Chapter 1

General introduction

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8

Introduction

In our society, data and statistics have become pervasive (Gravemeijer,

Stephan, Julie, Lin, & Ohtani, 2017). These data can be used for inferential

reasoning (i.e., the process of drawing a conclusion based on evidence or

reasoning). However, for drawing adequate conclusions, it is fundamental that

the evidence is credible and the inferential reasoning process sound (Dewey,

1910/1997). In daily life, this might not be the case; for example, a newspaper

reports that “commuters sick of crowded, dirty train” (Pel, 2016). The

newspaper thus makes a suggestion about Dutch commuters in general, while

the sample contains only people who filed a complaint about train hygiene. In

some instances, the role of empirical evidence in drawing conclusions seems to

be left out altogether (Viner, 2016). In 2016, the Oxford English Dictionaries Word

of the Year was “post-truth,” where “objective facts are less influential in

shaping public opinion than appeals to emotion and personal belief” (Word of

the Year 2016, 2016). In the Netherlands, for example, encountered crime

decreases at a slower rate than recorded (Statistics Netherlands, 2018). Thus, in

a time where the importance of data and inferential reasoning increases,

citizens need to be able to assess the validity of inferences. Also, in the

workplace, the importance of inferential reasoning is likely to increase because

such reasoning is a skill not easily substituted by automated technologies (Liu

& Grusky, 2013).

One of the primary purposes of education is to help children become

qualified members of society, both today and in the future (Biesta, 2009). Since

inferential reasoning is an important skill in this society, primary education

could introduce children to inferential reasoning. To this end, teachers could

introduce primary school students to informal statistical inference (ISI) (Groth

& Meletiou-Mavrotheris, 2018). ISI can be defined as “a generalized conclusion

expressed with uncertainty and evidenced by, yet extending beyond, available

data” (Ben-Zvi, Bakker, & Makar, 2015, p. 293).

If primary school teachers are to introduce their students to ISI, they need

to have appropriate content knowledge of ISI (ISI-CK) themselves. However,

research suggests that many primary school teachers lack the content

knowledge to teach ISI (Batanero & Díaz, 2010; Groth & Meletiou-Mavrotheris,

2018). Moreover, the literature provides few successful examples of how the

INTRODUCTION

9

ISI-CK of pre-service primary school teachers1 can be fostered (Groth &

Meletiou-Mavrotheris, 2018; Leavy, 2010). In particular, no research has been

reported about contexts where limited time was available for interventions to

foster pre-service teachers’ ISI-CK. This restriction in intervention length is

relevant for our setting of Dutch teacher college education, where it is typical

that little time is reserved for statistics. Therefore, the aims of this thesis are to

extend the evidence about what ISI-CK pre-service primary school teachers

actually have and to investigate how interventions of limited length can foster

the ISI-CK of pre-service primary school teachers. The overall research

question of this thesis is: What is the ISI-CK of pre-service primary school

teachers, how does their ISI-CK develop during an intervention of limited

length, and what ISI-CK do they show during teaching ISI in primary school?

Informal Statistical Inference

ISI could be a suitable form for introducing primary school students to

inferential reasoning because the statistical reasoning involved in ISI tends to

be of lower complexity than in formal statistical inference. This difference in

complexity varies in a number of applications (Makar & Rubin, 2018). For

instance, while formal statistical inference might use quantitative expressions

of uncertainty, such as p-values, ISI permits qualitative expressions of

uncertainty, such as “I am quite sure.” Second, inference in ISI can be based on

(hands-on) simulations rather than closed-form formulas based on probability

theory (J. D. Mills, 2002). For example, an ISI activity could have students first

draw samples individually and then have them compare their results, thus

simulating a sampling distribution. Third, ISI allows for a diversity of images

of distributions, such as dot plots that highlight individual samples in

sampling distributions (Makar & Rubin, 2018).

The following is an example of how, in ISI, multiple statistical concepts are

used as statistical arguments to support the inference (Makar & Rubin, 2009;

Watson, 2001): Confronted with a particular sample and its distribution,

1 This dissertation concerns pre-service primary school teachers, which we often abbreviate as

“pre-service teachers.”

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10

someone notes the high level of variance in the sample and argues that the

sample is too small and possibly biased and, therefore, expresses the inference

with a high degree of uncertainty. In this example, the statistical concepts of

sample size, bias, sampling methods, and variance are arguments in the

inference.

For our studies among pre-service teachers, we used the ISI framework of

Makar and Rubin (2009), which covers the key themes of ISI identified in the

literature (Makar & Rubin, 2018; Rossman, 2008; Zieffler, Garfield, delMas, &

Reading, 2008). We conceptualized the three components of this framework as

follows:

1. Data as evidence, subdivided into two subcomponents:

a. Using data: The inference is based on available data and not on

tradition, personal beliefs, or personal experience.

b. Describing data: Before the data can be used as evidence

within ISI, one first needs to descriptively analyze the sample

data by, for instance, calculating a mean (Zieffler et al., 2008).

The resulting descriptive statistic then functions as an

evidence-based argument within ISI (Ben-Zvi, 2006).

2. Generalization beyond the data: The inference goes beyond a

description of the sample data by making a probabilistic claim about a

population, an individual not included in the sample, or a mechanism

that produced the sample data2.

3. Probabilistic language: Due to sampling variability and the degree of

sample representativeness, the inference is inherently uncertain and

requires using probabilistic language. For the correct usage of

probabilistic language, the origins of uncertainty in inferences must be

understood. Therefore, we divided this component into four

subcomponents:

a. Sampling variability: The inference is based on an

understanding of sampling variability; it is expressed from an

understanding that the outcomes of representative samples are

similar and that, therefore, under certain circumstances, a

sample can be used for an inference.

2 In this dissertation, “to make inferences” and “to make generalizations (beyond the data)” are

used interchangeably.

