thinking of a number that if you divide by three and addfive, the result is eleven

After several minutes, Mary Ann asked Evelynne to explain howshe got her (correct) solution.

1 Evelynne: Well, I started out with what you told us and then I put 3 asthe divisor. And then you said the answer plus 5 and it equals 11. So,I put 5 plus something equals 11 and that was 6, and so 6 times 3 is18, and that’s how I got it.

Mary Ann repeated Evelynne’s comments, then continued her lesson onworking backwards.

2 Mary Ann: O.K. We’re going to look at something a little different.. . . We’re going to use a method called working backwards. O.K.


(referring to Problem 1) we said that the result is 11, so we have 11here (Mary Ann writes this on the overhead projector [OP]). Thenwe said add 5 and divide by 3, and the result is 11. O.K, when wedecide to work backwards, we have to go through and [change everyoperation] to its inverse. Remember we’ve been talking about inverseand opposites. So what’s the inverse of add?

3 Class: Subtract.

4 Mary Ann: Subtract. So we’ll subtract 5 [from 11]. What’s the inverseof divide?

5 Class: Multiply.

6 Mary Ann: Multiply. So we’ll multiply [11 – 5] by 3.

After Mary Ann got a correct response of 18, she wrote a similar problemon the OP for the class to collectively solve:

Problem 2: I’m thinking of a number that if you divide it by 3 and then add5, the result is 13.

7 Mary Ann: So what should I do first just to get an idea of what we’retalking about?

There was a long pause during which no one responded.

8 Mary Ann: Does anybody know how we did the last one?

Again, no one responded.

9 Mary Ann: O.K., we want to work backwards. So what have we got todo when we work backwards? What was the word that we used whenwe talked about what we’ve got to do with all of these?

10 Student: Inverse.

11 Mary Ann: Inverse. O.K., so we have to take the inverse of the oper-ations. . . . O.K.? So what are we going to do with the 5? Add it orsubtract it?

12 Class: Subtract.

13 Mary Ann: Subtract. We’re working with the inverse. We’re workingwith opposites. O.K., then are we going to divide or multiply by 3?


14 Class: Multiply.

15 Mary Ann: Multiply by 3. O.K., that’s step two, to write downeverything that’s the inverse. And it’s very important that you keep thesame order. You have to keep the same order as the problem. (MaryAnn continues to write the mathematical pieces of this conversationon the OP.) O.K., step three is to actually solve the problem. What arewe going to do? Somebody tell me the first step. Terence?

16 Terence: Thirteen minus 5.

17 Mary Ann: Thirteen minus 5. O.K., and 13 minus 5 is?

18 Terence: Eight.

19 Mary Ann: Eight. Now what am I going to do? Andrea?

20 Andrea: Eight times 3.

21 Mary Ann: Eight times 3, and what is 8 times 3?

22 Andrea: Twenty-four.

23 Mary Ann: Twenty-four. So that’s my answer. I ask you what numberdid I start with [and] you’ll say what?

Students offered no response, which prompted Mary Ann to repeat herquestion:

24 Mary Ann: The problem says “I am thinking of a number.” Whatnumber am I thinking of?

At this point, students called out several different answers, leading toMary Ann’s frustration that they, with the exception of a few students,did not seem to understand despite their responses to her questions. Inexasperation, she tried to get students to respond:

25 Mary Ann: What did we just solve? What answer did we just get?Look up here (indicating the OP).

It was this classroom experience that the supervisor used to drawMary Ann into a discussion about her practice. From the discourse, thesupervisor inferred that Mary Ann’s focus was on students’ proceduralunderstanding of the task, and this focus had determined the way thewhole-class discussion played out. It was the supervisor’s intent to use thisevent to initiate Mary Ann’s sense making about her practice. The excerptchronicled below occurred in the subsequent teaching episode and refersto the whole-class discussion about Problem 1 and Problem 2.


26 Supervisor: Once you began talking with students, they began toanswer your questions. At first, they seemed hesitant. You had gonethrough the problem (Problem 1) and then you started asking them“What do you do next?”.

27 Mary Ann: Right.

28 Supervisor: Then they started answering your questions (see, e.g., 3, 5,10, 12).2 Do you think they understood the problem or the steps theywere supposed to do?

29 Mary Ann: I think they understood the steps. I totally agree. Because Ireally didn’t focus on understanding the problem as much as I shouldhave, and now that you bring it up, I know I didn’t. It was prettymuch the steps. But I don’t . . . these kids just have a really hard time.If you don’t know how to divide, it’s going to be hard for your levelof thinking to [figure out how to solve the problem]. So, really, I justdid [this lesson] to get them by these two sections, because they’re notgoing to see this. They’re not going to use that [technique of “workingbackwards”].

30 Supervisor: So apparently you think there’s a better way to solve thistype of problem?

31 Mary Ann: Now I think that [Problem 1] makes more sense than“Johnny went to the store and had, you know, twice as many . . . aslast year.” I think they were beginning to see [how to use “workingbackwards” as a problem-solving technique], because it was prettymuch everything [Evelynne] did when she showed me how she solvedher problem. I didn’t bring that out but I should have. . . . She workedbackwards in her mind, but she just didn’t realize she was workingbackwards.

Later, we will explore more from this teaching episode. For now, ourpoint is to illustrate the use of a classroom event to draw Mary Ann into aconversation that focused on her thinking, not the supervisor’s conclusions.We found that classroom experiences became the nexus between theoryand practice in teaching episodes with Mary Ann, effectively opening herZPD and drawing her into conversation with the supervisor. In particular,bringing specific classroom experiences to Mary Ann’s attention duringthe teaching episodes became a hook (Gallimore & Goldenberg, 1992) forher to openly analyze her practice. Conversely, she became visibly passivewhen referents beyond the scope of her own experiences, for example, the


supervisor’s analysis of Mary Ann’s teaching or the supervisor’s experi-ences as a student teacher, were introduced. As a caveat, we emphasize thatthe student teacher’s classroom experiences are merely a point of departurefrom which the supervisor may solely evaluate that student’s practice ofteaching. However, we contend that this alone can engender a passiveenvironment with no certainty that the student teacher will subscribe toour theories of teaching.

Open-Ended Questions as Prompts for Sense Making

We further found that the nature of the supervisor’s questions about MaryAnn’s classroom experiences prompted her sense making. Gallimore andGoldenberg (1992) argued that “when known-answer questions are asked,there is no need to listen to a child or to discover what the child mightbe trying to communicate” (p. 209). We assert that in the context ofsupervision, the exclusive use of evaluative comments and known-answerquestions constrains the need to listen to the student teacher. Thus, animperative of teaching episodes with Mary Ann became to avoid thesingular use of these types of questions and comments. As a result, thebalance of questions posed to Mary Ann were open-ended, with theexpectation that Mary Ann would justify her thinking about teachingmathematics and her consequent actions in the classroom, not passivelyrespond to a supervisor’s prompts. We include here representatives ofthe types of exploratory questions that evolved throughout the teachingepisodes. Although we do not suggest that the list is exhaustive or entirelyoriginal, we share these questions as significant in the supervisor’s effortto maintain the role of facilitator and Mary Ann’s role as sense maker. InTable 1, we contrast the tenor of these questions with what we see as theirpossible evaluative counterparts.

The Supervisor’s Voice

Our emphasis on open-ended questioning necessarily limited instancesof direct teaching by the supervisor. For us, this balance was rooted inthe belief that students are more likely to teach in ways they are taught(Borko & Mayfield, 1995; Feiman-Nemser, 1983). In practice, this meantthat we could not simply tell Mary Ann how to change her teaching inorder to move beyond the show-and-tell paradigm we had observed (e.g.,7–22). In this, we did not assume that Mary Ann would necessarily makea connection between the supervisor’s own practice during the teachingepisodes and her resulting classroom practice. The goal was instead forthe supervisor’s practice to be intellectually honest. Thus, moving awayfrom an authoritative voice, the supervisor used “prompting, modeling,


TABLE 1

Open-Ended Questions That Emerged During the Teaching Episodes and PossibleEvaluative Counterparts

Open-Ended Questions Evaluative Counterpart

“You taught this lesson twice today. Whatchanged about the way you taught it thesecond time?”

“The second time you taught this lesson,you did the following differently . . . ”

“How are you going to teach this lessondifferently?”

“You need to teach this lesson differentlyin the following way . . . ”

“What did you learn about your studentsfrom teaching this lesson?”

“You seem to have the following beliefabout students . . . ”

“Did students have any difficulties thatyou did or did not anticipate?”

“Students didn’t understand what youwere doing in this part of your lesson.”

“How did that [classroom experience]affect your teaching?”

“I noticed that you did the following as aresult of this [classroom experience].”

“How do/would you handle that type ofsituation?”

“The next time this occurs, you should dothe following . . . ”

“What do you see as your biggest diffi-culty?”

“Your biggest difficulty is . . . ”

“What was the most memorable thing thathappened in class today?”

“The most significant thing I observedtoday about your teaching is . . . ”

“Do you think that technique waseffective? Why?”

“Your technique seemed ineffective. Iwould recommend that you try thefollowing . . . ”

“How do you balance the use of hands-onactivities with whole-class discussion?”

“You need to use more hands-on activ-ities. There’s too much lecture.”

“What did you learn today about mathe-matics?”

“I observed that you have thefollowing (incorrect) conception aboutmathematics.”

“How would you handle a similar situ-ation in the future?”

“In the future, you should do this instead. . . ”

“Do you think there’s a better way forstudents to solve these problems? If so,what?”

“I think you should use the followingapproach in problem-solving . . . ”

“Why do you think this [belief aboutmathematics] is true?”

“You should have the following percep-tion about mathematics . . . ”

“At one point in your lesson, you said . . . .What were you thinking?”

“At one point in your lesson, instead ofdoing X, you should have been doing Y.”

“How would you describe a successfulclassroom?”

“I think a successful classroom has thefollowing characteristics . . . ”

“What did you want students to under-stand in this lesson?”

“Your goals for this lesson seemed to be. . . ”


Figure 1. Conversational time used by participants in the teaching episodes (Note: Figure1 represents percentages of time a given participant spoke during a teaching episode.Percentages are based on word counts.).

explaining, . . . discussing ideas, [and] providing encouragement” (Jones,Rua & Carter, 1997, p. 4) to give structure to teaching episodes. Thus,Mary Ann was encouraged to construct her own solutions to conflicts in herpractice (e.g., 32–39). In fact, throughout this episode and in the analysis ofother episodes, we observed shifts in the supervisor’s practice away fromevaluative conclusions about Mary Ann’s classroom experiences. Instead,the supervisor used Mary Ann’s experiences as a catalyst for engagingher sense making about teaching. What seemed to emerge for Mary Annwas a sense of ownership that heightened her willingness to implementalternative approaches.

The supervisor’s voice shifted not only in demeanor and intent, but alsoin quantity. Figure 1 illustrates the amount of conversational time used byMary Ann during the teaching episodes with the supervisor. The resultsprovide evidence of the supervisor’s intent to maintain a facilitory rolethat kept Mary Ann at the center of discourse. That is, although the datadepicted in Figure 1 do not address the nature of the teaching episodesper se, they do confirm that the supervisor did not control discourse inconversations with Mary Ann.


An Educative Focus for the Teaching Episodes

There is a risk that perfunctory evaluative visits with a student teacher willlack the depth of focus engendered through instances of genuine teaching.Gallimore and Goldenberg (1992) argued that “to open a zone of proximaldevelopment . . . , a teacher has to intentionally plan and pursue an instruc-tional as well as a conversational purpose” (p. 209). During the initialvisits with Mary Ann it became clear to us that the supervisor neededto establish a teaching focus for the episodes throughout the practicumthat would directly address Mary Ann’s specific needs as a novice teacher.By the third visit (the problem-solving day), the supervisor identified athematic focus regarding the nature of discourse in Mary Ann’s classroomthat emerged after a mathematical task or question had been posed.3 Thesupervisor’s observations prior to this visit revealed classroom discoursewhich served predominantly as a “passive link in conveying some constantinformation between input (sender) and output (receiver)” (Lotman, 1988,p. 36). Through these univocal interactions (see Wertsch & Toma, 1995),Mary Ann funneled students toward her interpretation of the problem athand.

This focus for supervision grew out of the supervisor’s effort to listento and reflect on Mary Ann’s teaching and find a focus issue for medi-ation relevant to her practice. Although there might have been other areasfor focus, it became clear from the supervisor’s classroom observationsthat the nature of classroom interactions, particularly what mathematicalproblem-solving looked like in Mary Ann’s classroom, would be an appro-priate area in which to challenge her practice. What was significant, atthis point, was that Mary Ann’s practice of problem solving with herstudents reflected what we found to be her thinking about the nature ofmathematics:

I know that math is one big word problem because one thing builds on another. But I don’tlook at it like that. I look at math as just operations you go through, just like a series ofsteps. You have to step on this step before you get to the next one.

Given this, it was not surprising to us that she enacted a step-by-stepapproach in her teaching (e.g., 7–23). What ultimately became the focusof supervision, namely, how to verbally engage students in mathematicalproblem solving, was seen as intrinsically bound to her knowledge aboutthe nature of mathematics.

The educative focus on the problem-solving day. The third visit presentedan opportunity for assisting Mary Ann in cultivating dialogic classroomdiscourse, that is, discourse that could serve as a thinking device by whichnew meaning could be generated (Lotman, 1988; Wertsch & Toma, 1995).


As we described earlier, during the supervisor’s first observation on thisparticular visit Mary Ann began a lesson on working backwards by givingstudents a problem to work individually. After a short pause, Mary Annbegan to dole out hints until a correct solution appeared. After a studentshared a procedure for obtaining this solution (1), Mary Ann began a step-by-step account of how to work backwards to find the answer (2–6, 7–23). Elsewhere, our analysis of the full classroom observation showed thatshe had interpreted students’ questions univocally, that is, as a result of abreakdown in communication (see Blanton, Berenson & Norwood, 2001).Moreover, she asked cognitively small questions (e.g., [What is] thirteenminus five? What is eight times three?) in order to align students’ thinkingwith her own. As Mary Ann equated student feedback with understanding,her frustration surfaced when the class unsuccessfully attempted to solve asimilar problem (24–25).

The perturbation that Mary Ann experienced from this interactionseemed to grow out of her confusion that students did not understandwhat she had carefully explained. In our interpretation, this left her ata pedagogical impassé. The challenge for the supervisor in the teachingepisode that followed and in future episodes was to use this experienceto scaffold Mary Ann’s incipient notions of mathematics as a problem-solving activity in which students struggled with unfamiliar problems andjustified their ideas through mathematical discourse with each other andMary Ann. In essence, the challenge was to help Mary Ann create aclassroom discourse in which dialogic and univocal interactions dualist-ically existed. Throughout the teaching episode on the problem-solvingday, the supervisor focused on how Mary Ann might alter the way the classapproached problem solving (42–49). When it seemed that the supervisor’sprompts were beyond Mary Ann’s ZPD, that is, when Mary Ann couldnot be hooked into the conversation and instead shifted the topic to oneof classroom management (see 42–45), the supervisor steered to relatedsubjects (e.g., Mary Ann’s perception of problem solving in mathematics)that would probe Mary Ann’s thinking and draw her back into conversationabout engaging students in problem solving. This type of threaded conver-sation seemed to scaffold Mary Ann’s thinking about teaching because itallowed her to appropriate an idea through multiple interactions over thelength of the teaching episode.

Multiple interactions on the educative focus for supervision. The educativefocus provided an instructional purpose in supervision that continuedthroughout the practicum. Having identified the thematic focus of supervi-sion on the problem-solving day to be the nature of mathematical discourse


in Mary Ann’s classroom, the supervisor’s goal was to build on thattheme and see how or if Mary Ann subsequently attended to mathematicaldiscourse in her classroom. The pattern-finding day, which occurred fourweeks after the problem-solving day, represented one of the more signi-ficant moments when this discourse took place. For the lesson on this day,students worked in dyads to investigate the number of diagonals in variouspolygons. Geoboards were used to model the specific cases. After tabu-lating students’ results for a triangle, quadrilateral, pentagon, and hexagon,Mary Ann asked students to find a pattern in order to predict the number ofdiagonals in a heptagon. Here, our purpose is to share an excerpt from theteaching episode on that day in order to analyze the continued interactionson the selected educative focus.

32 Supervisor: What was the point [of the lesson]?

33 Mary Ann: The main point is that this was a problem-solving strategy.It was to be able to set up a table, which is what we did up here. Itwas to be able to do a diagram, which is pretty much what they did.And then it was to see a pattern. Remember, we saw the pattern aboutthe difference, you know, increased by one.

34 Supervisor: This is a very important process.

35 Mary Ann: Right. The two problems they were assigned tonight weredealing with this same thing.

36 Supervisor: I really liked this activity. You’re using manipulatives,you’re gathering information, and you’re developing patterns. Goingback to what we’ve been talking about with mathematics as problemsolving and letting [students] struggle with the problem, you got tothe point where you were going to find the pattern, and you saidsomething like, “OK, we’re going to find the pattern”, and then youasked . . . leading questions such as “What is the difference betweenthese two numbers?”.

37 Mary Ann: What I meant to do, and I didn’t write it on my lessonplan, and I knew I’d forget it if I didn’t write it, is I was going totry and have them . . . like, we had the object here, the sides here, andthe diagonals here. (Mary Ann drew a polygonal figure on a piece ofpaper, recalling what she had written on the OP earlier.) You have tokind of narrow things down for them sometimes, so I was going toask them to look at this column (indicating the column containing thenumber of diagonals) and form some kind of prediction of what the


next numbers were going to be based on these numbers. And then Iwould have to give them some type . . . I would say, “There’s a patternforming in this column, can you find out what it is?”.

38 Supervisor: That would be super.

39 Mary Ann: Right, and I just didn’t because I didn’t write it on mylesson plan and I kept looking at the time and I was like, we have gotto get through this. So that would probably be a better way, becausethen they could come up with . . . because it’s just like that day in thirdperiod (referring to the problem-solving day) that I tried it and the kidsolved it!

40 Supervisor: Is this something you might try in the next period?

41 Mary Ann: Yeah.

We infer from Mary Ann’s comments (37, 39) that her intent was tocenter students in the activity of problem solving, but she was detractedby extraneous factors, e.g., time. Instead, as alluded to by the supervisor’sobservation (“you asked . . . leading questions such as ‘What is the differ-ence between these two numbers?’ ”), Mary Ann had enacted a whole-classdiscussion about the number of diagonals in a polygon that was moreunivocal than dialogic in essence. The supervisor’s purpose was to revisitthe educative focus and draw Mary Ann away from this univocal exchange.Now that Mary Ann already had experiences in her repertoire, such asthe problem-solving day, less intentional instruction was needed by thesupervisor. For instance, it was Mary Ann who suggested the alternativeapproach to finding the pattern for the subsequent lesson (37). Addition-ally, her comment “because it’s just like that day (the problem-solvingday) in third period” suggests to us that she was connecting her classroompractice on these two days in her own thinking. It is significant for us thatshe made these connections during the teaching episode.

It seemed to us that, through the interchange (32–41) that continuedto build on the educative focus, the supervisor was able to reinforceMary Ann’s commitment to a kind of classroom practice that wouldinclude meaningful mathematical discourse. In the lesson subsequent tothe teaching episode on the pattern-finding day, Mary Ann worked toinclude more dialogic discourse in her exchange with students, encour-aging them to “hypothesize and justify their thinking with mathematicalevidence in order to solve a non-routine problem, . . . (positioning herself)as an arbiter of students’ ideas, obligated to solicit students’ strategiesand explanations as a platform for resolving mathematical questions


Figure 2. Conversational time given to subject code during teaching episodes (Note:Figure 2 represents percentages of time the specified subject code was discussed in ateaching episode. Percentages are based on word counts.).

and extending students’ mathematical thinking” (Blanton, Berenson &Norwood, 2001, p. 239).

More generally, Figure 2 depicts the emphasis in the teaching episodesplaced on conversations about various aspects of Mary Ann’s teaching.For example, conversations that addressed principles of teaching notspecific to mathematics were coded as general pedagogy; conversationsthat addressed Mary Ann’s understanding about teaching mathematics aswell as how she taught mathematics were coded as mathematics pedagogy.In particular, these data illustrate that mathematics pedagogy domi-nated the supervisor’s conversations with Mary Ann during the teachingepisodes. Conversely, Figure 2 suggests that discussions about the peri-pheral issues of school bureaucracy (e.g., classroom management) receivedlittle emphasis in the teaching episodes. Although it might be argued thatsuch issues may of necessity dominate supervision in the case of otherstudent teachers, we point out that our conceptual focus throughout thepracticum was to influence Mary Ann’s teaching, specifically, to help hercultivate dialogic classroom interactions. Had our purpose in supervisionbeen solely evaluative, we would have missed the opportunity to challengeMary Ann’s practice of classroom discourse. As such, peripheral issueswere addressed as needed, but not at the expense of our purpose – toeducatively supervise Mary Ann’s practice.


Sensitivity to Mary Ann’s ZPD

We take the amount of conversational time used by Mary Ann (seeFigure 1) to indicate that her contributions were a priority in supervision.Moreover, in order to be responsive to her ideas we needed to be sensitiveto her ZPD as well. The following dialogue was excerpted from an instanceof intentional instruction in the teaching episode on the problem-solvingday. We include it as an illustration of the supervisor’s effort to maintainsensitivity to Mary Ann’s ZPD while guiding her thinking.

42 Supervisor: Is this the kind of problem (i.e., Problem 1) where youcould let two or three [students] work together, and try to figure outhow to do it, and see what kind of method they come up with?

43 Mary Ann: That could be an idea. Maybe I could let them work withthe person beside them.

44 Supervisor: Do you think that is even feasible? If so, why or why not?

45 Mary Ann: Two heads are always better than one, and the kid next toyou might be thinking of one way, but might be stumped on how todo the next. But you might be able to help him figure that out. Theonly thing is that I don’t know if they (her voice trails off). We’ll see,though. That might be a way to try. I don’t know if they can handlethat, talking to each other. . . . They’re just talkers, all the time. Maybeif I show them that they can have some freedom like that.

Mary Ann’s uncertainty toward this suggestion was manifested as concernover classroom management. The supervisor’s role then became to redirectthe conversation so that it was within Mary Ann’s ZPD. In other words, thesupervisor inferred that the notion of students working in groups of two orthree and solving a problem without step-by-step directives from MaryAnn did not seem to be in her conception of teaching at this point becauseMary Ann was unable to engage in a discussion about it (the conversationshifted to classroom management – (45)). Thus, maintaining sensitivityto Mary Ann’s ZPD involved connecting her concerns about students’behavior with the alternative instructional approach being negotiated (46).

46 Supervisor: Do you think they can handle working with a problem thatthey can’t figure out, trying to solve a problem in that sense?

47 Mary Ann: I think they would be more apt to keep their attention onthat problem if they’re working with somebody rather than workingby themselves.


Again redirecting the conversation, the supervisor probed Mary Ann’sunderstanding of the role of problem solving in mathematics. We maintainthat this activated her background knowledge of and about mathematicsand was therefore within her ZPD and continued to draw her into conver-sation. Later in the episode, the supervisor revisited the previous topic(42).

48 Supervisor: Would you be comfortable, for example, if you came [inclass], . . . threw out a problem, and [let] students work it for a while,and try to figure out how to come up with a solution?

49 Mary Ann: Yeah. That’s how I’m thinking about starting the next class.We’ll have to go over homework first because they’re having a quiz onthat tomorrow. And then just have [Problem 1] up on the board, andthen tell them to solve it. Don’t introduce anything about workingbackwards.

In this case, transcripts strip the dimensionality of dialogue. MaryAnn’s claim, “That’s how I’m thinking about starting the next class” (49),was spoken with a sense of reflection and ownership which stood in sharpcontrast with her initial reticence (45). It should also be noted that thisremark (49) occurred over halfway through the one-hour teaching episode,after much attention had been given to Mary Ann’s thinking about mathe-matics as problem solving and the nature of interactions that surroundeda problem posed in class. Although one might argue that a didacticalapproach to supervision would have been more efficient, we seriouslyquestion if it would have led to Mary Ann’s commitment to try an alterna-tive strategy. However, the nature of the teaching episode seemed to openher ZPD cognitively and affectively, thereby producing at least a short-termcommitment to change.

After the Teaching Episode

On the problem-solving day, Mary Ann seemed to extract from ourapproach of facilitating rather than directing her development a commit-ment to modify her practice (49). Moreover, we sensed that the super-visor’s presence in Mary Ann’s classroom after the teaching episode wasan additional support (if the plan did not work, Mary Ann could share theresponsibility with the supervisor). Mary Ann began her lesson as we hadplanned (42–49). Departing from her previous strategy, she placed studentsin dyads in order to solve the problem that had been assigned as individualseatwork in her earlier class. Removing herself as the sole authority, she


delayed closure so that students would attempt to communicate mathemat-ically with each other. As one of the students began explaining her group’sstrategy for solving the problem during the subsequent whole-class discus-sion, Mary Ann commended the student, “You just taught our lesson fortoday!” Mary Ann’s expression told the story that her journal reflectionlater confirmed.

Teaching this [to the first period class] was a real eye-opener for me. I think I totallyconfused my students completely. I tried to show them steps without letting them thinkabout the problem themselves. . . . [The next class] was different. After [the universitysupervisor] and I talked about the lesson and going over several suggestions, things seem[sic] to run much smoother. Instead of throwing information out, I let them figure theproblem out in their own style. . . . To my surprise, one of my students performed theproblem exactly as the strategy suggested. Boy, was this a memorable event. The pressurewas lifted off of me. . . . Once the students saw how one of their peers was able to solve theproblem, things were a lot more clear to all. I learned that having a student come up withthe solution means more to the others than the teacher giving a long, drawn-out lecture.

From our observations, the problem-solving day was a first step inMary Ann’s attempts to interact dialogically with her students. Moreover,we take her reflection as evidence that the teaching episode on that dayhelped to mediate her development within her ZPD. The reflection showsnot only a clear shift in Mary Ann’s thinking about what role studentsshould play in solving a mathematical problem, but also that mediation bythe university supervisor was a factor in this. In particular, Mary Ann’scomments confirm what we described earlier concerning her view ofmathematics as a step-by-step process that was ultimately reflected in herteaching (“I tried to show them steps without letting them think about theproblem itself”). The reflection also indicates a shift in what Mary Annwas beginning to value in how students did mathematics, that is, lettingstudents “figure out the problem in their own style” rather than the teacher“throwing information out.” Although the supervisor did not have accessto this reflection on the problem-solving day, the shared events of thisday mirrored Mary Ann’s reflection and served to focus the supervisor’sattention on challenging Mary Ann’s conception of what the classroomactivity of mathematical problem solving might resemble.

INSTRUCTIONAL CONVERSATION AS A PEDAGOGYFOR EDUCATIVE SUPERVISION

We view as a significant finding in our study that what seemed to emergeas a pedagogy of supervision with Mary Ann is coincident with thenotion of instructional conversation (IC) advanced by Gallimore and


Goldenberg (1992). In an investigation of elementary students’ readingcomprehension, Gallimore and Goldenberg mutually negotiated ten char-acteristics of IC: (a) activating, using, or providing background knowledgeand relevant schemata; (b) thematic focus for the discussion; (c) directteaching, as necessary; (d) promoting more complex language and expres-sion by students; (e) promoting bases for statements or positions; (f)minimizing known-answer questions in the course of the discussion; (g)teacher responsivity to student contributions; (h) connected discourse, withmultiple and interactive turns on the same topic; (i) a challenging butnonthreatening environment; and (j) general participation, including self-elected turns. It was consequential for us that IC was not a predeterminedpedagogy for our teaching episodes with Mary Ann, yet many of thesecharacteristics mirrored our own approach.

In addition, IC is rooted in a sociocultural perspective, which reflectsour own assumptions about teaching and learning. In particular, IC stemsfrom a cultural ethos that emphasizes the use of narrative in an individual’sdevelopment. Gallimore and Goldenberg (1992) and Rogoff (1990)described it as a primary means of assisted performance in preschooldiscourse between parent and child. One’s way of life, embedded in picturebooks and bedtime stories, is taught through conversation in the context offamilial relationships. Although formal schooling may seem far removedfrom this setting, the essence of IC is a promising technique in that contextas well. Gallimore and Goldenberg (1992) recognized that, traditionally,this form of teaching abates in school because teachers are more likely todominate interactions and students are less likely to converse with theirteacher or peers. They maintained that part of the difficulty of IC in formallearning contexts is that it requires teachers to shift from an evaluative rolegrounded in “known-answer” questioning, to a facilitory role in whichthey elicit students’ ideas and interpretations. As we experienced, suchdifficulties are no less present in the case of university supervision, wherethe supervisor is cast in the role of teacher and the student teacher becomesthe learner. Thus, although it is difficult to establish a direct link betweenIC and conceptual development (Gallimore & Goldenberg, 1992), weconclude that IC does suggest an alternative pedagogy for educative super-vision in that it captures the essence of the type of supervision that emergedin this study.

CONCLUSION

We have investigated what it means to educatively supervise one mathe-matics student teacher. In the process, we found that it is possible to


effectively challenge a student teacher’s practice of teaching, and wesuggest that the nature of the teaching episodes in supervision becamea conduit for change in Mary Ann’s practice. In particular, the teachingepisodes required that the supervisor move beyond the type of practicedescribed in the literature (see, e.g., Ben-Peretz & Rumney 1991; Borko& Mayfield, 1995) in which supervisors focus on superficial aspects ofteaching, assume an authoritative demeanor with the student teacher, andgenerally do not provide the type of professional support that is essentialfor a student teacher’s development. Instead, we found that by replacingdirect, authoritative evaluations of Mary Ann’s practice with more open-ended questioning that remained sensitive to her ZPD, and by pursuing aparticular teaching focus that derived from her own classroom experiences,the supervisor was able to support Mary Ann’s development. Weaver andStanulis (1996) argue from a sociocultural perspective that mentors should

provide opportunities for a student teacher to drive lessons, shift instructional strategies,and alter content; . . . [they should] encourage a student teacher to engage in dialogue aboutteaching practice; . . . and [they should] work hard to elicit a student teacher’s understandinginstead of relying on [their] own understanding (p. 28).

As we reflect on Mary Ann’s case, these words further capture our intentand our practice in educative supervision.

We note that in exploring a pedagogy for the teaching episodes withMary Ann, it seems that our findings were less about content thanpedagogy. Shulman (1987) defined pedagogical content knowledge as “theblending of content and pedagogy into an understanding of how particulartopics, problems, or issues are organized, represented, and adapted to thediverse interests and abilities of learners, and presented for instruction”(p. 8). In Mary Ann’s case, we see this knowledge domain as includingher understanding of how to take a mathematical problem or task andorchestrate a class discussion in a way that promoted the activities of argu-mentation, conjecture, and student justification. In linking mathematicsand pedagogy, the supervisor tried to mediate what Mary Ann saw as thestep-by-step nature of mathematics and teaching mathematics to an under-standing of teaching mathematics that included those processes originallymissing from her repertoire of classroom discourse. In general, it could bethat the novelty of the classroom setting for student teachers forces moreof the attention of the supervisor and student teacher on the pedagogicalside of this knowledge domain rather than the mathematical side. In MaryAnn’s case, content alone did not become a critical point of discussion. Herteacher education program included a heavy emphasis on mathematics,with courses in discrete mathematics, statistics, the development of proof,


geometry, and calculus (two semesters). Instead, the focus was on how tohandle that content in class discussions.

As we have described, our approach to supervision (a) supports a focusfor supervision related to the student teacher’s developing practice; (b)emerges from the student teacher’s conflicts in practice; (c) provides forsuccessive transformations of a concept through multiple interactions ona topic; (d) allows the student teacher ownership of solutions; and (e)encourages the student teacher’s risk-taking in his or her practice. In a morecritical treatment of our approach, we characterize here what might be seenas its more problematic aspects. First, identifying the focus for intentionalinstruction during supervision should perhaps have been an explicit partof the supervisor’s conversations with Mary Ann. Although this was notan intentional oversight by the supervisor (especially given that the prac-tice of identifying a thematic focus itself emerged through supervision), itseems that the student teacher would have a deeper sense of ownership inchange if he or she had a greater role in deciding about a focus for instruc-tion. Further, knowing how to pursue that focus in terms of the timing ofinterventions and the structure of the teaching episodes was a continualdilemma. It was not always clear when to direct Mary Ann’s thinking andin what way that should occur. More research is needed to understand thetrajectories of learning that characterize a student teacher’s development inthe context of university supervision. We also recognize that more researchis needed to determine if change in a student teacher’s practice can begenerative and self-sustaining. That is, would change continue beyond thestudent-teaching semester?

From the supervisor’s experience, shifting the focus to listening toMary Ann was much more challenging and time-consuming than a typicalsummative evaluation of her practice would have been. Moreover, itwas difficult to accept the tenuous nature of her development withoutcrowding it. It struck us that there first needs to be a serious commit-ment by the supervisor to relinquish an authoritative role. In essence,this requires a paradigm shift that is no less important or difficult thanwhat we ask of classroom mathematics teachers in shifting from tradi-tional forms of teaching mathematics to reform-minded practices. As wefound, part of this shift will require the use of open-ended questioning.Although we have included the open-ended questions that evolved throughthe teaching episodes (see Table 1), more research is needed on the natureof questioning associated with educative supervision.

Although the supervisor met with Mary Ann on a regular basis, therewas still much occurring in Mary Ann’s practice during the interim periodswhich made it difficult to establish a link with previous lessons that could


be used to move Mary Ann’s practice forward. Nevertheless, the weeklyintensity and focus of the supervisor’s interactions with Mary Ann and theparticular structure of a three-hour visit – a classroom observation beforeand after the teaching episode – did provide a sense of continuity that wespeculate does not occur through random visits by a supervisor. Moreover,the supervisor inferred through informal exchanges with Mary Ann thatthe supervisor’s presence in her classroom as she implemented instructionplanned during the teaching episodes encouraged her efforts. Because timeconstraints imposed on supervision make such a process arguably quixotic,we question if a professional semester is an optimal time frame in whichto effect long-term change in a student teacher’s practice. We concur withthe growing belief that student teachers are better served by teaching fewerclasses over a longer period of time. This would not only structure time forreflection, but it might address the difficulties supervisors face when theytry to impact the practices of numerous student teachers in a brief periodof time.

In retrospect, the dual nature of the role of the supervisor/researcher inthis study did affect the dynamic of the supervisory relationship. In partic-ular, our dependency on Mary Ann’s participation caused the supervisorto be less authoritative (and thus to experience the value of this). Addi-tionally, the inquisitive nature of research induced a more probing stancethan the supervisor might otherwise have had. Ultimately, the approachhere required the supervisor to draw simultaneously on multiple roles asa researcher, interviewer, and teacher. In general, supervisors might notbe prepared to do this. Thus, we must make decisions as a professionalcommunity as to what role we expect supervision to play in a studentteacher’s development and how we can build the structure that will providethis.

Certainly, factors other than university supervision contribute to astudent teacher’s professional growth. Moreover, these factors might limit,or even negate, the influence of the supervisor. Understanding how theycoalesce in the making of a teacher is at best a delicate process. As such,this investigation reflects a first attempt to explore that process through thesupervisor’s lens.

NOTES

1 The study is based on the dissertation research of the first author under the direction ofSarah B. Berenson and Karen S. Norwood.2 Numbers indicate paragraphs in the protocol.3 Blanton, Berenson and Norwood (2001) provide a more exhaustive analysis of the natureof discourse in Mary Ann’s classroom.


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Maria L. BlantonDepartment of Mathematics,University of Massachusetts Dartmouth,285 Old Westport Road,North Dartmouth, MA 02747-2300USAE-mail: [email protected]

Sarah B. BerensonCenter for Research in Mathematics and Science Education,North Carolina State University

Karen S. NorwoodMathematics, Science, and Technology Education Department,North Carolina State University

KAYE STACEY, SUE HELME, VICKI STEINLE, ANNETTE BATURO,KATHRYN IRWIN and JACK BANA

PRESERVICE TEACHERS’ KNOWLEDGE OF DIFFICULTIESIN DECIMAL NUMERATION

ABSTRACT. In this study we investigated preservice elementary school teachers’ contentknowledge and pedagogical content knowledge of decimal numeration. The preserviceteachers completed a decimal comparison test, marked items they thought would bedifficult for students, and explained why. Only about 80 percent of the sample tested asexperts, indicating that a significant proportion of preservice teachers have inadequatecontent knowledge of decimals. Confusion about the size of decimals in relation to zerowas a significant and unexpected difficulty, leading to concerns about the fragmentarynature of the preservice teachers’ knowledge. Most preservice teachers were aware oflonger-is-larger misconceptions in students, but had little awareness of shorter-is-largermisconceptions. Preservice teachers’ explanations for the reasons students might havedifficulty demonstrated that many are good at identifying features that make comparisonsdifficult but less able to explain why these cause trouble. Results point to the needfor teacher education to emphasise content knowledge that integrates different aspectsof number knowledge, and pedagogical content knowledge that includes a thoroughunderstanding of common difficulties.

In this study we investigated preservice elementary school teachers’content knowledge and pedagogical content knowledge of decimal numer-ation. We chose to focus on decimal numeration because it has beenrecognised for some time to be a significant source of learning and teachingdifficulties, and there is evidence that teachers themselves have difficultieswith decimals.

KNOWLEDGE FOR TEACHING MATHEMATICS

There is some variation in the way various components of teacher know-ledge are described and delineated, but most researchers agree that contentknowledge and an understanding of students’ learning and thinking arecritical aspects (Brophy, 1991; Fennema & Franke, 1992). Shulman (1986)highlighted the importance of two aspects of pedagogical knowledge heconsidered necessary for effective teaching, and which formed the basisof the current investigation: content knowledge and pedagogical contentknowledge. By content knowledge Shulman meant expertise in the partic-


206 KAYE STACEY ET AL.

ular content or discipline to be taught. He conceived pedagogical contentknowledge in terms of two interrelated components which comprise “theways of representing and formulating the subject that make it comprehen-sible to others . . . (and) an understanding of what makes the learning ofspecific topics easy or difficult: the conceptions and preconceptions thatstudents of different ages and backgrounds bring with them” (Shulman,1986, p. 9). The present article is concerned with content knowledge andthe second aspect of pedagogical content knowledge.

Content Knowledge

The vital role of teachers’ content knowledge in mathematics learninghas been highlighted in Ma’s recently published comparative study ofChinese and U.S. elementary teachers (Ma, 1999). She argued thateffective teaching is based on a “profound understanding of fundamentalmathematics” (p. xxiv) which comprises a thorough and well-connectedunderstanding of elementary mathematics and includes understanding howeach topic fits into the overall conceptual structure of the discipline, aswell as the relationship between elementary topics and more conceptuallyadvanced mathematical ideas. This knowledge enables teachers to drawfrom a wide variety of linked concept areas – or knowledge packages – tohelp students understand new ideas.

Ma’s study contributed to an already large body of research thatindicates that many preservice and practising teachers have impover-ished understanding of many of the mathematics concepts and processesthat they will be required to teach. Systemic reports claim that teachereducation courses are not successfully redressing the problem and revealthat it is long-standing and ongoing (Cockroft, 1982; Willis, 1990). Inthe US, for example, the NCTM Professional Standards for TeachingMathematics (1991) called for major shifts in teacher education andprofessional development, noting that teachers “must have access . . . toeducational opportunities that focus on developing a deep knowledge ofsubject matter, pedagogy and students” (p. 177). In their review of mathe-matics and science teacher education in Australia, Speedy, Annice andFensham (1989) argued that preservice education should better “enablestudents to understand the fundamental concepts which underlie the majorstrands of the mathematics curriculum K-10 and thus recognise the link-ages between early mathematical ideas and higher order mathematicalknowledge” (p. 23).

