Mathematics Education Research Journal, Vol.4, No.!, 1992.

BOOK REVIEWS

Mathematics in a Cultural Context: Aboriginal Perspectives onSpace, Time and Money.Pam Harris.Geelong, Deakin University, 1991, 168pp.

Australians, as a community, are seeking to identify what being anAustralian means. Part of this identity relates to the part that Mathemematicsplays within our culture as surely as it does for the Greeks, Italians,English....for any culture. At this time there comes Pam Harris' book titledMathematics in a Cultural Context Aboriginal Perspectives on Space, Timeand Money. The reader read this book to learn more about the country inwhich he lives. I was not disappointed.

This is a book compiling Pam Harris' work begun in 1977 and carriedout during the 1980's through the Mathematics in Aboriginal Schools (MAS)Project. The project investigated "aspects of the linguistic, cultural and socialbackground of Aboriginal children" in remote areas of Australia "which wouldinfluence the learning of mathematics in primary school" (p. 10). The purposeof the book " is to make available to teachers a body of information related toschool mathematics which will illustrate some of the differing points of viewof Aboriginal and other Australians" (p.l ).

This is not a book only about Aboriginal perspectives onmathematics.Throughout its pages the author presents information andchallenges that caused this reader to ask himself questions about his teachingof mathematics in the cross-cultural situations in which he works.

The Importantie of this RookThe importance of this book that concerns a relatively small number of

people (Aborigines living in remote Australia) is threefold. Most importantlythe book provides cultural based questions for teachers to ask themselveswhen teaching mathematics in their own situations. Further, it is the Aboriginepeople living in remote Australia who are seen by others as holding traditionalculture and law. They are of great importance to all Australians and increasingour knowledge of aspects of their way of life can only help in thecommunity's search for identity. Finally, there is no group of people nomatter how small that the quality of education given shouldn't 'be investigatedand made the best.

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An OverviewThe book begins with an Introduction that defines the terms remote

Australia and world view, two terras that need to be understood by the readerfor they form "the backdrop against which the material has to be viewed". Theauthor defines world view as "the different ways in which individuals andgroups of people view the world" (p. 12). This idea is central to the book forfrom it will come the development of vocabulary and a higher degree ofabstract language for those things that a cultural group views to be ofimportance.

The introduction is followed by 3 chapters that report on AboriginalPerspectives on space... .time. . . .money. The Aboriginal perspective on spacewas investigated because of its importance in the Aboriginal people's worldview; time and money because of their relevance to everyday life.Within thesechapters each concept is placed into the cultural context of Aboriginalcommunities in remote Australia.

Space. This chapter concerns the Aboriginal view of space and place,as well as the importance of mental maps. The author also discusses thesemantic differences between Aboriginal people and Anglo-Australians relatedto the description of shapes and lines. Highlighting that it is WesternEuropean culture which has classified and named 2-D and 3-D shapes and thatthis holds a low priority in an Aboriginal understanding of space, their worldview . Finally the author discusses the language problems faced byAborigines using English prepositions when postpositions are common intheir languages. This discussion emphasises the need for Aboriginal teacherswho share the students' language, experiences and culture.

Already many questions are raised in the reader's mind regarding hisown teaching of mathematics in a cultural context..

Time. Time is developed from the Aboriginal view of present-pastDreaming, through the use of natural measures to the need to cope with clocksand the artifical units used to measure time.

The author impresses upon the reader that white Australians have had along history of measuring time with hours and minutes, but that Aborigineshave no such tradition.This is not to say that there existed no time measuringbut that in the world view of the Aborigines their measuring was based on thenatural units of the seasons, phases of the moon, day and night, and thesignificant times in a person's life: "It is the events, and their significance forthe life of the people now, that are important, not the precise time at whichthey happened" (p. 66). Time is very much part of the Aborigine's life withthe stories of events.being passed down through the culture's oral history.Time for white Australians has always been specific with dates and timesusually in a written recorded history.

An intriguing part of the chapter is the discussion on the cultural base ofattitudes to time. Of how being on time has only been part of the Westernworld since the Industrial Revolution. It came about in order to get the most

Book Reviews 97

efficient factory production. How relevant was this to traditional Aboriginalpeople?

The discussion of the development and views of time measurement indiverse cultures provides the reader with a wealth of information. It also raisesfurther questions as to how to best teach time in a cross cultural situation.