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11

b. Sampling method: The inference includes a discussion of the

sampling method and the implications for sample

representativeness.

c. Sample size: The inference includes a discussion of the sample

size and the implications for sample representativeness.

d. Uncertainty: The inference is expressed with uncertainty and

includes a discussion of what the sample characteristics, such

as the sampling method employed and the sample size, imply

for the certainty of the inference.

We used this framework in several ways. First, the framework was used to

analyze and describe the pre-service teachers’ ISI-CK. For example, Chapters 2

and 3 show that many pre-service teachers appear to be able to use common

descriptive statistics (subcomponent “describing data”), such as the mean,

while at the same time, many tend to only describe the data without engaging

in making generalizations beyond the data (component “generalization”). This

finding led us to design tasks that put less emphasis on the subcomponent

“describing data” and more emphasis on the component “generalization” in

Chapters 4 and 5. The “describing data” subcomponent, therefore, no longer

features in the ISI framework of the later chapters. Second, the framework was

used was to operationalize the learning objectives for pre-service teachers

(Chapters 3 and 4). The attainment of these learning objectives was expected to

give them sufficient ISI-CK to teach ISI in the upper grades of primary school.

Third, the framework’s components helped us to formulate conceptual

mechanisms that explain how the activities used in the intervention may

scaffold the participants’ reasoning toward the attainment of the learning

objectives (Simon, 1995). For example, the first activity of the teacher college

intervention in Chapter 4 had the pre-service teachers search the media for a

news item that made a claim about a population based on a sample in order to

create awareness when inferential reasoning is used (component

“generalization”).

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ISI in Primary Education

In most countries, as in the Netherlands, ISI is not part of the (mathematics)

curricula of primary schools (Makar & Rubin, 2018). It is hypothesized,

however, that an introduction to ISI in the upper grades of primary school has

several potential benefits (Makar, 2016; Makar, Bakker, & Ben-Zvi, 2011). A

first benefit might be that, because it has been shown that understanding the

statistical concepts underpinning statistical inference is challenging

(Shaughnessy, Garfield, & Greer, 1996), an early introduction of ISI allows

students to be repeatedly exposed to the statistical concepts involved in ISI

over an extended period of time (Watson & Moritz, 2000). Second, it is

conjectured that ISI can facilitate the transition to formal statistical inference, as

students will gradually understand the processes involved in inferential

reasoning (Zieffler et al., 2008). Third, ISI can offer a unified framework for

learning multiple statistical concepts because ISI involves a reasoning process

in which multiple statistical concepts are used as reasoning tools, as the

example of ISI in the section on ISI shows. So it is conjectured that, by being

engaged in ISI, students learn statistics in an integrated way (Bakker & Derry,

2011).

The potential benefits of introducing ISI in primary school can only be

reaped if ISI is in the zone of proximal development of primary school

students; that is, if primary school students can be successfully engaged in

meaningful ISI activities while being supported by a more knowledgeable

other, which, in a school setting, is often the teacher (Vygotsky, 1978).

Successful engagement of students in ISI requires them to have some intuitive

notions of chance, as one to needs to have a sense that sampling results usually

vary from sample to sample (Chance, delMas, & Garfield, 2004; Saldanha &

Thompson, 2002). Piaget and Inhelder (1975/1951) showed that, around seven

years old, primary school students start to show intuitive notions about

uncertainty. Recent evidence suggests that even babies as young as eight

months “are sensitive to statistical patterns” (Gopnik, 2012, p. 1625). Babies

tend to watch for a longer period of time when a sample of mostly white balls

are drawn from a box of mostly red balls than when the sample is composed of

mostly red balls. The researchers thus suggest that babies possess intuition

about the likelihood of events.

INTRODUCTION

13

Also, recent research evidence in the field of statistics education suggests

an early introduction of ISI is possible (Ben-Zvi, 2006; Ben-Zvi et al., 2015;

Makar, 2016; Meletiou-Mavrotheris & Paparistodemou, 2015; Paparistodemou

& Meletiou-Mavrotheris, 2008). Leavy (2017) showed that kindergarten

children demonstrated some initial ability to make inferences based on given

data. Although far from being perfect, some inferences based on the data

(component “data as evidence”) were expressed tentatively, using expressions

such as “I think,” “maybe,” and “probably” (component “probabilistic

language”). While it may be that primary school students cannot be expected

to develop mature understandings of inferential reasoning, their emergent and

incomplete understandings can provide a valuable foundation for later

education (cf. the spiral curriculum Bruner, 1960; Makar et al., 2011). Chapter 4

provides an example of a meaningful introductory ISI activity that served as an

example for pre-service teachers how ISI could be introduced to primary

school students. These were hands-on activities that required little descriptive

analyses (Arnold, Pfannkuch, Wild, Regan, & Budgett, 2011) and involved a

tangible population and sample.

ISI and Pre-Service Primary School Teachers

Pre-service teachers’ knowledge of ISI

If primary school students are to be introduced to ISI, their future teachers

must be well prepared to conduct this introduction (Batanero & Díaz, 2010)

and, therefore, need to possess a thorough knowledge of ISI (Groth &

Meletiou-Mavrotheris, 2018). Teachers’ content knowledge impacts their

students’ learning achievements (Fennema & Franke, 1992; Rivkin, Hanushek,

& Kain, 2005) and may facilitate the development of the teachers’ pedagogical

content knowledge, which is, in this case, the knowledge of how to teach ISI

(Burgess, 2009; Groth, 2013; Leavy, 2010; Shulman, 1986). This content

knowledge must extend beyond what their pupils will learn. Teachers need to

have knowledge about the place of the particular content in the curriculum

(horizon content knowledge) and specific knowledge that is relevant for

teaching the content (specialized content knowledge) (Ball, Thames, & Phelps,

2008). Given the importance of teachers’ ISI-CK for primary school students’

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learning, our studies aimed to contribute to the scientific evidence about how

to foster pre-service teachers’ ISI-CK.