PRESERVICE TEACHERS’ KNOWLEDGE OF DECIMALS 207

Pedagogical Content Knowledge

As Shulman (1986) noted, knowledge of students’ conceptions andmisconceptions is an essential component of pedagogical content know-ledge. A number of recent studies have, however, found shortcomings inteachers’ understanding of students’ thinking and problem solving. In asmall but fine-grained study of students’ ideas about mathematical func-tions, Even and Markovitz (1993) found teachers’ awareness of students’conceptions to be inadequate, and their responses to students’ answerseither too general or giving too much emphasis to procedure at theexpense of meanings. Leu (1999) studied teachers’ understandings aboutstudent cognition in fractions, and found a disturbing mismatch betweenteachers’ judgments of students’ ability and students’ actual performancein solving certain kinds of fraction problems. The source of the mismatchappeared to be teachers’ underestimation of the extent to which studentsuse intuitive rather than rule-based or analytical thinking. In a study ofalgebraic reasoning, Nathan and Koedinger (2000) found that students’problem-solving strategies differed from those predicted by teachers andresearchers. These studies highlight the need for teaching that is based onaccurate knowledge of students’ difficulties and ways of thinking.

The Present Study: Teachers’ Knowledge of Decimal Numeration

We chose to focus on decimal numeration for a number of reasons. First,decimal numeration has been recognised for some time to be a significantsource of learning and teaching difficulties. Bell, Swan and Taylor (1981)noted that many pupils lacked an understanding of decimal place value;for example, a common misconception was that students thought that 0.8was equal or somehow analogous to an eighth. This is one of the formsof thinking underlying shorter-is-larger misconceptions, discussed later inthis report. There is now a detailed body of knowledge concerning a varietyof decimal misconceptions and the thinking that underpins these (Baturo,1998; Hiebert & Wearne, 1986; Resnick et al., 1989; Sackur-Grisvard &Leonard 1985; Stacey & Steinle, 1998).

Second, there is evidence to suggest that teachers themselves havedifficulties with decimals. Recent studies of preservice elementary schoolteachers’ understanding of decimals (Putt, 1995; Thipkong & Davis, 1991)identified significant difficulties in interpreting decimals and solving wordproblems involving decimals. Putt (1995) found that 52% of his samplewere unable to place the numbers 0.606, 0.0666, 0.6, 0.66 and 0.060 incorrect order from smallest to largest. Most of the incorrect responses(74%) involved the selection of 0.6 as the largest number, indicating


that most of these preservice teachers held one or more of the cluster ofthinking patterns associated with shorter-is-larger misconceptions.

Third, we are able to compare preservice teachers’ predictions ofstudents’ difficulties with existing data on students’ actual difficulties,as reported by Steinle and Stacey (1998) and thus gain some under-standing of preservice teachers’ pedagogical content knowledge in decimalnumeration.

The present study investigates preservice elementary school teachers’content knowledge and pedagogical content knowledge of decimals andis framed by four research questions. Questions 1 and 2 concern contentknowledge, whereas Questions 3 and 4 pertain to pedagogical contentknowledge:

1. How much do preservice teachers know about decimal numeration?2. To what extent are preservice teachers aware of their own difficulties

in decimal numeration?3. What do preservice teachers think makes decimal comparisons diffi-

cult for students?4. What are the characteristics of preservice teachers’ explanations of

students’ likely difficulties?

METHODOLOGY

Participants and Procedure

The sample for the study consisted of 553 preservice elementary schoolteachers from four universities in Australia and New Zealand (labelled A,B, C and D). No two universities were in the same educational jurisdic-tion. The sample came from all stages of preservice teacher education,and also included 25 practising teachers enrolled in a course to improvetheir initial qualifications. These 25 teachers were elementary educationcollege students (as were the rest of the sample), but are not strictly preser-vice teachers, because they had some experience in the field. However, inthis paper, we have used the term preservice teachers to clearly differen-tiate our adult sample from elementary school students. All participantscompleted a Decimal Comparison Test (DCT) where the task was to circlethe larger number from pairs of decimal numbers. This test was used asa quick and reliable indicator of understanding decimal numeration and isdescribed below. On completion of the test, preservice teachers at univer-sities A, B, and C were asked to asterisk the comparison items that were


likely to be difficult for students and to explain in writing on the backof the test paper where the difficulty may lie. We used this procedure fortwo reasons; first, to investigate preservice teachers’ ideas about the diffi-culty of the domain for students and second, as a methodological deviceto identify whether preservice teachers recognised where their own under-standings were wrong or incomplete. We assumed that preservice teacherswho were unsure about a question were likely to identify it as somethingthat students may find difficult.

Volunteer teaching staff from universities A, B, C and D collectedthe data. Amongst the large number of separate groups involved, therewere two variations to the testing procedure. At University D, preserviceteachers did not complete the asterisk task. They are therefore includedonly in the first part of this study to give an indication of the vari-ation of preservice teacher’s thinking between universities. At anotheruniversity, one group completed the DCT and then did the asterisktask in the next mathematics session. We judged that this variation inthe data collection procedure would have no effect on the results andso this group was fully included. The classes sampled were, as far aspossible, complete course groups, except for absentees on the day(s) oftesting. However, not all enrolled preservice teachers at each universitywere sampled. For example, University A tested only first-year preser-vice teachers from its four-year preservice undergraduate program forelementary teachers. There was an attempt to investigate separately theresults of preservice teachers according to the stage of their course, butthis was not practical given the wide variety of teacher training arrange-ments. However, there appeared to be no clear improvement in the resultsas preservice teachers progressed through their courses, probably due tothe different amount and placement of mathematics education generally,and decimals specifically, in the teacher education courses. Some preser-vice teachers would have touched briefly on decimal misconceptions butthis would not have been a major focus of the teaching in any setting.Some groups had studied decimal numeration and base ten place-valueproperties recently, and some groups had studied it in previous years,whereas others had had no prior instruction in their teacher educationcourse. The intention of this paper is not to relate performance to instruc-tion but to show a general picture of preservice teachers’ interpretations ofdecimals.


Type Number of Example Brief description

items

1 5 0.75/0.8 Unequal length. The larger decimalis shorter.

2 5 0.426/0.3 Unequal length. The larger decimalis longer.

3 4 3.72/3.073 A zero is in the tenths column of onenumber, which would otherwise belarger.

4 4 8.245/8.24563 One decimal is a truncation of theother.

5/6 6 0.3/0.4 Equal length decimals

7 3 0/0.6 A comparison of a positive decimalwith zero.

Figure 1. Types of decimal comparison items.

The Decimal Comparison Test (DCT)

The DCT has been developed from earlier research instruments as aquick and reliable method of identifying how a student is thinking aboutdecimals. A description and results on a sample of 2517 students fromgrades 5 to 10 are reported in Steinle and Stacey (1998). In addition tosome warm-up items, the DCT consists of 27 decimal comparison items,in seven carefully matched types. Items within Types 1 to 6 have beenempirically tested to perform homogeneously (Stacey & Steinle, 1998).For the purposes of this report two types (5 and 6) were amalgamated, asthere were no performance differences in the preservice teacher sample.Figure 1 shows the number of items of each type in the test and gives abrief description. No decimal numbers contained in the test ended in zero,and all numbers in the test and referred to in this paper are non-negativeunless specified.

Patterns of consistent errors on the test can be used to classify in detailthe way in which a person thinks about decimal numbers. For the purposeof this paper the most important pattern was related to what are calledshorter-is-larger misconceptions. Following Stacey and Steinle (1998),three different forms of thinking can be identified in this group of miscon-ceptions, according to scores (high or low) on different comparison types(see Figure 2). All those with shorter-is-larger misconceptions tend toscore high on Types 1 and 3 (where the shorter is indeed larger) and score


Underlying thinking Score on comparison type

1 2 3 4 5/6 7

Reciprocal high low high low low high

Negative high low high low low low

Denominator focussed high low high low high high

Figure 2. Identifying the thinking underlying shorter-is-larger misconceptions.

low on Types 2 and 4 where the shorter is not larger. Scores on Types 5, 6,and 7 suggest different reasons for shorter-is-larger misconceptions.

Reciprocal thinkers make consistent errors on Type 5/6 comparisons.They associate decimals with reciprocals so that, for example, 0.3 isthought to be bigger than 0.4 by analogy with the fact that 1

3 is largerthan 1

4 . Negative thinkers make consistent errors on Types 5/6 and 7. Theymay associate decimals with negative numbers, so that, for example, 0.6is perceived as smaller than 0, just as –6 is smaller than 0. Denominatorfocussed thinkers get Types 5, 6 and 7 correct. They use the place valuecolumn names to decide on the size of decimals. Thus they would judge0.3 to be bigger than 0.426 because they assume that any number of tenthsis greater than any number of thousandths.

The DCT also identifies other types of misconceptions that are notas relevant to this age group. The most common in young students is acluster of thinking patterns that underlie longer-is-larger misconceptions.Those with longer-is-larger misconceptions tend to score low on Type 1comparisons and high on Type 2 comparisons. Approximately six formsof thinking underlying this misconception have been identified. For details,see Steinle and Stacey (1998).

Data Analysis

Responses to test items were classified according to the way in whicheach student thought about decimals, following the procedures outlinedabove. Most of the subsequent analysis was undertaken at the level ofcomparison type, given that Types 1 to 6 had been previously shown tobe homogeneous. In order to explore whether the sample could be treatedas one in the data analysis, and also to inform future collaboration onteaching between the four universities, the data were examined for differ-ences between universities in the types of thinking and error patterns. Thetypes of items for which preservice teachers made errors were noted andthese were matched with the comparison types that preservice teachershad marked with an asterisk. Finally, preservice teachers’ explanations for


TABLE 1

Percentage of Specified Thinking About Decimals

University

A B C D Average

N = 185 N = 128 N = 154 N = 86 N = 553

Experts (%) 83 77 86 76 81

Shorter-is-larger misconceptions (%) 3 3 0 8 3

students’ difficulties were examined and classified according to the extentto which they demonstrated understanding of decimal numeration and thereasons for students’ difficulties.

RESULTS AND DISCUSSION

Preservice Teachers’ Thinking About Decimals

Classification of thinking patterns. The thinking of preservice teacherswas classified according to the procedures outlined above. As shown inTable 1, the results for each university were generally similar. Most preser-vice teachers answered the items expertly: from 76 to 86% of preserviceteachers at each university scored high on all seven comparison types,leading us to classify them as experts.

There were almost no preservice teachers who held any of the clustersof thinking patterns that lead to consistent choice of longer decimals aslarger (only 3 of 553 preservice teachers fit this pattern). This is consistentwith the trends identified by Steinle and Stacey (1998) who found thatlonger-is-larger misconceptions decline with age (from 32% in Grade 5 to5% in Grade 10), and probably disappear in educated adults.

On average, 3% of the preservice teachers held one of the severalbeliefs that led them to choose shorter decimals as the larger ones. Thecriteria are stringent: in order to be classified in this way they must haveconformed strictly to the error patterns outlined in Figure 2. Althoughpreservice teachers in this study performed much better than those in Putt’s(1995) study, both studies identified shorter-is-larger as the predominantmisconception. These findings extend the trends identified by Steinle andStacey (1998), who found that shorter-is-larger misconceptions persisted


TABLE 2

Number of Preservice Teachers Who Made at Least One Error on Items ofSpecified Type

Type Number of Example Percent by university

items A B C D Average

1 5 0.75/0.8 6 5 5 12 6

2 5 0.426/0.3 6 6 5 22 8

3 4 3.72/3.073 3 2 0 5 2

4 4 8.245/8.24563 6 6 5 10 7

5/6 6 0.3/0.4 10 9 3 19 9

7 3 0/0.6 15 21 6 9 13

throughout junior secondary school (about 7% at Grade 10) and predictedthat they are likely to continue into adulthood.

Errors by comparison type. Table 2 reports, for each comparison type, thepercentage of preservice teachers from each university who made at leastone error on that type. Note that a student who made more than one erroron a type is counted only once.

The average error rates on different comparison types were generallysimilar, with the exception of Types 3 and 7. Type 3 had the lowest errorrate of 2 percent, whereas Type 7 had the highest error rate of 13 percent,an unexpected result. The very low error rate for Type 3 items is consistentwith the results of Sackur-Grisvard and Leonard (1985) who found that azero in the tenths place marks the decimal as “small”.

The three Type 7 items (0/0.6, 0.22/0 and 0.00/0.134) all involved thecomparison of zero with a decimal between zero and one. The error ratewas markedly lower on the third comparison, probably because the extrazeros in this item triggered a digit-by digit comparison strategy. Also, somepreservice teachers commented that 0 was a whole number, whereas 0.00was not.

Differences among universities. In the classification of thinking patterns,some differences were observed among universities. Table 1 shows thatthere were variations in the frequency of shorter-is-larger misconcep-tions from 0 percent to 8 percent. University D, which had the highestfrequency of shorter-is-larger misconceptions, also appeared to have signi-


ficantly higher error rates in all comparison types, especially Types 2 and5/6 (see Table 2). Errors on these comparison types are consistent withthe relatively high proportion of shorter-is longer misconceptions at thisuniversity. Because these preservice teachers did not complete the aster-isking task we were unable to examine the possible reasons for this. Errorson Type 7 comparisons were especially prevalent at University B. Presentlywe cannot explain these differences; the differences may be due to differentteaching at the four universities, different experiences prior to university –the four universities draw preservice teachers from different geographicalareas and educational jurisdictions – or differing mathematics abilities ofthe preservice teachers in each course.

Preservice Teachers’ Awareness of Difficulties in Decimal Numeration

Do preservice teachers identify their own errors as difficulties for students?It seems reasonable to assume that items that left a preservice teacherpuzzled would be recognised as potentially puzzling for students and henceasterisked. The association between errors and asterisks was thus used asan indicator of the degree of confidence preservice teachers had in theiranswers. It is also hoped that preservice teachers who recognise their owndifficulties may take extra steps to clarify the issues involved when plan-ning their teaching. For these reasons, a high correspondence between thetypes of errors made and the placement of asterisks would be desirable.

The data for this question consisted of the test papers of the 78 preser-vice teachers at Universities A, B and C who made at least one error andwho also asterisked at least one item. (Note that preservice teachers whomade errors on more than one comparison type were counted more thanonce.) These papers were examined for correspondence between errors andasterisked items. Because items were homogeneous within each compar-ison type (except for Type 7 as noted above), the analysis in this sectionwas undertaken at the level of comparison type. Thus if a student madean error and also asterisked an item within the same comparison type, thiswas considered to be a matched error. If an error was made, and no itemswere asterisked within the same comparison type, this was considered tobe an unmatched error.

The results are shown in Table 2 and can be interpreted as follows.For the 20 preservice teachers who made errors on one or more Type 2comparisons, 12 preservice teachers marked at least one Type 2 compar-ison with an asterisk. Thus 60 percent of Type 2 errors were matched. Thedata in Table 3 indicate that particular comparison types were not randomlyselected for an asterisk by preservice teachers; Type 5/6 items (the least


TABLE 3

Number and Percentage of Matched and Unmatched Errors

Comparison type

1 2 3 4 5/6 7 Total

Number of matched errors 4 12 7 17 5 36 81

(44%) (60%) (78%) (65%) (24%) (69%) (57%)

Number of unmatched errors 9 8 2 9 16 16 60

Total 13 20 9 26 21 52 141

frequently asterisked) were the most frequently occurring items on the testwhereas Type 7 items, the least frequent on the test, were very frequentlyasterisked by students who made errors.

Overall, 57% of errors were matched by asterisks at the level ofcomparison type, which suggests that preservice teachers have moderateawareness of their own difficulties with decimals. The proportion ofmatched errors varied considerably across comparison types, from 24%(Type 5/6, the equal length decimals) to 78 percent (Type 3, zero in thetenths place). Many comments made by preservice teachers with errorsconfirmed their awareness of their difficulties:

The ones with zeros are a bit difficult to work out. Is a fraction bigger or smaller than zero?

In some cases comments shed light on preservice teachers’ own (unre-cognised) misconceptions. This was particularly evident for Type 7comparisons:

Comparing 0 with a decimal may also lead to confusion as to which is the smallest, as 0represents ‘nothing’ to young students and it may be difficult for them to understand thatsomething can be ‘smaller’ than this.May be confused as to whether 0 is larger than 0.134 which has more numbers but is infact smaller.The concept that 0 is larger than 0.22 may be difficult to recognise.

None of the comments by preservice teachers with Type 2 and 4 errors(associated with shorter-is-larger misconceptions) showed that they wereaware of this issue. Instead they were more likely to attribute longer-is-larger misconceptions to students, which are in fact unrelated to thesecomparison types:


The longer the number the bigger the students think it is.Students will have difficulty working in the opposite direction to whole numbers (i.e., themore numbers added to the right of the decimal, the smaller the number is broken into).

These comments demonstrate the usefulness of this technique for identi-fying preservice teachers’ confusions and provide a useful starting pointfor addressing their difficulties with decimal numeration.

TABLE 4

Percentage of Preservice Teachers Who Placed an Asterisk on at Least One Item perComparison Type

Type Number of Example Percent by university

items A B C Average

N = 185 N = 128 N = 154 N = 467

1 5 0.75/0.8 53% 32% 10% 33%

2 5 0.426/0.3 22% 9% 3% 12%

3 4 3.72/3.073 79% 42% 35% 54%

4 4 8.245/8.24563 53% 30% 34% 41%

5/6 6 0.3/0.4 14% 8% 1% 8%

7 3 0/0.6 54% 46% 10% 37%

Preservice Teachers’ Thinking about Students’ Difficulties with DecimalComparisons

In this section, the explanations given by preservice teachers at Univer-sities A, B, and C about students’ difficulties are examined. Table 4reports for each comparison type, the percentage of preservice teacherswho placed an asterisk on at least one item, regardless of whether thepreservice teacher made errors. If a preservice teacher placed an asterisk atmore than one item per type, this was counted only once. When they gavetwo different explanations for different items, both of them were takeninto account. Once again, the data in Table 4 do not simply replicate therelative proportion of items by type in the test, which would have occurredif preservice teachers had chosen items for asterisks at random.

Although the numbers of asterisks differed across universities, theranking of frequencies was very similar for each university. Table 4 showsthat Type 3 items received the most asterisks, followed by Types 4, 7 and 1,then Types 2 and 5/6. The preservice teachers at University C gave fewer


asterisks than the others, but these were still similarly distributed acrosscomparison types. The data in Tables 2 and 4 show that, for UniversityB, a high error rate for Type 7 comparisons was associated with a highnumber of asterisks, suggesting that preservice teachers at this universitywere aware of their confusion with respect to these comparisons. On theother hand, the relatively smaller number of asterisks given to Type 2 andType 5/6 comparisons at all universities suggests that preservice teachershad little awareness of shorter-is-larger misconceptions.

From the data in Table 3, it can be seen that the percentage of matchederrors (57% on average) was generally higher than the percentage ofpreservice teachers who placed an asterisk at least one item in a compar-ison type, as shown in Table 4. This indicates that preservice teachers whomade errors were also more aware of difficulties with decimal comparisonsthan was the sample as a whole.

Preservice teachers commonly identified four features of the numbersused in the comparisons which make them difficult for students: length,comparison with zero, presence of a zero digit, and similarity. Thesefeatures are discussed below, and representative comments are shown inTable 5.

TABLE 5

What Makes Decimal Comparisons Difficult?

Feature Representative comments

Length 1. I believe children may have trouble with some of thenumbers with 4 or 5 or more decimals as they areprobably not numbers which make a lot of sense to them.

2. Children tend to think that the more numbers there are,the larger the amount must be. Therefore children maychoose the number which has the most numbers in it,regardless of whether it is actually bigger in value.

3. A single decimal number would be considered larger thanseveral decimals e.g., 0.4 > 0.416.

Comparison with zero 4. Children are taught that the ones column is larger thanthe tenths column so assume 0 is bigger than a decimal.

5. 0.22 may be mistaken for a negative number below zero.

Presence of zero digit 6. They might disregard the zero and think 73 is larger than72 (3.72/3.073).

Similarity 7. These ones I think begin to become more difficult asthey start with all the same numbers and only a few aredifferent or extra at the end . . . Children have to look verycarefully at this.


Length. Preservice teachers commented on two aspects of length whichthey believed made decimal comparisons difficult, namely, long decimalnumbers and decimals of unequal length. Types 3 and 4 (the mostfrequently asterisked) contained the longest decimals as well as decimalsof unequal length. The majority of comments about decimals of unequallength described the misconception that a longer decimal is always largerthan a shorter decimal. It was rare to find comments that students mightmistakenly believe that the shorter decimal is larger. This is consistent withthe data in Table 4, which show that Type 2 items were asterisked by only12 percent of the sample, whereas Type 1 comparisons were asteriskedby 33 percent. One of the few such comments is reproduced in Table 5(Comment 3), and may refer to denominator-focussed thinking.

Comparison with zero. The high proportion of preservice teachers whoasterisked Type 7 comparisons (37%) and the considerable number ofcomments about these comparisons indicate that many preservice teachersconsidered them difficult, for themselves as well as for students. This wasparticularly the case for students from University B. Comments focussedon preservice teachers’ own difficulties, as well as attempted to explainstudents’ difficulties. Explanations included the possibility that studentsmight think that zero is bigger than a decimal number, either becauseit could be classified as a whole number (Comment 4), or that studentsmight think that decimals are negative (Comment 5). Students’ difficultiesmaking comparisons with zero have been documented elsewhere. Irwin(1996) reported that some 11- to 13-year-old students placed decimalnumbers starting with zero (e.g., 0.5 and 0.1) to the left of zero on a numberline. Irwin concluded that these attempts at ordering were consistent witha system that pivots around zero as equivalent to the decimal point,rather than pivoting around one. Some preservice teachers in the presentstudy who experienced difficulty with Type 7 items were interviewed inorder to explore the issue further. Preservice teachers who appeared tobe confusing decimals with negative numbers may have been doing sobecause decimals and negatives are both “opposites” of positive integers.We plan to investigate these findings further.

Presence of zero digit. Many preservice teachers noted that a zero in thetenths column of one of the two numbers made the comparison more diffi-cult, but without explaining why (Comment 6). Type 3 comparisons, themost frequently asterisked, contained a zero in the tenths column of onenumber, which would otherwise have been larger (e.g. 4.08/4.7). However,as discussed above, many students do understand that a zero in the first


column after the decimal point makes a number smaller, which makesType 3 items easier than the otherwise parallel Type 1 items. The lowerror rate for Type 3 comparisons (see Table 2) indicates that these itemsare easier for adults too. None of the comments however indicated aware-ness of students recognising that “zero makes small”. The large numberof asterisks given to Type 3 items is most likely because they containedthe longest decimals. Preservice teachers’ knowledge of other problemswith zero, such as comparisons involving decimals ending with zero (notrepresented on the test, e.g. 0.3/0.30), may also have contributed to thelarge number of asterisks for Type 3 comparisons.

Similarity. Similar decimal numbers were identified as a major source ofconfusion and difficulty, given the large number of comments about thedifficulty of comparing numbers that differed in the third or fourth decimalplace only. Preservice teachers’ comments expressed the view that studentsmight not know the effect of an extra digit in the third or fourth place onthe size of a number as in Type 4. Given the high number of asterisks forcomparison Types 1 and 3, and the high number of comments about longer-is-larger misconceptions, we can conclude that preservice teachers are wellaware of these misconceptions in students. On the other hand, there islittle awareness of shorter-is-larger misconceptions, and of the thinkingunderlying these.

Characteristics of Preservice Teachers’ Explanations

As discussed above, preservice teachers had been asked to asterisk compar-ison items that students might find difficult and explain why this may bethe case. Their comments ranged from non-engagement with mathematicalcontent and the reasons for students’ difficulties to thoughtful explanationsthat provided evidence of thorough and well-connected content knowledgeand pedagogical content knowledge. To illustrate these differences, wegrouped comments into three broad categories: non-engagement, surfaceengagement, and deep engagement. Note that the categories may overlap;we are more concerned with illustrating differences in degree of engage-ment with content rather than defining mutually exclusive categories ofresponse. Representative examples of explanations are shown in Table 6.Numbers in brackets refer to the comments in Table 6.

Non-engagement. These comments did not refer to or explain anythingabout the features of the decimals in the comparison task. Some preser-vice teachers only confirmed the item’s difficulty for students (4) or for


TABLE 6

Range of Explanations for Students’ Likely Difficulties With Decimals

Non-engagement 1. Because I was confused.2. Because I had trouble with zero myself in this exercise.3. Children might find this one interesting to discuss.

(4.4502/4.45)4. Children find it difficult to compare numbers with

different amounts of decimal places.5. Large amounts of numbers.

Surface engagement 6. Have to look a long way down the line to work out whichone is bigger.

7. These are virtually the same number, just a bit more onthe end of the number could be confusing.

8. A single decimal number would be considered larger thanseveral decimals e.g. 0.4 > 0.416.

9. 0.6 looks smaller than 0.10. I believe children may have trouble with some of the

numbers with 4 or 5 or more decimals as they areprobably not numbers which make a lot of sense to them.

11. (Children) would need to have good understanding ofplace value to realise that physically bigger N is notactually a larger number (8.052573/8.514).

12. I think that children would think that more numbersautomatically means larger.

13. Most of the ones I have asterisked are ones wherethe number following the decimal point is larger thanits partner’s . . . I think children would automaticallypresume that 0.36 is larger than 0.5 even though theopposite is true.

Deep engagement 14. These might be difficult because the larger number isshorter. With whole numbers, the number which is longeris larger.

15. Children are taught that the ones column is larger thanthe tenths column so assume 0 is bigger than a decimal.

16. When students see 0.3 they may think 13 but it is actually

310 so they may confuse 0.3 thinking it is bigger than 0.4.

17. Some children . . . might assume that because 3 is closerto zero, thus closer to a whole number, 0.3 would bebigger than 0.4.

18. When you see a number with a digit in more columns,you know those columns represent smaller parts themore they go to the right. Therefore you’re inclined tothink it might be a smaller number, but that’s not thecase.(original underlined)


themselves (1, 2). Others apparently adopted the role of teacher whilstfailing to engage with the content (3). Responses of this type suggest thatmany preservice teachers have not yet given much thought to decimalnumeration and the difficulties it presents for students.

Surface engagement with content. Responses in this category identifiedsomething about the surface features of the problems that related to theirdifficulty, such as unequal length decimals (8, 11, 12), long strings ofdecimals (8, 10), similarity (6, 7), and comparison with zero (9), withoutgiving an explanation of why this feature may be influential, or how thechild might be thinking.

Comments that included predictions of students’ behaviour (10), orassumptions they might make (12, 13), did so without elaborating on whythis might occur. For example, comment number 10 mentioned difficultiesthat students might have reading and making sense of long strings ofdecimal digits, but did not explain why this would be the case.

Deep engagement with content. These comments explained why certainfeatures made the comparisons difficult, usually by pointing to the falseanalogies that are at the heart of many decimal misconceptions. This wasparticularly evident in preservice teachers’ explanations of longer-is-largermisconceptions, demonstrating that, at least for some preservice teachers,the reasons for this are well understood. They recognised, for example, thatthe analogy with whole numbers might cause students to think that longerdecimals represent larger numbers (14). As discussed above, commentsrecognising that other analogies lead students to think that longer decimalsare smaller were rare but are included here to illustrate some of themore insightful comments (16, 17, 18). There were many comments inwhich preservice teachers attempted to explain why students might havedifficulty with comparisons with zero. As discussed earlier, explanationscanvassed two possibilities, (a) that students may think zero is bigger thana decimal due to its position in the ones column (15) and (b) students’thinking decimals are associated with negative numbers. One of the fewexplanations for students’ predicted difficulties with equal length decimalcomparisons also invoked the idea that students may associate decimalswith negative numbers (17).


CONCLUSION

In this study we explored the content knowledge and pedagogical contentknowledge of elementary school preservice teachers. The findings aresummarised below, followed by recommendations for teacher education.

Content Knowledge

We examined preservice teachers’ understanding of decimal numeration,and their awareness of their own difficulties in this domain. We rated only80 percent of preservice teachers as experts in decimal numeration. Twomajor difficulties were identified. First, we found that an unexpectedlyhigh proportion of preservice teachers (13%) made errors when comparinga decimal number with zero. Second, a small proportion of preserviceteachers showed evidence of shorter-is-larger misconceptions, consistentwith expectations for the general population, but clearly not a goodresult for teachers. Taken together, these errors indicate that many preser-vice teachers do not understand the relationships among decimals, wholenumbers, fractions, zero, and negative numbers. Only 57% of the preser-vice teachers reported that students might have difficulty with the items ofthe type that they got wrong themselves, indicating that quite a sizeableproportion of preservice elementary school teachers may not suspect theyare making errors. This was particularly the case for preservice teacherswith shorter-is-larger errors.

Pedagogical Content Knowledge

In view of Shulman’s (1986) call for teachers to understand students’conceptions and preconceptions, we examined preservice teachers’ know-ledge of students’ difficulties, and the nature of their explanations for these.Preservice teachers demonstrated a high degree of awareness of longer-is-larger misconceptions, but there was little evidence that preservice teacherswere aware of shorter-is-larger misconceptions. Of concern to us was thetendency of some preservice teachers to attribute longer-is-larger miscon-ceptions to students whilst unknowingly making shorter-is-larger errorsthemselves. Preservice teachers’ explanations for the reasons studentsmight have difficulty demonstrated that they are good at identifyingfeatures that make comparisons difficult but less able to explain why.


Implications for Teacher Education

The results discussed above indicate that at least one in five preserviceteachers in this study do not have a well-integrated knowledge of decimalnumeration, with the associated risk of transferring their own misconcep-tions to students. Indirect evidence of this risk is given by the substantialvariation in expertise between classes of students in Grades 5 to 10reported in Steinle and Stacey (1998) and the clustering of misconceptionsin classes with particular teachers. The percentage of experts in Grade 6classes, for example, ranged from 0 percent to 82 percent, with a mean of52%.

Because of the great variations within and between universities inpreservice teachers’ prior knowledge of decimals, the study was not ableto link the results with teaching at particular universities. However, wewere able to identify areas of difficulty common to all universities (partic-ularly shorter-is-larger misconceptions and comparisons with zero). TheDCT provided a relatively quick and reliable means by which this couldbe done. Given that these misconceptions arise from poorly integratedknowledge of fundamental number concepts, a strong case can be madefor teacher education to strengthen its efforts in developing in preser-vice teachers a well-connected understanding of decimals, fractions, andintegers, including zero.

Given the scant pedagogical content knowledge of many preserviceteachers identified in this study, we recommend that teacher educationprograms give more attention to developing this aspect of teacher know-ledge. The DCT has been shown to be a useful tool for initiating discussionof students’ likely difficulties and the reasons underlying these. We recom-mend this methodology to teacher educators as a first step towards athorough understanding of students’ difficulties with decimals and thedevelopment of appropriate ways of addressing them.

REFERENCES

Baturo, A.R. (1998). Year 6 students’ cognitive structures and mechanisms for processingtenths and hundredths. Unpublished doctoral dissertation, Queensland University ofTechnology, Brisbane, Centre for Mathematics and Science Education.

Bell, A., Swan, M. & Taylor, M. (1981). Choice of operation in verbal problems withdecimal numbers. Educational Studies in Mathematics, 12(3), 399–420.

Brophy, J.E. (1991). Conclusion to advances in research on teaching: Teachers’ knowledgeof subject matter as it relates to their teaching practice. In J.E. Brophy (Ed.), Advances


in research on teaching; teachers’ subject matter knowledge and classroom instruction(Vol. 2, 347–362). Greenwich, CT: JAL Press.

Cockcroft, W.H. (1982). Mathematics counts. London: Her Majesty’s Stationery Office.Even, R. & Markovitz, Z. (1993). Teachers’ and students’ views on student reasoning. In

I. Hirabayashi, N. Nohda, K. Shigematsu & F-L. Lin (Eds.), Proceedings of the 17thconference of the International Group for the Psychology of Mathematics Education(Vol. 2, 81–88). Tsukuba, Japan: Program Committee.

Fennema, E. & Loef Franke, M. (1992). Teachers’ knowledge and its impact. In D.A.Grouws (Ed.), Handbook of research on mathematics teaching and learning (147–164).New York: Macmillan Publishing Company.

Hiebert, J. & Wearne, D. (1986). Procedures over concepts: The acquisition of decimalnumber knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The caseof mathematics (199–223). Hillsdale, NJ: Lawrence Erlbaum.

Irwin, K. (1996). Making sense of decimals. In J. Mulligan & M. Mitchelmore (Eds.), Chil-dren’s number learning: A research monograph of MERGA/AAMT (243–257). Adelaide:AAMT.

Leu, Y.-C. (1999). Elementary school teachers’ understanding of knowledge of students’cognition in fractions. In O. Zaslavsky (Ed.), Proceedings of the 23rd conference ofthe international group for the psychology of mathematics education (Vol. 3, 225–232).Haifa, Israel: Program Committee.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understandingof fundamental mathematics in China and the United States. Mahwah, NJ: LawrenceErlbaum.

Nathan, M.J. & Koedinger, K.R. (2000). Teachers’ and researchers’ beliefs about thedevelopment of algebraic reasoning. Journal for Research in Mathematics Education,31, 168–190.

National Council of Teachers of Mathematics. (1991). Professional standards for teachingmathematics. Reston, VA: Author.

Putt, I.J. (1995). Preservice teachers ordering of decimal numbers: when more is smallerand less is larger! Focus on Learning Problems in Mathematics, 17(3), 1–15.

Resnick, L.B., Nesher, P., Leonard, F., Magone, M., Omanson, S. & Peled, I. (1989).Conceptual bases of arithmetic errors: The case of decimal fractions. Journal forResearch in Mathematics Education, 20, 8–27.

Sackur-Grisvard, C. & Leonard, F. (1985). Intermediate cognitive organization in theprocess of learning a mathematical concept: The order of positive decimal numbers.Cognition and Instruction, 2, 157–174.

Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching. EducationalResearcher, 15(2), 4–14.

Speedy, G., Annice, C. & Fensham, P. (1989). Discipline review of teacher educationin science and mathematics. Canberra: Department of Education, Employment andTraining.

Stacey, K. & Steinle, V. (1998). Refining the classification of students’ interpretations ofdecimal notation. Hiroshima Journal of Mathematics Education, 6, 49–69.

Steinle, V. & Stacey, K. (1998). The incidence of misconceptions of decimal notationamongst students in Grades 5 to 10. In C. Kanes, M. Goos & E. Warren (Eds.), Teachingmathematics in new times (548–555). Brisbane, Australia: Mathematics EducationResearch Group of Australasia.


Thipkong, S. & Davis, E. (1991). Preservice elementary teachers’ misconceptions ininterpreting and applying decimals. School Science and Mathematics, 91(3), 93–99.

Willis, S. (Ed.) (1990). Being numerate: What counts? Hawthorn, Vic: Australian Councilfor Educational Research.

Kaye StaceyDepartment of Science and Mathematics Education,University of Melbourne 3010,Australia

Sue HelmeVicki SteinleUniversity of Melbourne

Annette BaturoQueensland University of Technology,Australia

Kathryn IrwinUniversity of Auckland,New Zealand

Jack BanaEdith Cowan University Australia

RON TZUR, MARTIN A. SIMON, KAREN HEINZ and MARGARET KINZEL

AN ACCOUNT OF A TEACHER’S PERSPECTIVE ONLEARNING AND TEACHING MATHEMATICS:

IMPLICATIONS FOR TEACHER DEVELOPMENT1

ABSTRACT. This report presents an account of one teacher’s mathematics teaching anda perspective that underlies his teaching. Nevil was a fifth grade teacher participating incurrent mathematics education reforms in the United States. Through the account, wemake distinctions about teachers’ thinking and practice that can inform teacher educationefforts. We constructed an account by analyzing four sets of classroom observations andinterviews. We observed that Nevil decomposed his understandings of the mathematicsinto smaller components and connections among those components. He created situationsthat he believed made those components and connections transparent and attemptedto elicit those connections from the students. This account illustrates a practice that isdifferent both from traditional practice and the type of practice that we would envision as agoal for teacher development. We contribute two important aspects of mathematics teacherdevelopment from traditional to reform-oriented teaching. In particular, we describeteachers’ perspectives – assimilatory structures that constrain and afford (a) the sense theymake of professional development opportunities and (b) their potential learning in teachereducation settings.

Mathematics teacher education is a primary force in current efforts toreform mathematics teaching. Ball (1988) pointed out that for mathematicsteacher education to be effective, teacher educators must understand andbe responsive to teachers’ thinking. Based on an analysis of case studydata from work with a fifth grade teacher, Nevil, we postulate perspectivesthat underlie the practice of teachers participating in current reforms. Wethen examine how these perspectives define the challenges facing teachereducation and how understanding these perspectives can inform teachereducation efforts.

Reform documents (National Council of Teachers of Mathematics,1989, 1991, 2000) in the United States encourage mathematics teachersto decrease traditional activities of telling and showing the mathematicsstudents need to know. Instead, teachers are encouraged to increaselearning experiences that promote problem solving, reasoning, commu-nicating, and making meaningful connections among mathematical ideas.However, these documents “do not provide guidance on the specifics ofday-to-day, minute-to-minute practice” (Ball, 1996, p. 502). Quite the


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contrary, the “vision of the mathematics curriculum and the goals in theStandards are offered by NCTM without a prescription for achieving them.The approach taken is one that will empower teachers . . . to make thechanges” (NCTM, 1989, p. 251). Thus, teachers are challenged to adapttheir mathematics teaching by creating alternative, reform-oriented prac-tices. This study is part of research efforts to understand the perspectivesof teachers in transition from traditional to reform-oriented teaching.

CONCEPTUAL FRAMEWORK

Because this research involved an investigation of a teacher’s perspectiveon mathematics, mathematics learning, and mathematics teaching, webriefly describe our conceptual framework in these areas. In the context ofour research, this framework affects what we notice (Mason, 1998), whatwe take to be significant, and what we identify as challenging our currentunderstandings. For this same reason, we also articulate our understandingof teacher education and of teaching the particular mathematical contentinvolved in the case study data that we present.

A Perspective on Knowing and Learning

Our conceptual framework is based on a social constructivist perspective, acoordination of cognitive and social perspectives on knowing and comingto know (Cobb & Yackel, 1996). From a social perspective, learningis a process of enculturation to the communities in which one partici-pates (Cobb & Bauersfeld, 1995). We use this perspective to identifynorms and practices in mathematics and mathematics teacher educationclassrooms and to understand how new and modified norms and practicesare constituted. Understanding norms and practices helps us explain theinterplay between the nature of the classroom microculture (Voigt, 1995)and students’ learning.