Money. This chapter begins with a discussion of the traditionaleconomy of Aboriginal Australia which focussed on trade, barter andexchange as compared to the cash economy tradition of non-AboriginalAustralia. In doing this the reader becomes more sensitised to the difficultiesfaced by Aboriginal people with the controlled political introduction of money.

The historica) picture painted of the last 150 years beginning withpayment in kind, lack of access to wages, followed by the inequality of cashwages, high levels of unemployment and the current dependent economies ofthe welfare state raise many questions for the reader concerning the need toteach about the cultural differences between Aboriginal and non-Aboriginalattitudes towards money and "to the whole education system which supportsthe national economy"(p. 120).

The final section of this chapter reports on the current use of money byAborigines in remote areas. Discussion includes the tendency of bit-at-a-timebuying; getting change and being cheated; budgeting and saving; the problemsof budgeting on a low income; the place of gambling and the introduction ofWestern-style banking and its services including credit cards and chequeaccounts.

Future DirectionsAlready Pam Harris' work has been used by many people as they

endeavour to add an Aboriginal perspective to their mathematics teaching, andgain a personal understanding of how groups within another culture viewmathematics concepts.

In writing this book Pam Harris has set future challenges for herselfand others. Personally, she hopes to include a chapter in a future editionrelated to Aboriginal perspectives on number. For the education researchcommunity she perceives a need for there to be "work for long periods inclose cooperation with groups of Aboriginal people to discover together waysof conducting schooling and of teaching mathematics so that the integrity ofboth the Aboriginal and the White Australian world views are preserved"(p12). For the Aboriginal community there is the stated hope that this research"wil) encourage more Aboriginal educators to become involved indocumenting their own people's conceptualisation of mathematics" (p. 2).

Pam Harris concludes that there are difficult challenges for the teacherto meet when teaching mathematics in cross-cultural situations; but they had tobe met and welt answered for the continued empowerment of Aboriginalpeople. For this to happen there needs to be more research into the "culturaland linguistic starting points and present-day needs of students.....greaterflexibility in curriculum design and implementation....and teachers need better

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basic training and more inservice support to meet the demands of their task"(p. 125).

As teachers, we are becoming more and more aware of the cross-cultural situations that exist in our classrooms from pre-school to tertiarylevel. This book can only add to our sensitivity and knowledge when teachingmathematics. It.has to be one of the most relevant books on the teaching andleaming of mathematics to have been printed within Australia in recent years.It should be available in libraries for all teachers as they develop a Perspectivein Aboriginal Studies across their school. Further it should be part of any pre-service/inservice course for teachers of mathematics which considers thecross-cultural classrooms within which we teach.

Peter Howard.Australian Catholic University, NSW.

Communication in the Classroom: The Importance of GoodQuestioning.Peter Sullivan and David Clarke.Making Sense of Algebra: Cognitive Processes InfluencingComprehension.Mollie MacGregor.Bilingualism and Mathematics Learning.Philip Clarkson.Geelong, Deakin University, 1991, 73pp, 139pp, 60pp respectively.

An Emerging Picture of Research on Communication inMathematics Education: A Review of Three Efforts

The traditional interactions that take place in the mathematics classroommay be likened to a "painting by the numbers". Teachers direct students to fillin blanks as questions are asked, much in the same manner colors are matchedto numbers on a picture. Mathematics education is moving away from such anemphasis on only giving students discrete pieces of information and skills.The NCTM Standards (1989) stress a shift to an environment in whichstudents assume more responsibility for validating their own thinking. Thefocus is on leaming mathematics through open interactions between teacherand students. Accordmg to Silver, Kilpatríck and Schiensinger (1990) theteaching and leaming of mathematics "is like a conversation that is periodicallyinterrupted, but that is gradually shaped by the participants into somethingthey mutually share" (p. 8).

From a research perspective we need to focus our inquiries on severalquestions. These include: How do students and teachers communicate ideas?What are obstacles to effective communication? How can teachers engineerclassroom discourse to help students think about mathematics?

This paper reviews three monographs which address aspects of theabove questions. Sullivan and Clarke deal with questioning strategies for

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shaping discourse and student thinking. MacGregor attempts to describe andexplain obstacles and difficulties students have in learning mathematics. Adifferent, but related facet of communication is examined by Clarkson. Heexplores the interaction between the language of the students and the languageof instruction when the home and school languages are different.