For teachers, the foremost reason for learning ISI-CK is to use this

knowledge in their teaching. It is, therefore, important to investigate how

teachers’ ISI-CK “plays out” in teaching (Rowland, Turner, Thwaites, & Hurst,

2009), as “teacher knowledge can only be understood in the context in which

they work” (Petrou & Goulding, 2011, p. 20). So, while Chapters 2 and 4

describe ISI-CK using an online questionnaire, and Chapters 3 and 4

investigate pre-service teachers’ ISI-CK when engaged in ISI activities in

teacher college sessions, the focus of Chapter 5 is on the ISI-CK of three pre-

service teachers “in action” while teaching an ISI lesson.

A review of the literature suggests that few studies have reported on pre-

service teachers’ ISI-CK (Ben-Zvi et al., 2015). For example, the knowledge of

pre-service teachers in the (sub-)components “using data” and “generalization

beyond the data” have hitherto not been investigated. This implies that there is

also no overview available that integrates pre-service teachers’ knowledge of

all ISI components. Therefore, our first study, reported in Chapter 2, aimed to

comprehensively describe pre-service teachers’ ISI-CK by surveying 722 first-

year pre-service teachers from seven teacher colleges across the Netherlands.

The limited available evidence about pre-service primary school teachers’

ISI-CK suggests that many pre-service teachers have difficulty making

inferences and lack an understanding of representativeness and sampling

variability (Groth & Meletiou-Mavrotheris, 2018). The evidence on the

“probabilistic language” component suggests that many pre-service teachers

show a limited understanding of sampling methods, sample size,

representativeness, and sources of bias in the case of self-selection (Groth &

Bergner, 2005; Meletiou-Mavrotheris, Kleanthous, & Paparistodemou, 2014;

Watson, 2001). Concerning the “sampling variability” subcomponent, Mooney,

Duni, VanMeenen, and Langrall (2014) report that a substantial proportion of

the teachers they examined understood that sample distributions are likely to

be different from the population distribution. However, an understanding of

the implications of variability for the actual sample proved to be more difficult.

Watson and Callingham (2013) found that only half of the teachers they

interviewed could conceptualize that a smaller sample has greater variability.

INTRODUCTION

15

No studies have so far been conducted on how pre-service teachers use

descriptive statistics in the context of ISI. However, the literature on pre-service

teachers’ understanding of descriptive statistics in general has shown this

understanding to be generally superficial (Batanero & Díaz, 2010;

Chatzivasileiou, Michalis, Tsaliki, & Sakellariou, 2011; Garfield & Ben-Zvi,

2007; Jacobbe & Carvalho, 2011; Koleza & Kontogianni, 2016). More

specifically, only a minority of teachers attend to both center and spread when

comparing data sets (Canada & Ciancetta, 2007), while understanding of the

mean, median, and mode is mostly procedural (Groth & Bergner, 2006; Jacobbe

& Carvalho, 2011). In sum, this research suggests there is a need to develop at

least some components of pre-service primary school teachers’ ISI-CK.

Interventions to foster pre-service teachers’ ISI-CK

In order to investigate how to foster pre-service teachers’ ISI-CK, we aimed to

design an intervention that would be useful in mathematics education

curricula of Dutch teacher colleges. In these curricula, there is usually little

time reserved for statistics, and attention to ISI is likely to be absent altogether

(Van Zanten, Barth, Faarts, Van Gool, & Keijzer, 2009). We, therefore, did not

expect that teacher colleges would consider implementing lengthy ISI courses

in their mathematics education courses. To obtain ecological valid results, we

designed relatively short interventions (Chapter 3: one session; Chapter 4: five

sessions) that could be implemented in the current mathematics education

curriculum.

To date, the literature has not provided other examples of such short

interventions, and very few studies have investigated how to foster pre-service

primary school teachers’ ISI-CK (Ben-Zvi et al., 2015; Groth & Meletiou-

Mavrotheris, 2018). The only study to date, Leavy (2010), showed a tendency

among pre-service teachers to focus on data collection and analysis at the

expense of inferential reasoning. This tendency is ascribed to the pre-service

teachers’ inexperience with conducting statistical investigations themselves.

Furthermore, pre-service teachers have severe problems designing lessons that

provide opportunities to discuss ISI with students (Leavy, 2010; Makar &

Rubin, 2009).

In designing the interventions reported in Chapter 4, we incorporated

these findings by using activities that elicit awareness of inferences and by

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providing pre-service teachers with a pre-designed lesson plan rather than

having them design their own.

Theoretical Framework

As the above discussion revealed, there are no examples of interventional

studies that show how to foster the ISI-CK of pre-service teachers within a

limited time frame. Therefore, the second aim of this thesis is to design such

interventions and study the development of pre-service teachers’ ISI-CK. This

section sets out which theories on learning and teaching were used in

designing the interventions.

The activities used in our studies are predominantly based on inquiry-

based teaching, where the learning involved is defined as guided co-

construction of knowledge (Dobber, Zwart, Tanis, & van Oers, 2017). For

students in general, inquiry-based learning environments have been shown to

be effective in developing their ISI knowledge (Makar et al., 2011). In such

learning environments, a driving question is typically posed to motivate

students to engage in inquiries (Garfield & Ben-Zvi, 2008; Groth, Bergner,

Burgess, Austin, & Holdai, 2016). In the case of ISI, such a question would elicit

the need to conduct statistical investigations as a means of finding an answer to

the driving question (Garfield & Ben-Zvi, 2008; Groth et al., 2016). It is expected

that such environments help to construct knowledge of key statistical topics

(Ben-Zvi, Gravemeijer, & Ainley, 2018; Cobb & McClain, 2004). Chapters 3

through 5 report on pre-service teachers or primary school students being

engaged in statistical investigations.