Our cognitive perspective draws on a constructivist epistemologicalstance and theory of learning (Dewey, 1938; Dewey & Bentley, 1949;Piaget, 1970; von Glasersfeld, 1995). In particular, we conceive of know-ledge as a person’s conceptual structures and operations that are used tomake sense of and organize her or his experiential world. For abbrevi-ation purposes, we refer to both conceptual structures and operations asconceptions. We use the term perspective to postulate a broad pedago-gical structure composed of multiple conceptions that collectively organizesome aspects of a teacher’s practice. In characterizing a teacher’s pedago-gical perspective, we make no claim of having analyzed the component

A TEACHER’S PERSPECTIVE 229

conceptions. In our work, this difference of grain size is common aswe consider mathematical understandings (conceptions) and pedagogicalunderstandings (perspectives).

Our understanding of the coordination between social and cognitiveanalyses is consistent with Cobb and Bauersfeld’s (1995) position:

This coordination does not . . . produce a seamless theoretical framework. . . . When thefocus is on the individual, the social fades into the background, and vice versa. Further,the emphasis given to one perspective or the other depends on the issues and purposes athand. (p. 8)

In our study, we focused on the perspectives underlying a teacher’s prac-tice. Therefore, we elaborate below only the cognitive component of oursocial constructivist perspective.

Our cognitive perspective on knowing and learning builds on Piaget’s(1980) fundamental notion of assimilation. This notion implies that anindividual’s current way of thinking affords and constrains (a) the senseone makes of events and processes in which one participates and (b) theconceptual advances that are possible. This view is consistent with Varela,Thompson and Rosch’s (1991) rejection of the Cartesian view of humanknowing as a connected set of mental representations of a world, includingthe world of mathematics, that exists independently of the people whoknow it. In other words, we think of the mathematics or pedagogy onesees in the world as being afforded and constrained by one’s assimilatorystructures and operations.

From this perspective, a paradox emerges with regard to how learningtakes place (Fodor, 1980; Bereiter, 1985). In order to recognize something,one must already have a structure into which that something can be assimi-lated. For example, Cobb, Yackel and Wood (1992) noted that a set ofbase ten blocks might be a clear representation of place-value relationshipsto a teacher. However, learners who do not have a concept of compositeunits (Steffe & Cobb, 1988) of ten cannot see place-value relationships inthe blocks. So how does one ever develop a more advanced assimilatorystructure?

We address the learning paradox on the basis of a constructivist view oflearning as transformation (accommodation) in the learner’s current waysof thinking (Piaget, 1985). Conceptual transformation is the result of threeinterrelated processes in the learner (Dewey, 1933; Simon et al., 1999):

1. Carrying out goal-directed activities – mental operations and physicalactions – that he or she can already perform.

2. Discriminating effects of those activities that are more and lesssuccessful relative to the goal.

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3. Abstracting new regularities in activity-effect relationships throughreflection on those efforts that brought about desired or undesiredeffects.

As Bickhard (1991) noted, such an explanation of learning does not requirethe learner’s use of the more advanced structure to recognize a situationprior to constructing that structure. It only requires the learner’s abilityto (a) use activities available through current structures that bring abouteffects, (b) reflect on and differentiate between those effects, and (c) reflecton and differentiate among classes of activity-effect relationships. Thisview of learning can be illustrated in people’s independent constructionof increasingly sophisticated strategies while playing games. An exampleis learning to play the game “24.” In this game, one is competing withothers to quickly figure out how to use four given numbers and the fourbasic operations to arrive at 24. Through reflection on previous attempts,a player comes to realize the usefulness of considering factors of 24. Forexample, given the numbers 2, 3, 4, 9 one uses 9

3 and 4 × 2 to produce 3and 8, respectively. Having described the cognitive component of our viewof knowing and learning, we turn to key features of our understanding ofteaching.

A Perspective on Teaching

Our perspectives on teaching build on the perspectives on learning we havearticulated. A significant part of the teacher’s responsibility is to promoteand support the students’ development of more powerful mathematicalconceptions. This is accomplished through a reflection-interaction cycle(Simon, 1995). We assume that the teacher must make a clear distinctionbetween her or his knowledge and the learners’ conceptions in order togenerate useful hypotheses about students’ learning. Through reflection onstudents’ actions and language, the teacher infers students’ current concep-tions. In particular, the teacher attempts to understand students’ currentunderstandings of the mathematical area to be addressed. Based on theseinferences, and on the teacher’s own understanding of the mathematics, theteacher specifies a hypothetical learning trajectory (HLT).

The HLT is composed of three interrelated parts: the teacher’s goals forplausible advances in the students’ current conceptions, problem situationsthat can create opportunities to promote such advances, and hypothesesas to the processes by which the learning might occur (Simon, 1995). Togenerate or adjust learning opportunities, the teacher designs activities thatshe or he hypothesizes students can initiate and carry out on the basis oftheir current conceptions (Tzur, 1999). When the goals for the studentsare conceptual, the teacher engages them in activities available to them


already; in particular, he or she encourages and orients their reflection onactivity-effect relationships as a means to promote the intended conceptualadvance.

We take for granted that the teacher’s hypotheses as to the path ofstudents’ learning are only that, hypotheses. Thus, while interacting withstudents the teacher continually modifies the goals and/or learning oppor-tunities on the basis of ongoing reflections on students’ evolving activity.When a period of teacher-student interactions (e.g., a lesson) is over, theteacher reflects on the students’ current conceptions and plans for thenext period of interactions. This perspective of teaching as a reflection-interaction cycle implies that the teacher’s knowledge – conceptions ofmathematics, perspectives on mathematical activity and representations,and teaching-learning processes of particular mathematical content – isconstantly changing.

A Perspective on Teacher Education

Our perspectives on mathematics teacher education are based on ourperspectives on mathematics teaching in two ways. First, mathematicsteaching as we conceive of it defines the goal for our teacher educa-tion efforts. Second, we attempt to adapt the processes that we havearticulated for mathematics teaching to the teaching of mathematicspedagogy (Simon, 2000a). From our perspective on mathematics teaching,a teacher’s understanding of students’ current understandings plays acrucial role in the teacher’s ability to promote learning. Cooney andKrainer (1996) emphasized that this role applies equally to promotingdevelopment of children’s mathematics and of teachers’ teaching.

A central problem for research, then, is to articulate develop-mental landmarks in teachers’ practices. This orientation “necessitatesan emphasis on understanding teachers’ thought processes” (Cooney &Krainer, 1996, p. 1162), that is, the complex perspectives that structureteachers’ attention and awareness (Mason, 1998). In line with this orienta-tion, recent research focused on certain aspects of those perspectives.For example, Steinbring (1998) identified three components of teachers’epistemological knowledge about mathematics. Grant, Heibert and Wearne(1998), Lloyd and Wilson (1998), and Vacc and Bright (1999) studiedthe relationship between teachers’ beliefs about and understanding ofmathematics and the nature of change in the teachers’ practice. Ourstudy was designed to further our understanding about the interconnectednature of mathematics teachers’ practices and the complex perspectivesthat underlie their practices. At this time of radical shift in teaching, thisstudy informs a reconceptualization of mathematics teacher education by

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exploring pedagogical perspectives of a teacher in transition. In particular,this work contributes to articulating goals for teacher education and tounderstanding assimilatory structures that teachers use to make sense ofteacher development opportunities.

A Perspective on Knowing and Learning Division

In the analysis section, we focus on data from a set of lessons in whichNevil’s goal for the students was for them to understand the long-divisionalgorithm most frequently taught in the United States. We articulate ourthinking about teaching this mathematical content to make explicit theconceptual framework that structured our analysis of the data.

Before an understanding of the long-division algorithm can be fostered,students must have a conception of division. Because this conception isprerequisite to understanding long-division, we provide a brief outline forthis part of our framework. A conception of division may be developedin the context of realistic problems that can be solved by separation of anoriginal quantity into equal groups. For example, the teacher might presenttasks such as “How many boxes are needed to pack 32 candies, if each boxcontains exactly 8 candies?” Such tasks do not require a conception of divi-sion. Students can solve them using their established conception of number– composing and counting same-size groups – and manipulative tools suchas Unifix� cubes. Students’ reflection on such grouping activities can leadthem to distinguish a set of problems for which the grouping activity isappropriate. Initially, students would abstract two separate relationships,partition and quotition. We use the two terms to indicate the actions usedby the learner. Partition indicates distribution into a given number of sets;quotition indicates the making of given-size sets. Later, the students wouldconstruct connections among these ideas into a concept of division.

To foster students’ (re)invention of computational algorithms for divi-sion, we would engage them in solving division problems with base tenblocks. An example of such problem would be, “If you want to place192 marbles in 6 boxes equally, how many marbles will each box hold?”In using base ten blocks we assume that the students already understandhow these blocks represent a place-value, base ten number system. As thestudents create the initial quantity and begin to separate it into 6 equalgroups, we would engage them in writing down their steps. After theactivity with the blocks had become a routine, we would also ask themto predict the result of their next step. For example, we would ask, “Couldyou tell how many you will have left after equally distributing the 19 “tens”into the 6 boxes?” Through reflection on their division activities with baseten blocks, and their activities to predict the next steps, we expect the


students to abstract relationships among the quantities involved. Relation-ships between the product of the divisor and the partial quotient, or thedifference between that product and the dividend can serve as the basis forunderstanding the canonical algorithm.

METHODOLOGY

This study was part of the Mathematics Teacher Development (MTD)Project, a 4.5-year inquiry into the development of prospective and prac-ticing elementary mathematics teachers (henceforth referred to as the“teachers”). By development, we mean professional development towardsthe establishment of teaching practices consistent with current reformrecommendations. We structured the project using the teacher develop-ment experiment (TDE) methodology articulated by Simon (2000b). Thismethodology involves promotion of teacher development and study of thatdevelopment in a cyclical fashion similar to the constructivist teachingexperiment (Cobb & Steffe, 1983). Thus, each intervention with theteachers is followed by an analysis of the teachers’ current perspectives,and each analysis session leads to a formulation of the next intervention.

The TDE consists of two components: whole class teaching experi-ment(s) (Cobb, 2000) and accounts of practice (Simon & Tzur, 1999). Thewhole class teaching experiment of the MTD Project consisted of 5 coursesin consecutive semesters. Each course was taught by the second author andinvolved 9 teachers and 10 prospective teachers. The first course and partof the second focused on promotion of the teachers’ conceptions of keyideas in geometry and ratio. The last three-and-a-half courses focused onpromotion of the teachers’ pedagogical understanding. In particular, wewanted them to learn how to analyze children’s thinking and how to deviseand implement tasks that promote certain conceptual advances in students.

The study reported in this paper derived from the account of practicecomponent of the MTD research program. An account of practice is anadaptation of case study methodology and is the researchers’ explicationof a teacher’s practice. Practice refers to what the teacher does and theperspectives that underlie what he or she does. Using several data sets, wegenerate an account through intensive team-work of line-by-line analysisof transcripts while listening to the interviews or observing classroomvideotapes, as well as reflecting on our own field notes regarding teachers’participation in the courses and work in their own classrooms. We identifysections of data that pertain to what we consider to be the central challengefacing teachers who are participating in the reform. This challenge is tocreate alternatives to direct instruction that can promote students’ learning

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of particular mathematical ideas. With this focus in mind, MTD researchteam members focus on how the teacher promotes students’ learningof particular mathematical ideas, contributing either tentative inferencesabout the teacher’s practice or questions about possible interpretations ofthe data. Those inferences and questions serve as the basis for hypothesesthat we generate and continually revise to explain the ensemble of the data(Glaser & Strauss, 1967).

We consider that our hypotheses constitute an account of practice whenthey offer a coherent explanation for what previously had been, for theresearch team, a diverse and often unrelated set of observations. Eachaccount represents our commitment to comprehend how the teacher organ-izes her or his experiential reality with respect to teaching mathematics.We assume that everything a teacher does makes sense from her or hisperspective. It is our challenge to infer from our data a perspective fromwhich this might be so. An account pertains to a period of time in theteacher’s teaching during which we consider her or his practice to beessentially consistent.

In an account of practice, we explain the teacher’s perspective fromthe researchers’ perspective, a stance that involves a subtle but importantdistinction. Our accounts may differ significantly from what teacherswould articulate about their practices. We structure our accounts usingparticular conceptual lenses, often not shared by the teachers, that defineour focus and guide our interpretations. We recognize the paradox withinwhich we work. To conceptualize the teacher’s practice we must “setaside” our current view of practice to consider perspectives different fromour own. Yet, our current view of practice is fundamental to what wenotice. For a detailed discussion of accounts of practice see Simon andTzur (1999).

In the MTD Project, we typically generated 3 data sets per semesterfor each teacher. The account of Nevil’s practice, reported here, emergedfrom our analysis of four data sets. In particular, we analyzed one data setgenerated before the instructional program and three data sets generatedearly in the program, prior to observable impact of the program on hispractice. In a single data set, we interviewed Nevil about his plans priorto the observed lessons, observed and videotaped at least two consecutiveand related mathematics lessons, and interviewed him after each lesson todocument his reflection about that lesson and plans for the next lesson.Most of the data discussed in this paper were taken from the extendedfourth set that consisted of four consecutive observations and five inter-views. We chose this data set because it was central to our ability to make


sense of Nevil’s practice and because we considered it to represent andillustrate his teaching over the period of time spanning all four data sets.

We present this account of Nevil’s practice because we believe thiscareful analysis can contribute to understanding the perspectives ofteachers involved in the reform. As stated above, we consider under-standing teachers’ perspectives at different points in their developmentto be a critical component of the knowledge base for effective teachereducation.

ANALYSIS OF NEVIL’S PRACTICE

This section consists of four parts. First, we provide an overview of Nevil’sunderstanding of mathematics and his orientation to teaching. Next, wesummarize the sequence of his four lessons on the long-division algorithm.In the third part, we identify five aspects of the lesson sequence thatinitially we found difficult to explain. Finally, we articulate our account ofNevil’s practice and indicate how it explains the five aspects of the lessonsequence that we could not explain initially.

Overview of Nevil’s Mathematics and Orientation Toward Teaching

Nevil taught in a district that provided a list of desired grade-leveloutcomes in mathematics; the identification and creation of learning activ-ities for the students was left to the teacher. Nevil was pleased to workunder these guidelines. Nevil’s mathematical understanding was strong. Inparticular, his work in the first MTD course showed a solid understandingof multiplicative reasoning related to ratio. He utilized his considerablemathematical knowledge in designing instruction for his fifth-graders.His instruction usually was preceded by careful analysis of the intendedmathematics.

Nevil was committed to minimizing his telling and showing studentsthe mathematics to be learned. He strove to create a collaborative climatein which students participate in all activities. Rather than focusing onstudents’ abilities to execute mathematical procedures, Nevil strove toreveal basic ideas and connections between these ideas and correspondingpaper-and-pencil algorithms. For example, in Excerpt 1 Nevil articulatedhis goals for the division unit. In Excerpt 2 he explained the need foranalysis of the content to be taught.

Excerpt 1 (Pre-set interview)

N: [I want my students to] understand what the [division] algorithmis all about, that it is not just, we put the 2 up here, we multiply,

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we subtract, we get an answer, we bring something down, we dothis. I want them to see why we are doing it that way, that thereis a purpose to this algorithm, . . . that there is something actuallygoing on with these numbers and how they relate to each other.

Excerpt 2 (Interview about lesson 1)

N: Last year I didn’t [teach division] this way. I did the, “here’s a[long] division problem. Let’s review how to do it, and let’s doa bunch of them, and then let’s have a test on Friday” kind ofapproach . . . Some of the kids could do it, others never got it. . . Then the next time we did division was with fractions . . . and Itried desperately to help the kids understand what was going on.. . . I didn’t do any of the stuff that I learned where you [just] invertthe fraction and multiply. So I spent two or three weeks goingback to division with whole numbers and trying to re-teach theconcept of division . . . and they never understood. . . . So I foundthat I had to go back and really understand what division was allabout. . . . [This is how] I came up with . . . the rationale for this[year’s] approach – to examine what division is first, and try some[division situations] with the Unifix cubes and to see it, and then tobegin analyzing this long-division algorithm that they have alreadylearned and to see if they have been making connections and if Ican help strengthen those connections

In order to monitor his students’ progress, Nevil attended to their contri-butions and noted differences in their mathematical competence. Thesedifferences were rooted in his analysis of the intended mathematics. Forexample, in one interview he clearly distinguished between the mathe-matical understandings of two students. He said that one student could useUnifix cubes to represent both partitive and quotitive division; the othercould show a partitive representation only and was unable to make senseof her partner’s quotitive representation. In another interview, Nevil madefiner distinctions among four different quotitive solutions that some of hisstudents used and two different partitive solutions.

In all, we conceived of Nevil as a skillful mathematical thinker. Hewanted students to make connections between their everyday experiencesand mathematical algorithms, and he was competent in making distinc-tions among individual students’ mathematical actions. He reflected onhis teaching and endeavors to help his students develop understanding.Towards this end, he also chose to participate in the MTD teacherdevelopment program.


Nevil’s Division Unit

In this part we present our analysis of the data set from the division unit.The four lessons that we observed are numbered 1–4. In the interview priorto Lesson 1, Nevil described the previous lesson in which the studentsworked on an introductory task.

Excerpt 3 (Pre-set Interview)The kids started out with 16 Unifix cubes and I asked them . . . to showtheir partner what 16 divided by 2 would look like. . . . I am trying toget the students to understand that division has to do with beginningwith a number of things and then dividing it into sets. . . . Some of thestudents have a really good understanding of what division is all about,but I feel that there are still some that might not quite get it.

Nevil had a clear goal for his students’ learning: He wanted them toform a specific idea about division that can support their understandingof the algorithm. The first lesson that we observed proceeded along theplan that Nevil described to the researcher during the pre-set interview(last part of Excerpt 2). First, he engaged pairs of students in workingwith Unifix cubes on the task, “Could you show 20 divided by 4 with thecubes?” Based on Nevil’s responses in both interviews, before and after thelesson, his goal was that the students come to see, or revisit, both aspectsof division: quotitive and partitive. Nevil interacted with the pairs, both toassess the aspects of division they used and to promote their ability to useone, and preferably both, aspects. After 25 minutes, Nevil concluded thatmost students were able to see at least one of the two aspects of division theway he did – breaking a set into smaller equal sets. He put the Unifix cubesaway and moved to the second part of the lesson. Assuming that studentshad already been taught the algorithm in fourth grade, he asked them tosolve, individually, three long-division examples that he wrote on the board(140 ÷ 7, 568 ÷ 8, 5454 ÷ 6) and observed their execution of the algorithmfor about 10 minutes. Nevil had volunteers put their computations for thethree examples on the board. He then initiated the third part of the lesson,a 30-minute, whole-class discussion about the meaning of the steps of thealgorithm as performed in the example 140 ÷ 7. In summary, the lessonconsisted of a task with the cubes, pencil-and-paper computation, and adiscussion of the meaning of the steps used in the computation.

Nevil began the whole-class discussion by eliciting students’ reasoningabout the location of digits in the quotient of the first computation (140 ÷7), that is, why they put the 2 above the 4. Many of them explained thatthe 2 stood for 2 tens, hence 20. This explanation seemed to fit with hisexpectations. He then asked them to discuss the reason for the subtraction

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step (14 – 14 = 0). Students’ responses included, “to find the remainder,”“to find what is left,” “to check the answer,” or “to make the problemeasier, because the number is smaller.” The students’ responses did notseem to be what he wanted. After asking them several times to visualizethe absent Unifix cubes in order to explain, “Why do we subtract?” Nevilasked them to continue thinking about the reason for the subtraction stepfor homework.

In the interview after Lesson 1, Nevil explained that he was frustratedwith the students’ inability to grasp what for him seemed obvious. Forhim, one uses subtraction in the long-division algorithm to account for thequantity that had already been distributed into equal groups. Specifically,he explained to the interviewer that in the example 140 ÷ 7 one subtracts14 – 14 and gets zero because by distributing two tens (20) to each of theseven groups one had accounted for all 14 tens (140) of the initial quantity.In his reflecting on the lack of success of the lesson, Nevil consideredthat students might not recognize the relationship between the algorithmand the solution with cubes because of how place value is handled in thealgorithm.

Excerpt 4 (Interview about Lesson 1)

N: If we are going to multiply maybe it would make more sense tothem to see that we are really multiplying by 20 . . . that we writeout the whole number and then we are comparing. . . . I think Iwould ask them to explain to me visually what we’ve done so far,with – if they could use Unifix cubes or something.

R: So [at] that point you might bring the Unifix cubes back for themto work or you just [ask them to] imagine it?

N: I think I would ask them to imagine it. And to tell me that we’vemultiplied 20 times 7, and we’ve gotten 140, well that means thatwe’ve accounted for everything that was in that original set. Andin this next step, what we are doing is comparing the original setto the, the first number which [in this case] is 20 times 7, 140.

Nevil planned and implemented Lesson 2 along the lines of Lesson 1,trying to get the students to see the reason for the subtraction step. Hebegan by reviewing the initial steps of the algorithm. This time he empha-sized the multiplication step 20 × 7 = 140 and even recorded the productas 140 rather than as 14. From there, he moved on to the subtraction step.However, when the students’ responses to his repeated questions, “Why arewe subtracting?” did not satisfy him, he changed the manipulatives that heasked them to visualize, from Unifix cubes to base ten blocks. In the nextinterview, he explained that such a change might help the students see the


base ten aspect of distributing the initial quantity into a certain number ofgroups, that is, the distribution first of the hundreds, then the tens, and thenthe ones.

Following this change, Nevil continued his inquiry about the subtrac-tion step for the two first examples (140 ÷ 7, 568 ÷ 8), but still thestudents did not seem to respond as he expected. Toward the end of thelesson, he posed a new example, 100 ÷ 6. In the interview that followedLesson 2, Nevil explained why he posed the new example. He said thateven with the change to base ten blocks, “We had gotten as far as we weregoing to get with this idea of subtracting and what does it mean, and weweren’t going to get beyond from the students.” He made it clear that hisintention in posing the new example was to begin working on the issue ofremainder, not to inquire further into their understanding of the subtractionstep. Nonetheless, we observed that he incorporated his interpretation ofthe subtraction step into the discussion of the new example:

Excerpt 5 (Lesson 2)

N: So what did we take care of? What shape of base ten blocks didwe take care of?

S: 100.N: Yeah, we couldn’t do the 100 right. . . . So we need to trade it in

and get rows. Yeah?S: Um, since there is [inaudible] 60 minus 100 is forty and forty

[inaudible]N: Okay 40 is still in my bag. Still got 40 left. Yes?S: Yeah.N: Okay, now what do I do? I figured out how many tens will be in

each of the 6 groups, right? How many rows. And there is onlyone [row] in each, okay. But that accounts for 60 of the original100. Agreed? So I have 40 left in my original pile. That is why Isubtracted, to find that out. Okay. Because I don’t have the cubesin front of me. I have to figure it out with the numbers so now Ihave 40 left.

In Excerpt 5, we noticed that once Nevil decided he could not elicitfrom the students the reason for the subtraction step, his teaching changed.He explicitly presented his interpretation of that step through the nextexample. At the end of Lesson 2, Nevil assigned for homework the taskof explaining the connection between the algorithmic solution for a newexample (200 ÷ 6) and the process with base ten blocks.

Nevil planned to begin Lesson 3 with students using base ten blocksto demonstrate 100 ÷ 6. Nevil explained that he wanted the students to

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actually use the blocks. His reason was that the blocks serve “as a concreteway of showing what division is all about . . . a kind of a concept builder.. . . Instead of [focusing] on the algorithm and the numbers [the kids are]actually dividing something by something else.” Then, he would again putthe blocks away and ask them to visualize their solution with the blockswhile connecting this solution to steps in the algorithm, particularly thesubtraction step. As it turned out, Lesson 3 consisted mostly of actualmanipulation of the base ten blocks in small groups and of groups reportingtheir solutions to the class. Contrary to Nevil’s intention that students showthe connection to the algorithm via partitive solutions, a majority of thestudents solved the example in a quotitive way. Nevil tried to lead thosestudents to consider a partitive solution by asking them, “Is there any otherway to do this problem?” However, they responded with more quotitivesolutions. Toward the end of the lesson he called on a student (Carol)whom he knew had used a partitive solution. Through leading questions,he elicited a description of the process of trading base ten units and puttingthem into 6 equal groups, while he specified the connection between herdescription and the algorithm, orally and by writing on the board.

Nevil began Lesson 4 by asking students to work the example 100 ÷5 using base ten blocks. He then asked them to consider the connectionbetween division with the blocks and the algorithm, first in small groupsand later in a whole class discussion. Again, the first two groups presentedquotitive solutions, so he called on Carol, the student who presented thepartitive solution in the previous lesson:

Excerpt 6 (Lesson 4)

N: Okay. Now I am asking the people who did it in five groups,and you found 20 in each group. I am asking you. Did youfind any relationship to this algorithm?

Carol: None.N: None. I don’t believe that. I believe that if you would have

thought – you would have found some kind of connection.You are all very intelligent people and we have been doingthis [for] several days and you have been doing a lot ofthinking about this. And I don’t believe that you didn’t findany kind of connection.

Excerpt 6 indicates Nevil’s expectation that students who used apartitive solution would see the connection to the algorithm. We note thatNevil seldom blamed students. He usually responded to students’ diffi-culties by taking responsibility for the difficulties and striving to figure outan alternative way of teaching that would reveal the intended mathematics


to the students. This rare response seemed to indicate a high level of frus-tration. Nevil had tried all the alternatives that he could generate, includingthe work with the base ten blocks that he believed made the connectionsobvious. Yet, in spite of all his attempts to reveal to the students whatseemed so obvious to him, they did not “get it.”

Following the exchange presented in Excerpt 6, Nevil asked thestudents to work on another example (97 ÷ 5) with the base ten blocks.Then, by heavily paraphrasing contributions from a small group that usedthe partitive solution, he explained in detail the process of trading base tenblocks, while pointing to specific connections to steps in the long-divisionalgorithm. For example, he pointed out that algorithmically asking howmany times 5 goes into 9 corresponded to putting one ten in each of thefive groups and that the number 1 above the 9 represented that ten. Whentalking about the connection to the subtraction step, Nevil repeated severaltimes the idea that the subtraction did not mean taking something away, butrather that portions of the initial quantity were moved into the five groups.

Developing an Account of Nevil’s Practice

As we initially examined the data set described above, we noticed partic-ular aspects that we could not explain with the perspectives that we hadavailable. A useful account of Nevil’s practice would have to provide asound interpretation of the following:

1. Nevil began the division lessons by asking the students to show 16divided by 2 and 20 divided by 4 with Unifix cubes. His goal was“to get the students to understand that division has to do with begin-ning with a number of things and then dividing it into sets.” From ourperspective, in order to complete the task of representing division withthe cubes, the students needed to have already constructed the meaningof division.

2. During the first two lessons, Nevil deliberately chose not to makethe manipulatives available to students during the discussion ofthe long-division algorithm. Rather, he invited students to visualizemanipulative solutions to inform their interpretation of the algorithm.

3. For homework, Nevil assigned the same question the students couldnot answer during Lesson 1, and proposed and implemented essen-tially the same plan for Lesson 2.

4. Initially, Nevil did not focus on the multiplication step, but proceededdirectly to the subtraction step.

5. In the latter stages of the lesson Nevil began to use leading questionsand explicit statements of what he previously tried to elicit from thestudents.

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In our first attempt to make sense of these aspects of Nevil’s practice,we observed the following characteristic that seemed to remain unchangedthroughout the lesson sequence. Nevil expected the students to see whatseemed obvious to him, the relationship between dividing blocks andthe long-division algorithm. This was extremely clear when he becamefrustrated about Carol and her classmates’ inability to see the connectionbetween a partitive solution with the blocks and the steps in the algorithm.He did make modifications in his approach: He changed the manipulativesto be visualized, changed from visualizing to manipulating base ten blocks,changed the example, and moved from eliciting ideas to stating his ideas.However, all of these efforts seemed to be grounded in the notion that if thestudents had experienced division using the physical manipulatives, theyshould see how the long-division algorithm paralleled the manipulativesolution. We came to understand Nevil’s perspective as follows.

Nevil attempted to deeply understand the mathematics that he teaches.He viewed mathematics as a connected, logical set of ideas. He wasaware of those situations that clearly demonstrated, for him, particularmathematical relationships. Cobb (1989) pointed out that an experi-ence of perceiving mathematical truth is an inherent part of developingmathematical understandings.

Once we have made a mathematical construction and have used it unproblematically, weare convinced that we have got it right – it is difficult to imagine how it could be any otherway. Mathematical objects are, for all intents and purposes, practically [emphasis original]real for the experiencing subject. (p. 33)

From our perspective, Nevil saw the world through his own mathe-matical conceptions, unaware of the role that his conceptions played inhis experience of the world. For Nevil, his identification of division inthe world around him was a perception of one aspect of objective reality.Thus, he assumed that division could be seen by all in the distributionof physical objects into groups. Further, for him, the connection betweendivision in physical representations and the steps of the algorithm wasperceivable and apparent. Note that we do not claim that Nevil held anyformulated positivistic view of mathematics; we only infer that a positiv-istic view was implicit in his perspective on mathematics learning whichmight have been partly or wholly implicit as well. That is, learning meantcoming to see first-hand particular aspects of mathematical reality and theconnections among them. Our sense is that for Nevil, coming to understandmathematics was often like identifying difficult-to-see physical entities inthe environment. For example, to find a house from the top of the localmountain, one needs to have particular reference points that orient one’sperception, such as locating the nearby water tower. In this view, then,


learning is a gradual process of coming to see additional aspects of the webof mathematical relationships; aspects seen previously serve as points ofreference for what is yet to be seen. Accordingly, first-hand manipulationof physical objects is critical for learning, because it shows the intendedmathematics in the concrete relationships among objects and affords theconnection to the abstract, symbolic operations with numbers.

Consistent with his perspective of mathematical knowing and learningpostulated above, Nevil considered teaching to be about assisting studentsin seeing the intended mathematics. Towards this end, he employed threeinterrelated activities. He decomposed his mathematics into perceivablepieces and connections. He created conditions that afforded students’perceptions of these pieces and connections. He monitored whether ornot the students perceived the mathematics by determining whether whatthey reported matched the mathematics as he knew it. Nevil continuallyadjusted the first two activities based on the results of the previous activity.We illustrate the activities with examples from the data presented earlier.

In planning the lesson sequence, Nevil decomposed the long-divisionalgorithm into sub-procedures to be considered. Underlying this strategyseemed to be his appraisal that the students would not see the whole rela-tionship, but that breaking the task into smaller steps would make therelationship to the blocks solution easily perceivable. His monitoring ofthe students’ contributions led him to decompose the mathematics in thealgorithm further. For example, he decided that a discussion of the multi-plication step and a greater focus on the place-value meaning would makeit easier for students to see the relationship to the subtraction step.

We can see Nevil’s second strategy, the creation of conditions thatafforded students’ perceptions, in his initial activity with Unifix cubesand the subsequent assignment of three long-division computations. Byjuxtaposing the two situations, he intended to allow the students to seethe correspondence between the concrete situation and the algorithm. Hismonitoring of the students’ contributions led also to a revision of the initialconditions. Instead of asking them to visualize Unifix cubes, he asked themto visualize base ten blocks. He later modified the conditions further byengaging the students in actually working with the base ten blocks.

The examples above highlight that Nevil’s focus in teaching was onorienting students’ perceptions. This is illustrated further in Excerpt 7.

Excerpt 7 (Interview after Lesson 3)

N: I’m wondering if they are not understanding what I want them tounderstand because I haven’t come up with a way [that allows]them to see it. . . . I’m approaching it from the way I saw it, and Isaw it [only] recently. . . . At some point trying to come up with a

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way to teach this I had to understand it more than I already did.. . . So I’m trying to come up with a way that will help them be ableto see what I want them to understand.

Excerpt 7 indicates that, for Nevil, what was problematic in teachingwas how to create situations in which students could see the mathematicsfor themselves. This approach is different from traditional teaching interms of types of learning situations the teacher considers to be appro-priate. In this case, the teacher focused on students’ active involvement inthe exploration of concrete representations of the intended mathematics sothey would discover the mathematics rather than being told or shown.

In Nevil’s assessment activity, we identified two related key features:carefully listening to students’ responses and analyzing what they learnedof the intended mathematics. Nevil’s listening was oriented to whetherstudents’ contributions matched his way of understanding the mathe-matics. If there was a match, he concluded that the students had grasped theintended mathematics, as his acceptance of their responses to the meaningof the “2 above the 4” (in 140) indicated. If there was no match, heconcluded that they had not grasped it; hence more decomposition andrefining of conditions was needed.

Nevil’s decisions about what and how to teach next were based on whathe had identified as lacking. Thus, he monitored students to determine whatthey still needed to learn and not to identify what they did know. Thismakes sense from the perspective of the account that we have articulated.If the mathematical relationships are pre-existing and perceivable as is, byall, Nevil was concerned with what was not yet seen. A focus on whatand how the students already understood fit with a different perspectivein which perception is understood as afforded and constrained by extantconceptions.

Having articulated our account of Nevil’s practice, we now use it toexplain the five observations presented at the beginning of this section.The first observation referred to Nevil’s task of using Unifix cubes to show16 ÷ 4 and 20 ÷ 4 as a means to promote students’ understanding of divi-sion. This puzzled us because of our perspective that a student must haveabstracted a rudimentary notion of division in order to be able to representit on the basis of a context-free division expression. However, that wouldnot be a goal that Nevil would formulate. His implicit sense that divisioncould be seen by all, as is, led him to inquire not how a concept of divisioncould first be constructed, where no such concept previously existed, butrather how he could orient the students to see division more fully. Thus, byworking with the Unifix cubes he was giving them an opportunity to seedivision.


The second observation referred to Nevil’s deliberate choice to putaway the Unifix cubes before the discussion of the meaning of the stepsin the algorithm. Nevil’s goal was to promote a particular connection,the connection between the idea of division and the algorithm. For Nevil,the cubes provided a context from which the idea of division could begleaned. Once he believed that this had occurred, he worked on a connec-tion between the idea of division and the algorithm. Only a realizationthat students had not gleaned the idea from the cubes was grounds forbringing out the concrete materials once again. We infer such a realizationto be implicated in his decision to bring out the base ten blocks in the thirdlesson.

The third observation was about Nevil’s plan for the second lesson andthe questions he posed to students. The plan and questions were essen-tially the same as what, by his judgement, had not worked in the firstlesson. To explain this, we focus on what we infer to be Nevil’s implicitpresumption that the intended connections were visible to all. Accordingto such a perspective, the modification needed was one designed to makethe connections more obvious. As discussed above, for Nevil this involvedfurther decomposition of the mathematics and modifications in the condi-tions created to foster the perception. When these changes met withlimited success, he was unable to question his overarching perspective,that is, to consider that the students were unable to see the mathematicalrelationships that were apparent to him.

The fourth observation referred to Nevil’s initial lack of focus on themultiplication step. Our interpretation is that Nevil was using his ownanalysis of division, with blocks, as the basis for teaching about themeaning of the algorithm. In his own exploration with the blocks, themultiplication step was assimilated into his conception of multiplicationand division as being an organization of quantities into equal-size groups.For Nevil, arranging the 14 stacks of 10 blocks into groups was recorded byplacing the digit in the quotient and by multiplying 14 x 10. It did not occurto him to ask about the meaning of the multiplication because the studentshad just explained (in the context of explaining the 2 in the quotient) thelink to arranging the 14 stacks of blocks into groups. Therefore, it wassufficient to go directly to the subtraction step as a comparison between theoriginal amount and the amount already distributed to groups. Note that weare not claiming that Nevil consciously rejected a focus on the multiplica-tion step. Rather, due to the way he had assimilated the algorithm into hisconception of multiplication and division, it was not a salient part of hisoriginal decomposition of the mathematics. After the students repeatedlyfailed to give responses that he considered satisfactory, he re-examined the

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intended mathematics for opportunities for further decomposition. This re-examination led to an explicit focus on the multiplication step before thesubtraction step.

The fifth point was that Nevil became more explicit in indicating tothe students the ideas he previously tried to elicit from them. We suggestthat because of his feeling that he had done all in his power to decomposethe mathematics and to create proper conditions for perceiving it, Nevilresorted to showing them directly what he wanted them to see. He wasfrustrated with his inability to indirectly orient their seeing. However, hewas committed to revealing to them what he considered to be an importantand accessible piece of mathematics.

DISCUSSION AND IMPLICATIONS

Characterizing a Perspective

The development of the account of Nevil’s practice has contributed to ourpostulation of a more general construct, a perception-based perspective(Simon et al., 2000), which we take to be fundamental to Nevil’s practice.This perspective consists of four interrelated stances:

1. A platonic view of knowledge, that it exists as part of an objectivereality accessible to all.

2. A view that mathematics makes sense and that mathematical ideas areinterconnected.

3. A view of mathematics learning as an active process of first-handexperiencing of mathematics as it exists.

4. A view of mathematics teaching as creating situations that reveal themathematical ideas and as orienting students’ attention to key aspectsof those situations.

Thus, the teacher’s role is to create opportunities for students to perceive,first-hand, intended aspects of mathematics.

Contrasting Three Perspectives

A perception-based perspective is a complex set of pedagogical concep-tions. For further clarification, we contrast it below with traditionalperspectives and with perspectives that we refer to as conception-based.We emphasize that traditional, perception-based, and conception-basedare our characterizations of teachers’ practices, and not how the teacherswould describe their practices.


Traditional approaches to mathematics teaching can be characterizedby teachers’ attempts to transmit particular mathematical ideas to students.Like a perception-based perspective, traditional approaches include aplatonic view of knowing. However, in contrast to traditional approaches,teaching based on a perception-based perspective emphasizes studentscoming to see mathematical ideas and relationships through their ownexperience of the mathematics. Hence, the teacher’s primary role is notto directly transmit the intended ideas to students, but to orchestrateconditions that engage students in actively seeing and connecting thoseideas. We conjecture that transition from a traditional to a perception-based perspective can be an important step toward development of aconception-based perspective. We will return to this point in the nextsection.

A conception-based perspective is based on the notion of assimila-tion, that is, that humans have no access to a reality independent of theirways of experiencing it. One can only perceive what is part of one’sexperiential reality (von Glasersfeld, 1995). Thus, individuals’ conceptionsplay a major role in what they perceive, what they learn, and how theylearn it. The perspective that we described in the conceptual framework isan example of a conception-based perspective. From such a perspective,mathematics is thought of as a web of conceptions that humans abstractthrough reflection. Learning is the building up and the continual trans-formation of one’s conceptions. Teaching, promoting intended conceptualadvances, entails two major components. The first is the creation oflearning tasks for which students set goals and use activities available tothem to accomplish the task. The second is orienting students’ reflectionto identify patterns in their activity and the effects of that activity. Theseideas are further elaborated in Dewey (1933) and Simon et al. (1999).

A perception-based perspective on teaching contrasts with aconception-based perspective. Whereas the former assumes access to apre-existing, objective mathematical reality, the latter explains learning interms of building on the distinctions that learners can make. These distinc-tions are afforded and constrained by their current conceptions and theirgoals for the activity in which they are engaged.

The construct of perception-based perspective that we postulated didnot result only or directly from the account of Nevil’s practice. Yet, thisaccount helps to elucidate the construct. Because we could not attributelimitations of Nevil’s teaching to some lack of mathematical understand-ings, we were compelled to reflect on and explain aspects of his practicethat initially did not make sense to us. By doing so, we realized thatamong the four stances of the perception-based perspective, the last two

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– learning as an active process and teaching as decomposing the mathe-matics and creating conditions to reveal it – serve an epistemologicalcore similar to the core of a traditional approach. In other words, theaccount of Nevil’s teaching accents the nature of constraints put forthby the epistemological stance of a perception-based perspective and thelimitations of an approach to teacher development that focuses on fosteringbetter mathematical understandings among teachers.

