The Montana Mathematics Enthusiast

ISSN ****-****

VOL. 6, NOS.1&2, January 2009, pp.1-294

Editor-in-Chief

Bharath Sriraman, The University of Montana

Associate Editors:

Lyn D. English, Queensland University of Technology, Australia

Claus Michelsen, University of Southern Denmark, Denmark

Brian Greer, Portland State University, USA

Luis Moreno-Armella, University of Massachusetts-Dartmouth

International Editorial Advisory Board

Miriam Amit, Ben-Gurion University of the Negev, Israel.

Ziya Argun, Gazi University, Turkey.

Ahmet Arikan, Gazi University, Turkey.

Astrid Beckmann, University of Education, Schw bisch Gm nd, Germany.

Morten Blomh j, Roskilde University, Denmark.

Robert Carson, Montana State University- Bozeman, USA.

Mohan Chinnappan, University of Wollongong, Australia.

Constantinos Christou, University of Cyprus, Cyprus.

Bettina Dahl S ndergaard, University of Aarhus, Denmark.

Helen Doerr, Syracuse University, USA.

Ted Eisenberg, Ben-Gurion University of the Negev, Israel.

Paul Ernest, University of Exeter, UK.

Viktor Freiman, Universit de Moncton, Canada.

Fulvia Furinghetti, Universit di Genova, Italy

Anne Birgitte Fyhn, Universitetet i Troms, Norway

Eric Gutstein, University of Illinois-Chicago, USA.

Marja van den Heuvel-Panhuizen, University of Utrecht, The Netherlands.

Gabriele Kaiser, University of Hamburg, Germany.

Tinne Hoff Kjeldsen, Roskilde University, Denmark.

Jean-Baptiste Lagrange, IUFM-Reims, France.

Stephen Lerman, London South Bank University, UK.

Frank Lester, Indiana University, USA.

Richard Lesh, Indiana University, USA.

Nicholas Mousoulides, University of Cyprus, Cyprus.

Swapna Mukhopadhyay, Portland State University, USA.

Norma Presmeg, Illinois State University, USA.

Gudbjorg Palsdottir,Iceland University of Education, Iceland.

Jo o Pedro da Ponte, University of Lisbon, Portugal

Demetra Pitta Pantazi, University of Cyprus, Cyprus.

Linda Sheffield, Northern Kentucky University, USA.

Olof Bjorg Steinthorsdottir, University of North Carolina- Chapel Hill, USA.

G nter T rner, University of Duisburg-Essen, Germany.

Renuka Vithal, University of KwaZulu-Natal, South Africa.

Dirk Wessels, UNISA, South Africa.

Nurit Zehavi, The Weizmann Institute of Science, Rehovot, Israel.

The Montana Mathematics Enthusiast is an eclectic internationally circulated peer

reviewed journal which focuses on mathematics content, mathematics education research,

innovation, interdisciplinary issues and pedagogy. The journal is published by Information

Age Publishing and the electronic version is hosted jointly by IAP and the Department of

Mathematical Sciences- The University of Montana, on behalf of MCTM. Articles

appearing in the journal address issues related to mathematical thinking, teaching and

learning at all levels. The focus includes specific mathematics content and advances in that

area accessible to readers, as well as political, social and cultural issues related to

mathematics education. Journal articles cover a wide spectrum of topics such as

mathematics content (including advanced mathematics), educational studies related to

mathematics, and reports of innovative pedagogical practices with the hope of stimulating

dialogue between pre-service and practicing teachers, university educators and

mathematicians. The journal is interested in research based articles as well as historical,

philosophical, political, cross-cultural and systems perspectives on mathematics content, its

teaching and learning.