Inquiry-based teaching is well aligned with the Dutch mathematics

primary school curricula given that these curricula are typically based on

realistic mathematics education (RME), which originates from the work of

Hans Freudenthal (1973). For example, guided reinvention, which is one of the

central tenets of RME, resembles the idea of guided co-construction of

knowledge: Students reinvent mathematical and statistical tools for solving

problems under the guidance of the teacher and the materials (Freudenthal,

1991). An educational designer could incorporate activities that use students’

informal solution strategies as starting points for the reinvention process. In

INTRODUCTION

17

this thesis, the principle of guided reinvention underlies the growing samples

activity (Bakker, 2004) employed in Chapter 3 as well as the activities in

Chapter 4.

For effective inquiry-based learning, appropriate teacher guidance is

essential (Dobber et al., 2017). In particular, the teacher plays an important role

in providing scaffolds, in being a role model for conducting inquiries, and in

establishing the cultural norms of statistical inquiry (Makar et al., 2011). An

important task of the teacher is, thus, to enculturate students into such

practices. Although stemming from a different theory on learning, direct

explanations can be useful for teachers to employ in inquiry-based teaching

when such explanations build on pre-existing knowledge and are provided

when students are aware of what they do not know (Hmelo-Silver, Duncan, &

Chinn, 2007). The demonstration of a simulated sampling distribution in

Chapter 4 provides an example of such a “just-in-time” explanation.

Inquiry-based learning can be linked to sociocultural learning theories such

as situated cognition theory, which posits that knowledge is contingent upon

how and where it was learned (Putnam & Borko, 2000). From this theory, we

highlight three issues that are relevant for our research. First, we assume that

learning activities, or practices, need to be authentic and make sense to the

students. We follow the view of Brown, Collins, and Duguid (1989) of

authentic practices in the sense that these are practices that professionals

would be engaged in as well. From the perspective of ISI, this implies that

students conduct statistical investigation themselves (Garfield & Ben-Zvi, 2008)

and work with real, or realistic, datasets (Cobb & McClain, 2004) to experience

real statistical research (see Chapters 3 and 4). From the perspective of teacher

education, teaching internships in placement schools are used to offer

authentic practices (Putnam & Borko, 2000). In our research, we had the

participating pre-service teachers teach an ISI lesson based on meaningful

problems (Chapter 5).

Second, we acknowledge the situated nature of knowledge. Therefore, we

assume that what pre-service teachers can do and how they think depends on

the context (Ball, 1997). As mentioned in Section 3.1, we, therefore, do not look

only at the knowledge in contexts outside the practice of teaching (during tests

and during teacher college sessions, in Chapters 2, 3, and 4) but in Chapter 5,

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we also investigate how their ISI-CK is displayed in their teaching (Rowland et

al., 2009) – which is the ultimate goal of learning ISI.

A third and related point is that successful use of knowledge in new

situations is facilitated by a high degree of similarity with the context where

the knowledge was learned (Carter & Doyle, 1989; Putnam & Borko, 2000). So,

to facilitate the use of the pre-service teachers’ ISI-CK in primary classrooms,

we aimed for close links between the teacher college activities and the teaching

practice in primary school by, for example, modeling a lesson that the pre-

service teachers could teach themselves in their placement schools (Chapter 4)

and by integrating content knowledge and pedagogical content knowledge

(Groth, 2017).

Finally, our research was also informed by cognitive learning theories such

as conceptual change theory. We follow Cobb (1994), who argued that

enculturation of practices presupposes an actively constructing student and

that different constructions can result for different students. We assume that

learners have to actively construct their knowledge (Ben-Zvi et al., 2018;

Freudenthal, 1973; Mickelson & Heaton, 2004). Therefore, all interventions are

primarily activity based.

Methodological Approach

The following methodological tenets guided the design and legitimatization of

the research and the interventions.

Agency

The concept of agency was important in designing the research, in particular in

terms of the content of Chapters 4 and 5. Agency can be defined as “the actual

ways situated persons willfully master their own life” (van Oers, 2015, p. 19).

For our research, agency implies that the pre-service teachers are motivated to

participate in the intervention and experience a sense of ownership about their

own learning and use of ISI and the task of teaching an ISI lesson. Pre-service

teachers who express such agency could, for example, prepare the ISI lesson in

such a way that they tailor the lesson to their students’ needs while

simultaneously ensuring the underlying ideas of ISI are maintained. Such an

INTRODUCTION

19

attitude is opposed to pre-service teachers who teach as executors of a course

of actions prescribed in a lesson plan.

A first reason why agency is important in our research is that the pre-

service teachers’ motivation to participate in the intervention was necessary for

a proper functioning of the intervention’s activities and, thus, for obtaining

good research data. A second and more important reason is that, in their role

as teachers, they need to be able to show their primary school students the

relevance of ISI and motivate them to engage in ISI. This task can only be done

successfully if the teachers are motivated to engage in ISI themselves.

Dutch pre-service primary school teachers are likely to be unfamiliar with

ISI and inquiry-based teaching, which may have affected their development of

agency. Historically, Dutch primary mathematics curricula spent little time on

statistics, and to date, ISI is not part of Dutch primary education curricula (Van

Zanten et al., 2009). Moreover, an inquiry-based teaching approach appears to

be at odds with mainstream teaching practices in Dutch primary schools.

Although Dutch mathematics curricula are mostly based on RME (Freudenthal,

1973), Gravemeijer, Bruin-Muurling, Kraemer, and van Stiphout (2016)

speculate that actual teaching practices are characterized by “the tendency to

link learning to the ability of students to successfully carry out well-defined

tasks and by ‘getting right answers’” (p. 36). Most pre-service teachers will

have encountered this procedure-oriented mathematics in their placement

schools. These schools have been argued to be “powerful discourse

communities” in which “patterns of classroom teaching and learning have

historically been resistant to fundamental change” (Putnam & Borko, 2000, pp.

8-9). As a result, we expected that the historical context of procedure-oriented

mathematics teaching and the institutional context of placement schools could

lead to doubts about the relevance of ISI and lead to teacher reluctance to

adopt the openness and uncertainty of an inquiry-based teaching approach

(Engeström, 2011; Penuel, 2014; Putnam & Borko, 2000).