A Perception-Based Perspective as a Step Toward a Conception-BasedPerspective

There are three parts to our reasoning behind the conjecture that aperception-based perspective can be a step towards a conception-basedperspective. First, a platonic epsitemological core is shared by both tradi-tional and perception-based perspective. Hence, the change from tradi-tional to perception-based perspective does not require a major paradigmshift. Second, our analysis of the MTD course data (Simon et al., 2000)has yielded the following conclusion. In the MTD courses, the design ofinstruction for the combined group of practicing and perspective teacherswas based on our conception-based perspective. However, the courseexperiences seemed to be assimilated and accommodated by the teachersinto perception-based perspectives. These first two points suggest thattraditional perspectives might evolve into perception-based perspectives.The third point involves consideration of a perception-based perspective asa step towards a conception-based perspective. From a conception-basedperspective, seeing mathematical relationships is key. If teachers can cometo explore why seeing particular relationships is problematic for somestudents and not for others, they may begin to develop understandings ofassimilation that can support a conception-based perspective.

Implications of a Perception-Based Perspective

We present three implications of a perception-based perspective for teachereducation. First, the construct provides teacher educators with a frame-work within which to (a) anticipate how teachers might interpret partic-ular teacher development opportunities and (b) understand the teachers’responses after the fact. We illustrate this point with two examples. As partof teacher development programs, opportunities to interview and reflecton students’ mathematical activity might be interpreted by teachers as achance to look closely at what mathematical relationships students havenot perceived, as opposed to an opportunity to focus on students’ under-standings of the mathematics. Another type of opportunity for teachersis a course in which they learn mathematics in a reform-based situation.


Teachers who experience a new-found understanding of the mathematicsmight conclude certain problems, manipulatives, or software environmentsmake particular mathematical relationships obvious. Thus, because theydo not make a distinction between their assimilatory conceptions and thestudents’, they may attempt to simply transport what was done with themto their own classrooms.

Second, as a description of teachers’ perspectives on mathematicsteaching, the construct can inform teacher education instructional designby specifying broad understandings of teachers – understandings they canuse to engage in teacher development tasks. This point is illustrated in thenext section.

Third, the construct perception-based perspective and its contrast withconception-based perspective define a direction for teacher development.That is, if we take perception-based and conception-based as descriptionsof two locations on a developmental spectrum, we can take as a goal forteacher development the change from the former to the latter. Thinkingof a transition from perception-based to conception-based as a develop-mental step depends on a claim that a conception-based perspective is morepowerful for mathematics teaching than a perception-based perspective.We now offer a brief argument to justify this claim. The reader shouldkeep in mind that our conception-based perspective, as outlined in theconceptual framework for this article, structures not only our postulationof these perspectives, but also our understanding of the limitations of aperception-based perspective.

In order to discuss the relative power of these two perspectives,it is necessary to revisit the fundamental difference between them. Aperception-based perspective is based on a view of knowledge as a“mirror” of a reality that is external and universal (Cobb, Yackel & Wood,1992). A conception-based perspective is based on the notion that wecan only know our own experiential reality, which is determined by ourassimilatory conceptions to that point. However, even those who we wouldcharacterize as having a conception-based perspective operate, at times, asif there were a universally accessible reality. For example, people usuallyassume that others see the same physical objects that they do, or give thesame meaning to commonly used words, unless there are some indicationsthat this is not the case (von Glasersfeld, 1995). What is different aboutthe conception-based perspective is that individuals who have developedthat perspective have the possibility, at any time, to step back from thisassumption of a universally accessible reality to question the differencesin learners’ experiential realities.

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In mathematics education, a perception-based perspective may beadequate for thinking about some situations. When learners have alreadyabstracted two potentially related conceptions, a teacher who holds aperception-based perspective like Nevil’s might be able to design alesson to promote connections between the two conceptions. That is, theexpectation that students see the connections in a well-chosen situationis reasonable, because, from a conception-based perspective, the ideas tobe connected are available in the learners’ experiential reality. However,a perception-based perspective does not address the challenge of engen-dering a new conception. For example, a perception-based perspectivedoes not explain why a child does not see division in an arrangementof objects, nor how the child comes to understand division. Moreover,a perception-based perspective does not assist the teacher in thinkingabout how such conceptions might be promoted when, as in Nevil’s case,students’ do not perceive the intended mathematics despite the teacher’sdeep understanding and decomposition of the mathematics. In contrast, aconception-based perspective can be the basis for inferring how learners’current conceptions limit the mathematics they can see in certain situ-ations, and for thinking about how these conceptions might afford activitiesand goals that can foster intended abstract ideas.

Summary

Teachers’ perspectives consist of intricately related webs of understand-ings of which little is known, particularly as these perspectives haveevolved in relation to current mathematics education reform efforts. In theaccount of Nevil’s practice we postulated an assimilatory perspective thatmay be useful to consider in teacher development efforts with teachers whoare participating in current reforms. By focusing on teachers’ assimila-tion, we invite rigorous consideration of how teachers make sense of whathappens in their classrooms and what they encounter in teacher develop-ment situations. Research of this type can contribute to the establishmentof a broader set of distinctions about teachers’ conceptions, and henceserve as a significant part of a knowledge base for mathematics teacherdevelopment.

The account of practice presented in this report describes one adapta-tion of teaching that is based on, but not determined by, a perception-basedperspective. In our work, we have observed that teachers develop diversepractices that can be characterized as perception-based (Heinz et al., inpress). An important implication for teacher education of the particularpractice characterized in the account of Nevil is the need to help teachersdistinguish between their mathematics and students’ mathematics. Nevil


represents a subset of teachers who are at least beginning to have somepowerful mathematical understanding. Nevil used his mathematical under-standings in a pedagogically significant way to set goals for students’learning. On the other hand, teachers must set aside their mathematicsin an attempt to see the mathematical situation through the eyes of theirstudents.

The account of Nevil is particularly illustrative because Nevil under-stands the mathematics he is teaching, is able to set mathematical goalsthat focus on understanding, and is aware of his students’ failure to under-stand – all key attributes for carrying out a reform agenda that emphasizesproblem solving, reasoning, and making connections. The account demon-strates the limited range of options available to Nevil and his inability toquestion the fundamental assumptions of a perception-based perspective.As we suggested earlier, the ability to identify these assumptions and tosee them as modifiable is the beginning of a paradigm shift.

Looking Ahead

Based on the work that we have been describing, we would argue theneed for research that investigates the promotion of teacher developmenttowards a conception-based perspective. Promoting such a paradigm shiftis an ambitious undertaking even in teachers who use significant mathe-matical understandings, like Nevil. In our own research, we have onlybegun to explore this possibility. Here we describe the two approaches thatwe have used in our initial attempts in hope that it will spark other ideasand further work.

One approach has been to create opportunities for teachers to comparetwo students in the context of similar mathematical tasks. We use thisapproach to orient teachers’ thinking to what students bring to the task.Working with teachers such as Nevil, who deeply understand the mathe-matics involved, enables them to potentially make useful comparisonsbetween the students’ thinking and their own.

Our second approach has been to set up a task environment that wouldbe used with elementary students and ask the teachers to anticipate howthe students’ work in that environment might progress. The teachers areencouraged to role-play the students. In the MTD Project, this activity withteachers builds on a shared notion that has developed during earlier coursework, that it is important to monitor whether what is attributed to studentsis the students’ voice or the teacher’s voice.

Although we are cautiously optimistic that such approaches can effectsome change, we do not know what reasonable expectations are nor howto maximize the impact of our efforts in this direction. Currently, we are

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analyzing more data from case studies and from the whole class teachingexperiment that might shed light on activities of the teacher educator thatcontributed to promotion of teachers’ sensitivity to students’ thinking. Thedata consisted of teachers’ struggle to understand why, for example, astudent could not make sense of a certain situation that they took forgranted. We think that through this work some teachers, such as Nevil,might have begun to develop a distinction between the child’s and theteacher’s voice, that is, their different mathematical experiences.

NOTE

1 The research was conducted as part of the activities of the NSF Project No. REC-9600023, Mathematics Teacher Development. All opinions expressed Are solely those ofthe authors.

REFERENCES

Ball, D.L. (1988). Unlearning to teach mathematics. For the Learning of Mathematics,8(1), 40–48.

Ball, D.L. (1996). Teacher learning and the mathematics reforms: What we think we knowand what we need to learn. Phi Delta Kappan, 77, 500–508.

Bereiter, C. (1985). Toward a solution of the learning paradox. Review of EducationalResearch, 55, 201–226.

Bickhard, M.H. (1991). The import of Fodor’s anti-constructivist argument. In L.P. Steffe(Ed.), Epistemological foundations of mathematical experience (14–25). New York:Springer-Verlag.

Cobb, P. (1989). Experiential, cognitive, and anthropological perspectives in mathematicseducation. For the Learning of Mathematics, 9(2), 32–42.

Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In R.Lesh & E. Kelly (Eds.), Research design in mathematics and science education (307–334). Hillsdale, NJ: Lawrence Erlbaum Associates.

Cobb, P. & Bauersfeld, H. (1995). Introduction: The coordination of psychological andsociological perspectives in mathematics education. In P. Cobb & H. Bauersfeld (Eds.),The emergence of mathematical meaning (1–16). Hillsdale, NJ: Lawrence ErlbaumAssociates.

Cobb, P. & Steffe, L.P. (1983). The constructivist researcher as teacher and model builder.Journal for Research in Mathematics Education, 14, 83–94.

Cobb, P. & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives inthe context of developmental research. Educational Psychologist, 31, 175–190.

Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist alternative to the representationalview of mind in mathematics education. Journal for Research in Mathematics Education,23, 2–33.

Cooney, T.J. & Krainer, K. (1996). Inservice mathematics teacher education: The import-ance of listening. In A.J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde


(Eds.), International handbook of mathematics education (1155–1185). Dordrecht,Netherlands: Kluwer.

Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to theeducative process. Lexington, MA: D.C. Heath.

Dewey, J. (1938). Experience and education. New York: Collier.Dewey, J. & Bentley, A.F. (1949). Knowing and the known. Boston: Beacon.Fodor, J. (1980). On the impossibility of acquiring “more powerful” structures. In M.

Piattelli-Palmarini (Ed.), Language and learning: The debate between Jean Piaget andNoam Chomsky (142–162). Cambridge, MA: Harvard University Press.

Glaser, B.G. & Strauss, A.L. (1967). The discovery of grounded theory: Strategies forqualitative research. New York: Aldine De Gruyter.

Grant, T.J., Hiebert, J. & Wearne, D. (1998). Observing and teaching reform-mindedlessons: What do teachers see? Journal of Mathematics Teacher Education, 1, 217–236.

Heinz, K., Kinzel, M., Simon, M. & Tzur, R. (in press). Moving students through steps ofmathematical knowing: An account of the practice of an elementary mathematics teacherin transition. Journal of Mathematical Behavior.

Lloyd, G.M. & Wilson, M. (1998). Supporting innovation: The impact of a teacher’sconceptions of functions on his implementation of a reform curriculum. Journal forResearch in Mathematics Education, 29, 248–274.

Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness andstructure of attention. Journal of Mathematics Teacher Education, 1, 243–267.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1991). Professional standards for teachingmathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (2000). Principles and standards for schoolmathematics. Reston, VA: Author.

Piaget, J. (1970). Genetic epistemology. New York: Columbia University Press.Piaget, J. (1980). The psychogenesis of knowledge and its epistemological significance. In

M. Piattelli-Palmarini (Ed.), Language and learning: The debate between Jean Piagetand Noam Chomsky (23–34). Cambridge: Harvard University Press.

Piaget, J. (1985). The equilibration of cognitive structures: The central problem of intel-lectual development (Trans. Terrance Brown and Kishore J. Thampy). Chicago: TheUniversity of Chicago.

Simon, M.A. (1995). Reconstructing mathematics pedagogy from a constructivistperspective. Journal for Research in Mathematics Education, 26, 114–145.

Simon, M.A. (2000a). Constructivism, mathematics teacher education, and research inmathematics teacher development. In L.P. Steffe & P. Thompson (Eds.), Radicalconstructivism in maths and science: Essays in honor of Ernst von Glasersfeld (213–230). London: Falmer.

Simon, M.A. (2000b). Research on mathematics teacher development: The teacher devel-opment experiment. In R. Lesh & E. Kelly (Eds.), Research design in mathematics andscience education (335–359). Hillsdale, NJ: Lawrence Erlbaum.

Simon, M.A. & Tzur, R. (1999). Explicating the teacher’s perspective from the researchers’perspective: Generating accounts of mathematics teachers’ practice. Journal forResearch in Mathematics Education, 30, 252–264.

Simon, M.A., Tzur, R., Heinz, K., Kinzel, M. & Smith, M.S. (1999). On formulating theteacher’s role in promoting mathematics learning. In O. Zaslavsky (Ed.), Proceedings of

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the 23rd annual meeting of the International Group for the Psychology of MathematicsEducation (Vol. 4, 201–208). Haifa, Israel.

Simon, M.A., Tzur, R., Heinz, K., Kinzel, M. & Smith, M.S. (2000). Characterizing aperspective underlying the practice of mathematics teachers in transition. Journal forResearch in Mathematics Education, 31, 579–601.

Steffe, L.P. & Cobb, P. (1988). Construction of arithmetical meanings and strategies. NewYork: Springer-Verlag.

Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers.Journal of Mathematics Teacher Education, 1, 157–189.

Tzur, R. (1999). An integrated study of children’s construction of improper fractions andthe teacher’s role in promoting that learning. Journal for Research in MathematicsEducation, 30, 390–416.

Vacc, N.N. & Bright, G.W. (1999). Elementary preservice teachers’ changing beliefsand instructional use of children’s mathematical thinking. Journal for Research inMathematics Education, 30, 89–110.

Varela, F.J., Thompson, E. & Rosch, E. (1991). The embodied mind: Cognitive science andhuman experience. Cambridge, MA: The Massachusetts Institute of Technology.

Voigt, J. (1995). Thematic patterns of interaction and sociomathematical norms. In P. Cobb& H. Bauersfeld (Eds.), The emergence of mathematical meaning (163–201). Hillsdale,NJ: Lawrence Erlbaum.

von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning.Washington, DC: Falmer.

Ron TzurMartin A. SimonCurriculum and Instruction Department,The Penn State University,270 Chambers Bldg,University Park, PA 16802,USAE-mail: [email protected]

Karen HeinzCollege of Science and Technology,Department of Computing and Mathematical Science,Texas A&M University,Corpus Christi, TX 78412,USA

Margaret KinzelDepartment of Mathematics and Computer Science,Boise State University,Boise, ID 93724-1555,USA

EDITORIALTHEORIES, OPPORTUNITIES, AND FAREWELL

The Journal of Mathematics Teacher Education (JMTE) is now fourvolumes, 13 issues old with contributions by authors from 20 differentcountries. About 300 articles have been processed. How might we assessthe value and contribution of JMTE at this early age? What challenges lieahead?

Recently, Marty Simon expressed concern about the quality of researchin mathematics education; he will convene a discussion group at PME-NA as a first step in fostering an examination of the purposes, means,and products of conducting high-quality mathematics education research.He is not the first, nor will he likely be the last, to express this concern.In some sense, the quality of research is a methodological issue thatmore recently has become blurred because of the increasing popularityof qualitative methodologies. How is it that we define rigor? But theproblem goes beyond that of methodology. It is also a matter of how weexplain what it is that we observe. The field of mathematics educationis inherently a field of practice, particularly so for mathematics teachereducation. The many interviews that my colleagues and I have conductedover the years with preservice and inservice teachers reveal teachers’ deep-seated commitment and professionalism generally housed in a pragmaticworld. Teachers are concerned with reaching their students as much from ahumanistic perspective as from a mathematical one, often creating a cleardichotomy between the two. Teachers at all levels consistently express theview that students should feel comfortable and not be frustrated by theirstudy of mathematics. Although this perspective is laudable in many ways,it tends to promote an atomistic approach to the teaching of mathematicsthat runs counter to those reform efforts that promote problem solving,reasoning, and communication. It is not uncommon for teachers to usethe phrase “step-by-step” when describing their teaching of mathematics.This should be no surprise given the pressures teachers face to finish agiven curriculum, to promote student achievement defined narrowly asperformance on high stakes tests (especially in the United States), and to


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attend to the many pressures from a variety of sources that often “deskill”their teaching. Given this circumstance and the fact that those of us whoregularly work with teachers desire to help them reconceptualize whatconstitutes “good” mathematics teaching, it should not be surprising thatwe, too, get entangled in the practical. I suspect that Simon’s concernabout the quality of research, its lack of rigor, and its sometimes athe-oretical nature is likely rooted in the very nature of the practice itself.Our propensity to deal with the practical snares us in the practical aswell.

Nevertheless, research by its very nature seeks to explain, not simplydescribe. As my major professor, Ken Henderson, once said, “An engineeris one who has the theory to determine if the bridge will collapse before thetruck crosses it. Any fool can determine whether the bridge will collapseafter the truck crosses it.” The distinction between explanation and descrip-tion lies in the generality of the explanation, that is, its ability to capturethe web in which the particulars are located. In an effort to develop webs,that is, theoretical constructs that help explain, it is interesting to note therebirth of deep thinkers like Dewey and Vygotsky. No doubt our increasedattention to reflection and context as two pillars of our explanations hascontributed to this rebirth.

What, then, might the contribution of JMTE be toward the develop-ment of constructs that help us explain? A review of the articles containedin the first four volumes of JMTE reveals that considerable attention hasbeen paid to a broad array of constructs. Still, one has the feeling thatthe literature review in many of the articles is intended more for thepurpose of providing a longitude and latitude of the study rather thanbuilding schemes or constructs per se. If so, then the question arises asto whether our collective research can provide a foundation necessary forsubstantial progress other than providing better locations for subsequentresearch. The problems we study may be inherently practical but surelytheir solutions (in the sense of greater wisdom) lie in our ability to seeteaching and teacher education outside the confines of the acts themselves.This makes life difficult. It is relatively easy for us to see teaching froma more abstract level if we are not part of the act. In teacher educa-tion, however, teacher educators are indeed part of the act. So how is itthat we can educate ourselves to engage in Dewey’s notion of reflectivethinking or to follow von Glasersfeld’s notion of reflection which requiresus to step out of ourselves and see our actions from a vantage pointbeyond our actions themselves? It is only from this meta-vantage pointthat we can begin developing schemes and constructs that capture the

EDITORIAL 257

webs in which we are all entangled. Simply put, we cannot engage inan analysis of our own activity using only the language of that activityitself.

I leave it to others to resolve the question of whether JMTE has contrib-uted to the kind of theoretical dialogue of which Simon and others speak.What surely is the case is that JMTE provides at least the opportunity tocontribute to the desired dialogue. It is my fervent hope that when thenumber of volumes of JMTE has doubled, we can sense movement towardour goal of better explanations about the practice of teaching and teachereducation.

Whatever credit is due JMTE for contributing to the current dialoguebelongs to a collection of individuals including: (a) authors who haveexpended considerable energy in writing articles that address issues inmathematics teacher education, (b) reviewers who have so faithfully sharedtheir insights and expertise with authors, (c) Editorial Board members whohave doubled as reviewers and contributors to the policies that guidedJMTE, and (d) Associate Editors who have been so instrumental in shapingthe early direction of JMTE as evidenced by the statement of intentfound in the inside back cover of this and every issue. A special thanksgoes to Heide Wiegel for her work and unwavering support as ManagingEditor of JMTE. Authors know firsthand of her expertise and dedication infinding the structure and words that better enabled them to communicateeffectively with their peers.

Time flies. It seems but such a short time ago that Peter de Liefde fromKluwer Academic Publishers first contacted me about the possibility ofinitiating a journal devoted solely to mathematics teacher education. Itwas decided that a book on mathematics teacher education would firstbe published as a forerunner to the establishment of a journal. Chapterswere solicited with a deadline of fall, 1996. We expected approximately40 submissions but received nearly 100, which subsequently led to thedecision to launch the journal sooner rather than later. It took six monthsto process those 100 submissions which resulted in the first issue of JMTEpublished in early 1998. It has been a short five years but a time thatI cherish greatly for it provided me the opportunity to work with manytalented and dedicated professionals. And so I now pass the editorshipof JMTE to my esteemed colleague, Dr. Barbara Jaworski, whose careerhonors both the practice and the theory to which I have previously referred.May her journey with authors and reviewers be as rewarding as mine. Iappreciate the opportunity to have been a part of JMTE’s maiden voyage.Barbara, may all of your authors be as professional as the ones with whom I

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have worked. Let us move onward to even better explanations that enhancethe theory and practice of mathematics teacher education.

Thomas J. CooneyEditor, JMTE

RON TZUR

BECOMING A MATHEMATICS TEACHER-EDUCATOR:CONCEPTUALIZING THE TERRAIN THROUGH

SELF-REFLECTIVE ANALYSIS

ABSTRACT. My purpose in this article is to contribute to the conceptualization ofthe complex terrain that often is indiscriminately termed mathematics teacher educatordevelopment. Because this terrain is largely unresearched, I interweave experiencefragments of my own development as a mathematics teacher educator, and reflectiveanalysis of those fragments, as a tool to abstract notions of general implication. Inparticular, I postulate a framework consisting of four stages of development that aredistinguished by the domain of activities one’s reflections may focus on and the natureof those reflections. Drawing on this framework, I present implications for mathematicsteacher educator development and for further research.

During the third year of my work as an assistant professor I encountered aproblem that is not uncommon. A prospective doctoral student canceledher participation in the program. Thus, we needed an instructor for amathematics education methods course for preservice teachers. I askedKelly, a graduate student who had been working for two years as a researchassistant in the Mathematics Teacher Development (MTD) project, if shewould take on the responsibility. On the intuitive basis of knowing Kelly asa developing researcher and as a student in courses I had taught, I expectedthat she would be able to take on the responsibility. Most important, myidea was that adding teaching experience to Kelly’s research work wouldpromote her development.

While talking about my idea with Kelly and my colleagues, it soonoccurred to me that I had never before articulated the perspective under-lying the idea of combining teaching and research. In fact, I had not artic-ulated the perspective underlying my practice as a mentor of developingmathematics teacher educators, nor had I articulated my own developmentas a novice teacher educator. Therefore, I decided to search the literature.To my surprise, I could find generic studies on development of teachereducators (Diamond, 1988; Ross & Bondy, 1996) or on such develop-ment in other disciplines like English teaching (Farrell, 1985), but onlyone (Onslow & Gadanidis, 1997) that focused on development of mathe-matics teacher educators. The void in the literature played a key role in my


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decision to write this article. In it, I report my attempt to begin conceptu-alizing the terrain of mathematics teacher educator development throughself-reflective analysis.

The problem of how mathematics teacher educators develop is animportant aspect of current reform in mathematics education (NationalCouncil of Teachers of Mathematics, 1991, 2000). This reform callsto question assumptions about the nature of mathematics, mathematicslearning, and mathematics teaching that underlie mathematics teachers’traditional practices. Such reform depends on “the individual teacherin the classroom and the body of teachers within a country” (Howson,1992, p. 9). It requires extensive work with teachers who learned throughand are using traditional methods of teaching. This, in turn, implies thecritical need for a large body of mathematics teacher educators who under-stand and can foster change in traditional practices of prospective andpracticing mathematics teachers. But how do these educators develop?In my conceptualization of mathematics teacher educator development Ifollowed Hamilton and Pinnegar’s (2000) plea to hear the “passionatevoice from the heartland of teacher education, the voice of the teachereducator reflecting upon and critiquing the possibilities for a foundation forteacher education” (p. 234). My goal is to suggest a way of thinking aboutand carrying out desired teacher-educating practices, that is, for clarifyinggoals for and potential processes of mathematics teacher educator devel-opment. To this end, I reflected on, made explicit, and elaborated on theimplicit and informal base of my personal story of moving from the roleof student to the role of teacher, teacher educator, and mentor of teachereducators. The premise is that through self-study one’s private theory canbe “transformed to public theory that could then support the study ofteacher education by others as well” (p. 239).

I organized the article as follows. First, I present a conceptual frame-work that revolves around the core idea that development takes placevia reflection on activities. Second, I describe key aspects of the methodused for the study, that is, the presentation and analysis of fragmentsof experience. Third, I interweave narratives and analyses of experiencefragments of my own development. Building on that story, I present away of conceptualizing mathematics teacher educator development thatconsists of four interconnected foci of reflection: (a) learning mathematicsas a student, (b) learning to teach mathematics as a teacher, (c) learningto teach mathematics teachers as a teacher educator, and (d) learningto teach mathematics teacher educators as a mentor. Finally, I delineateimplications of the four-foci model for mathematics teacher educatordevelopment.

BECOMING A TEACHER-EDUCATOR 261

CONCEPTUAL FRAMEWORK

The framework that structures my thinking about teaching-learningprocesses is a social constructivist one (Cobb & Bauersfeld, 1995). Inparticular, two foundational constructs of the works of Dewey (1933,1938), Piaget (1970a, 1985), and Schön (1983, 1987), reflection and inter-action, guided my study. I consider reflection on one’s actions and on thoseof others while interacting with people and objects in the environmentto be the mental root of conceptual development. I am aware that theterm social constructivism was not used at the time of Dewey and Piaget.However, I consider their work as social constructivist for two reasons:(a) the centrality of social interaction to the very process of reflection and(b) the rejection of a positivistic view of the mind and the epistemologicalemphasis on the role of human experience in the formation of knowledge.Below, I present a brief summary of Dewey’s, Piaget’s, and Schön’s views.

As a philosopher and educator, Dewey (1933) introduced the constructof reflection on the relationship between activity and consequences toexplain how, building on their experiences, human beings both thinkand advance their thinking. He asserted that the human mind continu-ally identifies particular consequences to follow particular activitiesand explained the abstraction of new conceptions as the formation ofnew activity-consequence relationships via reflection on unanticipated,surprising consequences of known, goal-directed activities. Accordingly,Dewey asserted that the art of teaching consists of the teacher’s ability toanalyze students’ current conceptual state, to stimulate perturbing expe-riences, and to direct their reflective thinking on activity-consequencerelationships.

Trained in biology, Piaget (1970a) took on the epistemological task ofexplaining how the human creature as an organism – a body capable ofsensual experience – is capable of forming logical thinking. Like Dewey,Piaget opposed the view of knowledge as a copy of the real world. Rather,he asserted that “human knowledge is essentially active. To know is toassimilate reality into systems of transformations. To know is to transformreality in order to understand how a certain state is brought about” (p. 15).To construct knowledge means, then, to regulate, reflect on, and coordinaterelationships among actions that succeed or fail to bring about the goaldetermined by a subject’s scheme. The coordinated relationships are inter-iorized to become abstract operations and structures that can be carriedout mentally, in anticipation. Accordingly, Piaget (1970b) emphasized theteacher’s need to carefully observe students’ actions as a means to analyzetheir current assimilatory schemes and to engage the students in activitiesand reflection on them appropriate to transforming their current schemes.

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Further, he suggested that a similar approach of inquiry and reflection mustbe employed by those who educate teachers in order for the teachers toconstruct and utilize a method of active teaching.

Schön (1983, 1987), a professor of urban studies and education,attempted to explain and provide guidance to professional practitioners’development. Like Dewey and Piaget, he proposed that at the heart of thatdevelopment is the process of reflection. In particular, Schön proposedwhat he called an epistemology of practice, where people constructprofessional knowledge by recursively applying reflection-in-action whileparticipating in a community of practice. Recursive means that reflection-in-action can be turned into the object of the reflective process. One canreflect on the result of the reflection-in-action and produce a good descrip-tion of it; then one can reflect on the result of the latter reflection by givingmeaning to that description. It is noteworthy that by carefully observingprofessionals in diverse fields to explain how they become practitioners,Schön abstracted the very notion of reflection as the core process of devel-opment. That is, the learning of highly specialized types of knowledge,such as teaching, is made possible via active engagement in tasks of thetrade and reflection, reflection that can be intentionally guided by expertsin the field, on one’s goal-directed actions.

In recent years, the notion of reflection became central to thinkingnot only about mathematics learning but also about teacher education(Edwards, 1996; Richert, 1993; Smith, 1994). For example, Cooney andKrainer (1996) contended that “teacher education programs are becomingless technically oriented and more process or constructivist oriented withan emphasis on reflection and self analysis” (p. 1159). A specific approachthat views teaching as a practice and hence advocates the integration ofreasoning and knowing with action (Ball, 2000) is the so-called actionresearch. As proponents of that approach, Crawford and Adler (1996)suggested that research should be woven into teaching and learning activ-ities, where research is thought of as an inquiry process of reflection onquestions and dilemmas that are personally meaningful to the teacher. Itis in this sense that the next two sections, the method of reflecting onexperience fragments and the reflection-based analysis, are organized.

METHOD: REFLECTING ON EXPERIENCE FRAGMENTS

To conceptualize the process of becoming a teacher educator I used themethod of narrating and interpreting fragments of experience. In thissection I briefly present key aspects of this method and its appropriatenessfor that purpose. I build on Polkinghorne’s (1988) work regarding the study


of human experience because “narrative is the fundamental scheme forlinking individual human actions and events into interrelated aspects of anunderstandable composite” (p. 13).

A narrative research report recreates a history that leads to a story’send by pointing to significant factors that brought about that end. Thus,fragment-based narrative research is “retrodictive rather than predictive,that is, it is a retrospective gathering of events into an account that makesthe ending reasonable and believable” (p. 171). It should be noted thatmy use of the term “fragment” only partly overlaps with Mason’s (1988)use of it. Like Mason, I think of a fragment as “a recallable incidentwhich spans a single block of time” (p. 198), a brief-but-vivid account thatcaptures an essential aspect of one’s experience. However, I do not requirethat different observers can agree upon the event described in a fragment.The reason is that, like Polkinghorne (1988), I conceive of fragments andnarratives as necessarily an individual’s construction at all three levels ofnarrative: living, telling, and interpreting the experience.

The use of fragments as tools to conceptualize teacher-educator devel-opment is appropriate and valuable because the fragments serve as a meansto recall and reconstruct past experience into a meaningful story (Polking-horne, 1988; Mason, 1988). Such reconstruction is based on the premisethat people organize distinct temporal experiences into meaningful wholes.Or, as Clandinnin and Connelly (1994) succinctly put it, “when personsnote something of their experience, either to themselves or to others, theydo so not by the mere recording of experience over time, but in storiedform” (p. 15). The researcher makes use of this by working backward,from identifying the purpose of telling the story to capturing its theme todeciding on its beginning, middle, and end.

As a type of qualitative research, the authority of narrative research isnot judged by some technical, statistical tests, but by four general criteria:plausibility, credibility, relevance, and importance of the topic (Altheide& Johnson, 1994). That is, in narrative research the term valid retains itsordinary meaning of well-grounded and supportable; the term significancerefers to the importance of the issue at the story’s focus (Polkinghorne,1988). Yet, to be of general significance beyond an idiosyncratic story, thefragments chosen need to have a clear and significant face value that isreflected in and resonates with the experience of others in the community(Jaworski, 1999). Thus, the researcher has to select and weave past eventsinto a thick description (Geertz, 1988) of context, intentions, and meaningsso that the events reveal the experience as a process – the genesis of theoutcome. It is out of such thickly described process that the text’s claimfor truth, or its verisimilitude, arises (Denzin, 1994).

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REFLECTING ON EXPERIENCE FRAGMENTS

In this section I interweave narratives and analyses of experience frag-ments that greatly impacted my current work as a teacher educator and amentor of teacher educators. The experiences are ordered according to myunderstanding of the sequence in which the four foci were developed. I willelaborate on relationships among those foci later. However, at this point itis critical to note that I view the four foci as highly interconnected anddistinguish among them on the basis of what issues serve as material onwhich the learner reflects and on differences in the nature of the reflectiveprocess.

I am aware that as one reads and reflects on the fragments, questions andthoughts about their meaning and implications arise. However, due to thesequential nature of language, one has to choose an order to ease the flowand enhance clarity. Instead of addressing the meaning and implication ofeach fragment separately, I organized the story as follows. First, I presenta series: fragments, followed by an analysis of their significance, followedby additional fragments and their analyses. Then, I reorganize the storyinto a four-foci model of development. Finally, by revisiting examplesfrom the story in light of the model, I address implications for teachereducation in terms of how to think about and devise activities appropriatefor promoting development.

Learning Mathematics

I began learning mathematics at my kibbutz elementary and high schools.A kibbutz is a small community operated by its members on the basis ofsocialist ideals such as equality, cooperation, and mutual aid. To educatethe young toward daily realization of those ideals, the kibbutz schoolwas operated as a mini-community. Guided by adult educators, studentsnegotiated norms and practices that allowed them to gradually take respon-sibility for academic, social, and physical aspects of their lives. Forexample, as a capable student I was encouraged to tutor my peers andto serve as a leader for younger students. Below, I present a fragmentpertaining to my learning through peer tutoring.

My peers knew of my affection for mathematics and frequently asked me to help themdoing their homework. My usual technique was to work one or two problems step-by-step,then ask my peers to try the next problems, problems that for me seemed just the same.For example, I remember how I tried to help Tammy solve pairs of two-variable linearequations. Time and again I would show her how I multiplied each equation by a numberand then added the equations to cancel one of the variables and solve for the other variable.Moreover, I explained to her what I saw as the reasons behind the procedure we were taughtin class, e.g., which numbers would be useful in the multiplication, and enjoyed how all the


pieces fit in the process of solving the problem. However, Tammy was stuck and frustratedany time she faced a new pair of equations.

Often, the struggle to figure out how to explain the particular material to my peersbrought about new insights for me. In Tammy’s case, I realized that systems of equa-tions are invariant under certain operations. My peers’ struggles also made me curiousabout their understanding and learning, which fostered my decision to major in educationduring my junior and senior years in high school. I took courses that introduced me to theeducational and psychological thought of Dewey, Freud, Piaget, and Vygotsky. Despitemy learning in those courses, however, at the time I continued to use the same tutoringtechniques.

In reflecting on these fragments, I noticed three significant aspects: (a)the social context in which I learned, (b) the advancement of my mathe-matical understanding, and (c) the lack of change in my ways of tutoring.Regarding the social context, it seems that my tutoring was encouraged andaffected not only by the generic social norm of helping others, but also byour classroom socio-mathematical norms (Yackel & Cobb, 1996) of beingexpected to justify a solution and to make connections between resultsof different problems. My individual participation in class and homeworkactivities was guided by such expectations. It was that socially-rootedgoal that oriented my interactions with peers and the articulation of themathematics to be taught via reflections on those interactions.

Regarding my learning, it seems that my continuous reflections onthe activities that I used to explain my solutions to others played a keyrole in advancing my mathematical understanding. To solve problems andreason about my solutions, I had to (a) continually interpret the informationprovided in the problem, (b) set some image of what would constitute asolution to the problem, (c) carry out activities to work toward the solution,and (d) relate the effects of my activities to the solution image. Throughcycles of such activities I noticed new mathematical regularities in activity-effect relationships, such as invariance of systems of equations. A keyaspect of this learning is that it was the tutor who learned mathematicsvia reflection on teaching activities and on learners’ work in response tothese activities.

I did not understand that many of my peers could not see what I sawbecause their mathematical conceptions were different than mine. Thus,I did not change my ways of tutoring. My reflection was focused on myown understanding of the mathematics, not on others’ thinking and howit might be changed via their activities and reflective processes. Simplyput, I used activities in an attempt to promote others’ mathematics, but Idid not reflect on the relationship between those tutoring activities and theeffects in terms of my peers’ learning. In retrospect, my tutoring attemptscan be characterized as being rooted in what Cobb, Yackel and Wood

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(1992) termed the representational view of the mind. This view assumesthat people come to see identical mathematical ideas because these ideasare formed as a mirror of some external reality that exists independent ofthe people’s experience. I saw the mathematics in the situation, assumedmy peers could also see it, and did not question what they were seeing inthat situation. Altogether, as the teacher I learned mathematics but verylittle about teaching; it is not clear what learning took place on the part ofthe students.

Learning to Teach Mathematics

A few years after high school I decided to pursue my dream of becomingan educator and a mathematics teacher at the kibbutz high school. Istarted my formal studies in 1982, at the kibbutz branch of Haifa Univer-sity School of Education (“Oranim”). I chose a program that combinedhigher-level mathematics including algebra, calculus, differential equa-tions, and geometry with foundations of education and psychology,and with increasing responsibility for teaching students in experiencedteachers’ classrooms. Being intrigued by educational questions, I added tothe program a minor in social education that addressed issues in cognitiveand social psychology, sociology, pedagogy, and the kibbutz educationalthought.

Two aspects of that period stand out as significant to my development.First, my teacher educators were supportive of my decision to take on thefull responsibility as a teacher at the kibbutz high school while completingmy studies as a part-time student. Second, they encouraged me to engage inconducting mini-research projects on issues as diverse as specific historicaldevelopments in mathematics, gender differences in kibbutz labor distribu-tion, or the chemical effects of drug abuse on the brain. This combinationof teaching, studying, and researching led me to actively search for andexperiment with activities that might promote my students’ understandingand to inquire into theories that could shed light on regularities I noticedin their difficulties. Specifically, I became deeply involved in caring forand exploring the learning of failure-experienced students who struggledto understand basic mathematical ideas and to stay in mathematics courses.Below, I present fragments pertaining to that period.

Teaching failure-experienced students was challenging and rewarding. I was pleased whentheir work indicated progress and felt somber when they were frustrated and stuck. To helpmy students understand key mathematical ideas that an expert might consider rather simpleand unproblematic, I would take those ideas apart and look for the underlying pieces andconnections. I selected or created activities that the students would use to understand thepieces and connections I saw.


An example I remember vividly was my effort to help the students learn about theCartesian coordinate system. I did not want them to merely execute procedures such asplacing points of given (x,y) values on a graph or reading such values from a given graph.Rather, I wanted to initiate my students’ learning in a way that would give them a sense ofthe problem for which the coordinate system was a solution. After some intensive searchfor such a method, I created a navigation problem that allowed for giving oral directionsonly. They were to take turns in guiding a volunteer, whose eyes were closed, to walk safelyfrom the far end of the classroom and precisely put her or his finger on a point marked onthe blackboard. As students struggled to communicate directions to the volunteer, theyrealized the need to establish a common frame of reference in the classroom space, andparticularly on the blackboard. Most importantly, it was the reflection on my students’activities, not just my mini-research on the topic of Cartesian systems, that fostered mydeep understanding of the random nature of the origin point of the coordinate system.

I noticed three significant aspects of my early teaching experiences– my continuous learning of mathematics and how it differed from mystudents’ learning, my instructors’ expectation that preservice teachersshould conduct research, and my learning to teach as a result of reflectingon related activities of teaching, course work, and research. Just like intutoring, the first aspect highlights my significant mathematics learningwhile being involved in preparing for and implementing the teaching ofspecific ideas to others. In the example of the coordinate system, I had agoal – figuring out ways to engage my students in actively experimentingwith methods of locating points in the plane. In turn, the students’ activity,and particularly their arguments about where to locate the origin of thecoordinate system, challenged me to re-examine my own view of the issue.Through listening to and reflecting on their arguments, I re-establishedfor myself the random nature of that location, a concept that I previouslyunderstood rather superficially.