The journal also includes a monograph series on special topics of interest to the

community of readers The journal is accessed from 110+ countries and its readers include

students of mathematics, future and practicing teachers, mathematicians, cognitive

psychologists, critical theorists, mathematics/science educators, historians and philosophers

of mathematics and science as well as those who pursue mathematics recreationally. The 40

member editorial board reflects this diversity. The journal exists to create a forum for

argumentative and critical positions on mathematics education, and especially welcomes

articles which challenge commonly held assumptions about the nature and purpose of

mathematics and mathematics education. Reactions or commentaries on previously

published articles are welcomed. Manuscripts are to be submitted in electronic format to

the editor in APA style. The typical time period from submission to publication is 8-11

months. Please visit the journal website at http://www.montanamath.org/TMME or

http://www.math.umt.edu/TMME/

Indexing Information

Australian Education Index (For Australian authors);

EBSCO Products (Academic Search Complete);

EDNA;

Cabell s Directory of Publishing Opportunities in Educational Curriculum and Methods

Directory of Open Access Journals (DOAJ);

Inter-university Centre for Educational Research (ICO)

PsycINFO (the APA Index);

MathDI/MathEDUC (FiZ Karlsruhe);

Journals in Higher Education (JIHE);

Ulrich s Periodicals Directory;

Zentralblatt MATH

THE MONTANA MATHEMATICS ENTHUSIAST

ISSN 1551-3440

Vol.6, Nos.1&2, January 2009, pp.1-294

TABLE OF CONTENTS

Editorial Information

0. TO PUBLISH OR NOT TO PUBLISH?- THAT IS THE (EDITORIAL) QUESTION

Bharath Sriraman (USA) ... ..pp.1-2

FEATURE THEMES

STATISTICS EDUCATION/MER1 IN THE SOUTHERN HEMISPHERE

1. TEACHER KNOWLEDGE AND STATISTICS: WHAT TYPES OF KNOWLEDGE

ARE USED IN THE PRIMARY CLASSROOM?

Tim Burgess (New Zealand) ..pp.3-24

2. WHAT MAKES A GOOD STATISTICS STUDENT AND A GOOD STATISTICS

TEACHER IN SERVICE COURSES?

Sue Gordon, Peter Petocz and Anna Reid (Australia) .. .pp.25-40

3. STUDENTS CONCEPTIONS ABOUT PROBABILITY AND ACCURACY

Ignacio Nemirovsky, M nica Giuliano, Silvia P rez, Sonia Concari, Aldo Sacerdoti and

Marcelo Alvarez (Argentina pp.41-46

4. UNDERGRADUATE STUDENT DIFFICULTIES WITH INDEPENDENT AND

MUTUALLY EXCLUSIVE EVENTS CONCEPTS

Adriana D'Amelio (Argentina) pp.47-56

5. ENHANCING STATISTICS INSTRUCTION IN ELEMENTARY SCHOOLS:

INTEGRATING TECHNOLOGY IN PROFESSIONAL DEVELOPMENT

Maria Meletiou-Mavrotheris (Cyprus), Efi Paparistodemou (Cyprus) & Despina Stylianou(USA)

.. pp.57-78

1

Mathematics Education Research

6. TEACHING STATISTICS MUST BE ADAPTED TO CHANGING

CIRCUMSTANCES: A Case Study from Hungarian Higher Education

Andras Komaromi (Hungary) .pp.79-86

7. STATISTICS TEACHING IN AN AGRICULTURAL UNIVERSITY: A Motivation

Problem

Klara Lokos Toth (Hungary) ..pp.87-90

8. CALCULATING DEPENDENT PROBABILITIES

Mike Fletcher (UK) ..pp.91-94

9. FOR THE REST OF YOUR LIFE

Mike Fletcher (UK) ..pp.95-98

10. LEARNING, PARTICIPATION AND LOCAL SCHOOL MATHEMATICS

PRACTICE

Cristina Frade (Brazil) & Konstantinos Tatsis (Greece pp.99-112

11. IF A.B = 0 THEN A = 0 or B = 0?

Cristina Ochoviet(Uruguay) & Asuman Okta (Mexico) .. pp.113-136

FEATURE ARTICLES

12. THE ORIGINS OF THE GENUS CONCEPT IN QUADRATIC FORMS

Mark Beintema & Azar Khosravani (Illinois, USA pp.137-150

13. THE IMPACT OF UNDERGRADUATE MATHEMATICS COURSES ON

COLLEGE STUDENT S GEOMETRIC REASONING STAGES

Nuh Aydin (Ohio, USA) & Erdogan Halat (Turkey) .. . ..pp.151-164

14. A LONGITUDINAL STUDY OF STUDENT S REPRESENTATIONS FOR

DIVISION OF FRACTIONS

Sylvia Bulgar (USA) .pp.165-200

15. ELEMENTARY SCHOOL PRE-SERVICE TEACHERS UNDERSTANDINGS

OF ALGEBRAIC GENERALIZATIONS

Jean E. Hallagan, Audrey C. Rule & Lynn F. Carlson (Oswego, New York)