Another factor that could lead to reluctance to participate in the

intervention might be that when pre-service teachers were themselves primary

school students, they may have been taught with a similar procedure-oriented

approach. Since their experience as learners influences their perceptions of

what good teaching entails (Lortie, 1975), they may be reluctant to accept an

inquiry-based teaching approach.

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The possible resistance of pre-service teachers to ISI and inquiry-based

teaching made it essential to search for ways to foster the participants’ agency.

We first approached this possibility with the initial activity used in the teacher

college intervention (Chapter 4), which had the participants search the media

for reports where claims were made based on a sample. It was hypothesized

that this activity would help them see the relevance of ISI. Second, we stressed

their position as key actors in translating what was offered at teacher college to

their own classrooms. Third, we emphasized their role as research informants,

whose information was necessary for our research purposes. For example, we

stressed that the purpose of teaching an ISI lesson was not that their students

would learn ISI, but rather that we could investigate the possibilities of having

inexperienced teachers teach ISI. We hypothesized that this role of test case

would alleviate possible feelings of anxiety. In Chapter 6, I will discuss to what

extent we were successful in fostering the pre-service teachers’ agency.

Practice-oriented research

As mentioned above, we aimed to design interventions that could be

incorporated with relative ease into the mathematics education curricula of

primary education teacher colleges. As such, the interventional studies

discussed in Chapters 3 through 5 are examples of practice-oriented research

(A. J. Mills, Durepos, & Wiebe, 2009). The first approach to increase the

usability of the interventions was by embedding relatively short interventions

in existing teacher education courses. This stands in contrast with other

interventional studies in the domain of statistics education for teachers, which

often involved semester- or summer-long interventions (e.g. Groth, 2017;

Leavy, 2010). Second, the intervention was implemented in existing classes of

pre-service teachers enrolled in a regular curriculum for primary education

rather than in classes with teachers specializing in mathematics or statistics

education or among high-ability pre-service teachers. Third, the activities were

designed in such a way that teacher educators, as well as teachers, could apply

the activities with relative ease in their own classrooms. The activities require

little organization and mostly involve hands-on activities (Pratt & Kazak, 2018)

in lieu of using educational statistics software such as Tinkerplots (Konold &

Miller, 2011). The use of such software requires a considerable time and

financial investment, which could make teacher educators and teachers

INTRODUCTION

21

reluctant to use the activities in their own education. These methodological

choices were expected to foster the ecological validity and transferability of the

results (Lewis & Ritchie, 2003).

Qualitative and mixed-methods research

Given that there is limited previous research related to our research aims, the

study in this thesis is predominantly explorative. Qualitative and mixed-

methods research designs are particularly well suited (Ritchie, 2003) for

exploratory research. Both types of designs are also commonly used in

statistics education research (Petocz, Reid, & Gal, 2018).

The studies reported in Chapters 2 and 4 used mixed-methods designs. The

questionnaire study in Chapter 2 used open questions that allowed the

respondents to give their own answers without steering them in a particular

direction, while the subsequent statements to be evaluated allowed for probing

for information that was not provided through the initial questioning. These

different perspectives thus yield complementary information with the effect of

describing a more complete picture of pre-service teachers’ ISI-CK

(Schoonenboom & Johnson, 2017). Chapter 4 covers a study using an

abbreviated and slightly adapted version of the questionnaire from Chapter 2

as a pretest and posttest. The results from these sources were primarily used

for providing quantitative overviews of the development of the participants’

ISI-CK. Qualitative data from the open-ended questions and from classroom

observations helped us gain deeper knowledge of pre-service teachers’

reasoning and argumentation about ISI. The latter data also helped us gain

insight into which, and in what way, activities were useful for fostering the

pre-service teachers’ ISI-CK (Garfield & Ben-Zvi, 2008).

First-person perspective

In the two interventional studies (Chapters 3 and 4), I combined the roles of

designer, researcher, and teacher educator. This combination of roles was

expected to have several benefits. First, as a researcher, I was well acquainted

with the ISI literature, and as a teacher educator, I was familiar with the

particular teacher education context. This combination helped me design ISI

activities that were framed by the results in the literature but that were also

tailored to the specific context. Second, because ISI is not part of Dutch teacher

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college curricula, Dutch teacher educators are usually not familiar with ISI and

would have to be instructed about it. This might lead to a loss of instructional

quality. Moreover, it would make the study unnecessarily complex by

introducing a third layer of interaction between researcher and teacher

educator, on top of the interactions between teacher educator and pre-service

teachers, and between pre-service teachers and primary school students. Third,

being also a teacher educator helped me to experience from the inside the

workings and effectiveness of the activities (Ball, 2000). Furthermore, I knew

the pre-service teachers I was working with well because I was their regular

mathematics teacher educator. This helped me to interpret their discussions

and mannerisms, and it allowed me to use our shared history and experiences

in teaching ISI.

However, this combination of roles could threaten the reliability of the

observations and the validity of the conclusions. Therefore, in Chapter 6, I will

discuss in what ways I tried to separate the roles and to what extent the threats

to the reliability and validity were counteracted.

Outline of this Thesis

The overall research question was, What ISI-CK do pre-service primary school

teachers have and how does their ISI-CK develop during interventions of

limited length? To answer this question, four studies were conducted:

Chapter 2 reports on a study that surveyed 722 first-year pre-service

primary school teachers from seven teacher colleges across The Netherlands in

order to describe the ISI-CK of first-year pre-service teachers. The results

informed the design of the interventions to foster pre-service teachers’ ISI-CK.

Research question, Chapter 2: To what extent do first-year pre-service

primary school teachers have appropriate ISI-CK?

Chapter 3 reports on an exploratory study of pre-service teachers’

reasoning processes about ISI when they were engaged in a growing samples

activity, to study how this heuristic supported pre-service teachers’

development of reasoning about ISI.