Common to my learning and my students’ learning were the activitiesused in class. However, the students’ learning was different than minebecause the focus of their reflection was different than mine. My students,it seems, were focusing on the task of leading a peer to a certain point.Via reflection on incidents of not getting the peer to that place the studentsabstracted the need to establish a common frame of reference with thepeer. Having formed this goal they could then easily make sense of theCartesian coordinate system that I presented because it reoriented andreorganized reflection on their own attempts. Unlike my students, the focusof my reflections was not on creating a frame of reference but on, amongother things, the different points of reference they were using on the board.Thus, I articulated further the previously loosely noted idea that the originof a coordinate system is a matter of choice and convenience. I conjecturethat for my students to learn what I had learned, their reflection wouldhave to be refocused. Once they had established the use of a coordinate

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system we could have gone back to the activity from which it originatedand reoriented their reflection.

With respect to the two other aspects, the program seemed to emphasizethe importance of research and the integration of the diverse learningexperiences. Being simultaneously engaged in the activities of teachingmathematics, conducting research, taking courses, and working withcolleagues provided me with many situations in which to reflect on myown teaching, as well as the teaching of others. The fragments above illus-trate my active and systematic experimentation with tasks and problemsthat I found in various sources and adjusted to my students’ experience,or generated on my own. When the activities failed to bring about theintended learning, I felt responsible for adjusting them. The continualadjustment process focused my reflection on relationships between myteaching activities and the effects of those activities in terms of students’learning. This was key to my learning to teach. I tried to explain to myselfwhy specific teaching interventions did or did not advance my students’understanding. Consequently, in my view of teaching I differentiated andreorganized aspects of my teaching that were useful to my students’ under-standings from those that were not useful. In this sense, it seems that myundergraduate program encouraged a process of learning to teach that iscompatible with Schön’s (1983, 1987) notion of the reflective practitioner,and with recent trends of teacher action research (Crawford & Adler, 1996;Jaworski, 1998). Or, as my students sometimes prudently commented, Iwas improving as a teacher. My improvement also resulted from guidingother teachers, an experience I report on next because it highlights mylimited understanding of teachers’ learning at the time.

Learning to Teach Teachers

While deliberately committing myself to full-time teaching of mathematicsto noncollege-bound students, I began my studies toward a master’s inmathematics education at the Technion-Israel Institute of Technology. Ijoined a research and curriculum development project (Movshovitz-Hadar,1992) responsible for creating materials for noncollege-bound studentsacross Israel, and I used those materials in my own teaching. In the project,I focused on developing an alternative assessment method that would beintegrated into the learning materials and might promote successful expe-riences in those students. Concurrently, I studied the impact of using thatassessment method on students and teachers (Tzur & Movshovitz-Hadar,1998). Below, I present fragments pertaining to that period.

In addition to using the project materials in my teaching, I traveled every week to oneof six high schools and met with teachers to promote their understanding and use of the


materials. During those visits, I collected data from observations, interviews, and surveysfor my thesis. The teachers’ use of the curriculum often indicated misunderstandings ontheir part. By communicating how I thought of the materials and demonstrating ways Iwould use them, I tried to encourage the teachers to not only use the materials as given, butalso to understand the underlying pedagogical reasoning and to tailor teaching activities tostudents’ work and understanding.

In working with the teachers, I observed that quite often they seemed to miss points thatI considered essential. For example, I remember how I tried to convince Joe to let studentssolve problems before telling them procedures and rules of solution, as a way to learn aboutsine relationships in right triangles. I also suggested specific activities, which I was usingwith my students, such as figuring out measures of the sides of large triangles. Yet, Joecontinued to first teach the procedures and rules as a means to obtain students’ mastery, atwhich point he considered them ready to solve problems.

In reflecting on these fragments, I noticed two significant aspectsin addition to the obvious continuation of my learning of mathematics– my learning to teach as a result of integrating activities of researchand teaching, and the lack of significant changes in my attempts topromote reasoning about teaching-learning processes in teachers like Joe.Regarding the first aspect, my engagement in writing my thesis and inreflecting on my activities with teachers enabled me to notice relation-ships between specific teacher interventions and students’ learning. Whilereflecting on my own teaching activities and on activities used by teacherslike Joe, I abstracted, for example, relationships between specific teacherquestions and students’ ability to (a) interpret the questions and (b) initiatesome solution activities. The work on my thesis greatly enhanced myrecognition of their relationships, particularly as I interacted with the ideasof Piaget and Vygotsky. Consequently, I began using interaction as a keyconstruct for integrating teaching-learning processes, and I distinguishedassessment as a specific case of interaction.

Regarding the lack of change in my work with teachers, it seems thatI did not understand the conceptions about mathematical knowledge andlearning underlying the teachers’ work. That is, my view of mathematicsteaching was not similar to the teachers’, but just as in my work as a tutor Ifailed to make this distinction and to form a model of how the teachersmight have seen it. For example, at the time I did not understand theproblematic nature of my attempts to “give” my ideas to Joe. I failed toconsider his understanding of learning and teaching and how it affordedand constrained the sense he made of my suggestions. Moreover, I didnot think of learning as problematic and I did not have an explanatorymechanism as to how one can see a mathematical or pedagogical conceptin particular situations, let alone how one might develop new concepts.

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Learning to Mentor Teacher Educators

In 1992 I started my doctoral program at the University of Georgia.Although my previous work as a high school teacher and as a guide toothers’ mathematics teaching included the teaching of teachers, it was onlythen that I began to seriously consider issues of teacher education. Aftera year of working primarily as a research assistant on two major projects,I took on a teaching role. I was asked to team-teach, with a new doctoralstudent (Lori) who had little teaching experience, a mathematics courseand two methods courses for elementary preservice teachers. Upon gradua-tion in 1995, I assumed an assistant professor position in the College ofEducation at the Pennsylvania State University. Below, I present fragmentsthat pertain to my learning to teach teachers and teacher educators.

Entering a doctoral program in the U.S. was a major cultural transition. My mentorsencouraged me to pursue my interest in articulating relationships between several theore-tical approaches and the real-life experiences of students and teachers. I clearly rememberhow I was stuck when Les Steffe asked me: “When interacting with children, how doyou know that you met a scheme?” Such questions sparked periods of intensive thinking,reading, and conversations with my peers and mentors. In particular, while reflecting onschool teachers’ assessment activities or struggling to generate useful fractional tasksthrough reflection on children’s activities, I developed a deep interest in the teacher’s roleand ways of thinking.

Team-teaching with Lori turned out to be a significant experience. We shared fullresponsibility in planning, implementing, and reflecting on our teaching of reform-orientedmathematics and pedagogy to the prospective teachers. Having excitedly worked withthe students throughout the course, I was startled when I received their low rating ofmy teaching. Their feedback threw me into an intensive period of reflection. I painfullyrecognized my failure to understand the teachers’ experience as a critical first step toteaching them anything. I discussed those issues with Lori and listened to her intuitions.For example, she told me that many of them were confused about the reform-orientedlearning situations in which we engaged them. She said students were concerned aboutsituations that seemed to promote learning experiences yet were perhaps frustrating. I nowknow that what I failed to consider was their traditionally-oriented experience, that is, theirway of making sense of the reform-oriented goals and activities I was so excited about.

At Penn State, I tried to translate the “Aha” experiences of my research into newor revised methods courses for teachers and to actively engage graduate students inthe revision process. In the last five years, I have been struggling with Marty Simon’squestion, “Do you have a way of thinking, a theory, about graduate students’ learningprocesses?” For example, halfway into a graduate seminar I realized that just readingand conversing about children’s mathematics resulted only in students’ superficial under-standings. Consequently, I engaged the graduate students in activities of preparing for andinterviewing a child on which they could reflect to make sense of a theory about children’searly number conceptions. Such adjustments in my teaching intensified the question I cameto view as preliminary to Marty’s question: “What are graduate students’ perspectives ofmathematics education and, in particular, of mathematics teacher development and theways one might promote such development?” In part, that question provoked my work onthis paper.


In reflecting on these more recent experiences, I noticed two significantaspects. First, I noticed the great impact that culture has on one’s learning.Second, I noticed how one’s explicit ways of thinking about learning,and particularly about the development of teachers and teacher educators,empowers one’s acting as a teacher educator or a mentor.

The first aspect seems to be a case of the general notion of situ-ated learning (Lave & Wenger, 1991). Three examples in my personalexperience highlight this notion. First, the continual comparison betweenteaching in Israel and in the U.S. was affording a new awareness almostdaily. Culture change intensified my learning by pushing me to see oldexperiences in new light, hence to abstract new ideas about teaching andlearning to teach. Second, Lori’s cultural background and her ability tosense the students’ culturally rooted concerns and interests afforded mygrowing sensitivity to cultural issues. In this sense, team-teaching inten-sified my development because it constantly brought forth reflection onrelationships between my teaching activities and their results throughobserving Lori’s teaching and receiving her feedback on my teaching.Third, as my experience indicates, the activity of writing serves as a mainvehicle for constant, rigorous, and deep reflection on and reorganizationof thinking about teaching and teacher educating. Consequently, I am nowmuch more aware of the need to analyze the sense that my students makeof the cultural contexts in which they work. This analysis greatly impactsthe second aspect of my learning.

The second aspect, the empowering impact my growing understandingof teaching-learning processes had on my work, was indicated in twoshifts: from teacher to teacher educator and from teacher educator tomentor of teacher educators. My understanding grew out of a rigorousarticulation of perspectives of teaching (Simon et al., 2000). In partic-ular, we postulated a new construct – conception-based perspective – toemphasize how a teacher thinks about the ways extant knowledge affordsand constrains what learners can perceive, understand, and do in givensituations. I now realize how the lack of such an articulation limitedthe repertoire of plans and interactions that I or other teacher educatorsand mentors could generate and use in teaching. For example, whenteaching a course on research literature in mathematics education, I noticedthat my students experienced great difficulty in reading studies based onVygotsky’s theory. I inquired into this and found out they never studiedVygotsky nor conceptualized his key constructs, such as Zone of ProximalDevelopment or Mediation by Tools. Through a dialogue with my studentswe together decided to change the course and to focus on learning aboutVygotsky’s work. I did this both by selecting appropriate readings and by

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engaging them in solving mathematical problems in a computer learningenvironment so that they might reinvent key constructs of Vygotsky’stheory out of their solution processes.

CONCEPTUALIZING THE TERRAIN OF MATHEMATICSTEACHER EDUCATOR DEVELOPMENT

In this section, I first present a four-foci model of teacher educator devel-opment. Then, by revisiting examples from my own story, I discussimplications of the model for the development of teacher educators.

A Four-Foci Model of Teacher Educator Development

In reflecting on and reorganizing my story, I tried to differentiateconstructive activities that are idiosyncratic from those that seem to betypical of many teacher educators. For example, specific constraints andneeds during graduate school may have been unique to my story. Yet, theimpact of the interplay between teaching and research seems to be thegeneral case.

A key point I noticed in my story is the recursive, non-linear nature of ateacher educator’s development, which I abstracted into an interconnectedfour-foci model. I distinguish among foci according to the explicit roleof the learner, a student, a teacher, a teacher educator, and a mentor ofteacher educators, in terms of the materials one reflects on and the natureof this reflection. Advancement to a higher -level focus proceeds throughreflecting on activity-effect relationships at the lower level(s). A level isconsidered higher in that the reflective process engenders a conceptualreorganization of practices used at the lower level(s). Thus, each higherlevel focus embodies the lower level foci; it encompasses new, explicitlyintegrated ways of thinking of what at the lower level was used implicitlyand/or locally. For example, a mathematics teacher carries out activities tounderstand students’ learning of mathematics, which may result in relatedmathematics learning in the teacher.

The first focus. Students develop mathematical ways of thinking and parti-cipating in community discourse and activity. Through reflection, studentsmay become aware of the perspectives that underlie their mathematicalpractice in terms of what it means to (1a) solve problems, reason, andcommunicate mathematically, and (1b) connect mathematical and non-mathematical experiences (NCTM, 2000). In this sense, a teacher educatorcan be a student of mathematics while teaching students or while workingto promote teachers’ mathematical and/or pedagogical understandings.


Figure 1. Focus 1.

Figure 2. Foci 2 and 1.

The second focus. Mathematics teachers develop ways of thinking aboutand intentionally participating in their students’ mathematics learning.Through reflection (indicated by the arrow in Figure 2), mathematicsteachers may become aware of the perspectives that underlie their teachingpractice in terms of (2a) what it means to know mathematics, (2b) howsomeone comes to know mathematics, and (2c) how teacher activitiespromote mathematics learning in others. In this sense, a teacher educatorcontinues to be a student of mathematics while teaching mathematics toteachers and a student of mathematics teaching while teaching pedagogyto other teachers.

The third focus. Mathematics teacher educators develop ways of thinkingabout and intentionally participating in others’ learning to teach mathe-matics. Through reflection (indicated by the arrows in Figure 3), teachereducators may become aware of the perspectives that underlie their teachereducation practice in terms of (3a) what it means to teach mathematics,(3b) how someone comes to know how to teach mathematics and (3c) howsomeone’s activities promote others’ learning of mathematics teaching.In this sense, being teacher educators themselves, mentors of teachereducators continue to develop as students of mathematics and/or pedagogy.

The fourth focus. Mentors of mathematics teacher educators develop waysof thinking about and intentionally participating in others’ learning to

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Figure 3. Foci 3, 2, and 1.

Figure 4. A four-foci model of teacher education.

educate mathematics teachers. Through reflection (indicated by the arrowsin Figure 4), mentors may become aware of the perspectives that underlietheir practices in terms of (4a) what it means to educate mathematicsteachers, (4b) how someone comes to know how to educate mathematicsteachers, and (4c) how someone’s activities promote others’ learning ofhow to educate mathematics teachers.


A key to understanding this four-foci model is that development froma lower to a higher level is not a simple extension, that is, doing more andbetter of the same thing. On the contrary, development entails a conceptualleap that results from making one’s and others’ activities and ways ofthinking at a lower level the explicit focus of reflection (cf., Cooney &Krainer, 1996; Edwards, 1996). Through such reflection, the developingteacher educator may construct conceptions about learners and learning atthe lower level(s), conceptions that become the theoretical ground for one’spraxis. One may question the four-foci distinction by noting, for example,that all foci include mathematical thinking. However, I emphasize that evenin the case of mathematical thinking, let alone pedagogical and epistemo-logical thinking, each focus is qualitatively different from the previous fociin that it embodies the lower level as an explicit way of thinking abstractedvia reflecting on lower level ways of interacting in communities of prac-tice. This is why, for example, proficient mathematics teachers often feellimited in their ability to teach pedagogy to other mathematics teachers onthe basis of what they came to experience as empowered understandingsof mathematics. That is, being a good teacher does not necessarily implybeing a good teacher educator.

Implications for Teacher Educator Development

In this section I discuss theoretical and practical implications of the four-foci model for mathematics teacher educator development. That is, I tryto suggest how the model, which grew out of my personal story, cancontribute to producing useful insights and sound practices in attempts tofoster development in others.

Theoretical implications. The model points at two important theoreticalimplications, a goal for teacher-educator development and the natureof such development. Let us first consider the goal, to foster teachereducators’ development of inquiry-based practice, where the inquiry isrooted in a problematic view of mathematics learning and, hence, teaching.To this end, a sub-goal is to support the evolution in teacher educatorsof what Jaworski and Watson (1993, cited in Jaworski, 1999) called aninner mentor. To develop an inner mentor means to continually orientteacher educators to question aspects of their practice and how this practicebrings the ends towards which their activities of educating teachers aredirected. That is, one needs to orient the teacher educator’s reflection onrelationships between their attempts to transform teachers’ practices andthe effects of those attempts. For example, a mentor of a teacher educatorshould foster the teacher educator’s questioning of how a mathematics

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teacher’s way of seeing the world of mathematics learning might differfrom how the teacher educator sees it.

The key theoretical construct for the goal of fostering conceptualdevelopment in teacher educators is to orient reflection on activity-effectrelationships. This construct is key because it has a potential to advance anunderstanding of how the social and the cognitive domains in the processof becoming a reflective practitioner (Schön, 1983) might be related.Moreover, this construct can shed light on the critical role that mathematicsteaching plays in the development of mathematics teacher educators. Ielaborate on both below.

To explain the potential power of this construct, one is reminded ofthe two fundamental constructs used to explain any type of develop-ment, reflection and interaction (Dewey, 1933; Piaget, 1985; Schön, 1983).Reflection is a cognitive process in which the learner continually comparesthe goal toward which her or his activities are directed and the effects ofthose activities. The goal and the activity belong in the cognitive domainproper; they can only be set forth internally, via conceptions that areavailable to the learner. However, they are not set forth in a vacuum.Rather, both the goal and the activity are deep-seated in the social domain;they are continually oriented and re-oriented by the learner’s interactionwith others in her or his environment. That is, a specific, reflexive, andinseparable relationship between the social and the cognitive domains ispostulated; through social inter-action people continually orient the focusand content of one another’s reflection on activity-effect relationship ,andthrough available activity-effect relationships they set forth inter-actionswith others.

The power of this view for fostering the development of mathe-matics teacher educators lies in the need to intentionally orient educators’reflection on the relationship between their activities to foster learningexperiences in others and on the effects of those activities. Such a reflec-tion, which a mentor of developing teacher educators can foster, mightbring about in them an inquisitive frame of mind regarding the prob-lematic nature of the learning process, particularly learning to teach tosomeone else conceptions the teacher already knows. The point is thatteacher educators do not have direct access to the ideas a mentor intendsfor them to develop, nor could he or she simply show or tell these ideasto the teacher educators. However, through interaction the mentor canengage them in situations, relevant to the practice of educating mathe-matics teachers, which are likely to orient the teacher-educators’ reflectionon relationships between activities of inferring students’ available concep-tions and effects of such activities. It is through such reflection that teacher


educators may develop an inquisitive frame of mind and specific ideasabout the internal learning experience of mathematics pedagogy in others.

The second theoretical implication of the model refers to the postulatedstage-like nature of the developmental process. That is, new understand-ings grow out of reflection on and reorganization of conceptions regardingwhat previously had been used in practice. The key is that in order to moveto a higher stage, one needs to experience activities of the previous stageso that one has suitable material on which to reflect. For example, considerteacher educators who, as mathematics teachers, do not seem to havearticulated the problematic nature of students’ reinvention of mathematicalconcepts because they do not consider how their own mathematics differsfrom the students’ mathematics. In working with such teacher educators,one will have to engage them in reflection on activities of teaching mathe-matics that will foster their development of what Steffe (1995) called asecond order model of students’ mathematics. This is not a trivial task,because, as Holstein and Gubrium (1994) asserted regarding how peopleexperience the world, “We take our subjectivity for granted, overlooking itsconstitutive character, presuming that we intersubjectively share the samereality” (p. 263). The theoretical claim is that unless teacher educatorsconstruct second-order models of their students’ mathematics they cannothave appropriate experiences on which to reflect in order to conceptualizeteachers’ development of practices that build on a problematic view oflearning and teaching.

The stage-like nature of development implies an asymmetry between astudent and a teacher, between a teacher and a teacher educator, or betweena teacher educator and a mentor. The examples in my story indicate thata person operating at the higher level can learn more about a certainconcept than learners who do not yet have the intended concept available.The reason is that the person at a higher level already has some activity-effect relationship available for reflection and thus the reflection can becontinually reoriented via interaction with those to be taught. However,the learners at the lower level lack the intended conception, they do nothave access to the goal and activity of the person who tries to teach them,and thus cannot learn by simply reflecting on that person’s activities. Thisseems to explain why showing and telling rarely works as a teachingmethod at any level of the model.

Combined, the view of the indirect, orienting role one might have inthe learning of others and the stage-like nature of development point toan important difference between teacher educators who themselves weremathematics teachers and those who were not. The issue is that for bothtypes of teacher educators a mentor must find situations that will call

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upon available activity-effect relationships in them on which to reflect.For example, the mentor might like the teacher educators to bring fortha mathematics teacher’s experience of setting a goal for students’ learningof a particular concept, selecting teaching activities, and relating that goalto the effects of those activities. To this end the mentor might engage theteacher educators in analyzing real or videotaped teaching sessions in someclassrooms. However, in those who are not teachers, such an activity islikely to bring forth very different internal experiences of and meanings forsetting a conceptual goal for students, selecting teaching activities suitablefor that goal, or relating activities of teaching and their effects. This differ-ence would likely be a difficult but perhaps fruitful problem for furtherresearch.

Practical implications. I address practical implications of the four-focimodel by considering a novice teacher educator similar to me when Ibegan my master’s program. In such a case, the development sought is ashift from the second level of a mathematics teacher to the third level of amathematics teacher educator. The work with beginning teacher educatorsrequires, first, an analysis of their ways of thinking about how peoplelearn mathematics and on the teacher’s role in promoting such learning. Amentor can present such teacher educators with situations where children’sknowledge is clearly different than that of adults and inquire into howthe teacher educators would explain that difference and how would theyact. For example, one can present a student who was shown two identicalsquares (cookies), each cut into two equal parts, one diagonally and onehorizontally. The child could clearly state and explain that each part washalf of the cookie from which it was cut; yet, when asked which part shewould rather eat if she liked to eat more, she said she would rather havethe diagonally-cut piece because it was larger. The reactions of teachersto such a situation can be telling. For example, the teachers with whomwe worked at Penn State dismissed the problematic issue. The teachers,referring to their own knowledge rather than the child’s, claimed that thechild basically knew what half is, and simply confused the size relation-ship because the two halves looked different. The key is that it did notoccur to the teachers to even ask themselves why they did not considerthe appearance as a factor and how they immediately knew that the twohalves were of the same size. That is, the teachers used their first ordermodel of mathematics and were unable to even question if such a modelwas appropriate to explain the child’s mathematics. Guided analysis ofsuch teachers’ activities and ideas can orient teacher educators’ reflectionon understandings the teachers bring to learning to teach situations.


When novice teacher educators’ conceptions of mathematics learningare analyzed, one also needs to formulate clear goals for their developmentin terms of the interconnected experience and knowledge of mathematics,mathematics learning, and mathematics teaching they need to have avail-able for further reflections and inquiry. In the above example, it seemsclear that the novice teacher educators have to become reflective practi-tioners. In particular, they need to learn to independently pay attentionto their own ways of thinking and to be able to distinguish between theirmathematics and their students’ mathematics. In turn they can begin to alsoreflect on the difference between their thinking as teacher educators and theteachers with whom they work. It is this last point that I missed when I wasworking with Joe; I did not focus my reflection on how he saw learningand teaching. A particular criterion for selecting appropriate goals is onhow teacher educators’ mathematical understandings should be integratedwith their understandings of mathematics teaching and learning. This isimportant because, as Ball (2000) noted, “teacher education throughoutthe 20th century has consistently been structured across a persistent dividebetween subject matter and pedagogy” (p. 242).

With an analysis of the teacher educators’ current understandings andclearly stated goals to direct their development, one can create or selecttasks and means of working that are likely to foster that development. Forexample, one can present novice teacher educators with experiences likethe one I had when teaching the coordinate system and ask them to recallsuch experiences of their own. Then, one can orient their reflection onthose experiences by asking them, for example, in what way their learningwas similar to or different from the students’ learning. This might allowteacher educators to “understand the learning-to-teach experience fromthe perspective of their students” (Hamilton & Pinnegar, 2000, p. 236).Another example can be the use of technology in the attempt to makesense of theories about learning, like my work with graduate students onthe theory of Vygotsky. Having analyzed their knowledge-base, I createda learning experience in which they integrated learning of a mathematicalidea with an improved understanding of a view of how learning takes place.In this context, they were then creating research problems and designs tofigure out how tools mediate students’ or teachers’ learning of a specificmathematical concept.

An important implication of my story is that advancing from the teacherstage to the teacher educator stage and then to the mentor of teachereducator stage was greatly enhanced via team teaching. Jaworski (1999)noted that team-teaching might foster co-learning because of the reflex-ivity established between co-mentoring others and the inner mentor. This

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was clearly the case when I was working with Lori, as she provided mewith culturally appropriate feedback that called for further reflections inmy inner mentor domain. In fact, team-teaching is but a case of a largercategory implied by my story, that of learning via comparison. Comparisonbetween expected and actual consequences, viewed both by oneself and byothers, is a critical aspect of the reflection process (Dewey, 1933). With orwithout moving to another country, teacher educators can be engaged inculturally different settings that are likely to foster comparison and hencereflection between their ways of functioning as teachers in those settings.

CLOSING COMMENTS

The four-foci model presented in this paper is a work in progress. It high-lights the long and complex process required to reflect upon componentsat one level and develop, first locally and then globally, a new, integratedpraxis at a higher level. In particular, the model captures a primary goalfor teacher educators and their mentors to promote teacher educators’appreciation of the different foci of reflection. Key to this appreciationis to understand that the move from the role of a mathematics classroomteacher to the role of educator of mathematics teachers requires much morethan a shift in the curriculum. It requires a shift in the kind of reflectiveanalysis in which the educator and teachers are engaged. Such a shift callsinto question the context and the content of teaching and its impact on therespective learners’ development.

This article is but an incomplete, self-analysis of one teacher educator’sstory. Thus, it is limited to an individual’s evolving perspective of teachereducator development, including recent learning resulting from reflectingon literature and reviewers’ feedback while preparing the article. Guil-foyle et al. (1996) suggested that self-reflective analyses should not beunderestimated as a way to study teacher educator development. Althoughthey emphasized the need to extend studies of the meaning that teachereducators give to their development beyond this method, they saw thiskind of research as appropriate because it provides access to importantaspects of the teacher educator’s past and current experiences. A supportiveexample of this point is the interesting correspondence between Zaslavskyand Leikin’s (1999) article and my self-reflective analysis. Zaslavsky andLeikin started with articulating a framework for teacher-leaders’ devel-opment, and their analysis of case studies resonates with key aspects ofmy development. I started with an analysis of my personal path, and theconclusions resonate with key aspects of their framework. In this sense,the hope is that both articles provide stimulating and relevant material for


further collective and individual reflection among mathematics educatorson the phenomenon loosely called teacher educator development.

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Department of Mathematics, Science, and Technology EducationNorth Carolina State UniversityBox 7801, 502 Poe HallRaleigh, NC 27695-7801USAE-mail: [email protected]

MARGARET TAPLIN and CAROL CHAN

DEVELOPING PROBLEM-SOLVING PRACTITIONERS

ABSTRACT. This article presents a research project on the development of pre-servicemathematics teachers’ skills and understanding of themselves as pedagogical problemsolvers. The problems were similar to those they are likely to encounter in their futuremathematics classrooms. The project took place within a Bachelor of Education program.The article describes changes in the students’ attitudes towards problem-based learningand examines the critical incidents that were catalysts for these changes to occur. Theproject addressed an important issue in the current Hong Kong context, with the emphasison quality learning and instruction. With curriculum reforms in primary education,teachers are required to respond to changes and implement recommendations within theconstraints of day-to-day classroom management. They need to be critical and informedprofessionals. Therefore, we argue that by adopting a problem-solving approach toteaching, teachers would be better able to view themselves as competent problem solverswho are able to develop various strategies to deal with change.

Ongoing reforms in mathematics education require teachers to engage incontinuous professional growth and adjustment to change throughout theircareers in ways that were unprecedented in the past. Teachers who donot adapt successfully to change will more likely produce students whocan only “follow the rules and procedures and conventions specified inthe textbook” (Gregg, 1995). Several reasons have been put forward forteachers’ reluctance to embrace reform. One is that the teachers may lackthe pedagogical skills and/or confidence to overcome obstacles to changes(Gregg, 1995). Also, according to Gregg, in many cases teachers believethat these obstacles are insurmountable. Many feel unable to be innova-tive because they are effectively isolated in a sink-or-swim atmosphere inwhich they are subject to accountability pressures (Gratch, 2000). Gratchalso suggested that teachers simply lack understanding of what they aresupposed to do. Whatever the reason, there is certainly evidence of a dearthof teachers who are implementing current reform ideas in mathematics(Simon, 1995).

Of particular concern are beginning teachers who, despite having hadrecent instruction about up-to-date methods of teaching mathematics, oftenrevert to teaching styles similar to those of their own teachers (Brown,Cooney & Jones, 1990); they show little or no change in their concep-tions of mathematics teaching despite their methods courses (Thompson,


286 MARGARET TAPLIN AND CAROL CHAN

1985). When teachers become caught up in the “ ‘reality shock’ at thecomplexity of professional work” (Myint, 1999, p. 18), they find it difficultor impossible to implement in the classroom even those practices that theyhave come to believe in during their methods courses (Brown & Borko,1992).

Consequently, it is especially important for pre-service programs toplay a role in preparing prospective teachers to adapt to continuingprofessional change, rather than just to be “technicians” (Gratch, 2000)or to “paint by numbers” (Thiessen, 2000). Often teacher educationprogrammes fall short in this mission because they are too abstract andacademic and provide insufficient opportunities for students to confronttheir beliefs about teaching (McDiarmid, 1990). For mathematics teacherssuch a confrontation is particularly important because there is a tendencyamong them to believe that learning mathematics is equated with remem-bering rules, procedures, and facts primarily through practice (Ball, 1989;McDiarmid, 1990).

If pre-service programs are to do more in preparing teachers to beproblem-solving practitioners who can adapt to change, it is important toestablish just what it is that teachers need to be able to do this. There areseveral kinds of knowledge with which teachers need to be competent. Oneis procedural knowledge, which refers to knowledge of the rules, proce-dures and symbols needed to complete a task (Eisenhart, Borko, Underhill,Brown, Jones & Agard, 1993; Simon, 1993). A second kind of knowledgeis conceptual or content knowledge, which refers to the ability to under-stand the concept and connect or apply the different parts of knowledge(Leinhardt, 1988; Sullivan, Clarke, Spandel & Wallbridge, 1992; Eisenhartet al., 1993; Simon, 1993). For mathematics teachers, conceptual know-ledge includes the ability to make generalisations, describe relationships,and demonstrate higher order reasoning skills (Sullivan et al., 1992). Inaddition, teachers need to be competent with general pedagogical know-ledge of how to teach and specific content pedagogical knowledge, that is,knowledge of how to teach specific content in their subject area. It is notenough, however, to simply impart this knowledge – it is also importantto challenge the students to examine their fundamental beliefs about thisknowledge and to help them discard or revise some of these beliefs (Ball,1988). In other words, it is important for teacher education programs toaddress the issue of how to cope with the complexities of professionalchange.

Peterson, Williams, Dock & Dunham (1998) suggested that a problem-solving approach to teaching can be better realized by beginning teachersthrough collegial problem solving rather than through advice on crisis

DEVELOPING PROBLEM-SOLVING PRACTITIONERS 287

management or tricks of the trade meant to provide immediate solutions.An increasingly popular approach for creating this kind of environmentis problem-based learning (PBL). The literature describes various modelsof PBL (e.g., Kingsland, 1994; Ryan & Koschmann, 1994; Fogarty, 1997).An accepted model is one in which an initial problem situation is presentedto pre-service teachers as it would be to real practitioners of the discip-line. With guidance, the pre-service teachers identify areas of skills andknowledge that they need to acquire and apply in their solution attempts.Pre-service teachers are required to reason and apply this knowledge; thelearning that occurs is integrated into their body of skills and knowledge(Schiller, Ostwald & Chen, 1994, pp. 301–302). The major differencebetween this and teacher-centred learning is that students have greaterresponsibility for generating their own learning issues (Dolmans, 1992).

There are, of course, some disadvantages to the use of PBL, oneof which is that it can be time consuming, particularly in early stageswhen students are still uncertain how to go about the process. Manylecturers have expressed concern about whether sufficient knowledge canbe conveyed through a PBL format and whether students have sufficientprior experience to be able to benefit from the problem-solving situa-tion (Trevitt & Grealish, 1994). Another concern is that when studentsare initially confronted with this approach many are suspicious about itsvalue, particularly if they have previously been used to teacher-centredapproaches (Ulmer, 1994; Felder & Brent, 1996). Nevertheless, researchhas indicated that students in general are usually more motivated to acquireknowledge in the context of solving a problem than they are if the contentis delivered out of context (Idrus, 1993; Schiller et al., 1994). They alsoare more likely to be able to generalise that knowledge to new situations(Chappell & Hager, 1992). PBL approaches have successfully promotedhigher-order thinking skills, such as analysis and synthesis (Idrus, 1993).Kemp (1995) reported that students were able to benefit from being able toapply and test the theory they had learned in real-life situations. Further-more, Schiller et al. (1994) reported that students seemed to think theirlong-term goals, including employment and success, are better met by PBLthan by more teacher-centred approaches. Moreover, the advantages arenot only to the students; there is also evidence to suggest that lecturers canbe enriched by learning from the students’ experiences and discoveries(Kemp, 1995).

We are concerned that teachers develop positive, creative attitudestowards change and be open to change throughout their careers. Thus, wedesigned the project to investigate the effectiveness of a PBL mathematicsmethod course affecting pre-service teachers’ feelings about their ability


to solve problems related to content pedagogical knowledge. The rationalewas that prospective teachers should be taught in a manner similar to howthey are to teach, that is, “by exploring, conjecturing, communicating, andreasoning” (NCTM, 1989, p. 253).

Like most teacher educators, we wanted to help our pre-service teachersto be able to make effective use of approaches to teaching mathematicssuch as problem solving, investigation, discussion, and catering to indi-viduals. Although we encouraged our pre-service teachers to learn aboutcurrent theories of learning and teaching, and to actually consider usingthese in their own future classrooms, we knew all too well that therewere obstacles which would probably prevent this from happening. InHong Kong primary schools, these obstacles include large class sizes, timeconstraints, pressure to cover the syllabus and achieve high examinationresults, and the fact that each mathematics teacher is responsible for severalclasses. We were confident that the student teachers understood what wewere teaching them; and most of them probably developed a genuine beliefin the value of the teaching approaches we were discussing. But we knewthat, as Brown et al. (1990) found, when it came time to assume respon-sibility for their own classes, many of them would simply revert to thetraditional methods of teaching by which they had been taught. This rever-sion is not a problem unique to Hong Kong – both authors had experiencedsimilar situations while working, respectively, in Australia and Canada.However, at the time of this project, the introduction of curriculum reformsthat included an emphasis on problem solving, discussion, constructivism,and catering to individual pupils’ needs were still very new in Hong Kong.Most of the student teachers had only experienced a very traditional,whole-class approach, in which the teacher was the expert conveyor ofknowledge and the pupils were encouraged to be passive recipients. Inother words, these beginning teachers were being asked to adopt changesthat were vastly different from their existing conceptions of mathematicseducation.

Research on PBL can be classified into three groups. The first isconcerned with the effectiveness of this mode of learning in promo-ting higher-order thinking skills and facilitating transfer to problemsencountered in the workplace (c.f. Nuy & Moust, 1990; Kingsland, 1994).The second group is concerned with learning issues, such as how to helpstudents adapt to a problem-solving mode of thinking (Schiller, Ostwald& Chen, 1994), and the effectiveness of problem-based tasks to enablestudents to generate skills and knowledge consistent with the objectivesof the lecturers (Dolmans, 1992; Kingsland, 1994; Duek and Wilkerson,1995). The third area is students’ perceptions of themselves as problem-


solving learners. Our project investigated all of the above issues, but thisarticle focuses specifically on the third aspect.

METHODOLOGY

Context

The trial was implemented in 1996 and 1998, respectively, with two groupseach of 14 final-year Bachelor of Education pre-service teachers training tobe primary school mathematics teachers. This meant that they would mostlikely teach mathematics exclusively, rather than teach a range of subjects.It should be noted, however, that despite the fact that they were majoring inmathematics teaching, there was little evidence that their actual mathema-tical skills and knowledge were any different from primary school teacherspreparing to teach a variety of subjects.

The course we used for the project was the 11-week module Plan-ning and Teaching Primary Mathematics. This module was the finalmodule in the Bachelor of Education programme and focused primarilyon knowledge about how to teach mathematics; other types of knowledgerelevant to a mathematics education programme had been the focus ofearlier studies. At the time of this project, most mathematics teaching wasexamination-driven and delivered by a strict adherence to a curriculumdetermined in a set text. Therefore, the pedagogical topics covered in thismodule were very unfamiliar to the student teachers, even though theseteachers formed the basis of teaching reforms that were being introducedat the time. Our intention was to explore the extent to which the pre-serviceteachers developed in their ability to solve pedagogical problems, theirmetacognitive awareness of their problem-solving strategies, and theirperceptions of themselves as problem solvers.

In the place of more traditional lectures, the pre-service teachers werepresented with a series of content-pedagogical problems, typical of thekind that practicing teachers encounter in their planning and classroommanagement. Because of the commitment to cover quite a large amount ofcontent, each problem task was self-contained so that it could be addressedin one or two class sessions, with pre-service teachers being required todo further work outside of class to prepare their decisions and reflectionsfor assessment purposes. To facilitate this process, each task was accom-panied by a set of readings provided by the lecturer. The structure of thisprogramme, with the corresponding tasks, is shown in Figure 1.

The pre-service teachers were asked to discuss each problem in groupsof four, and to reach a practical, feasible solution. They were provided with


Topic Task

Planning a well-balancedmathematics programme

What do you hope that the children in your class next year willgain from having been taught mathematics by you? Make a list.Devise a check-list system to be included in your planning soyou can check that you are addressing each of these goals inyour planning.

Using evaluations as a basisfor future planning

You have been given some examples of a teacher’s lessonevaluations [one of a lesson on perimeter, one on areasof “awkward shapes” such as parallelograms, triangles andtrapezia, and one on subtraction involving zeros]. Your task is tohelp the teachers to plan the next day’s lessons, so that the chil-dren who need help are receiving it, and the ones who alreadyknow the concept are not wasting their time. Use the guidelineson the attachment as a basis for your discussion.

Is it realistic to expect a teacher to use evaluations as a basis forfuture planning?

Why or why not?

What do you think would be some of the major difficulties?

What solutions might you be able to suggest to overcome thesedifficulties, and make it more realistic?

Planning for individual dif-ferences

The attachment gives you a profile [of ability level, strengthsand weaknesses, and working habits] of a [real] class of Primary6 children. Look at the sample Area activity on page 75 of thecurriculum document, and use this activity as a basis for plan-ning a lesson for the class. You will need to decide how to caterto the different children’s strengths and weaknesses. You willalso need to think about how groups can be arranged to ensurethat the children work effectively.

Is it possible to plan a lesson which will cater to such a widerange of individual differences?

How can you make the most effective use of your time, so thatyou can give appropriate guidance to the children when theyneed it?

Strategies and criteria forassessment

You have a job in a TOC school [a school which has volunteeredto implement the new Target Oriented Curriculum Programmeof Study for Mathematics]. Your principal expects you to usea variety of assessment strategies, which include testing, childobservation records, conferencing, journal writing and portfo-lios. Read the attached articles about these various forms ofassessment. Develop a system which will enable you to use all ofthese strategies in a realistic way. Using the attached examplesas a model, design a record-keeping system which will give youmaximum information about the pupils, without becoming toocumbersome for you to keep up to date.

Planning for different typesof mathematics lessons:discovery, expository anddirected discussion

The attached handout gives a description of different types oflessons. Your task is to select a topic from the mathematicssyllabus and decide which of these appraoches would be the bestto use for this topic, and write a lesson plan.