.. pp.201-206

16. COMPARISION OF HIGH ACHIEVERS WITH LOW ACHIEVERS: Discussion

of Juter s (2007) article

T. P. Hutchinson (Australia) . .. pp.207-212

17. FOSTERING CONNECTIONS BETWEEN THE VERBAL, ALGEBRAIC, AND

GEOMETRIC REPRESENTATIONS OF BASIC PLANAR CURVES FOR

STUDENT S SUCCESS IN THE STUDY OF MATHEMATICS

Margo F. Kondratieva & Oana G. Radu (New Foundland, Canada) . pp.213-238

18. KOREAN TEACHERS PERCEPTIONS OF STUDENT SUCCESS IN

MATHEMATICS: Concept versus procedure

Insook Chung (Notre Dame, USA pp. 239-256

19. HOW TO INCREASE MATHEMATICAL CREATIVITY- AN EXPERIMENT

Kai Brunkalla (Ohio, USA) pp.257-266

20. CATCH ME IF YOU CAN!

Steve Humble (UK) . pp.267-274

21. A TRAILER, A SHOTGUN, AND A THEOREM OF PYTHAGORAS

William H. Kazez (Georgia, USA) . ..pp.275-276

MONTANA FEATURE

22. BOOK X OF THE ELEMENTS: ORDERING IRRATIONALS

Jade Roskam (Missoula, Montana) ...pp.277-294

TMME, vol6, nos.1&2, p. 1

TO PUBLISH OR NOT TO PUBLISH- the Editorial conundrum

Bharath Sriraman, The University of Montana

This editorial began in my mind (a mental blog if you will) as I was making my way from

Troms (Norway) to Montana late in December. As 2009 slowly rolls in, I am reminded of the

18th century Scottish bard Robert Burn s famous poem Auld Lang Syne for several reasons.

Should auld acquaintance be forgot,

And never brought to mind ?

Should auld acquaintance be forgot,

And days o' lang syne ?

This poem, typically sung on New Year s eve, has served as the backdrop for many important

events all over the world. Most recently it was played when the Pakistani president Pervez

Musharraf stepped down as the Army Chief, signaling a transition to an era of civilian

government in Pakistan. The heinous terrorist incidents that followed in Mumbai (Bombay),

which partly can be attributed to the turmoil caused by the artificial borders carved by the British

Raj in the wake of their departure from the Indian subcontinent, served as a reminder to the

tenuous nature of change . Yet we are hopeful that things are changing in a positive direction in

spite of the mess caused by post colonial geopolitics. After all politics and radicalism need not be

the lowest common denominator for communication between sides that share thousands of years

of common heritage, language and history (Yes we Can!).

What role, if any, does mathematics and mathematics education have in all this? If we claim to

live in a world where any two people can theoretically meet within 24 hours, or communicate in

real time thanks to the advances in information technology, then it only makes sense that

education instill in future generations of students a sense of shared heritage despite superficial

differences based on the Bismarckian notion of a nation-state.

The history of Central Asia, the Indian sub-continent, the Persian-Greco world and numerous

other regions when analyzed from the viewpoint of trade and the exchange of mathematical ideas

reveals an intricate shared heritage. The current day turmoil in the world based on ideology,

religion and artificially drawn post-colonial borders can very well serve as a focal point to

examine how culturally based studies of mathematics could serve as a vehicle for promoting

peace and discourse instead of economies that flourish under the politics of division and the

export of weapon s technology. I envision one of our goals should be to revisit fundamental

notions of what constitutes a culturally appropriate math curriculum, in a globally linked world

with shared problems and a collective future. For the last few decades many mathematics

educators have emphasized the place of critical mathematics education in order to better

understand problems plaguing society. The global fall out resulting from the unchecked greed of

Wall Street and the corporate world/mentality in general in numerous parts of the world, serves

The Montana Mathematics Enthusiast, ISSN 1551-3440, Vol. 6, nos.1&2, pp.1- 2

2009 Montana Council of Teachers of Mathematics & Information Age Publishing

Sriraman

as an important context to promote the basic principles of mathematics and the necessity to

revisit prevalent notions of consumerism and materialism in the West, which come at the

expense of other regions of the world. However as well intentioned an analysis of local socio-

economically and politically situated problems may be through the lens of critical mathematics

education, it is equally important to better educate young minds in critical history and geography.