In their monograph, Sullivan and Clarke deal with the general issue ofhow teachers can structure classroom discourse to assist studentscommunicate and think about mathematics. They focus on four questions:• What type of questions teachers ask?• What type of questions produce the quality of learning teachers seek?• What is the effect of the questions which teachers ask?• Are there ways teachers can improve on the quality of questions they ask?

The authors summarize the research relative to each of these questionsin a succinct and comprehensive manner. They describe that over half of whattakes place in classrooms is comprised of questions and instruction. Only 5per cent of the questions asked "required the pupils to think independently orto give more than one answer" (p. 9). They also relate that, even though theresearch is inconclusive in this area, asking higher order questions enhanceslearning.

The authors address the fourth question by presenting an instructionalstrategy to help teachers stimulate higher order thinking among students. Theyrefer to the strategy as asking good questions. The authors define goodquestions as having three features. Good questions require more than recall,they are open ended, and they promote active learning. Specific methods forconstructing good questions are given. This 'applied' aspect of the researchmonograph continues with a discussion of how good questions may be usedwith students from different achievement levels. The authors conclude withhow effectave questioning techniques promote problem solving skills, and therole such questions play in student assessment. The monograph provides thereader with an excellent integration of research reporting and classroomapplications. It gives teachers ways in which they can move away fromperceiving learning mathematics as "painting by the numbers".

The reform movement in mathematics education stresses the importanceof algebraic knowledge. Algebra is unique in the mathematics curriculum inthat it presents students the challenge to develop symbolic communicationskills. MacGregor's monograph focuses on the obstacles and difficultiesstudents have in using the symbolic language of algebra. She begins bysummarizing related research studies conducted in the United Kingdom andthe United States. MacGregor concludes from the review of studies thatstudents misuse the idea of variable and that linguistic factors act often timesas obstacles in the interpretation and application of algebraic equations.

MacGregor reports that Australian students make the same kinds ofmistakes produced by U.S. and British pupils. She also identifies severalfactors which may account for the errors in algebraic word problems. Theseinclude the students' misinterpretation of the meanings of certain words, andthe lack of knowledge conventions of mathematical notations. The author

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concludes by stressing that linguistic factors play a critical role in students'processing of algebraic problems. MacGregor gives no indications ofstrategies teachers can use to help students overcome such difficulties.Readers of this monograph are encouraged to review Kieran's (1992). Kieranpresents the same conciusions stated by MacGregor; But she follows themwith research-based suggestions for heiping students deal with difficulties inthe solution of algebraic word problems.

Australia as well as other developed countries in the world, face largenumbers of migrant students in their schools. This creates the challenge ofhaving to help these students bridge the language differences that existbetween their home language and the language of instruction. Schools haveattempted to address this challenge by implementing a variety of programs.These programs have ranged from providing instruction in two languages toan emphasis on the development of communication skills in English as asecond language. Clarkson's monograph explores these issues in the contextof Australia, and Papua New Guinea (PNG). The results of his researchreflect the `state-of-the art' in second language studies about factors whichrelate to the learning of mathematics. What we know about the interaction offirst and second languages with learning processes is increasing.Unfortunately, we still do not have empirically-based instructional strategiesto help students; (a) develop language skills and mathematical contentknowledge in two languages, nor (b) bridge the gaps between their homelanguage and the language of the schools.

. Overall, the monographs represent three different aspects of the generaltopic of communication in mathematics education. One deals very effectivelywith the enhancement of classroom discourse through the effective use ofquestions. Another explores difficulties students have with word problems inalgebra. And the third one examines the issue of bilingualism and learning inmathematics. All three reflect the current view of learning as a process inwhich students and teachers actively engage in mathematically interestingtasks. Classroom discourse plays an important role in such settings. Theefforts of the authors of the monographs reviewed contribute to the emergingpicture that reflects this view.

References

Kieran,C. (1992). The learning and teaching of school algebra. In D.Grouws(Ed.), Handbook of research in mathematics teaching and learning(pp.390-419). New York: Macmillan Publishing Company.

Silver,E. A., Kilpatrick,J., & Schlesinger,B. (1990). Thinking throughmathematics. New. York: The College Board.

Gil Cuevas.University of Miami.