Planning the daily routine[i.e. revision, maintenanceof mental strategies, intro-ducing new material, prac-tice activities], planning theteacher’s time

What are the important components of a daily routine, and howmuch time should be spent on these? How can you allocate yourtime to individuals and groups? How can you make the otherchildren in the class “teacher independent” when your presenceis needed with others?

Figure 1. The tasks.


Reflections on tasks 1, 2 and 3: Some questions to guide your journal writing

In week 1, we looked at the important goals for a mathematics programme and youdeveloped a method to make sure that a well-balanced programme was covered.

What, if anything, did you learn from this activity?

Do you think the matrix idea [an example shown to them at the end of the exercise]is a good way to record your long-term programme and make sure that it covers theimportant cognitive, affective and pedagogical domains? Would you use it in your ownteaching? What needs to be changed about this idea for you to use it in your ownteaching?

In weeks 2 and 3 we looked at a teacher’s evaluations of three lessons and you wereasked to make some suggestions about how the teacher could plan for the next lesson.Briefly describe what your group decided was important to take into account whenplanning follow-up lessons. Do you think you will be able to use this kind of planningin your teaching? Why or why not? If you have identified some difficulties associatedwith the use of evaluations for planning, what solutions can you suggest to overcomethese difficulties and make it more realistic?

General questions

After the first three weeks, how do you feel about the problem-based approach tolearning? What do you think are your own strengths and weaknesses as a pedagogicalproblem solver? (This means a teacher who is able to overcome the difficulties andconstraints of the “real” classroom situation, to adopt new or unfamiliar ideas that mightbe recommended by principals or curriculum planners). Has there been any experienceduring the past three weeks of activities that has influenced your feelings about yourselfas a pedagogical problem solver?

Figure 2. Sample of prompts for journal writing (first entry).

background information and readings to give them an adequate knowledgebase for the discussion. We acted as facilitators, asking appropriate ques-tions to direct their thinking rather than giving them direct answers. Allgroup discussions were recorded for later analysis of the problem-solvingprocesses that were used.

In addition, the pre-service teachers were asked to write three journalentries in which they reflected on themselves as problem solvers and onthe ways in which their thinking about mathematics teaching had beenchanged by the experiences in class. In each journal entry, they were askedto reflect on their skills as problem-solving teachers, particularly theirability to use problem-solving skills to adapt to classroom situations. Asample of the prompts for their journal entries is shown in Figure 2. It wasemphasised repeatedly, both before they submitted their journals and in thelecturer’s feedback, that marks would be awarded for honest reflectionsabout themselves, rather than whether they agreed or disagreed with thelecturer’s ideas. Other assessment criteria included their evaluation of what


TABLE 1

Pre-service Teachers’ Feelings About the PBL Approach

Feelings changed Feelings remained Feelings remained Noncommittal

from negative to positive negative about feelings

positive during throughout throughout

course

11 12 3 2 28

they had learned in relation to themselves and their own practice, prac-tical suggestions for overcoming obstacles they identified, links to theirown practice-teaching experience, and evidence of higher-level thinkingabout the issues. The discussion in this paper will focus primarily on thepre-service teachers’ responses to the second set, the general questions.

Analysis

In analysing the journal entries, we used the constant comparative method(Glaser and Strauss, 1968). Ten categories were identified and coded toindicate (a) expressions of negative/uncertain reactions; (b) satisfactionwith the experience; (c) confidence to put an idea into practice; (d) confu-sion/worry; (e) turning points in their feelings about PBL; (f) topics/issuesabout which they still felt unprepared; (g) conflicts with their previousteaching experience/thinking; (h) feedback from the lecturer; (i) relevanceof the tasks to the reality of teaching; and (j) their perceptions of them-selves as pedagogical problem solvers. A 3x10 matrix was constructedwith these 10 categories and the three groups of pre-service teachers:those whose feelings changed during the course, those whose feelingsremained positive throughout and those whose feelings remained negativethroughout. This matrix was used to identify patterns in the comments ofthe pre-service teachers in each of these three groups.

RESULTS

The analyses of the journal entries indicated that there were three distinctgroups of learners (Table 1). Firstly, there was a group who were initiallypositive about the PBL approach and remained positive throughout theeleven weeks of the course. A second group were those whose attitudesremained negative throughout the course. A third group consisted of those


who started out with negative attitudes but became positive about theapproach and about themselves as problem solvers as the course continued.It is interesting to note that there were no pre-service teachers whoseattitudes changed from positive to negative.

The pre-service teachers seemed to be fairly evenly divided in theirfeelings at the outset of the PBL experience. Twelve of the 28 were positiveabout the approach from the beginning, and 14 were negative, whereas twostudents were non-committal. By the end of their respective courses, 23 ofthe 28 pre-service teachers were recording comments that suggested theyhad reacted positively to the problem-based approach and were able todescribe, often with considerable enthusiasm, what they had gained fromthe experience. In order to find out more about the nature of the pre-serviceteachers’ feelings, their journal reflections were examined in detail. Someof their comments will be described below. It should be noted that thepre-service teachers wrote their reflections in English which, although itwas the medium of instruction for their BEd course, was their second,and in some cases third, language. Some spelling and grammatical correc-tions have been made, but the pre-service teachers’ expressions have beenunchanged.

Table 2 summarises the main categories that were related to the pre-service teachers’ feelings and beliefs, and the numbers of pre-serviceteachers who made comments related to each of these categories. It is alsointeresting to note that in their third entry, very few pre-service teachersmade any comments about their feelings, with the majority saying some-thing to the effect that their opinions had not changed since the previousentry.

The journal entries of the pre-service teachers who started out withnegative or uncertain reactions to the problem-based approach suggestedthat, at first, they did not like having to make their own decisions. Theywere worried they were not going to learn enough, and that they wantedmore guidance and “correct answers” from the teacher. In the first week,for example, Wai Yin1 wrote:

I am very surprised that the teacher has changed the approach of teaching . . . . I wonder ifit is a good way to learn.

It is interesting to note, however, that by the second journal entry therewas a decrease in the number of pre-service teachers expressing negativefeelings. From the start, pre-service teachers felt positively about theopportunity to share ideas and resources with their classmates in discussingthe problems. They regarded this as being an important element in solving

1 All names are pseudonymns.


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the problem, and some related this to their future careers, saying theyhad realised it would be important to have discussions with their futurecolleagues:

I was very glad that we could co-operate effectively in order to find out quite significantdecisions. We shared ideas and found solutions together. On the other hand, the givenmaterials also gave us a lot of ideas and suggestions in order to stimulate us to find outmore methods and solutions. I found that I learned a lot from my group members (YinLing).

A particular change in attitude occurred when the pre-service teachersrealised that the tasks were in fact realistic and relevant to the situa-tions they were likely to find themselves facing in their future teaching.Whereas only four pre-service teachers realised this by the first journalentry, another six had commented on it in the second entry and anotherone in the third. Typical comments about this realisation included:

We begin feeling that the tasks are quite practical and challenging to be coped with. It isbecause we perceived that those tasks will be very likely to occur in our future teaching. Itis good for us to think of ways to cope with them before we start to teach. So, we are morewilling to think and feel free to discuss (Mee Lin).I just did not understand what all these questions about being a problem solver had todo with my work in class and how they could connect to my future teaching. Then Irealised that for the past lessons, I had been given a lot of problem-solving tasks to do andI was actually learning through problem solving! And the most important of all, I realizedthat teaching was indeed a problem-solving career. For as teachers, we often have to useproblem-solving skills to adapt to our classrooms. Knowing these were very important, forsuddenly what I did in class became meaningful (Chi Kit).

Not surprisingly, when the pre-service teachers first started outwith the problem solving, they did not see themselves as pedago-gical problem solvers. In the first journal entry, only seven pre-serviceteachers commented that it was difficult to imagine the real situation inthe classroom, that their past experience was not enough to draw on,everything seemed to be impossible, and there were too many details tothink about each time:

Handling different ability students in one class is a difficult task for a teacher in real practicebecause of limited time available and management problems. In my first sight everythingseems to be impossible (Mee Lin).

In the second journal entry, only three pre-service teachers made suchcomments, with none in the third entry. On the other hand, there was quitea substantial increase in the numbers who expressed confidence and beliefin their ability to find solutions to pedagogical problems. Seven said thatthey felt this way in the first journal entry, although they had not neces-sarily started out with such confidence, with a further eight expressing theirgrowing confidence in the second entry and four in the third:


So now we find the problem solving easier . . . learned that we must break through ourtraditional minds and be open-minded to accept new ideas, even if they are not used widely(Mee Lin).For the second problem, I discovered that I began to adapt to the new approach. Thatprocess may be slow. But indeed, I gradually realized that it is natural to be unhappy orconfused when we face any difficulties or obstacles. So there is no need to care too much.Moreover, I found that we would become happier when we could find a solution. Thesatisfaction in some cases seems to cover the initial unhappiness or frustration (Kit Wan).

An attempt was made to identify patterns in the pedagogical knowledgeabout which the pre-service teachers had changed their beliefs. There werefew distinct patterns, with individuals mostly commenting on differentthings. Some of the more common ones included: the use of alternativeapproaches to assessing mathematics, particularly the use of journals andstudent portfolios (eight pre-service teachers), and the use of group activ-ities as an organisational strategy to cater to students of different abilities(four pre-service teachers). The classroom problems about which theywere still concerned they could not find solutions varied from studentto student. Some of the more common ones included that the physicalenvironment in Hong Kong schools is not conducive to group activities,that many principals still disapprove of noisy classrooms, and that the pre-service teachers still had not developed suitable strategies for monitoringindividual students’ progress (12 pre-service teachers). This latter problemis particularly difficult because, in Hong Kong, a primary school mathe-matics teacher typically teaches several classes and therefore has to assessas many as 200 pupils. One quite lengthy entry by Chi Kit, in her secondjournal, gives some interesting insight into the change in her beliefs aboutmathematics teaching:

I learnt not only some useful ideas about establishing goals, assessment criteria or usingevaluations as basis for future planning, etc. . . . most importantly the belief that the prob-lems could be solved! One will never know how great the impact was on me in suchchanging of my belief. A long time ago, I developed a perception that what we learnt inclass (teacher training), read in books, was too ideal to be actually used in school. I was justfollowing the rules, following the syllabus, following the principal, following the EducationDepartment, etc. I had never thought of what kind of education the children should receive,what the ideal education would be. I was just doing a job. I had no beliefs. Might be Iwas an active listener, but I did not have my “own opinion”. Everything changed since Irealized that “ideal education” was possible! It can be done by using problem solving inmy teaching: not only trying to adapt the teaching in the principal’s/parents’ acceptableway, but trying to change the principal’s/parent’s mind and make them support me. Thatis, overcome all the limitations to implement one’s ideal teaching instead of trying to teachbetter under all the limitations. It sounds hard, but it is a problem-solving task (Chi Kit).

It is important to note that Chi Kit was very excited to realize she hadactually changed an old belief that had been a part of her thinking for a


long time. In a later journal entry, she referred to this revelation as her“mindstorm.” For her, it was not necessarily the triggering incident, whichin this case was the use of portfolios for assessment, that was importantas much as the insight that something she had previously dismissed in hermind as being too idealistic could actually be applied in the classroom.Added to this was the excitement that if this idea could really be used, thenthere was also hope that some of the other idealistic teaching approachesshe had encountered could also be applied to classroom teaching. Theawakening of her sense of ownership and ability to control the situationfor herself seemed to be the critical turning point.

Experiences that Facilitated Positive Attitudes

In the analysis of the pre-service teachers’ journals, it became clear thatthere were certain change factors that most of the pre-service teacherscommented on as having influenced their feelings about themselves asproblem-solving teachers. Quite often, these were the factors that triggeredchanges in the pre-service teachers’ feelings and beliefs. The incidents aresummarised in Table 3.

There appears to be some similarity between the change factorsmentioned by those pre-service teachers who were positive right fromthe start and those whose feelings and beliefs changed from negative topositive during the course. The only difference was that some pre-serviceteachers recognised these benefits at the outset, whereas others came tothe realisation later, at different times and with different problem activitiesthat had some particular relevance for the individual. The issue mentionedthe most frequently by pre-service teachers in all categories was the reali-sation that they were learning something. The pre-service teachers whochanged described as a significant turning point when they felt they hadsome success with a problem, for example, by finding a solution that theyconsidered workable. The pre-service teachers who started out negativecommented on the influence of positive feedback and encouragement inchanging their feelings. This feedback was offered in the form of ongoinginput from the lecturer during small-group discussion, through generaldiscussion about the solutions at the end of each problem-solving sessionand the lecturer’s written comments in response to the solutions theysuggested in their journal entries.

Another time change of attitude occurred was when they found aproblem solution that enabled them to realise that, with a little flexibilityand adaptability, they probably could achieve teaching approaches theyhad previously regarded as “impossible.” Another statement indicated thatpositive attitudes began to develop when the pre-service teachers realised


TABLE 3

Experiences That Facilitated Positive Attitudes

Pre-service teachers whose Pre-service teachers whose Pre-service teachers whose

feelings changed during feelings remained positive feelings remained negative

course throughout throughout

• realised that they were • valued the learning / • did not regard selves as

learning from discussion exposure to new ideas good problem solvers

and sharing ideas with occurring throught group • began to see that they

peers discussion and sharing were learning something,

• saw that something ideas with peers but did not change feelings

previously regarded as • realised that they were of frustration and distress

“impossible” was widening their horizons associated with the

“parctical” and realised • recognised that their problem-solving

it was possible to adapt solutions were realistic • anxious

ideas to suit their contexts and practical

• become aware of issues • challenged their former

not previously considered belief that their teaching

• gradual acceptance that methods were already

tasks were relevant to adequate

future needs • relised that problem-

• had some success, i.e. solving activities would

found some “good”, help reduce feeling of

workable solutions to being “overwhelmed” by

problems (and received issues

positive feedback /

encouragement)

• realised / acceted that

sense of confusion /

frustration is a nautral

part of problem solving

the extent to which the problem-solving activities were helping them towiden their horizons about teaching. A common report amongst the pre-service teachers who changed their feelings was that one of the issuesthey had to come to terms with was their negative emotions of frustrationand confusion – the very things mentioned by Gregg (1995) as reasonsfor teachers not implementing change. When they could see that these


emotions are a natural part of the problem-solving process, they becamemore accepting and positive. The three whose feelings remained negativewere all locked into reflections about their negative emotions, even if theyreported that they could see that they were learning something.

SUMMARY

Our findings represent a small-scale action research project in which weexplored the extent to which a group of Hong Kong pre-service teacherscould be encouraged to change their feelings about themselves as problem-solvers, while engaged in student-centred, problem-based learning, and thefactors which contributed to the changes that occurred. At the end of the11-week course, the majority of the pre-service teachers made positivecomments about the knowledge and skills they had developed from theproblem-solving activities.

Some of the outcomes of this project suggest implications for mathe-matics teacher education programmes. It is not surprising that, similar tothe findings of Ulmer (1994) and Felder and Brent (1996), about half of thepre-service teachers were negative at first. They did not favour having totackle the pedagogical problems; they did not like having to make theirown decisions, but wanted guidance and correct answers. The findingsare consistent with Gratch’s (2000) suggestion that many teachers avoidchange because they lack understanding of what they have to do. In thiscase, the pre-service teachers certainly showed evidence that they lackedthis understanding at the beginning of the course. However, they alsoshowed that they were able to develop the problem-solving strategies andappropriate understandings in a short time. This is consistent with Idrus’(1993) and Schiller’s et al. (1994) suggestions that learners are more likelyto acquire knowledge if it is delivered in a problem solving context. Wetherefore suggest that, with similar opportunities for supported collegialdiscussion, more beginning teachers should be able to overcome the hurdleof not knowing what to do. The pre-service teachers particularly supportedas a favourable strategy the use of collegial discussions, which had alsobeen suggested by Peterson et al. (1998).

Some of the pre-service teachers’ fundamental beliefs about mathe-matics teaching as a drill-and-practice process assessed by tests andexaminations may have blocked their initial ability to view change assomething within their power. This is why they initially dismissed some ofthe suggestions as being impossible or “too much to think about” which, asGregg (1995) suggested, many mathematics teachers tend to do. However,


having been forced to confront these beliefs in considering the problems,many of the pre-service teachers came to realise that, even despite the“reality shock” (Myint, 1999) of real-life classroom, they could find waysto incorporate ideas such as group discussions, or subjective means suchas portfolios to assess children’s growth.

Several factors emerged as being significant in changing the pre-serviceteachers’ feelings about the problem-solving approach. One importantfactor was that they did have some success, that is, they valued the oppor-tunity to increase their confidence through constructive feedback abouttheir ideas. A lack of constructive feedback is a potential block to teacherswho want to implement change. Another important factor was realisationthat something impossible could in fact be solved creatively. Brown andBorko (1992) found one of the reasons teachers avoid change to be thatthey see it as too difficult, if not impossible. However, our pre-serviceteachers’ experiences suggest that if they are given time and space todiscuss possible solutions they could come to their own awareness that theseeming impossible can be overcome. A further important change factorwas that they realised, through discussions of their feelings with theirpeers and the lecturer, that negative feelings about change are natural andshould not be seen as a reason not to try. Although it was not a plannedaspect of the interaction between pre-service teachers and instructors orbetween pre-service teachers themselves to discuss this particular issue,we did utilise incidental opportunities to discuss their negative feelingsabout change. More useful, however, was that often when the pre-serviceteachers were first exposed to a new problem they would naturally discusstheir negative feelings with their group mates and, in doing so, came torealise that others felt the same initial confusion or panic and that it was notsomething unique to any one of them. It was interesting to note that thosewho did not show any changes in their feelings were those who remainedlocked into reflections about their negative emotions even if they had saidthat they were learning something about teaching mathematics from theprocess.

One factor mentioned by several of the pre-service teachers as beinga catalyst in changing their attitudes, although it was not deliberatelybuilt in as a part of the design, was the input they received from theinstructors. This input primarily took the form of (a) incidental questioningand comments from the lecturer during small-group discussion, (b) generaldiscussion about the solutions at the end of each problem-solving segment,and (c) and written comments made by the lecturer in response to thesolutions the pre-service teachers suggested in their journal entries. This


input was not designed specifically to change their attitudes. In fact, theinput focused not as much on their feelings as on the suggested solutionsand their practicality, on questions about unforeseen problems they hadnot encountered, and on comparison of their suggestions to those that hadbeen used successfully by experienced teachers. Therefore, it seems thatthe changes in attitude that occurred were not prompted externally byus, as much as they were derived internally by the pre-service teachersthemselves, from feeling that their ideas were possible to implement.

So far, the study has only been conducted with two small groups ofpre-service teachers. Nevertheless, it is encouraging to see, as in the studyby Schiller et al. (1994), that in both of these groups the majority of thepre-service teachers were reporting positive reflections about the problem-based approach and its impact on preparing them more effectively for theirfuture employment. Of course, the real evidence of the success of theproject will be if, in their early teaching careers, the beginning teachershave in fact demonstrated their developing confidence and skills in over-coming obstacles to change. In the meantime, we feel encouraged that theuse of the problem-solving approach had a substantial effect on changingthe pre-service teachers’ way of thinking about change, which in turncontributed to the quality of their thinking about learning and instruction.

Tentatively, we can suggest that the use of pedagogical problem solvingin teacher education programmes can be an effective way of facilitatingteachers in confronting the obstacles to change, but that it is important that

• the tasks have relevance and applicability to the classroom context inwhich the teachers will be working;

• the teachers have some early experience of success, to build theirconfidence and recognise that there are ways to solve even seeminglyimpossible situations;

• there is plenty of opportunity for collegial discussion, as alsosuggested by Peterson et al. (1998) to share ideas about approachesto the problem; and that

• some counseling or support is given when the teachers experiencenegative emotions in their attempts to implement new ideas, so theycan learn to accept this as a normal part of the process.

The findings of this study also raise some potential questions for furtherresearch. In particular, it is important to investigate any effects of thepre-service teachers’ increased confidence in themselves as pedagogicalproblem solvers as they progress through the early years of their careers.Also, it would be worthwhile to explore the use of this type of activity within-service programs and, particularly, to monitor the impact on teachers’


ability and willingness to incorporate change into their classroom prac-tices. Although we found that the pre-service teachers’ confidence in theirability to solve problems increased, it is also necessary to explore whetherthis strategy of challenging their beliefs about their ability to solve theproblems has any effect on their capacity to actually solve the problems.It may also be worthwhile to investigate further the negative emotions thatteachers can feel when they encounter obstacles while attempting to imple-ment change. Although none of these research issues is necessarily specificto mathematics teaching, it is possible that, as Ball (1989) and McDiarmid(1990) have suggested, many primary school teachers’ stereotyped beliefsabout what mathematics is and how it should be taught make it particu-larly difficult to implement changes in this area of the curriculum. Hence,the need exists to investigate these issues given that they are particularlyrelated to mathematics education.

REFERENCES

Ball, D. (1988). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1),40–48.

Ball, D. (1989). Breaking with experience in learning to teach mathematics: The role ofthe preservice methods course. (Issue Paper 89-10). East Lansing, MI: Michigan State.University, National Center for Research on Teacher Education.

Brown, C. & Borko, H. (1992). Becoming a mathematics teacher. In D. Grouws (Ed.),Handbook of research on mathematics teaching and learning (209–239). New York:Macmillan.

Brown, S., Cooney, T. & Jones, D. (1990). Mathematics teacher education. In W. R.Houston (Ed.), Handbook of research on teacher education (469–497). New York:Macmillan.

Chappell, C. & Hager, P. (1992, Nov.). Problem-based learning and competency devel-opment. Paper presented at the Australian Association for Research in Education/NewZealand Association for Research in Education Joint Conference, Deakin University,Geelong.

Dolmans, D. (1992). The relationship between student-generated learning issues and self-study in problem-based learning. Instructional Science, 22, 251–267.

Duek, J. & Wilkerson, L. (1995). Learning issues identified by students in tutorlessproblem-based tutorials (ERIC Document No. ED 394 986).

Eisenhart, M., Borko, H., Underhill, A., Brown, C., Jones, D. & Agard, P. (1993). Concep-tual knowledge falls through the cracks: complexities of learning to teach mathematicsfor understanding. Journal for Research in Mathematics Education, 24, 18–40.

Felder, R. & Brent, R. (1996). Navigating the bumpy road to student-centred instruction.College Teaching, 44(2): 43–47.

Fogarty, R. (1997). Problem-based learning and other curriculum models for the multipleintelligences classroom (ERIC Document No. ED 405 143).


Glaser, B. & Strauss, A. (1968). The discovery of grounded theory: Strategies forqualitative research. London: Weidenfeld and Nicolson.

Gratch, A. (2000). Teacher voice, teacher education, teaching professionals. High SchoolJournal, 82(3), 43–54.

Gregg, J. (1995). The tensions and contradictions of the school mathematics tradition.Journal for Research in Mathematics Education, 26, 442–466.

Idrus, R. (1993). Collaborative learning through teletutorials. British Journal of Educa-tional Technology, 24, 179–184.

Kemp, S. (1995). Innovation in teaching case studies in tourism management. In A. Zelmer(Ed.), Higher education: Blending tradition and technology. Proceedings of the 1995Annual Conference of the Higher Education and Research Development Society ofAustralasia (434–513). Canberra, ACT: HERDSA.

Kingsland, A. (1994, Nov.). Broadening the base and deepening the understanding inproblem-based learning. Paper presented at the Australian Association for Research inEducation, Newcastle, NSW.

Leinhardt, G. (1988). Getting to know: Tracing students’ mathematical knowledge fromintuition to competence. Educational Psychologist, 23, 199–144.

McDiarmid, G. (1990). Challenging prospective teachers’ beliefs during early fieldexperience: A Quixotic undertaking. Journal of Teacher Education, 4, 12–20.

Myint, S. (1999). Japanese beginning teachers’ perceptions of their preparation andprofessional development. Journal of Education for Teaching, 25(1), 17–29.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.

Nuy, H. & Moust, J. (1990). Students and problem-based learning: How well do they fitin? Journal of Professional Legal Education, 8, 97–114.

Peterson, B., Williams, S., Dock, T. & Dunham, P. (1998). Mentoring beginning teachers.Mathematics Teacher, 91, 730–734.

Ryan, C. & Koschman, T. (1994). The collaborative learning laboratory: A technology-enriched environment to support problem-based learning (ERIC Document No. ED 396678).

Schiller, J., Ostwald, M. & Chen, S. (1994). Implementing a problem-based, distanceeducation undergraduate course in construction management. Distance Education, 15,300–317.

Simon, M. (1993). Prospective elementary teachers’ knowledge of division. Journal forResearch in Mathematics Education, 24, 233–254.

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivistperspective. Journal for Research in Mathematics Education, 26, 114–145.

Sullivan, P., Clarke, D., Spandel, U. & Wallbridge, M. (1992). Using content-specificopen questions as a basis of instruction: A classroom experiment. Victoria, Australia:Mathematics Teaching and Learning Centre, Australian Catholic University.

Thiessen, D. (2000). Developing knowledge for preparing teachers: Redefining the role ofschools of education. Educational Policy, 14, 129–144.

Thompson, A. (1985). Teachers’ conceptions of mathematics and the teaching of problemsolving. In E. Silver (Ed.), Teaching and learning of mathematical problem solving:Multiple research perspectives (281–294). Hillsdale, NJ: Lawrence Erlbaum.

Trevitt, C. & Grealish, L. (1994). Learning to crawl: Development of problem-basedlearning as a teaching strategy. In M. Ostwald & A. Kingsland (Eds.), Research and


development in problem based learning (307–314). Newcastle, Australia: The Universityof Newcastle.

Ulmer, A. (1994). Overcoming problems with problem based learning expectations. In M.Ostwald & A. Kingsland (Eds.), Research and development in problem based learning(315–327). Newcastle, Australia: The University of Newcastle.

Centre for Research in Distance and Adult Learning Margaret TaplinThe Open University of Hong KongHong KongE-mail: [email protected]

Department of Education Carol ChanThe University of Hong KongPokfulam RoadHong KongE-mail: [email protected]

JANE M. WATSON

PROFILING TEACHERS’ COMPETENCE AND CONFIDENCE TOTEACH PARTICULAR MATHEMATICS TOPICS: THE CASE OF

CHANCE AND DATA

ABSTRACT. This report presents an instrument designed as a profile of teacherachievement and teacher needs with respect to the probability and statistics strands inthe mathematics curriculum. In developing the profiling instrument, I had two primaryobjectives. First, the instrument was to assist in the assessment of teacher achievement inthe context of the adoption of professional standards for mathematics teachers. Second,the instrument was to assess professional development needs for teachers in the lightof changes to the mathematics curriculum. The background for the development of theinstrument is presented, followed by a description of the instrument and the results ofresponses to the instrument from 43 Australian teachers. Uses for the instrument andfurther development possibilities are discussed.

Two significant developments in education provided the setting for thestudy: (a) the renewed general interest in teacher professionalism and theaccompanying calls for criteria to assess teacher competence; and (b) thechanges within the mathematics curriculum itself. Associated with thesechanges is the need to assess teachers’ current levels of competence forthe purposes of ensuring continued professionalism and suggesting profes-sional development programs for improving performance. The questionarises whether a single instrument can satisfy the requirements of bothdemonstrating teacher professional competence and indicating aspectsof performance that require further development. If teacher profession-alism is to be a meaningful goal in a constantly changing educationalmilieu, then both aspects of teacher practice must be acknowledged. Theprofiling instrument presented here was developed to address the issue ofteacher competence and curriculum change in relation to the reform of themathematics curriculum in the 1990s.

Although the issue of what constitutes a professional teacher wasdiscussed widely in the 1990s (e.g., Preston, 1996), professional stand-ards for teachers of specific subject areas were slow to develop. The workof the National Council of Teachers of Mathematics (NCTM, 1991) toset professional standards for mathematics teachers in the United Stateswas followed by that of the Australian Science Teachers Association and


306 JANE M. WATSON

the Australian Association of Mathematics Teachers, Inc. [AAMT]. TheAustralian Science Teachers Association set professional standards forscience teachers in Australia (Ingvarson, 1995), and AAMT published itsfirst draft of descriptors of excellence in teaching mathematics in 2000(AAMT, 2000).

Efforts were made to establish teacher professionalism firmly on theeducational agenda as curriculum reform was increasing the complexityof the teacher’s task. For mathematics this began in the United Statesin 1989 with NCTM’s Curriculum and Evaluation Standards for SchoolMathematics. The Standards were followed by similar documents in othercountries, including Australia. In 1991 the Australian Education Council[AEC] published A National Statement on Mathematics for AustralianSchools, which was followed in 1994 by Mathematics – A CurriculumProfile for Australian Schools. These documents proposed changes inprocess as well as content and spawned several professional developmentprojects to aid teachers in adapting to the new curriculum (e.g., Watson,1998). One of the significant changes to the mathematics content was theinclusion of probability and statistics as one of five general topics to becovered. Because most teachers of all levels in Australia had no priortraining in either content or teaching methodology in these two areas,teachers were placed under increased pressure by inclusion of probabilityand statistics at a time when their professionalism was being redefined andeven questioned (Sachs, 1997).

The study presented here contributes to the debate about the competen-cies from which the mathematics standards used to assess the professionalstatus of mathematics teachers should come. At the same time, it providesmeasures that can be used to determine professional development needs ofteachers in a new area of the curriculum. Such assessment of needs will berequired whenever revisions are made to the curriculum, and again aftereducational programs have been delivered, in order to provide evidencethat new levels of professionalism have been achieved. This cyclic natureof assessment-action-reassessment is inherent in the competency-basedmodel of education that is the basis of teacher professionalism in a timeof rapid educational change (Sachs, 1997).

Olssen, Adams, Grace, and Anderson (1994) described the Australianmathematics student profile (AEC, 1994) as “a framework for reportingstudents’ mathematics achievements” (p. 7). In the context of this study,I define profile as a framework for reporting teachers’ achievements andcompetencies for teaching the topics chance and data in the mathematicscurriculum. I devised the framework after considering the literature onteacher competencies as well as the demands of the curriculum. As a

PROFILING TEACHERS 307

vehicle for making judgements about teachers, a profile must be nonthreat-ening and encourage authentic teacher reflection on beliefs, knowledge,and practice (Valli, 1992). Such an approach itself contributes to therenewed sense of professionalism being advocated (Preston, 1996; Sachs,1997).

DEVELOPMENT OF THE PROFILING INSTRUMENT

Theoretical Rationale and Preliminary Work

The factors that affect teaching performance vary widely and includehow teachers are affected by the system in which they operate, how theyperceive the students they teach, how they plan for teaching, how theyassess difficulties, how they view knowledge, how they relate to theirprofessional community, and how they respond to curriculum change. Thestance of particular relevance to this study is that of Shulman (1987a,1987b). In considering the assessment of teachers he concluded that

teaching typically occurs with reference to specific bodies of content or specific skills andthat modes of teaching are distinctly different for different subject areas . . . the particularkinds of learners and the character of the setting also influence the kind of teaching . . . [and]most assessments must examine the applications of pedagogy to specific subject areas.(1987a, p. 41)

Shulman (1987b) identified seven types of knowledge: (a) content know-ledge; (b) general pedagogical knowledge; (c) curriculum knowledge;(d) pedagogical content knowledge; (e) knowledge of learners and theircharacteristics; (f) knowledge of education contexts; and (g) knowledgeof education ends, purposes, and values (p. 8). Shulman’s perspectiveis consistent with that of the NCTM in the United States in its Profes-sional Standards (1991) and with that of Fennema and Franke (1992) whosuggested detailing teachers’ context-specific knowledge of (a) mathe-matics, (b) pedagogy, and (c) learners’ cognitions in mathematics. Recentresearch has employed Shulman’s categories to explore teachers’ contentknowledge, pedagogic content knowledge, and curriculum knowledge(Kanes & Nisbet, 1996); detailed knowledge of students (Mayer &Marland, 1997); and definitions of professionalism (Preston, 1996).

Shulman (1987a) suggested that the assessment of teacher perform-ance requires several methods, including written assessments, assessment-centered exercises, documentation of performance during supervised fieldexperiences, and direct observation of practice by trained observers. Othermethods discussed by Fennema and Franke (1992) range from the use ofclosed scales to open-ended tasks. Valli (1992) emphasised that teachers’

308 JANE M. WATSON

reflections about practice need to be part of any assessment process. Pegg(1989), in his analysis of the levels of integration of ideas within teachers’lessons, described the observed outcomes in terms of a hierarchical struc-tural model. Pegg’s approach highlights the importance of an integratedview of the components that contribute to successful teaching. Although– in an ideal setting – it would be desirable to gather input from all ofthese standpoints in a triangulation process, this is not realistic in mosteducational systems. This project intended to provide as many aspects oftriangulation as possible within a profiling instrument that could be admin-istered in a limited, practical time frame. The work of Shulman (1987b)offered a useful framework for this objective as it incorporates the aimsof NCTM’s (1991) professional standards and Preston’s (1996) desire tobase professionalism on teachers’ work. From a general educational pointof view, Shulman’s (1987b) seven types of knowledge appeared to be asatisfactory starting point.

With respect to the implementation of the probability and statisticscurriculum, Shaughnessy (1992) noted the need to unravel teachers’conceptions of the topics: “The success of the NCTM’s ambitious stand-ards recommendations will ultimately depend upon teachers. What areelementary and secondary teachers’ conceptions of and attitudes towardsstochastics? . . . We need to gather information from teachers at . . . the . . .

in-service level” (p. 489). Shaughnessy highlighted stochastics as an areain which little or no research exists and for which data could be of assist-ance to those making decisions regarding the professional developmentneeds of teachers. Little work, however, appears to have followed Shaugh-nessy’s call for research. The survey work of Greer and Ritson (1993)found Northern Ireland teachers at all levels in need of in-service trainingto upgrade their understanding of probability and statistics and to becomeaware of appropriate teaching methodologies. Bright and Friel (1993)employed teacher concepts maps, which were useful in the exploration ofprimary school teachers’ perceptions of the relationships among statisticalconcepts. Their research indicated a disappointing lack of connectedness,with many isolated concepts. Associated with the same project, Bright,Berenson, and Friel (1993) conducted a brief pedagogy survey of teachers’knowledge and beliefs. The survey asked teachers to identify three statist-ical concepts important for third-grade students to know; how the mostimportant concept could be taught with manipulatives and/or calculators;and what students would need to know about statistics in order to besuccessful in middle school mathematics. The survey was administeredbefore and after a professional development program, and responses tothese items were used to assess the success of the program. Another


requirement for a profiling instrument was that it be nonthreatening tothose who agreed to respond to it. This was particularly important in a newarea of the curriculum where some teachers felt intimidated by content thathad not been part of their pre-service training.

Most, but not all, of the criteria discussed above were taken into accountin an initial teacher survey that was tested in 1994 with 72 teachers from14 Tasmanian government schools. The intent of the original survey wasto determine where teachers were starting with content and pedagogicalknowledge and to inform professional development activities in the nextfew years (Watson, 1998). Within a structured interview format, teachersfrom kindergarten to Grade 10 were asked about a variety of topics. Theinterviews lasted approximately forty-five minutes and were audiotaped.Selected elements of that survey, such as respondents’ rating of the import-ance of statistics in society and their confidence in teaching a varietyof topics, were analysed numerically (Callingham et al., 1995). Otherelements were analysed qualitatively, with input from the teachers whohad been interviewed, to help develop the final profiling instrument.

After conducting the initial trial and considering various perspectivesfrom which profiles could be developed, I decided to devise a protocolthat could be used by systems or schools to evaluate teacher performancerelated to the chance and data curriculum. I decided to place additionalemphasis on assessment of pedagogical content knowledge, reflection onpractice, and models of teaching. A more comprehensive coverage wasgiven to individual lessons in recognition of Pegg’s (1989) structuralapproach to planning and implementation.

Structure of the Profile Instrument

The final profiling instrument was designed to be administered as a ques-tionnaire or used as a basis for a one- to two-hour interview. This choiceof design for the instrument was made so it could be used with large-scale professional development programs and yet would result in adequatedata on teachers’ knowledge, practices, and beliefs. The profile needed toinclude Shulman’s (1987b) knowledge variables as well as specific infor-mation on teacher background and professional development in order toencourage positive teacher reflection on the relevant issues. The issuesspecifically related to the chance and data curriculum were adapted fromthe earlier interview instrument, the curriculum itself, and student surveys.

The final protocol for the profiling instrument is presented in theAppendix with all space for written responses deleted. The profile consistsof 10 sections outlined in the paragraphs below. Clarifying commentsare inserted with square brackets. The profile begins with open-ended

310 JANE M. WATSON

questions allowing teachers to discuss aspects of what they consideredimportant. The later sections address specific issues with directed ques-tions to ensure these aspects were covered. Table 1 contains a summaryof the objectives of each section of the profile categorised by Shulman’s(1987b) seven knowledge types; teacher background and professionaldevelopment (Preston, 1996); and an opportunity for reflection.

The initial section of the profile allowed teachers to brainstorm factorsthat they considered significant for the teaching of chance and data, eitherfor themselves or for others they might employ to teach the subject. Thepurpose of this section was two-fold: (a) to obtain information on teachers’views and (b) to encourage a reflective attitude that would be sustainedthroughout the profile. The item on significant factors was placed at thebeginning of the profile to allow for open-ended reflections before teacherswere influenced by the views of the researcher. It allowed for responseswith respect to all seven of Shulman’s (1987b) categories of knowledge.Section 2 of the profile presented questions related to teacher preparationat the unit level; to assist teachers with responses, several questions wereprovided about preparing a unit. The questions in Section 3, related toteacher preparation at the individual topic or lesson level, were similar tothose for the unit level but more detailed. It was intended that responsesin Sections 2 and 3 illustrate Shulman’s (1987b) first four categories andperhaps touch on education contexts (for units) and learners (for lessons).

Section 4 of the profile instrument gathered self-reports of teachingpractice in lieu of classroom observation. Although it may be claimedthat, compared to classroom observation, there are weaknesses in self-reporting, it should also be noted that there are strengths in this approach.Self-reporting should reflect patterns of behaviour over time, not just theobservable outcomes of a single lesson that may have been orchestratedto be part of an assessment process. It is also of interest to know whatteachers think is important over long periods of time when they are notbeing observed. Self-reporting is less time consuming than long-termobservation; it also engenders good habits of reflection. In this sectionteachers were asked to give overviews of their practices and to reflect ontheir years of teaching experience. The knowledge of student difficulties,which relates to Shulman’s (1987b) knowledge of learners and their char-acteristics, was further developed in later parts of the profile. Section 5considered two specific topics within the chance and data curriculum:average, a well-established topic, and sample/sampling, a less well-knowntopic. The questions in this section focused primarily on pedagogicalcontent knowledge, with obvious links to teacher content knowledge,general pedagogical knowledge, and knowledge of students as learners.