That is, not boring details and facts such as how high a mountain is, or how long a river is

(Dewey, 1927 as cited by Howlett, 2008, p.27), but a global awareness of peoples, cultures,

habits, occupations, art and societies contributions to the development of human culture in

general (Dewey 1939, as cited by Howlett, 2008, p.27) in addition to the contiguous

contributions of all cultures to the development of mathematics and science.

Edward Said (1935-2003), the Palestinian American literary /critical/cultural theorist redefined

the term Orientalism to describe a tradition, both academic and artistic, of hostile and

deprecatory views of the East by the West. The curricula used in many parts of the world today

is still shaped by the attitudes of the era of European imperialism in the 18th and 19th centuries

and conveys in a hidden way prejudiced interpretations of colonized cultures and peoples,

particularly indigenous peoples. These biases become apparent in the popular media s simplistic

and dichotomous view of problems in post colonial Asia (including the Middle East) where

oversimplification is often done on religious, nationalistic and ethnic terms, such as Hindu versus

Muslim, Arab versus Jew, Sunni versus Shia, Kurd versus Turk, Turk versus Greek, Irani versus

Iraqi, etc. This perpetuates the patronizing and overtly patriarchical view of colonized peoples

and indigenous cultures to justify external meddling in their political affairs.

What is the role of a math journal in all this? The Montana Mathematics Enthusiast aims to

publish critically oriented articles relevant for mathematics education in addition to striving to

represent under-heard voices in the larger debates characterizing mathematics education. The

journal is thriving with submissions from all parts of the world and we are delivering on our

promise to help non-English speaking authors from under-represented regions, to the extent we

can to publish their work, by finding appropriate reviewers and other means of support. The

present issue contains 22 articles with numerous authors from South America [Argentina, Brazil,

Uruguay] in addition to contributions from authors in Central Europe (Hungary) and the

Mediterranean (Cyprus, Greece, Turkey). Many of these articles are developed from papers

presented at the International Conference on Teaching Statistics in Brazil (ICOTS-7). Other

voices from Australia and New Zealand lend a nice representation to mathematics education

developing in the Southern hemisphere. As usual there is a nice synthesis of articles focused on

mathematics content, and those that focus on research of teaching, learning and thinking issues in

mathematics education, as well as a Montana feature on Book X of Euclid s Elements.

In 2009, the journal will publish its normal 3 issues in addition to publishing special

supplementary issues on inter-disciplinarity, mathematics talent development and at least three

new monographs! This hopefully answers the rhetorical question, to publish or not to publish

References

Howlett, C. F. (2008). John Dewey and peace education. In M. Bajaj (Ed). Encyclopedia of

Peace Education (pp. 25-32). Information Age Publishing, Charlotte, NC.

TMME, vol6, nos.1&2, p.3

TEACHER KNOWLEDGE AND STATISTICS: WHAT TYPES OF

KNOWLEDGE ARE USED IN THE PRIMARY CLASSROOM?

Tim Burgess1

Massey University, New Zealand

Abstract: School curricula are increasingly advocating for statistics to be taught through

investigations. Although the importance of teacher knowledge is acknowledged, little is known

about what types of teacher knowledge are needed for teaching statistics at the primary school

level. In this paper, a framework is described that can account for teacher knowledge in relation

to statistical thinking. This framework was applied in a study that was conducted in the

classrooms of four second-year teachers, and was used to explore the teacher knowledge used in

teaching statistics through investigations. As a consequence, descriptions of teacher knowledge

are provided and give further understanding of what teacher knowledge is used in the classroom.