INTRODUCTION

23

Research question, Chapter 3: What reasoning about ISI do first-year pre-

service primary school teachers display when they are engaged in a growing

samples activity, and what is the quality of their reasoning?

Chapter 4 reports on a case study investigating ISI-CK development in a

class of 21 pre-service primary school teachers who participated in a short

intervention consisting of several activities aimed at fostering their content

knowledge and pedagogical content knowledge of ISI.

Research question, Chapter 4: In what respect does the ISI-CK of pre-

service primary school teachers develop during a teacher college intervention,

and what role do the activities used during the intervention play in this

development?

Chapter 5 reports on an embedded case study of three pre-service teachers

who participated in the study discussed in Chapter 4 as they demonstrated

their ISI-CK by teaching an ISI lesson.

Research question, Chapter 5: What ISI-CK do three pre-service primary

school teachers express while teaching an ISI lesson in primary school?

The four studies were separately published or submitted in different

places. Therefore, each study necessarily had to discuss the practical relevance

and the theoretical background. As a consequence, the chapters are to some

extent repetitious with regard to these issues.

Expected Contribution to Research and Practice

This thesis aims to extend the body of scientific knowledge about ISI-CK of

pre-service primary school teachers and demonstrate how this knowledge can

be developed as a result of teacher college education. These results may inform

other researchers, instructional designers, and teacher educators who wish to

study ISI in the context of pre-service teacher education or to implement ISI

activities in teacher training.

Although this thesis offers grounded examples of ISI education in only one

Dutch teacher college, it may reveal characteristics of learning processes among

CHAPTER 1

24

pre-service teachers that could be applicable in other contexts as well. The

results may, thus, contribute to the formation of a domain-specific theory of

learning and inquiry-based teaching of ISI (Van den Akker, Gravemeijer,

McKenney, & Nieveen, 2006), in particular, for pre-service teachers, but also for

other types of learners. The data from the different studies in this thesis

resulted in a list of provisional design heuristics that could be part of such a

domain-specific theory of ISI (see Chapter 6).

As some of the teacher college activities used in our research are explicitly

designed to be used with relative ease in primary education, these activities

may be useful for those wishing to design and study ISI learning environments

for primary education. Furthermore, the results of our research may reveal

how recommendations for statistics education in general (Cobb & McClain,

2004) can be applied in the context of ISI and whether adaptations to these

recommendations are required when applied in ISI education. Finally, this

thesis also contributes to the conceptualization of ISI for pre-service teachers

and how ISI-CK can be measured (Chapter 2).

INTRODUCTION

25

References Arnold, P., Pfannkuch, M., Wild, C. J., Regan, M., & Budgett, S. (2011). Enhancing students’

inferential reasoning: From hands-on to “movies”. Journal of statistics Education, 19(2), 1–32.

Bakker, A. (2004). Design research in statistics education: On symbolizing and computer tools. Utrecht,

the Netherlands: CD-ß Press.

Bakker, A., & Derry, J. (2011). Lessons from inferentialism for statistics education. Mathematical

Thinking and Learning, 13(2), 5–26.

Ball, D. L. (1997). What do students know? Facing challenges of distance, context and desire

intrying to hear children. In B. J. Biddle, T. L. Good, & I. F. Goodson (Eds.), International

handbook of teachers and teaching (Vol. II, pp. 769-818). Dordrecht: Kluwer.

Ball, D. L. (2000). Working on the inside: Using one's own practice as a site for studying teaching

and learning. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and

science education (pp. 365–402). Mahwah, NJ: Lawrence Erlbaum.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it

special? Journal of Teacher Education, 59(5), 389–407.

Batanero, C., & Díaz, C. (2010). Training teachers to teach statistics: What can we learn from

research? Statistique et enseignement, 1(1), 5–20.

Ben-Zvi, D. (2006). Scaffolding students’ informal inference and argumentation. Paper presented at the

Seventh International Conference on Teaching Statistics, Salvador, Brazil.

Ben-Zvi, D., Bakker, A., & Makar, K. (2015). Learning to reason from samples. Educational Studies in

Mathematics, 88(3), 291–303.

Ben-Zvi, D., Gravemeijer, K., & Ainley, J. (2018). Design of statistics learning environments. In D.

Ben-Zvi, K. Makar, & J. Garfield (Eds.), International handbook of research in statistics education

(pp. 473-502). Cham, Switzerland: Springer.

Biesta, G. (2009). Good education in an age of measurement: On the need to reconnect with the

question of purpose in education. Educational Assessment, Evaluation and Accountability, 21(1),

33-46.

Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning.

Educational Researcher, 18(1), 32-42.

Bruner, J. S. (1960). The process of education. Cambridge, MA: Harvard University Press.

Burgess, T. (2009). Teacher knowledge and statistics: What types of knowledge are used in the

primary classroom? Montana Mathematics Enthusiast, 6(1&2), 3–24.

Canada, D., & Ciancetta, M. (2007). Elementary preservice teachers' informal conceptions of distribution.

Paper presented at the 29th annual meeting of the North American Chapter of the

International Group for the Psychology of Mathematics Education, Stateline, NV.

Carter, K., & Doyle, W. (1989). Classroom research as a resource for the graduate preparation of

teachers. In A. Woolfolk (Ed.), Research perspectives on the graduate preparation of teachers (pp.

51-68). Englewood Cliffs, NJ: Prentice Hall.

Chance, B., delMas, R. C., & Garfield, J. (2004). Reasoning about sampling distributions. In D. Ben-

Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp.

295–323). New York, NY: Kluwer.

Chatzivasileiou, E., Michalis, I., Tsaliki, C., & Sakellariou, I. (2011). In service elementary school

teachers’ conceptions of arithmetic mean. Paper presented at the 58th World Statistical Congress,

Dublin, Ireland.