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312 JANE M. WATSON

In Section 6, teachers were asked to mark their level of confidence intheir ability to teach nine topics from the chance and data curriculum.Responses were marked on a continuous scale from Low Confidence toHigh Confidence and translated into integers from 1 to 5, with an option tomark “would not be teaching.” Concerns expressed in this section reflectedboth teachers’ content knowledge and their pedagogical knowledge. It wasintended that interpretation of the confidence scale should identify topicsteachers considered more difficult to teach relative to other topics. Wherea topic was listed as a single phrase, level of sophistication and scope ofthat topic were left open. Although this resulted in some ambiguity in thequestion, it gave the scale flexibility as it could be used by teachers ofdifferent grade levels. The option of “would not be teaching” also allowedteachers to exclude topics not relevant to their grade levels; this yieldedinformation on which topics were not being taught at various levels. Theconfidence scale also provided a cross-check for assessing expertise forsome of the mathematical topics covered elsewhere in the profile, forexample in Section 8. For Section 7 of the profile, 10 items developed byGal and Wagner (1992) were adapted in order to assess teachers’ beliefsabout statistics in everyday life. Four items addressed personal confidencein interpreting statistics encountered in everyday life, four gauged beliefsin the social importance of statistics, and two items assessed understandingof the issue of sample size as used in social settings. As such, these itemsreflected the concerns of Shulman (1987b) on the need to assess knowledgeof education contexts related to the character of cultures and of educationalends, purposes, and values. They also recognised the dimension of civicand social responsibility noted by Sachs (1997). Responses were measuredalong a continuous scale which translated into a scale from 1 to 5, withreversal to take account of negatively worded statements.

Section 8 included six items used in student surveys (Watson, 1994).These items presented the opportunity to explore teachers’ understandingof common student responses, reflecting Shulman’s (1987b) concern thatknowledge of learners and their characteristics, as well as teachers’ levelsof content knowledge, be addressed so that appropriate responses mightbe discerned. Further, the different settings that were used for the sixitems gave an opportunity to assess teachers’ appreciation of the educa-tional contexts in which chance and data concepts are applied. Thebenefit of using items from previous student research was that teachers’views of common student responses could be referenced later against aresearch base of common student responses. The first question for eachitem concerned likely and appropriate student responses and the secondconcerned their potential use in teaching. If teachers considered an item


useful for teaching, the profile provided further information with respect tothe pedagogical content knowledge and curriculum knowledge of teachers,regarding how they might use the item to address student difficulties.

The questions on teachers’ backgrounds in Section 9 reflected areasof interest in relation to this new part of the curriculum. In some studiesa teacher’s background is taken as a measure of content knowledge. Inthis profile, however, a teacher’s background contributed to, but was notthe only measure of, content knowledge. The questions in Section 10sought information that could be used in planning development programs,particularly in relation to curriculum knowledge associated with importantcurriculum documents. They reflected to some extent teachers’ appre-ciation for the educational context within which teaching takes place(Shulman, 1987b) and, on a system basis, a renewed sense of profession-alism in the field (Sachs, 1997).

APPLICATION OF THE PROFILE INSTRUMENT

A group of 43 teachers completed the teacher profiling instrument. Theteachers came from primary schools (Grades 1 to 6, n = 15) and secondaryschools (Grades 7 to 12, n = 28) from public and private systemsthroughout Australia. Not all teachers responded to all parts, but allanswered Section 10 indicating that they had considered responding to allsections. Of the 43 teachers, 21 were interviewed over a 90-minute period,19 submitted the survey in writing by mail and 3 submitted the surveyin an internet hypertext form. The non-random sample was chosen to berepresentative of primary school teachers and high school mathematicsteachers, but it was voluntary in the sense that no pressure was exertedto participate. There was no indication that the teachers had any partic-ular expertise in probability and statistics, although some were planning toreview some learning material about these topics. In the following section,I describe the use of the profile with these 43 teachers and illustrate the typeof information that can be obtained from each section of the profile. Themajor concern addressed is whether the responses from teachers indicatethat the profile instrument satisfies the criteria outlined in Table I. Potentialuses of this type of information are considered in the Discussion.

Section 1: Significant Factors for Teaching Chance and Data

Teachers noted many different factors as significant for teaching chanceand data. These factors reflected the different perspectives of the teachersand can be clustered into four groups: the teachers themselves, the

314 JANE M. WATSON

students, the content, and school issues. The factors associated withteachers illustrated all seven of Shulman’s (1987b) categories as well asthe background and professional development concerns of Preston (1996).They were grouped into subcategories related to previous experience (S1,S2, S3, S4, P from Table 1), current experience (S3, S6, S7), and pedagogy(S2, S4, S5, S7). The factors associated with students reflected prior know-ledge, ability, interests, learning styles, attitudes, and literacy level (S5,S7). The factors noted for content were grouped into subcategories relatedto the subject matter (S1, S3, S6), such as specific topics, state syllabi,integration across the curriculum, the use of technology and news media;and to how the content is presented (S2, S4, S6, S7), for example, relevanceto the real world, practical nature of experiences, sequencing of topics,motivation, group work, and projects. The factors associated with schoolissues involved resources and budgeting, time tables, materials, views ofthe school system, and class size (S2, S6, S7).

In addition to factors mentioned by teachers as affecting the teachingof chance and data, it was possible to identify cases of teacher reflection.The following excerpt from a primary teacher addresses the preparation ofstudents for taking their places in society:

Comprehension of children about this area in today’s society: The importance of devel-oping an understanding for the children – but I guess that’s what teaching is all about;but children live in a world where data is [sic] flowing so fast that they must be able tocomprehend what is going on and that is a specific field I think.

This comment reflects the pressure on professional teachers to preparestudents to take civic and social responsibility as noted by Sachs (1997),as well as Shulman’s (1987b) concern for educational ends, purposes, andvalues.

Section 2: Preparing to Teach a Unit in Chance and Data

Teachers’ perspectives in addressing the question on teaching a unitvaried greatly. Teachers of senior school levels were more likely thanthose at primary levels to take a wider perspective and list topics withina unit related to the entire chance and data curriculum in the systemin which they taught. A typical high school teacher’s response listedtopics such as “DATA: graph interpretation/construction, central meas-ures, spread of data, links to standard deviation – practical applications.PROBABILITY: terms, how used by society, arrays/tree diagrams/sets,applications.” Teachers of Grades 11 and 12 made similar responsesincluding more sophisticated techniques. At the other extreme, upperprimary teachers were more likely to consider a more focused unit of work.One teacher responded with the following list. “Survey material about


Figure 1. One teacher’s concept map for teaching a unit on chance and data.

themselves, interests, hobbies, etc.; using computer software to do this;graphing; frequency of things happening: coin tosses, recording results ina table, tallying, etc.” A few teachers drew flow charts or created conceptmaps of the relationships among the topics in the curriculum. A conceptmap from a high school teacher is presented in Figure 1.

Section 3: Preparing to Teach a Lesson in Chance and Data

Two interesting aspects with respect to the preparation of an individuallesson surfaced from this section: (a) the topic chosen and (b) the detailthe outline provided. The topics chosen for lessons ranged widely, froma general outline with no specific topic to a single activity, such asconducting an experiment with drawing pins (thumb tacks) or doingpermutations. For primary teachers the most commonly mentioned topicswere surveys, graphing, chance generally, and probability in context (suchas dice). For secondary teachers the most commonly mentioned topicwas probability (often at Grades 7 and 8). Other topics mentioned bymore than one secondary teacher were associated with normal and otherprobability distributions; data collection and graphical presentation; andmeasures of central tendency and spread. These topics may have reflectedteachers’ favourite topics; this conjecture was checked by comparison ofthese responses with those in Section 4. Overall the responses representedmost areas of the chance and data curriculum and hence indicated thatgeneral coverage was taking place in schools.

The detail provided in the lesson descriptions again varied greatlyamong the teachers. With respect to Pegg’s (1989) model for identifyingincreasingly complex lesson structures, the five points in the questions

316 JANE M. WATSON

encouraged sequential responses but gave opportunities for relating thelesson to others in a unit or to students’ needs and interests. At the lowestlevel, a few teachers ignored the outline provided and indicated a broadsingle idea for a lesson; for example, a primary teacher focused onlyon dice, indicating he would introduce games related to fairness. Mostteachers, assisted by the outline, produced a sequenced account of theingredients of a lesson. The following outline is from the profile of a highschool teacher.

• Methods of displaying data – Frequency distribution, stem and leaf. . .

• I’d use one set of collected data, e.g., number of pets/family for eachstudent.

• Show on board various ways of displaying data.• Have students use their own set of data (or one from a textbook) to

practice.• Students would work individually or in the small groups they

collected their data in.• Would spend about 50 minutes.• Follow with graphing, extension to cumulative frequency (ogives),

etc.

Some teachers used the outline and reflected in a more integrated fashion.The following extract is part of a high school teacher’s description of anintroductory lesson on experimental probability.

To date I have spent about 3 lessons with each of 2 classes, but would have liked anotherone or two to facilitate more investigation rather than exam training . . . . I would like toconcentrate on the practical side of the unit, encouraging students to find out for themselvesrather than having to ‘take my word for it.’ As I work with older students I tend to be fairlyunrestrictive with student groupings, allowing groups of two or three to be organised bythe students. Only rarely would I choose groups myself.

Section 4: Teaching Practices

From the questions about which grades were taught and the number ofhours or lessons, it was possible to provide summary statistics to char-acterise the group being profiled. For example, nearly all teachers taughtsome chance and data concepts, an indication that the suggested nationalcurriculum was having some impact on classrooms.

In relation to the enjoyment associated with specific topics, for teachersand their students, 12 teachers explicitly mentioned the same topicsas most enjoyable for themselves and their students. These includedgraphing; normal distributions; surveys and data collection; analysing and


interpreting data; and various aspects of probability. Only four teachers didnot respond to this question, three because they were not currently teachingthe topics.

In the responses about enjoyable topics, 53% of the teachers chosethe same topic as the topics suggested for lessons (Section 3) whereas14% chose different topics. The remainder of the teachers did not respondor reported that they enjoyed all topics. These data suggest that manyteachers are either comfortable describing the teaching of topics theyenjoy or come to enjoy the topics they are more familiar with teaching.Many teachers specifically mentioned hands-on and practical activities orresearch/project work as those that their students enjoyed the most, withoutspecifically mentioning the subject matter associated with them. Many ofthe other topics mentioned as those enjoyed by students involved action,such as carrying out surveys, playing chance games, and doing projects.Several teachers admitted that they did not know what topics their studentsenjoyed, which may point to a more “telling” teaching style. On the otherhand, some teachers differentiated students who liked a topic and thosewho did not. Two primary teachers pointed out that some of their strongerstudents liked graphing whereas other students struggled. A high schoolteacher made a similar comment about the topic of correlation.

Four teachers could not suggest topics their students found difficult;one said students had no problems, and another said that difficulties variedamong his students. Teachers’ responses ranged from describing proce-dural problems to discussing conceptual difficulties. These problems aresummarised in Figure 2, along with a brief summary of the remediessuggested for each. The suggested difficulties demonstrate that teachersknew much about their students as learners and how to assist their thinking.The suggested remedies reflected varying teachers’ values, beliefs abouttheir roles as instructors, and beliefs about students’ roles in constructingtheir own knowledge.

With respect to specific materials used in teaching chance and data,most teachers used concrete materials, materials for exploring chanceoutcomes, various sources of data, and either computers or calculators.Between 70% and 81% of teachers acknowledged using each of thesefour categories. This section of the profile elicited the types of knowledgesuggested in Table I and was particularly useful in relation to knowledgeof learners and their characteristics.

Section 5: The Topic of Sampling

In general, responses keyed by the term sample displayed less sophistic-ation than responses to the previous sections or to the term average. This

318 JANE M. WATSON

Difficulties Remedies

Primary:

Procedural

• Graphing accurately (2) • Focus on basics, use grid paper

• Finding probabilities (2) • Be explicit, model, practice

• Mechanics of tables • Demonstrate

• Complex multiplication • Use calculators, simplify

Conceptual

• Organizing / presenting data (3) • Discuss, demonstrate, use group work

• Interpreting information / graphs /outcomes (3)

• Discuss, find different angles

• Formulating questions (2) • Question, focus, share

Secondary:

Procedural

• Calculating probabilities, permutations,combinations (7)

• Persevere, “come in at lunch,” “bepatient”

• Mean / median / mode (5) • “M–most,” “O–often,” vary approach,be concrete, remind of assessment

• Range / variation / standard deviation(3)

• Slow pace, break down, remind ofassessment

• Tree diagrams / Venn diagrams • Introduce easy problems

• Choosing form of graph • Give examples

Conceptual

• Interpreting data / results (4) • Ask leading questions, give examples

• Theoretical component / probability /inferential (3)

• Use diagrams, life-related examples,variety

• Normal distribution (2) • Slow pace, give practical examples

• Conditional probability • Be concrete

• Data analysis • Simplify

• Language (new English speakers) • Give individual help, use examples

Figure 2. Perceived student difficulties and remedies.

may have been due to either a lack of familiarity with the topic or a beliefthat it was peripheral to the curriculum. Some teachers had not taught alesson on sampling and hence hypothesised about how they would go aboutdoing so.

An example of a high school teacher’s low-level response with a singlebroad idea for a lesson is as follows: “A survey is often used to show howsample is a subset of population. This is an activity that kids like doing,usually working in pairs or small groups.” The following response from


a primary teacher illustrates the more complex sequential nature of theteacher’s planning:

• Intro – discuss current understanding.• Using a survey to take a sample, e.g., left/right handedness – any

relationship to numbers who are boys/girls.• Just survey own class group. Use figures to predict for school

population. Discuss.• Enlarge sample and compare predictions.• Paired or individual work. Combine results and discuss predictions as

a whole class. Just one possible situation – many various samplingscould be done concurrently.

A Grade 11–12 teacher who had not taught sampling produced thefollowing higher-level response indicating how the topic could be integ-rated with other work.

Some problem that could not realistically be answered without sampling, such as thepercentage of flathead in the Derwent River. Students would then need to come up witha suitable method for solving this. It could then be simulated by computer or via colouredtokens in a bag or similar . . . . Sampling would be a recurrent theme through a reasonablepart of the course, particularly in data representation, so it could cover 9 or 10 lessons intotal, but as a topic on its own it would receive maybe 2 lessons (3 hours) . . . . Experimentalprobability would be a useful precursor. It would then lead well into data representation.

Teachers were less familiar with the topic of sampling than withaverage. Some teachers were made aware of the possibility of teachingsampling only by the profile instrument; for example, one high schoolteacher, who had taught the topic only to a small degree, concluded, “Ihave taught them to some degree but it wasn’t until I read your questionsthat it encouraged me to think more about the individual aspects insteadof just a big picture view.” Other details of teachers’ knowledge about andattitudes toward sampling are discussed in Watson and Moritz (1997). Inrelation to the types of knowledge expected to be elicited by this section(see Table I), there was little evidence given of knowledge of learnersand their characteristics. Teachers may not have had enough experienceto comment or may not have thought it relevant to the question.

Section 6: Confidence

The lowest overall mean for the 42 teachers who expressed confidenceon teaching the nine topics listed in Section 6 was for odds and the highestwas for graphical representation. For the primary teachers the mean scoresranged from 3.00 for median to 3.92 for data collection. For the secondaryteachers the range was from 3.68 for odds to 4.59 for median. All means

320 JANE M. WATSON

for secondary teachers, except for odds, were higher than the highestmean (3.92) for primary teachers. The only topic for which males andfemales differed was average, with males being more confident. Similarly,there were six topics about which high school teachers were significantlymore confident (p = 0.05): equally likely outcomes, average, basic prob-ability calculations, median, graphical representation, and sampling. It isnot surprising that high school teachers were more confident than wereprimary teachers about some topics because they generally had strongermathematical backgrounds. This was confirmed by information collectedin Section 9.

Section 7: Beliefs about Statistics in Everyday Life

For the teachers in this study, the responses to the questions in Section 7were not of an extreme nature (strongly disagree to strongly agree). Whenthe four items associated with personal confidence in handling statistics insocial settings (2, 4, 6, and 9) were combined, there were no large differ-ences between males and females or between primary and high schoolteachers. Similar results were found for the four statements on the socialneed for statistics (1, 5, 8, and 10) and for the two items on sampling (3 and7). The largest difference between high school and primary teachers wasassociated with Item 2, with high school teachers agreeing more stronglyabout understanding statistical terms in the media (p < 0.01).

Section 8: Student Survey Items

I discuss two of the six student survey items: Item (3) on odds and Item(4) on sampling. I have chosen to focus on these items because of thedifferences in the information obtained and the opportunity to compareresults with student outcomes.

For Item (3) some teachers admitted that they did not know what “7:2”meant or were unsure of how odds ratios were to be interpreted. Forthe first question on likely student responses, most teachers could giveexamples of likely student responses that were inappropriate, whereasonly 15 teachers could think of appropriate student responses. All 15teachers were high school teachers. Among the likely but inappropriatestudent responses suggested for primary students by their teachers werethe following: “North is expected to win”; “North is 7 times more likelyto win”; and “North has 7 goals, South 2 goals.” High school teacherssuggested these and other inappropriate responses: “The chance of Northwinning is 7/9”; “You are 3.5 times more likely to win”; and “North has thebetter chance of winning because the coach says so.” Among the responsesindicated as appropriate were the following: “South more likely to win”;


“Only 2/9 chance of winning”; “If you bet $2 you will win $7, so Northisn’t likely to win”; and “If North and South played 9 games you wouldexpect South to win 7 times and North only twice.”

That the teachers’ suggestions of appropriate and inappropriate studentresponses represented a valid spectrum of what students would actuallysay can be confirmed if the responses from the two groups are comparedwith others in the literature. Moritz, Watson, and Collis (1996) reportedon responses to this item from 823 students in grades 6 to 10. They foundthat students’ responses came from three contexts: a chance context, forexample “a 2/9 chance of winning”; a frequency context, for example, apredicted score or “for every nine games played North wins two”; or asocial context, for example “bet $2 to win $7.” The responses suggested byteachers also fell into one of these three contexts. All responses suggestedby teachers can be represented by students’ responses cited by Moritz etal. (1996).

Of the 15 teachers who gave appropriate student responses to the oddsquestion, 10 reported the highest level of confidence (5) in teaching thetopic (mean = 4.4) in Section 6. None of the teachers who could notthink of an appropriate student response reported a confidence level of5. The levels of confidence for other responses ranged from 1 to 4 (mean= 3.2) with three non-responses to the confidence question. Hence there isevidence that teachers were realistic in evaluating their ability to teach thistopic.

For Item 4, three teachers did not respond, and of the remaining 39teachers, 24 (20 high school, 4 primary) could give appropriate criticismsof the article, based on the nonrepresentative sample from Chicago andthe claim for the entire United States. A number of teachers, both primaryand secondary, noted that this item would be difficult for their students;some indicated that their students would believe the article due to theirstereotypes about life in the United States.

Among the inappropriate or less appropriate responses suggested forstudents were the following: “I wish we could carry guns,” “How were thedata collected?”, “What is a fair sample size?”, “A third of 6 out of 10 or allstudents?”, “Gun-control/civil liberties comments,” and “They asked a lotof kids so it must be true.” Again, these responses reflected what studentswould actually say (see Watson & Moritz, 2000). The teachers’ successrate on the item (56% overall) was higher than that of grade 11 students(23%) surveyed by Watson and Moritz.

The comparison of teachers’ responses with their levels of confidencein teaching sampling showed a different pattern than the comparison forthe topic of odds. Although 7 of the 24 teachers who gave appropriate

322 JANE M. WATSON

responses rated their confidence in teaching sampling the highest (5), tworeported a low confidence level (2). For the teachers who failed to giveappropriate responses, two expressed confidence at the highest level andnone at level 2 or below. The average confidence level for the two groupswas very similar (3.9 for those who gave appropriate responses comparedto 3.7 for the others). In this case a context-based question focussing ona particular aspect of sampling (bias) did not appear to be associated withlevel of confidence in teaching sampling.

Many primary teachers felt that the topics of sampling and odds werenot accessible to their students due to either complexity or context. Theopinion of high school teachers was split on the odds item, with somethinking it would be a good example to use in their classes; others thoughtit was biased against girls and students who had English as a secondlanguage. There was more enthusiasm expressed by high school teachersabout the sampling item; some said it would be good to illustrate thesample-population distinction, although a few others were concerned abouttheir students’ understanding of the geography necessary to interpret theitem. Reflection by teachers on the place of both topics in the curriculumoccurred on several occasions in relation to educational contexts and/orvalues. Teachers were not inhibited in expressing their views about theseitems and, sometimes, their lack of confidence in relation to the topics,particularly the topic of odds.

Section 9: Teacher Background

The sample of 43 teachers consisted of equal numbers of male and femaleprimary teachers but contained more male than female secondary teachers.Primary teachers either taught a single grade or a combined class such asGrades 5 and 6. Secondary teachers ranged across several possibilities: oneteacher taught only one Grade 8 mathematics class, others taught Grades 7to 10, Grades 11 and 12 only, or all Grades 7 to 12.

The median number of years of teaching experience was 16 with themean of 16.7 and a standard deviation of 7.64. The range was from 1 to37. There were no differences for male and female teachers or betweenprimary and secondary teachers. In terms of previous exposure to statistics,79% of the 43 teachers had taken courses including probability and stat-istics in previous tertiary study. Of these, 88% had been exposed to topicsin a specialist mathematics or statistics course, 32% in a psychology oreducation course, 9% in a zoology course, 9% in an economics course,and 6% in a geography course. The percentages add to more than 100%due to multiple responses. Of the 33 who reported the length of timethey had studied statistics, 9% reported a few hours, 15% a few weeks,


36% a semester, and 39% more than a year. All teachers who had studiedstatistical topics for more than a year were secondary teachers.

Of the 33 teachers who responded to the request to list up to three topicsthey remembered studying, six could not remember any topics. The mostfrequently mentioned topics were general ones like probability and hypo-thesis testing. More specific topics mentioned by five or more teachersincluded permutations and combinations, normal distribution, chi-squaretest, t-test, and correlation.

Section 10: Professional Development

Exposure to the state curriculum documents and National Statement (AEC,1991) was similar for both primary and secondary teachers, but moreprimary teachers had made use of the Profile (AEC, 1994). Commer-cial and Australian Bureau of Statistics materials were favoured more bysecondary teachers. The question on professional development revealedthat 47% had not participated in any professional development relatedto chance and data. When asked the type of professional developmentthey would prefer, 40% expressed a preference for school-based sessions,19% selected either personal reading or a University course; the remainderoffered no clear preference or selected multiple options. In terms of peoplewho would be best to lead professional development, 21% suggestedanother teacher at the school, 21% a regional curriculum officer, 21% allthree of the options, and 51% an outside expert. Many teachers indicatedmore than one choice. Many also made the proviso that whoever did theprofessional development should understand what it is like to be in aclassroom.

A final question provided an opportunity for teachers to makeother comments about professional development in chance and data, assuggested by Edwards (1996). Many did not reply. Of those who did,many expressed a need for some or more professional development, orcomplained of the current situation in their localities. Specific topics, suchas the use of relevant computer software, were mentioned by some. Onlyone teacher felt that professional development was unnecessary becausechance and data represented such a small part of the overall curriculum.A few made comments on the general usefulness of chance and datafor students in preparation for participating in society adding an unex-pected value component of Shulman’s (1987b) knowledge not anticipatedin Table 1.

324 JANE M. WATSON

DISCUSSION

How the information obtained from the profile is integrated depends on thepurposes for which it is to be used. It may be used to assess an individualteacher’s performance, for example to obtain accreditation or promotion;to devise a personal professional development program; or to gain anoverall distributional picture of a group of teachers; to compare groupsof teachers, for example, to establish relative excellence criteria or neededchanges to practice. Groups that might be compared include primary andsecondary teachers, males and females, teachers from public or privatesectors, a certain group of teachers longitudinally over time, or groupsdefined by the profile itself, such as by years of teaching experience orexpertise on a particular topic. For some purposes individual sections of theprofile offer the relevant information, whereas for others, a combination ofinformation from several sections may be appropriate. Within the scope ofthis study it is not possible to illustrate all possible uses of the profile withthe small sample of 43 teachers. I first discuss the immediate uses to whichthe information gleaned from this sample could be put. I then comment onuses in the wider educational community and possible revisions to improveoutcomes.

Specific Outcomes for the Teachers Surveyed

Teachers were positive in their attitudes toward completing the profilinginstrument. Those who were interviewed often discussed ideas as well asprovided written responses. Although a few of the teachers interviewedpreferred to write only, many appreciated the opportunity to work throughthe questions orally. Teachers appeared to appreciate the necessity to docu-ment their background, understanding, and perceived needs, as well as theopportunity to present their own perspectives to someone who was goingto value and use the information. Instances of reflection by teachers wereobserved throughout the profile both in oral and written responses. Thosewho provided responses as a written survey generally gave more conciseanswers, with a few omitting some items, saying that they had run out oftime. Whether this was the case or whether they did not feel comfortableanswering a particular question is unknown. Respondents were asked todevote up to 90 minutes to the profile; some indicated that they had spentmore than 90 minutes.

No one objected to any questions as being intrusive. It appearedthat teachers felt free to agree or disagree with the designers of theprotocol. There were several places where teachers took positions poten-


tially different from those of the researchers. These included suggestionsthat some topics (e.g., odds) were not particularly important in relationto the overall curriculum, topics they did not like (e.g., probability) ormethods they would not use (e.g., calculators), and controversial viewson professional development (e.g., the presenter must understand theclassroom). Teachers also occasionally commented that the process ofcompleting the profile instrument had itself raised issues that they wouldthink about afterward. The presence of such comments suggests the instru-ment was exploring views that teachers actually held, rather than theviews they perceived the interviewer wanted them to hold. It must benoted, however, that teachers knew they were participating in a universityresearch project that, although it might inform professional developmentprograms affecting them, would have no effect on their future employment.It is impossible to say how responses might have been altered had theprotocol been administered by the teachers’ employers or by an accreditingagency.

Some of the critical issues related to the teaching of chance and datathat arose from the current study were related to teachers’ planning, confid-ence, content knowledge, and curriculum knowledge. At the primary level,although many activity-based chance and data lessons occurred, therewas little evidence of coherent program planning. This corroborates thefindings of Bright and Friel (1993). At the senior secondary level, wheresome teachers had previously taught theoretical aspects of probability andstatistics and traditional programs were well documented, there was arecognition of the difficulty of the topics for students but little effort tointroduce activity-based aspects, such as simulation or actual sampling,that would reinforce the theory. These suggestions are made in A NationalStatement on Mathematics for Australian Schools (AEC, 1991) but thisstudy revealed that less than a quarter of the secondary teachers usedthe document. These observations are consistent with those of Stigler andHiebert (1997).

Several specific proposals for professional development related tochance and data can be made for the teachers in this study. The lowermean confidence ratings for primary teachers suggest the need for specificexperiences related to equally likely outcomes, basic probability outcomes,odds, median, and sampling, and what to expect of student learning onthe topics. Given the lack of structure of many of the lesson and unitplans presented by teachers, it is advisable to provide activities to assistteachers with more integrated planning, as suggested by Pegg (1989). This

326 JANE M. WATSON

would incorporate pedagogical content knowledge, content knowledge,and curriculum knowledge, with an appreciation of learner characteristics.

The views expressed in Section 4 on enjoyable topics and remedialactions for students with difficulties were particularly illuminating andsuggest that many teachers need to become more aware of their students’needs. Section 8 was also useful in exploring teachers’ knowledge of theirstudents as learners, as well as their own content knowledge. Although itwas reassuring to find a strong correspondence between teachers’ assump-tions about their students’ responses and actual student responses, there isa need for professional development to increase not only teachers’ know-ledge of the content but also their appreciation of how their students learnthat content.

The fact that teachers were not familiar with important curriculumdocuments suggests the need to introduce sessions through which teachersmight become familiar with these documents and how the documents cancontribute to effective teaching. Finally, it is possible to base professionaldevelopment activities directly on the specific topic suggestions made byteachers, such as the need for relevant current examples, investigativeapproaches, and cross-curricular links.

Sachs (1997) viewed the recognition of the need for and the provi-sion of professional development as significant in reclaiming teacherprofessionalism in Australia.

One of the hallmarks of being identified externally as a professional is to continue learningthroughout a career, deepening knowledge, skill judgment, staying abreast of importantdevelopments in the field and experimenting with innovations that promise improvementsin practice . . . Here lies one of the paradoxes for teacher professionalism . . . While studentlearning is a goal, often the continuing learning of teachers is overlooked. (pp. 267–268)

These views reflect those of Stigler and Hiebert (1997) on the situation ofmathematics teachers in the United States. The cyclic nature of the useful-ness of a teacher profiling instrument to identify needs and to indicate thesuccess of professional development programs is evident.

General Use of the Profile and Possible Revisions

The concerns of Shulman (1987b) and Shaughnessy (1992) about docu-menting teachers’ knowledge bases have been addressed by the profilefrom several perspectives. The attempt to go beyond a documentationof number of courses as evidence of subject matter knowledge wasconsidered an important aspect of the profile, particularly for this new areaof the curriculum. In an interview setting it would be possible to includeprobing questions concerning types of knowledge not initially suggested


by teachers. Recent work of Ma (1999) suggests the potential usefulness ofa more direct approach to questioning teachers about the connectedness ofthe various concepts in the chance and data curriculum and about possiblemultiple perspectives to approaching topics.

The items in Sections 7 and 8 of the profile are potentially moreuseful with large, representative samples of teachers. For systems, theycan provide a quick overview of teachers’ confidence in the classroom andmore widely in social contexts, as well as of values associated with socialapplication of the concepts in the curriculum. The information gainedfrom the questions on teachers’ backgrounds may have general or specificrelevance, depending on the purposes of the developers of a profile. If arandom sample of teachers had been selected in this study (it was not),then the information collected could indicate, for example, a gender bias ina particular teaching area. The grade levels teachers are currently teachingmay be important for systems if not all teachers are from a common group.This may suggest educational policy changes. The information on previouscourses completed could be of interest to those who teach preparatorycourses for teachers, as well as those preparing in-service professionaldevelopment. Information on whether teachers have seen, read, or usedvarious important curriculum documents can be used by systems to provideinsight not only into the teachers’ habits but also into the availability orappropriateness of the resources for teachers at different levels.

If information from various sections of the profile is combined it ispossible to piece together a picture of how teachers are coping with aparticular content area such as the chance and data curriculum. For indi-vidual teachers it would be possible to use the responses to suggest aprogram of personal professional development in the areas deemed defi-cient. For those who would wish to obtain a single number to quantifya teachers’ performance in relation to the profile, it would be possible toattach a numerical measure to many responses, employing a model suchas that used by Pegg (1989). I do not favor such an approach but preferto focus on the qualitative power of the data to provide information aboutteachers’ practices. The reliability of the profile can be confirmed to someextent by cross-referencing responses in different sections. Examples givenhere were related to expressed confidence about a specific topic and theability to provide appropriate answers to questions posed on that topic.Other opportunities to check for the reliability of parts of the profile arepossible, as indicated by the multiple referencing in Table 1.

Several critical issues arose from the teachers’ responses to the instru-ment. Although at the beginning of the profile teachers were asked about

328 JANE M. WATSON

“significant factors for teaching chance and data” and at the end they wereasked for specific comments about professional development, they werenot specifically asked to think about critical issues. In some contexts thismay be a pertinent question to address, but care must be taken not to beemotive in soliciting opinions. Most of the factors in Section 1 of theprofile discussed by teachers could be considered critical issues dependingon the context in which they were discussed. There is the opportunity, withmore focused questioning, to compare critical issues as seen by teacherswith those deemed critical by those administering the profile.

The stated objective to take a middle path among profiling alternativeswas achieved in terms of the time spent by the teachers and researchersand the quality of the information obtained. Of course, other alternativesexist as well. Mayer and Marland (1997), for example, focused entireinterview sessions on five highly effective teachers to gain a deeper under-standing of their knowledge of students, including knowledge of classesand groups, knowledge of individuals, sources and means of acquisition,and classroom use of the knowledge. This research provided data on oneaspect of expert teacher competence. The profile developed here, althoughnot exploring teachers’ sources and means of gaining knowledge aboutstudents, did document a range of behaviours in relation to students, fromthose considered commendable to those that appeared to need change.

The necessity to obtain information related to the teaching of the chanceand data curriculum for purposes of comparison over time is evident,especially during the early years of its implementation. How many yearsconstitute an early phase of curriculum implementation is problematic. Inthe United States, for example, Stigler and Hiebert (1997) found littleevidence that teachers had changed their practices five years after thepublication of the NCTM Standards in 1989. Similarly, Watson and Moritz(1998) found no change in student understanding of chance measurementfour years after chance and data were introduced in an Australian state in1993. The potential need for monitoring is likely to exist for many yearsafter change is proposed.

CONCLUSION

In the wider educational context in which this profile was used, it appearsto offer a reasonable alternative to some of the more expensive andtime-consuming suggestions espoused by Shulman (1987a). The semi-structured interview format offers the flexibility that suits a wide range ofteachers’ reflective styles. The validity of a seriously considered reflection


apart from the classroom may be just as great as that of a lesson preparedas a single demonstration of brilliance for an external assessor. In takingthe middle ground between extensive observation and/or elaborate writingon the one hand and Likert-type surveys on the other, I hope to providea useful model for others who would wish to profile quality teaching,particularly in the newer areas of the school curriculum. As Australia andother countries begin a concerted effort to develop standards of excellencefor mathematics teachers, the richness of responses obtained in this studywould indicate that there is promise in such an instrument.

ACKNOWLEDGEMENTS

This research was funded by an Australian Research Council SmallGrant awarded at the University of Tasmania. The author wishes tothank Jonathan Moritz for assistance in preparing the profile and makingcomments on early drafts of the paper. Interviews were conducted by BobPrasad.

APPENDIX: CHANCE AND DATA TEACHER SURVEY

Significant Factors for Teaching Chance and Data

The questions through-out this survey are an attempt to identify factors which aresignificant for the teaching of Chance and Data.

BEFORE you go further, please brainstorm and write down what aspects you thinkmight/should be included in a survey of this type. It may help you to consider:

• factors which you consider influence your teaching of Chance and Data• factors which you would look for in employing a teacher to teach Chance and Data

Some factors may be particular to Chance and Data, while others may be general factorsin teaching which you feel have a significant impact in teaching Chance and Data.Feel free to write your answers in point form or any form you wish.

AFTER you have completed this survey, please return to this page, and write below anyaspects which you would like to include which are not present in your list or our survey,or any aspects which you think are under-represented.

Preparing to Teach a Unit in Chance and Data (1)

• If you were preparing to teach a unit (or sequence of lessons) in Chance and Data (toa grade level which you teach), how would you go about preparing?

• What resources would you refer to?• Would you consult anyone else? Whom?

330 JANE M. WATSON

• How long might you spend to prepare the overview of a unit plan?

• Briefly brainstorm topics which you might include in the unit in Chance and Data.Just spend a few minutes, and do not bother to consult other resources.

Arrange these topics into a rough overview plan to show the sequence of topics in the unit.

Preparing to Teach a Unit in Chance and Data (2)

• Now focus on one section or topic which you think is important. Briefly outline howyou might teach it.

• How would you introduce the topic?• What resources or materials would you use?• How much class time would you spend?• What teaching methods and groupings of students would you use?• What lessons might precede or follow the topic?

• Have you ever taught this topic?• Do you enjoy teaching this topic?• How do your students respond?

Teaching Practices

• Do you currently teach any topics in Chance and Data?

❐ No, because:❐ Yes:

• Which grades?• About how many hours or lessons?• Do you enjoy teaching topics in Chance and Data? Which topics?• Which topics do your students enjoy the most?• With which topics do they have the most difficulty?• What do you do when students have these difficulties?• What materials do you or your students use when studying Chance and Data?

• Calculators or computers?• Concrete materials?• Materials with chance outcomes?• Sources of data?

“Sample” in Chance and Data [repeated for “Average”]

Sample (if you haven’t used it as an earlier example)

• What do you think of when you hear the word “sample”?

• Consider its meaning in a variety of contexts.

• Briefly outline how you might teach “sample” to your students.

• How would you introduce the topic?• What resources or materials would you use?• How much class time would you spend?


• What teaching methods and groupings of students would you use?• What lessons might precede or follow the topic?

• Have you ever taught this topic?• Do you enjoy teaching this topic?• How do your students respond?

Confidence

Listed below are some of the topics which are included in the chance and data curriculum.Please mark your level of confidence in your ability to teach them to your class.You are free to place a mark anywhere on the scale to indicate your level of confidence.

My Ability to teach❐ ❐ ❐

Low Confidence High Confidence Would not be teaching

❐ ❐ ❐ Chance Language❐ ❐ ❐ Equally Likely Outcomes❐ ❐ ❐ Average

❐ ❐ ❐ Basic Probability Calculations❐ ❐ ❐ Odds❐ ❐ ❐ Median

❐ ❐ ❐ Graphical Representation❐ ❐ ❐ Data Collection❐ ❐ ❐ Sampling

❐ ❐ ❐

Low Confidence High Confidence Would not be teaching

Statistics in Everyday Life

Listed below are some statements concerning beliefs or attitudes about chance and data.Please mark your level of agreement with each statement.You are free to place a mark anywhere on the scale to indicate your level of agreement.

❐ | ❐

Strongly Neutral Strongly

Disagree Agree

❐ | ❐ 1. You need to know something about statisticsto be an intelligent consumer.

❐ | ❐ 2. I can easily read and understand graphs andcharts in newspaper articles.

❐ | ❐ 3. When buying a new car, it’s better to aska few friends about the problems with theircars than to read a car satisfaction survey ina consumer magazine.

332 JANE M. WATSON

❐ | ❐ 4. I can understand almost all of the statisticalterms that I see in newspapers or on TV.

❐ | ❐ 5. Understanding probability and statistics isbecoming increasingly important in oursociety.

❐ | ❐ 6. Statements about probability (such as theodds of winning a lottery) seem very clearto me.

❐ | ❐ 7. To learn about the side effects of a drug,it’s better to refer to the results of a medicalstudy that tested it on a few people, than totalk to someone who has taken the drug.

❐ | ❐ 8. People who have contrasting views can eachuse the same statistical finding to supporttheir view.

❐ | ❐ 9. I could easily explain how an opinion pollworks.

❐ | ❐ 10. Weather forecasts about the chances of rainare wrong so often that I don’t take themseriously.

❐ | ❐

Strongly Neutral Strongly

Disagree Agree

Student Survey Items (1)

The following questions come from Student Surveys used in research.Some involve newspaper extracts which include Chance and Data concepts.Read each question and suggest:

• Likely student responses. Use * to indicate appropriate response(s)

• Comments for use in teaching [Space provided for each question.]

A marketing research company wanted to find out how much money teenagers spend onrecorded music.The company randomly selected 20 shopping centers around the country.A field researcher stood in the middle of each shopping center and asked passers-by whoappeared to be the right age to fill out a questionnaire.A total of 2,050 questionnaires was completed by teenagers.From the survey, they reported that the average teenager in this country spends $95 eachyear on recorded music.Now imagine you were the researcher asked to find out how much money teenagers spendon recorded music. How would you have done it?



Explain the meaning of this pie chart.

Is there anything unusual about it?


What does “7-2” mean in this headline about the North against South football match? Giveas much detail as you can.From the numbers, who would be expected to win the game?


ABOUT six in 10 United States high school students saythey could get a handgun if they wanted one, a third of themwithin an hour, a survey shows. The poll of 2508 junior andsenior high school students in Chicago also found 15 percent had actually carried a handgun within the past 30 days,with 4 per cent taking one to school.

Would you make any criticisms of the claims in this article?If you were a high school teacher, would this report make you refuse a job offer somewhereelse in the United States, say Colorado or Arizona? Why or why not?

334 JANE M. WATSON


Family car is killing us, saysTasmanian researcher

Twenty years of research has convinced Mr. Robinsonat motoring is a health hazard. Mr. Robinson has graphswhich show quite dramatically an almost perfect rela-tionship between the increase in heart deaths and theincrease in use of motor vehicles. Similar relationshipsare shown to exist between lung cancer, leukemia. strokeand diabetes.