Cobb, P., & McClain, K. (2004). Principles of instructional design for supporting the development

of students’ statistical reasoning. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing

statistical literacy, reasoning and thinking (pp. 375-395). New York, NY: Kluwer.

Dewey, J. (1910/1997). How we think. Mineola, NY: Dover.

Dobber, M., Zwart, R., Tanis, M., & van Oers, B. (2017). Literature review: The role of the teacher in

inquiry-based education. Educational Research Review, 22, 194-214.

CHAPTER 1

26

Engeström, Y. (2011). From design experiments to formative interventions. Theory & Psychology,

21(5), 598-628.

Fennema, E., & Franke, L. M. (1992). Teachers' knowledge and its impact. In D. A. Grouws (Ed.),

Handbook of research on mathematics teaching and learning (pp. 147–164). New York, NY:

Macmillan.

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: Reidel.

Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht, The Netherlands:

Kluwer.

Garfield, J., & Ben-Zvi, D. (2007). How students learn statistics revisited: A current review of

research on teaching and learning statistics. International Statistical Review, 75(3), 372-396.

Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and

teaching practice. Dordrecht, The Netherlands: Springer.

Gopnik, A. (2012). Scientific thinking in young children: Theoretical advances, empirical research,

and policy implications. Science, 337(6102), 1623-1627.

Gravemeijer, K., Bruin-Muurling, G., Kraemer, J.-M., & van Stiphout, I. (2016). Shortcomings of

mathematics education reform in The Netherlands: A paradigm case? Mathematical Thinking

and Learning, 18(1), 25-44.

Gravemeijer, K., Stephan, M., Julie, C., Lin, F.-L., & Ohtani, M. (2017). What mathematics education

may prepare students for the society of the future? International Journal of Science and

Mathematics Education, 15(1), 105-123.

Groth, R. E. (2013). Characterizing key developmental understandings and pedagogically powerful

ideas within a statistical knowledge for teaching framework. Mathematical Thinking and

Learning, 15(2), 121–145.

Groth, R. E. (2017). Developing statistical knowledge for teaching during design-based research.

Statistics Education Research Journal, 16(2), 376–396.

Groth, R. E., & Bergner, J. A. (2005). Pre-service elementary school teachers’ metaphors for the

concept of statistical sample. Statistics Education Research Journal, 4(2), 27-42.

Groth, R. E., & Bergner, J. A. (2006). Preservice elementary teachers' conceptual and procedural

knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8(1), 37-63.

Groth, R. E., Bergner, J. A., Burgess, C. R., Austin, J. W., & Holdai, V. (2016). Re-imagining

education of mathematics teachers through undergraduate research. Council on Undergraduate

Research (CUR) Quarterly, 36(3), 41-46.

Groth, R. E., & Meletiou-Mavrotheris, M. (2018). Research on statistics teachers’ cognitive and

affective characteristics. In D. Ben-Zvi, K. Makar, & J. Garfield (Eds.), International handbook of

research in statistics education (pp. 327–355). Cham, Switzerland: Springer.

Hmelo-Silver, C. E., Duncan, R. G., & Chinn, C. A. (2007). Scaffolding and achievement in problem-

based and inquiry learning: A response to Kirschner, Sweller, and. Educational Psychologist,

42(2), 99-107.

Jacobbe, T., & Carvalho, C. (2011). Teachers’ understanding of averages. In C. Batanero, G. Burrill,

C. Reading, & A. Rossman (Eds.), Joint ICMI/IASE study: Teaching statistics in school

mathematics. Challenges for teaching and teacher education. Proceedings of the ICMI Study 18 and

2008 IASE Round Table Conference. (pp. 199-209). Dordrecht, The Netherlands: Springer.

Koleza, E., & Kontogianni, A. (2016). Statistics in Primary Education in Greece: How Ready Are

Primary Teachers? In D. Ben-Zvi & K. Makar (Eds.), The teaching and learning of statistics:

International perspectives (pp. 289-298). Cham, Switzerland: Springer.

Konold, C., & Miller, C. D. (2011). TinkerPlots: Dynamic Data Exploration (Version 2.0).

Emeryville, CA: Key Curiculum Press.

Leavy, A. M. (2010). The challenge of preparing preservice teachers to teach informal inferential

reasoning. Statistics Education Research Journal, 9(1), 46–67.

Leavy, A. M. (2017). Insights into the approaches of young children when making informal

inferences about data. In T. Dooley & G. Gueudet (Eds.), Proceedings of the tenth congress of the

INTRODUCTION

27

European Society for Research in Mathematics Education (CERME10, February 1-5, 2017) (pp. 788-

795). Dublin, Ireland: DCU Institute of Education and ERME.

Lewis, J., & Ritchie, J. (2003). Generalising from qualitative research. In J. Ritchie & J. Lewis (Eds.),

Qualitative research practice: A guide for social science students and researchers (pp. 263-286).

London, UK: Sage.

Liu, Y., & Grusky, D. B. (2013). The payoff to skill in the third industrial revolution. American

Journal of Sociology, 118(5), 1330–1374.

Lortie, D. (1975). Schoolteacher: A sociological analysis. Chicago, IL: University of Chicago.

Makar, K. (2016). Developing young children’s emergent inferential practices in statistics.

Mathematical Thinking and Learning, 18(1), 1-24.

Makar, K., Bakker, A., & Ben-Zvi, D. (2011). The reasoning behind informal statistical inference.

Mathematical Thinking and Learning, 13(1-2), 152–173.

Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference.

Statistics Education Research Journal, 8(1), 82–105.

Makar, K., & Rubin, A. (2018). Learning about statistical inference. In D. Ben-Zvi, K. Makar, & J.

Garfield (Eds.), International handbook of research in statistics education (pp. 261–294). Cham,

Switzerland: Springer.

Meletiou-Mavrotheris, M., Kleanthous, I., & Paparistodemou, E. (2014). Developing pre-service

teachers' technological pedagogical content knowledge (TPACK) of sampling. Paper presented at the

Ninth International Conference on Teaching Statistics (ICOTS9), Flagstaff, AZ.