Draw and label a sketch of what one of Mr. Robinson’s graphs might look like.What questions would you ask Mr. Robinson about his research?


During the recent Australian cricket tour of South Africa, the Hobart Mercury (6/4/1994,p. 52) reported that Allan Border had lost 8 out of 9 tosses in his previous 9 matches ascaptain. Imagine his situation at this point in time.

Suppose Border decides to choose heads from now on.For the next 4 tosses of the coin, what is the chance of the coin coming up tails (andhim losing the tosses) 4 times out of 4?Suppose tails came up 4 times out of 4. For the 5th toss, should Border choose

❐ Heads❐ Tails❐ Doesn’t matter

What is the probability of getting heads on this next toss?What is the probability of getting tails on this next toss?

Teacher Background

• Sex: ❐ Female❐ Male

• How many years have you been teaching?• Which grade levels have you taught in that time?• Which grade levels are you currently teaching?• During your teacher training or other tertiary study, did you study any courses which

included topics in probability and statistics?

❐ No❐ Yes: what sort of course?

❐ Specialist statistics❐ Maths ❐ Economics❐ Psychology ❐ GeographyOther:

• About how many years ago was this study?


• How much time did the probability and statistics subjects take?❐ A few hours ❐ A few weeks❐ A semester ❐ More than one year

• List three topics which you remember studying in probability or statistics

Professional Development

• Have you seen the following documents in your school?• Have you read parts of any of them?• Have you used any ideas from them in your classroom?

Not

Seen Seen Read Used

State Curriculum Document ❐ ❐ ❐ ❐

National Statement on Mathematics for Australian Schools ❐ ❐ ❐ ❐

Mathematics A Curriculum Profile for Australian Schools ❐ ❐ ❐ ❐

Chance and Data Investigations (Curriculum Corporation) ❐ ❐ ❐ ❐

MCTP Activity Books 1 and 2 ❐ ❐ ❐ ❐

Books from the Australian Bureau of Statistics ❐ ❐ ❐ ❐

Texts: ❐ ❐ ❐ ❐

❐ ❐ ❐ ❐

• Have you participated in any professional development related to chance and data?

❐ No❐ Yes If so, please detail:• Organised by school, university, or other body?• Participated with others from school, own initiative, etc?• How long did it last (hours)?

• What type of professional development would benefit you the most in your teachingof chance and data? Examples might include:

• School-based sessions• Personal reading• A university course, e.g. Graduate Gertificate

• In your opinion, who would be best to lead professional development?Examples might include:

• Another teacher at my school• A regional curriculum officer• An outside “expert”

• Do you have any other specific comments about professional development in relationto Chance and Data?

336 JANE M. WATSON

REFERENCES

Australian Association of Mathematics Teachers, Inc. (2000). Consultation draftdescriptors of excellence in teaching mathematics. Adelaide, SA: Author.

Australian Education Council. (1991). A national statement on mathematics for Australianschools. Carlton, Vic.: Author.

Australian Education Council. (1994). Mathematics: A curriculum profile for Australianschools. Carlton, Vic.: Curriculum Corporation.

Bright, G. W., Berenson, S. B. & Friel, S. (1993, February). Teachers’ knowledge ofstatistics pedagogy. Paper presented at the annual meeting of the Research Council forDiagnostic and Prescriptive Mathematics, Melbourne, FL.

Bright, G. W. & Friel, S. (1993, April). Elementary teachers’ representations of relation-ships among statistics concepts. Paper presented at the Annual Meeting of the AmericanEducational Research Association, Atlanta, GA.

Callingham, R. A., Watson, J. M., Collis, K. F. & Moritz, J. B. (1995). Teacher atti-tudes towards chance and data. In B. Atweh & S. Flavel (Eds.), Galtha. Proceedingsof the Eighteenth Annual Conference of the Mathematics Education Research Group ofAustralasia (143–150). Darwin, NT: MERGA.

Edwards, R. (1996). Teaching statistics: Teacher knowledge and confidence. In P. C.Clarkson (Ed.), Technology in mathematics education. Proceedings of the NineteenthAnnual Conference of the Mathematics Education Research Group of Australasia(178–185). Melbourne: MERGA.

Fennema, E. & Franke, M. L (1992). Teachers’ knowledge and its impact. In D. A. Grouws(Ed.), Handbook of research on mathematics teaching and learning (147–164). NewYork: MacMillan.

Gal, I. & Wagner, D. A. (1992). Project STARC: Statistical reasoning in the classroom.(Annual Report: Year 2, NSF Grant No. MDR90-50006). Philadelphia, PA: LiteracyResearch Center, University of Pennsylvania.

Greer, B. & Ritson, R. (1993). Teaching data handling with the Northern IrelandMathematics Curriculum: Report on survey in schools. Belfast: Queen’s University.

Ingvarson, L. C. (1995). Professional credentials: Standards for primary and secondaryscience teaching in Australia. Canberra: Australian Science Teachers Association.

Kanes, C. & Nisbet, S. (1996). Mathematics-teachers’ knowledge bases: Implications forteacher education. Asia-Pacific Journal of Teacher Education, 24, 159–171.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understandingof fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.

Mayer, D. & Marland, P. (1997). Teachers’ knowledge of students: A significant domainof practical knowledge? Asia-Pacific Journal of Teacher Education, 25, 17–34.

Moritz, J. B., Watson, J. M. & Collis, K. F. (1996). Odds: Chance measurement in threecontexts. In P. C. Clarkson (Ed.), Technology in mathematics education. Proceedingsof the Nineteenth Annual Conference of the Mathematics Education Research Group ofAustralasia (390–397). Melbourne: MERGA.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation stand-ards for school mathematics. Reston, Va.: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teachingmathematics. Reston, Va.: Author.

Olssen, K., Adams, G., Grace, N. & Anderson, P. (1994). Using the mathematics profile.Carlton, Vic: Curriculum Corporation.


Pegg, J. (1989). Analysing a mathematics lesson to provide a vehicle for improvingteaching practice. Mathematics Education Research Journal, 1(2), 18–33.

Preston, B. (1996). Professional practice in school teaching. Australian Journal ofEducation, 40, 248–264.

Sachs, J. (1997). Reclaiming the agenda of teacher professionalism: An Australianexperience. Journal of Education for Teaching, 23, 263–275.

Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and direc-tions. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching andlearning (465–494). New York: NCTM & MacMillan.

Shulman, L. S. (1987a). Assessing for teaching: An initiative for the profession. Phi DeltaKappan, 69(1), 38–44.

Shulman, L. S. (1987b). Knowledge and teaching: Foundations of the new reform. HarvardEducational Review, 57, 1–22.

Stigler, J. W. & Hiebert, J. (1997). Understanding and improving classroom mathematicsinstruction: An overview of the TIMSS video study. Phi Delta Kappan, 79(1), 14–21.

Valli, L. (1992). Reflective teacher education: Cases and critiques. Albany: State Univer-sity of New York Press.

Watson, J. M. (1994). Instruments to assess statistical concepts in the school curriculum. InNational Organizing Committee (Ed.), Proceedings of the Fourth International Confer-ence on Teaching Statistics (Vol. 1, 73–80). Rabat, Morocco: National Institute ofStatistics and Applied Economics.

Watson, J. M. (1998). Professional development for teachers of probability and statistics:Into an era of technology. International Statistical Review, 66, 271–289.

Watson, J. M. & Moritz, J. B. (1997). Teachers’ views of sampling. In N. Scott & H.Hollingsworth (Eds.), Mathematics creating the future. Proceedings of the SixteenthBiennial Conference of the Australian Association of Mathematics Teachers (345–353).Adelaide: AAMT, Inc.

Watson, J. M. & Moritz, J. B. (1998). Longitudinal development of chance measurement.Mathematics Education Research Journal, 10(2), 103–127.

Watson, J. M. & Moritz, J. B. (2000). Development of understanding of sampling forstatistical literacy. Journal of Mathematical Behavior, 19, 109–136.

Faculty of EducationUniversity of TasmaniaGPO Box 252-66Hobart, Tasmania 7001AustraliaE-mail: [email protected]

BOOK REVIEW

National Research Council, Mathematical Sciences Board, NationalAcademy Press (2001). Knowing and learning mathematics for Teaching:Proceedings of a workshop. Washington: National Academy Press. ISBN0-309-07252-2.

RE-EXPRESSING INSIGHTS IN THE DISCOURSE OFTHE TIMES

I have long thought that whereas mathematics progresses by proving newtheorems, defining new objects, and integrating new clusters of ideas,mathematics education progresses by re-expressing insights of the pastin the vernacular of the present, with wrinkles and additions cast in thecurrent discourse. We may feel that we have made fresh distinctions,achieved finer analysis, reached deeper insights, that through access tonew technology we have new opportunities, and that we are more sophis-ticated than those who went before us. But is this growth of knowledgein the community, or is it a sign of our own personal development? Howcan educational wisdom be communicated? Can we ever successfully tellothers what we really know?

This last question is addressed beautifully but subtly through the struc-ture of the event being reported in this book, and in the way in which thesessions are described so that others can try out the tasks for themselveswith colleagues. The event comprised 48 hours of workshops in March2001, held under the auspices of the Mathematical Sciences EducationBoard of the National Research Council in Washington. The descriptionof each session includes opening remarks, copies of the stimuli offered inthat session, and a summary of points made during discussion. The bookdemonstrates that while there is little new under the sun, every fresh insightis locally exciting and energising: Hope springs eternal that progress isbeing, and will be, made.


340 BOOK REVIEW

Central Issue

The central issue addressed in the book is what teachers need to knowin order to teach effectively: what mathematics, what mathematicalpedagogy, what didactical strategies, what support from professionaldevelopment. Alternating between panel discussions and workshop discus-sions, participants grappled with the following topics:

• Teachers’ understanding of fundamental maths (Liping Ma, MarkSaul, Genevieve Knight)

• Investigating teaching practice: What mathematical knowledge isentailed in teaching children to reason mathematically? (DeborahBall, Hyman Bass, James Lewis)

• Investigating teaching practice: What mathematical knowledge, skill,sensitivities does it take? (Virginia Bastable, Olga Torres, MichaeleChappelle, Erick Smith)

• What kinds of mathematical knowledge matter in teaching? (AlanTucker, Deborah Schifter, Gladys Whitehead, Gail Burrill)

• Investigating alternative approaches to helping teachers learn mathe-matics (Sin-ying Lee, Marco Ramirez, Carne Barnett, VirginiaBastable, Jill Lester, Deborah Schifter, Deborah Ball, BradfordFindell)

• Promising approaches to helping prospective elementary teacherslearn mathematics for teaching (Richard Askey, Carol Midgett, AliceGill, James Lightbourne, Joan Ferrini-Mundy)

Participants had a chance to examine in detail samples of Chinese andJapanese descriptions of systematic development of topics, to analyse anddissect both textbooks and video-taped portions of lessons in the US, and toengage in mathematical tasks in order to appreciate approaches to teacherdevelopment. There were sessions on managing discussion and on remod-elling classroom tasks – a topic of abiding interest in the U.K., recentlyarticulated at the secondary level by Prestage and Perks (2001). Therewere also sessions on analyzing student thinking and student work, leadingto concern about what a teacher could be expected to do with the results(p. 59) of such an analysis, even if the teacher had the time to undertake it.

I was particularly struck by the observation (p. 70) that it is possibleto attribute too much in the way of understanding to students who appearto be articulate: just because a student says something once, it does notmean that this particular thought will come to mind when it is neededin the future, as teachers and educators have found time and time again.This is, of course, a fundamental epistemological issue (Ryle, 1949):Knowing-that and knowing-to-act are entirely different. Even when people

BOOK REVIEW 341

know-about something (in the sense of writing essays or of using thediscourse), and even when they appear to know-how to do something (inthe sense of having previously acted appropriately, or written essays aboutit), it does not follow that “in the moment” (p. 61) an appropriate impulsewill come to mind. Awareness is much more complex than mere know-ledge, and certainly more complex than assessable knowledge (Mason,1998).

I was greatly encouraged by the use of active participation, for it hasbeen an established practice for many years in the U.K. to begin sessionsby engaging people in a task through which to become aware of processesand principles that can then be critiqued, developed, and applied to theclassroom. Indeed for our first course in mathematics education (OpenUniversity, 1980) we explicitly formulated this principle with the “snappy”label A-P-C for Adult-Process-Classroom as part of our communicationwith teachers about how the course was structured. I noticed that presentersnevertheless, at least in print, felt compelled to situate tasks at some lengthbefore inviting colleagues to participate. Yet sometimes it is even morepowerful to start with a task and then to provide context and reasons later asthe need for them arises. The temptation to tell is strong; what we learnedfrom constructivism is surely that what matters is not telling or not-telling,but whether the audience is in a position to hear and make sense of whatthey are being told. Starting with a shared task is a good way to prepare anaudience so that telling is indeed telling.

I was struck by the suggestion that video could be used to try to teachboth mathematics and pedagogy. Our experience is that video is mosteffective when it is used to locate incidents from one’s own experiencewhich can then be described and compared so that colleagues build up arich network of related but personal incidents. From these incidents canemerge both mathematical and pedagogic questions, and both mathema-tical and pedagogic strategies. If the video is perceived as displaying goodpractice, resistance sets in (“I wouldn’t let that teacher in my classroom”)and is difficult to dislodge.

Many discussants seemed to come to the conclusion that it is notenough to “know the maths” (p. 72). Something else is required, somethingwhich is very difficult to speak about without turning it into somethingextra to be imposed or trained. Knowledge may not be the same as aware-ness, and it is certainly distinct from wisdom. I had a sense from the bookthat the most appropriate goals are awareness and wisdom, but that theycannot be reached through mechanical methods. The more precisely oneformulates a curriculum, a teaching method, a topic-progression, or a tasksequence as knowledge to be conveyed or passed on, the more likely one is

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to be trapped in the transposition didactique (Chevellard, 1985), in whichexpert awareness is turned into instruction in behaviour: the relationalbecomes instrumental (Skemp, 1976).

The book contains a surprising number of shoulds rather than coulds,moral imperatives rather than conditionals, as some participants grappledwith trying to fix the situation as perceived (p. 96). In my experience,imperious imposition is much less likely to be successful than identifyingpossibilities. Much of the language was in terms of deficiency and howto remedy it (“tone of disparagement,” p. 71; “lament,” p. 68), rather thanproficiency and how to enhance it. These might entail only minor changesin wording perhaps, but could signal a major shift in perspective, andhence in action. Little subtleties in metaphors can make a big differencein how children experience mathematics and in how colleagues experienceworkshops, as we try to get both prospective and practising teachers torecognise.

The book is a beautifully and lavishly assembled account of whatappears to have been an event with powerful impact. There are brief-but-vivid accounts of the tasks with which workshops were initiated, andsummaries of ensuing discussion. I have to admit that this conforms withmy experience, that what is really effective is getting people working on atask together so that immediate shared experience, juxtaposed with expe-riences resonated from the past, leads to fruitful discussion, as long aspeople adopt a conjecturing atmosphere based around accounts of incid-ents rather than confuse discussion of incidents with theorising about themor explaining them away.

New Discourse

At least one presenter noted the lack of vocabulary for talking about“the mathematical knowledge that supports teaching.” This book certainlycontributes to that development! The very nature of text is to locate anddevelop ways of speaking about experience, with the result that certainphrases start to recur. I can warn colleagues in other countries that PUFM iscoming (“profound understanding of fundamental mathematics”) and thatit is something that some teachers are said to possess while other teachersdo not (p. 12), despite the common experience that to understand more isalso to appreciate how little you do understand. The issue is not so muchwhat any one teacher knows or does not know when tested, but the struc-ture of teachers’ awareness in the moment, and most especially, teachers’disposition to learn more themselves, about mathematical structure, aboutmathematical themes which link concepts, and about ways of working onmathematics with others.

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Knowledge packages for topics (a topic web for a central concept withlinks to other topics, language and images associated with the topic, tech-niques and classic confusions or errors) and systematic sequence charts(p. 80) are ways used in China and Japan for laying out the componentsof individual topics, and charting progression for learners. Other coun-tries use other formats, I suspect, but I wonder what happens if it allbecomes so crystallised and specified that there is no room for varia-tion and movement, for responding to the impulse “in the moment” tofollow students’ and teacher’s ideas. Certainly in the UK we recognise theendemic tension between excessive direction (which stultifies spontaneityand development) and insufficient support (which leads to impoverishedteaching).

I struggled for a while with the notion of “sites of practice” (p. 125)before realising that this refers to types of tasks offered to novice andexpert teachers in order to generate activity which can lead to the educa-tion of awareness, the development of sensitivities, the noticing of usefuldistinctions. It is refreshing to see that colleagues at the event came upwith lists which are similar to the practices to be found elsewhere: Nomatter what surface differences stand out between different articulationsof teaching and learning in different cultures and subcultures, when weventure beneath the surface, there is a great deal in common.

There are some interesting expressions which tempted me to readbetween the lines. The passing remark that “Thomas’s teacher should havebeen teaching in such a way that none of her pupils would make such anerror” (p. 96) suggested to me the presence of strong forces of mechanisticrationality (work out the cause-and-effect structure of learning, then trainteachers to teach properly using them) which includes a desire that learningbe smooth, constantly progressing, and inevitable. This despite each indi-vidual’s own experience that learning is messy and requires frequentback-tracking, re-adjustment, and re-construction, which in earlier gener-ations might have been called accommodation (Piaget, 1971), and whichwas charted in more detail recently by Pirie and Kieran (1994).

An associated concern is with coverage: If prospective teachers are notexposed to everything they will need to know mathematically and pedago-gically, aren’t we doing children and the teacher a disservice? “We can’tlet them be unprepared for whatever class they eventually may have toteach” (p. 48). But perhaps the preparation required is not to be taught orexposed to everything, but rather to be empowered and motivated to findout. Aiming to foster ways of working and enthusiasm for finding out seemmore important and lasting than trying to “cover everything.”

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Science and Engineering

I found the book at once profoundly inspiring and deeply disturbing. I wassimultaneously impressed and depressed. Inspiring and impressive was thesubtlety of the workshop constructions, and the sense that came across tome of colleagues collaborating and encountering new ways of workingand new distinctions to be made. I wished I had been able to participate.What disturbed and depressed me (until I remembered that each generationneeds to re-express for itself) was that much of what I encountered was,I thought, part of common practice, was part and parcel of how I and mycolleagues like to think we have worked with teachers over the last 20years, aware that we picked this up from yet other colleagues before usdoing the same. Of course one difference is that here there is the influenceand status of august national bodies, whereas in the past it has been lessovert, and perhaps confined to a few.

But this raises the thorny issue of science versus engineering in educa-tion. It is one thing to refine and develop one’s practices, to learn fromexperience so as to become more sensitive to making (and critiquing)possible distinctions which inform practice. It is quite another to arrangethat all, or even most teachers have similar experiences and adopt similaragenda. The analogy is with a scientific experiment that may work wellin the small scale in a laboratory, but which requires considerable engin-eering in order to develop a version which works on a commercial scale.The same seems to be true in education. Indeed it may be that large scaleengineering is not possible in education, for deep and endemic reasons:Until individuals want to develop their teaching, they are unlikely to bemoved by imposed demands, as the history of reform movements rathersuggests. The CGI project in the US (Carpenter, Fennema, Loef Franke,Levi, & Empson, 1999) CMIT in New Zealand (Thomas & Ward, 2001),and to a much lesser extent the NNS in the UK (Askew, Brown, Rhodes,Johnson, & William, 1997) have demonstrated that principled professionaldevelopment which informs as well as trains, can be effective in raisingachievement.

Themes

The book could be of immense value solely by virtue of the way it setsout not conclusions and exhortations, but a variety of exercises or taskswhich groups of colleagues could undertake together. In my experience,and that of colleagues around the world, it is from shared tasks that fruitfulnegotiation of meaning and exchange of insights into teaching strategiesemerge.

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I found strong resonances with the thread of “teaching mathematicsinvestigatively” which has had its ups and downs over 30 years in theU.K. For example, at the primary level, ATM (1967), ATM (1977), Fielker(1997), and Ainley (2000), and at the secondary level, Banwell, Saun-ders and Tahta (1972), DIME materials produced over many years byGeoff Giles in Stirling, Boaler (1997), and Watson and Ollerton (2001), tomention only U.K. based exemplars. A Canadian perspective can be foundat http://webstudio.educ.queensu.ca/faculty/tmc/ and associated pages. Butpointing to other writing relevant to the workshop themes is helpful onlyin reminding people that colleagues elsewhere have addressed and areaddressing similar questions. Being told that someone else has “lookedat this” is usually less than helpful, for when reading “the literature” it israre to find someone else asking and exploring precisely the question thatyou are asking.

One theme that came through sharply, possibly because it is an issuefor me as well, is “discerning the mathematics” in a classroom scene,whether live or on video, or even as tasks in a textbook (p. 71). Beingaware of mathematical potiential in tasks is not just a matter of “moreknowledge,” but rather of being sensitised to notice, to distinguish, todiscern, to recognise mathematical structures and pedagogic opportunities.To develop discernment requires, among other things, acting with “respectand generosity” towards teachers (p. 71).

Another theme which came through for me was that to become ateacher is to embark on a working-life journey of self-discovery, as sensi-tivities to notice, to make distinctions and to refine practices, are developedand honed. “The world is a mirror for the people,” as Hyemeyohsts Storm(1985) put it so beautifully in his succinct summary of a North AmericanIndian’s perspective. For a teacher, each student, each lesson, each topicprovides a mirror for seeing oneself (What is naturally stressed, whatignored?), and each experience of working with colleagues on mathe-matics and on mathematics teaching is an opportunity to become moresensitive to students through using the event as a mirror in which to seeoneself, and so to become more self-aware. For a mathematician, eachtheorem newly appreciated or newly proved is a mirror in which to detectpropensities and preferences, to detect modes and modalities of thinking.For a researcher, each observation, each analytic step, each insight, is aglimpse of oneself, of one’s own propensities and sensitivities in the mirrorof experience.

In his novel Mr. Palomar, Italo Calvino (1985) has his eponymous char-acter conclude that “It is only after you have come to know the surface ofthings . . . that you can venture to seek what is underneath. But the surface

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of things is inexhaustible” (p. 55). As was pointed out at the workshop,for students and for teachers, “It is easy to get lost in the doing and notpay attention to the actual mathematics involved” (p. 127). In a workshopsetting, being caught up in the mathematics is admirable, but surface, andcertainly not everything. Being caught up in managing the classroom whileignoring the mathematics is equally surface, and equally unhelpful. Putanother way, “doing �= construing.” Just because someone is engaged inactivity as a result of interpreting a task, it does not follow that learning istaking place, and certainly not specific learning intended or envisaged bythe task author. Students need to be encouraged and shown how to venturebeneath the surface of procedures and rules. By all accounts, some of thevideos shown to participants could be so interpreted.

Mathematics educators and researchers, while trying to probe beneaththe surface of children learning and being taught mathematics, often aretransfixed by surface phenomena. We think that by analysing videotapesin detail we will learn how children learn and how to teach effectively;that by testing students more and more frequently we will be in a positionto improve their learning; that by knowing more and more about childrenin general and specific children in particular, we will be able to adjustand modify what we teach so that learning comes about more smoothlyand effectively; and that by teaching teachers what they need to knowwe can make the whole process happen more efficiently. There are strongcultural forces to look for actions which will cause visible effects overshort-term scales (e.g., each lesson must have specifiable objectives orlearning outcomes stated in advance which can be verified by testing andmore testing), to which we conform and which we have helped strengthenby both our actions and our inaction. So we seek a philosopher’s stonewhich will unlock complexity and render the whole process simple andstraightforward.

And yet at the same time the book reveals (at least as I chose to read it)that it is possible to venture beneath the surface, that it is possible not to becontinually caught up in doing and teaching, and to experience, to becomemore aware, to become more sensitised.

In Conclusion

I had a strong sense of participants finding their perceptions, their attitudes,their weltanschauung being stretched and pressed. It is not, therefore, theconclusions reached, the summaries given which point the way forward,but the experience itself. As Deborah Ball is quoted as saying, “The work-shop was not designed to provide answers to the many questions aboutteacher content knowledge but to serve as an intellectual resource for the

BOOK REVIEW 347

participants to use in framing their own work” (p. 7). She encouraged parti-cipants to try some activities in their own context, based on the thinkingof each workshop, documenting results and sharing with colleagues. “Inthis way . . . the field can begin to move forward in a real analysis ofwhat mathematics teachers need to teach well and how they come to learnmathematics” (p. 7).

What teachers at every level need, in order to retain their profes-sionalism, in order to develop, is to work on themselves, on their ownawareness, sometimes under the guidance or inspiration of others. But careis needed: the rate of exposure to mathematics has to be adjusted carefully,lest the full flood of enthusiasm wash away the hand-holds of the timidexplorer (p. 19). What matters is not specific “knowledge” but “ways ofworking.” It is oh so tempting to participate in a workshop and then to tryto capture that experience by telling others, when at heart we know fromexperience that what is most effective is engaging others in similar work-shops. So I would echo Deborah Ball, but perhaps attenuate her clarioncall just a little: The field will continue to move forward only if we workon how we work with each other (and consequently with students), ondeveloping and refining tasks we can offer each other as stimulus forjuxtaposing immediate shared experience with experiences resonated fromthe past, producing heightened sensitivities to notice opportunities to actfreshly in the future. It is through sharing immediate experience in the formof common tasks, and reflecting on that experience both in the session andmore casually over time later, that the body-mathematic will deepen, that“PUFM” will arrive.

I expect that in a short time there will be many many well-thumbedcopies of the book on the bookshelves of teachers and educators at alllevels. But I caution those in authority who try to place the books there byforce. There is an Aesop fable about the sun and the wind vying to make atraveller remove his coat. Of course the wind fails while the sun succeeds,because compulsion fails in the end, while sensitivity and support succeed.Trying to control what teachers have access to, or worse, what they can do,can only lead to frustration and failure.

REFERENCES

Ainely, J. (2000). Constructing purposeful mathematical activity in primary classrooms. InC. Tikly & A. Wolf (Eds.), The Maths we Need Now (Bedford Way Papers 12). London:London Institute of Education.

Askew, M., Brown, M., Rhodes, V., Johnson, D. & Wiliam, D. (1997). Effective teachersof numeracy. London: King’s College.

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Association of Teachers of Mathematics (1967). Notes on mathematics in primary schools.Derby, UK: Author.

Association of Teachers of Mathematics (1977). Notes on mathematics in schools. Derby,UK: Author.

Banwell, C. Saunders, K. & Tahta, D. (1972). Starting points. Oxford, UK: OxfordUniversity Press.

Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex and setting.Buckingham, UK: Open University Press.

Calvino, I. (1985). Mr. Palomar. London: Harcourt, Brace, & Jovanovitch.Carpenter, T. Fennema, E., Loef Franke, M., Levi, L. & Empson, S. (1999). Children’s

mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.Chevellard, Y. (1985). La transposition didactique. Grenoble, France: La Pensée Sauvage.Fielker, D. (1997) Extending mathematical ability. London: Hodder & Stoughton Educa-

tional.Mason J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and

structure of attention. Journal of Mathematics Teacher Education, 1, 243–267.Open University. (1980). PME 233: Mathematics across the curriculum (Open University

Course). Milton Keynes, UK: Author.Piaget, J. (1971). Biology and knowledge. Chicago: University of Chicago Press.Pirie, S. & Kieren, T. (1994). Growth in mathematical understanding: How can we

characterise it and how can we represent it? Educational Studies in Mathematics, 26,165–190.

Prestage, S. & Perks, P. (2001). Adapting and extending secondary mathematics activities:New tasks for old. London: David Fulton.

Ryle, G. (1949). The concept of mind. London: Hutchinson.Skemp, R. (1976, Dec.). Relational understanding and instrumental understanding. Mathe-

matics Teaching, 77, 20–26.Storm, H. (1985). Seven arrows. New York: Ballantine.Thomas, G. & Ward, J. (2001). An evaluation of the count me in too pilot project: Exploring

issues in mathematics education. Dunedin, NZ: Dunedin College.Watson, A. & Ollerton, M. (2001). Inclusive mathematics 11–18. London: Continuum

Press.

JOHN MASONOpen University, Milton Keynes, UK

ACKNOWLEDGEMENT

The editors thank the following colleagues who reviewed manuscripts forJMTE in 2000. We appreciate our reviewers’ thoughtful critiques of themanuscripts and their contributions to the field of mathematics teachereducation.

Verna AdamsDeborah L. BallJerry BeckerSybilla Beckmann-KazezSarah BerensonMaria BlantonCarol BohlinRaffaella BorasiCathy BrownJinfa CaiOlive ChapmanDaniel ChazanAnna ChronakiVictor CifarelliCarne Barnett ClarkeSandra CrespoBeatriz D’AmbrosioBrent DavisDonald DessartPaul ErnestRuhama EvenMegan Loef FrankeJeffrey FrykholmToshiakira FujiiFulvia FuringhettiJoe Garofalo

Uwe GellertFred GoffreeElliot GootmanAnna GraeberTheresa GrantDoug GrouwsGuershon HarelLynn HartKath HartPatricio HerbstCelia HoylesDeAnn HuinkerBarbara JaworskiMartin JohnsonHenry KepnerTom KierenPeter KloostermanClifford KonoldKonrad KrainerColette LabordeDiana LambdinGilah LederSteve LermanFrank LesterMary LindquistRomulo Campos Lins

Gwendolyn LloydCheryl A. LubinskiJohn MasonJoão Filipe MatosSue MauKay McClainDouglas B. McLeodVilma MesaDenise MewbornL. Diane MillerTatsuro MiwaBarbara NelsonNel NoddingsDouglas T. OwensErkki PehkonenBarbara PenceRandy PhilippJoão Pedro da PonteDespina PotariNorma PresmegAnne RaymondBarbara ReysRobert ReysDeborah SchifterAnna SfardKenneth L. Shaw

Journal of Mathematics Teacher Education 4: 349–350, 2001.

350 ACKNOWLEDGEMENT

Anna SierpinskaMarty SimonJudy SowderLarry SowderKaye StaceyLynn StallingsLeslie P. SteffeHeinz SteinbringMarilyn StrutchensPeter SullivanJulianna SzendreiPaola SztajnAnne TeppoDenise ThompsonDina TiroshRon TzurDiane WearneNorman WebbMelvin R. WilsonTerry WoodErna YackelOrit ZaslavskyLaura van Zoest

Journal of Mathematics Teacher Education

INSTRUCTIONS FOR AUTHORS

EDITOR-IN-CHIEF

Thomas J. CooneyThe University of GeorgiaMathematics Education105 Aderhold HallAthens, GA 30602-7124, U.S.A.

AIMS AND SCOPEThe Journal of Mathematics Teacher Education (JMTE) is devoted totopics and issues involving the education of teachers of mathematics atall stages of their professional development. JMTE will serve as a forumfor research on teachers’ learning, for considering institutional, societal,and cultural influences that impact the education of mathematics teachers,and for creating models for educating teachers of mathematics. Criticalanalyses of development initiatives, technology, assessment, teachingdiverse populations, policy matters, and developments in teaching as thesetopics relate to educating mathematics teachers are welcome. Critiques ofreports or books that affect mathematics teacher education will appear asappropriate. In general, JMTE encourages the submission of articles thatidentify, examine, and develop areas of knowledge related to mathematicsteachers’ learning and development.

MANUSCRIPT SUBMISSIONKluwer Academic Publishers prefer the submission of manuscripts andfigures in electronic form in addition to a hard-copy printout. The preferred


352 INSTRUCTIONS FOR AUTHORS

storage medium for your electronic manuscript is a 3 1/2 inch diskette.Please label your diskette properly, giving exact details on the name(s) ofthe file(s), the operating system and software used. Always save your elec-tronic manuscript in the wordprocessor format that you use; conversionsto other formats and versions tend to be imperfect. In general use as fewformatting codes as possible. For safety’s sake, you should always retaina backup copy of your file(s). After acceptance, please make absolutelysure that you send us the latest (i.e., revised) version of your manuscript,both as hard copy printout and on diskette.

Kluwer Academic Publishers prefer articles submitted in wordprocessingpackages such as MS Word, WordPerfect, etc. running under operatingsystems MS DOS, Windows or Apple Macintosh, or in the file formatLATEX. Articles submitted in other software programs, as well as articlesfor conventional typesetting, can also be accepted.

For submission in LaTeX, Kluwer Academic Publishers have devel-oped a Kluwer LaTeX class files, which can be downloaded from:www.wkap.nl/kaphtml.htm/IFAHOME. Use of this class file is highlyrecommended. Do not use versions downloaded from other sites.Technical support is available at: [email protected]. If you are not familiarwith TeX/LaTeX, the class file will be of no use to you. In that case,submit your article in a common word processor format.

For the purpose of reviewing, articles for publication should initially besubmitted as hard-copy printout (4-fold) and on diskette to:

Kluwer Academic Publishers, P.O. Box 990, 3300 AZ Dordrecht, TheNetherlands.

MANUSCRIPT PRESENTATIONThe journal’s language is English. British English or American Englishspelling and terminology may be used, but either one should be followedconsistently throughout the article. Manuscripts should be printed ortypewritten on A4 or US Letter bond paper, one side only, leavingadequate margins on all sides to allow reviewers’ remarks. Please double-space all material, including notes and references. Quotations of more than40 words should be set off clearly, either by indenting the left-hand marginor by using a smaller typeface. Use double quotation marks for directquotations and single quotation marks for quotations within quotationsand for words or phrases used in a special sense.

INSTRUCTIONS FOR AUTHORS 353

Number the pages consecutively with the first page containing:

– running head (shortened title)– article type (if applicable)– title– author(s)– affiliation(s)– full address for correspondence, including telephone and faxnumber and e-mail address

AbstractPlease provide a short abstract of 100 to 250 words. The abstract shouldnot contain any undefined abbreviations or unspecified references.

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In addition to hard-copy printouts of figures, authors are encouragedto supply the electronic versions of figures in either EncapsulatedPostScript (EPS) or TIFF format. Many other formats, e.g., MicrosoftPostscript, PiCT (Macintosh) and WMF (Windows), cannot be used andthe hard copy will be scanned instead.

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ReferencesReferences to books, journal articles, articles in collections and conferenceor workshop proceedings, and technical reports should be listed at the endof the article in alphabetical order following the APA style (see examplesbelow). Articles in preparation or articles submitted for publication,unpublished observations, personal communications, etc. should not beincluded in the reference list but should only be mentioned in the articletext (e.g., T. Moore, personal communication).

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Mason, R. (1995). Using communications in open and flexible learning.London: Kogan Page.

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McIntyre, D.J., Byrd, D.M., & Foxx, S.M. (1996). Field and laboratoryexperiences. In J. Sikula, T.J. Buttery & E. Guyton (Eds.), Handbook ofresearch on teacher education (3rd. Ed.; pp. 171–193) New York: Simon& Schuster.

References to articles in conference proceedings should include theauthor’s name; year of publication; article title; editor’s name (if any);title of proceedings; first and last page numbers; place and date ofconference; publisher and/or organization from which the proceedings canbe obtained; place of publication, in the order given in the example below.

Yan, W., Anderson, A., & Nelson, J. (1994). Facilitating reflective thinkingin student teachers through electronic mail. In J. Willis (Ed.), Technologyand Teacher Education Annual, 1994: Proceedings of the Fifth AnnualConference of the Society of technology and teacher Education. VA:Association for the advancement of Computing in Education.

References to articles in periodicals should include the author’s name;year of publication; article title; full or abbreviated title of periodical;volume number (issue number where appropriate); first and last pagenumbers, in the order given in the example below.

Thomas, L., Clift, R.T., & Sugimoto, T. (1996). Telecommunication,student teaching and methods instruction: An exploratory investigation.Journal of Teacher Education, 46, 165–174.

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Hughes, M. (1975). Egocentrism in pre-school children, EdinburghUniversity, Edinburgh, unpublished thesis.

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OFFPRINTSTwenty-five offprints of each article will be provided free of charge.Additional offprints can be ordered by means of an offprint order formsupplied with the proofs.

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ADDITIONAL INFORMATIONAdditional information can be obtained from:

Journal of Mathematics Teacher Education, Kluwer AcademicPublishers, Attn Publishing Editor. Michel Lokhorst, P.O. Box 17,3300 AA Dordrecht, The Netherlands, tel.: +31-78-6392183; fax:+31-78-6392254. E-mail: [email protected]

CONTENTS OF VOLUME 4

Volume 4 No. 1 2001

GUEST EDITORIAL / ICME and Teacher Education 1–2

JEPPE SKOTT / The Emerging Practices of a Novice Teacher:The Roles of His School Mathematics Images 3–28

MURRAY S. BRITT, KATHRYN C. IRWIN and GARTHRITCHIE / Professional Conversations and Profes-sional Growth 29–53

GEOFFREY B. SAXE, MARYL GEARHART and NA’ILAHSUAD NASIR / Enhancing Students’ Understandingof Mathematics: A Study of Three ContrastingApproaches to Professional Support 55–79

Teacher Education Around the World

DESPINA POTARI / Primary Mathematics Teacher Educationin Greece: Reality and Vision 81–89

Volume 4 No. 2 2001

EDITORIAL / Appreciating the Challenge in Recognizing theObvious 91–93

ROBYN ZEVENBERGEN / Peer Assessment of StudentConstructed Posters: Assessment Alternatives inPreservice Mathematics Education 95–113

JANET BOWERS and HELEN M. DOERR / An Analysis ofProspective Teachers’ Dual Roles in Understandingthe Mathematics of Change: Eliciting Growth WithTechnology 115–137

DIANA F. STEELE / The Interfacing of Preservice andInservice Experiences of Reform-Based Teaching: ALongitudinal Study 139–172


360 CONTENTS OF VOLUME 4

Volume 4 No. 3 2001

HEIDE G. WIEGEL / Guest Editorial 173–175

MARIA L. BLANTON, SARAH B. BERENSON and KARENS. NORWOOD / Exploring a Pedagogy for the Super-vision of Prospective Mathematics Teachers 177–204

KAYE STACEY, SUE HELME, VICKI STEINLE, ANNETTEBATURO, KATHRYN IRWIN and JACK BANA /Preservice Teachers’ Knowledge of Difficulties inDecimal Numeration 205–225

RON TZUR, MARTIN A. SIMON, KAREN HEINZ andMARGARET KINZEL / An Account of a Teacher’sPerspective on Learning and Teaching Mathematics:Implications for Teacher Development 227–254

Volume 4 No. 4 2001

THOMAS J. COONEY / Editorial: Theories, Opportunities,and Farewell 255–258

RON TZUR / Becoming a Mathematics Teacher-Educator:Conceptualizing the Terrain Through Self-ReflectiveAnalysis 259–283

MARGARET TAPLIN and CAROL CHAN / DevelopingProblem-Solving Practitioners 285–304

JANE M. WATSON / Profiling Teachers’ Competence andConfidence to Teach Particular Mathematics Topics:The Case of Chance and Data 305–337

Book Review

National Research Council, Mathematical Sciences Board,National Academy Press (2001). Knowing andlearning mathematics for Teaching: Proceedings ofa workshop (JOHN MASON) 339–348

Acknowledgement 349–350

Instructions for Authors 351–358

Contents of Volume 4 359–360

journal of mathematics teacher education_4

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