Meletiou-Mavrotheris, M., & Paparistodemou, E. (2015). Developing students’ reasoning about

samples and sampling in the context of informal inferences. Educational Studies in Mathematics,

88(3), 385–404.

Mickelson, W. T., & Heaton, R. M. (2004). Primary teachers’ statistical reasoning about data. In D.

Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking

(pp. 327-352). New York, NY: Kluwer.

Mills, A. J., Durepos, G., & Wiebe, E. (2009). Encyclopedia of case study research. London, UK: Sage.

Mills, J. D. (2002). Using computer simulation methods to teach statistics: A review of the literature.

Journal of statistics Education, 10(1), 1–20.

Mooney, E., Duni, D., VanMeenen, E., & Langrall, C. (2014). Preservice teachers' awareness of

variability. In K. Makar, B. De Sousa, & R. Gould (Eds.), Proceedings of the ninth International

Conference on Teaching Statistics (ICOTS9). Voorburg, The Netherlands: International Statistical

Institute.

Paparistodemou, E., & Meletiou-Mavrotheris, M. (2008). Developing young students’ informal

inference skills in data analysis. Statistics Education Research Journal, 7(2), 83-106.

Pel, A.-F. (2016, November 4). Forens is volle, vieze trein nu echt zat. [Commuter is sick of

crowded, dirty train]. Metro, pp. 1-2.

Penuel, W. R. (2014). Emerging forms of formative intervention research in education. Mind,

Culture, and Activity, 21(2), 97-117.

Petocz, P., Reid, A., & Gal, I. (2018). Statistics Education Research. In D. Ben-Zvi, K. Makar, & J.

Garfield (Eds.), International handbook of research in statistics education (pp. 71-99). Cham,

Switzerland: Springer.

Petrou, M., & Goulding, M. (2011). Conceptualising Teachers' Mathematical Knowledge in

Teaching. In T. R. Rowland, Kenneth (Ed.), Mathematical Knowledge in Teaching (pp. 9-25).

London: Springer.

Piaget, J., & Inhelder, B. (1975/1951). The origin of the idea of chance in children (L.J. Leake, P.D. Burrell

& H.D. Fischbein, Trans.). London, UK: Routledge & Kegan Paul.

Pratt, D., & Kazak, S. (2018). Research on uncertainty. In D. Ben-Zvi, K. Makar, & J. Garfield (Eds.),

International handbook of research in statistics education (pp. 193–227). Cham, Switzerland:

Springer.

CHAPTER 1

28

Putnam, R. T., & Borko, H. (2000). What do new wiews of knowledge and thinking have to say

about research on teacher learning? Educational Researcher, 29(1), 4-15.

Ritchie, J. (2003). The applications of qualitative methods to social research. In J. Ritchie & J. Lewis

(Eds.), Qualitative research practice: A guide for social science students and researchers (pp. 24–46).

London: Sage.

Rivkin, S. G., Hanushek, E. A., & Kain, J. F. (2005). Teachers, schools, and academic achievement.

Econometrica, 73(2), 417–458.

Rossman, A. J. (2008). Reasoning about informal statistical inference: One statistician's view.

Statistics Education Research Journal, 7(2), 5–19.

Rowland, T., Turner, F., Thwaites, A., & Hurst, C. (2009). Developing primary mathematics teaching:

Reflecting on practice with the knowledge quartet. London, UK: Sage.

Saldanha, L., & Thompson, P. (2002). Conceptions of sample and their relationship to statistical

inference. Educational Studies in Mathematics, 51(3), 257–270.

Schoonenboom, J., & Johnson, R. B. (2017). How to construct a mixed methods research design.

Kölner Zeitschrift für Soziologie und Sozialpsychologie, 69(2), 107-131.

Shaughnessy, J. M., Garfield, J., & Greer, B. (1996). Data handling. In A. J. Bishop, K. Clements, C.

Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp.

205-237). Dordrecht, The Netherlands: Kluwer.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational

Researcher, 15(2), 4–14.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.

Journal for research in mathematics education, 26(2), 114-145.

Statistics Netherlands. (2018). Drop in recorded and encountered crime. Retrieved from

http://www.cbs.nl

Van den Akker, J., Gravemeijer, K., McKenney, S., & Nieveen, N. (2006). Educational design research.

London: Routledge.

van Oers, B. (2015). Implementing a play-based curriculum: Fostering teacher agency in primary

school. Learning, Culture and Social Interaction, 4, 19-27.

Van Zanten, M. A., Barth, F., Faarts, J., Van Gool, A., & Keijzer, R. (2009). Kennisbasis Rekenen-

Wiskunde voor de lerarenopleiding basisonderwijs. [Knowledge base mathematics for primary teacher

education]. Den Haag, The Netherlands: HBO-raad.

Viner, C. (2016, July 12). How technology disrupted the truth. The Guardian. Retrieved from

http://www.theguardian.com

Vygotsky, L. S. (1978). Mind in society: The development of higher mental process. Cambridge, MA:

Harvard University Press.

Watson, J. M. (2001). Profiling teachers' competence and confidence to teach particular

mathematics topics: The case of chance and data. Journal of Mathematics Teacher Education, 4(4),

305–337.

Watson, J. M., & Callingham, R. (2013). Likelihood and sample size: The understandings of

students and their teachers. The Journal of Mathematical Behavior, 32(3), 660-672.

Watson, J. M., & Moritz, J. B. (2000). Developing concepts of sampling. Journal for research in

mathematics education, 31(1), 44–70.

Word of the Year 2016. (2016). Retrieved from https://en.oxforddictionaries.com/word-of-the-

year/word-of-the-year-2016

Zieffler, A., Garfield, J., delMas, R., & Reading, C. (2008). A framework to support research on

informal inferential reasoning. Statistics Education Research Journal, 7(2), 40–58.