Keywords: cKc; elementary schools; mathematics teacher education; statistical investigations;

statistical thinking; teacher knowledge

INTRODUCTION

Statistics education literature in recent years has introduced the terms of statistical literacy,

reasoning, and thinking, and they are being used with increasing frequency. Wild and

Pfannkuch s (1999) description of what it means to think statistically has made a significant

contribution to the statistics education research field, and has provided a springboard for research

that further explores and contributes to an understanding of statistical thinking and its

application. Increasingly, it is recognised that statistics consists of more than a set of procedures

and skills to be learned. School curricula, including New Zealand s, advocate for investigations

to be a major theme for teaching and learning statistics.

Debate about teacher knowledge and its connections to student learning has had a long history.

An important question arises as to what knowledge is considered adequate and appropriate.

Although much is known about teacher knowledge pertinent to particular aspects of

mathematics, the situation for statistics is less clear. Arguably, the mathematical knowledge

needed for teaching and the statistical knowledge needed for teaching do share some similarities.

Yet, there are also differences (Groth, 2007), due in no small way to the more subjective and

uncertain nature of statistics compared with mathematics (Moore, 1990). Pfannkuch (2006,

personal communication) claims that, because of the relatively brief history of statistics

education research in comparison with mathematics education research, there is still much that is

unknown about the specifics of teacher knowledge needed for statistics.

1

abqnnf@r.postjobfree.com

The Montana Mathematics Enthusiast, ISSN 1551-3440, Vol. 6, nos.1&2, pp.3-24

2009 Montana Council of Teachers of Mathematics & Information Age Publishing

Burgess

This paper reports on a framework that was proposed and applied in a study that investigated

teacher knowledge needed and used by teachers during a unit in which primary school students

investigated various multivariate data sets. The focus here is on justifying the need for such a

framework in relation to teaching statistics, and on providing descriptions of teacher knowledge

as revealed in the classroom in relation to the framework for teacher knowledge that combines

statistical thinking components with categories of teacher knowledge. Examples from the

classroom are provided to support the knowledge descriptions in relation to some of the

components from the teacher knowledge framework. Finally, the conclusions consider some of

the implications of this research, particularly for teacher education, both preservice (or initial

teacher education) and inservice (or professional development).

LITERATURE REVIEW

Research on teacher knowledge is diverse. The thread of research from that of Shulman (1986)

who defined pedagogical content knowledge (as one category of the knowledge base needed for

teaching) provides a useful way of examining teacher knowledge. Shulman claims that a

teacher s pedagogical content knowledge goes beyond that of the subject specialist, such as the

mathematician. Subsequent research has attempted to clarify the differences between categories

of teacher knowledge, either using Shulman s categories, or others developed from Shulman s

categorisation.

Much of this research, although conducted with teachers, has not been conducted in the

classroom, the site in which teacher knowledge is used. Cobb and McClain (2001) advocate

approaches for working with teachers that do not separate the pedagogical knowing from the

activity of teaching. They argue that unless these two are considered simultaneously and as

interdependent, knowledge becomes treated as a commodity that stands apart from practice.

Their research focused on the moment-by-moment acts of knowing and judging. Similarly, Ball

(1991) discusses how teachers knowledge of mathematics and knowledge of students affect

pedagogical decisions in the classroom. For instance, the subject matter knowledge of the teacher

determines to a significant extent which questions from students should or should not be

followed up. Similarly, subject matter knowledge enables the teacher to interpret and appraise

students ideas. Ball and Bass (2000) argue strongly that without adequate mathematical

knowledge, teachers will not be in a position to deal with the day-to-day, recurrent tasks of

mathematics teaching, and as such, will not cater for the learning needs of diverse students.

A focus on the knowledge of content that is required to deliver high-quality instruction to

students has led to another model of teacher knowledge, which involves a refinement of the

categories of subject matter knowledge and pedagogical content knowledge. Hill, Schilling, and

Ball (2004) claim that teacher knowledge is organised in a content-specific way, rather than

being organised for the generic tasks of teaching, such as evaluating curriculum materials or

interpreting students work. Two sub-categories of content knowledge are further clarified by

Ball, Thames, and Phelps (2005): common knowledge of content includes the ability to recognise

wrong answers, spot inaccurate definitions in textbooks, use mathematical notation correctly, and

do the work assigned to students. In comparison, specialised knowledge of content needed by

teachers (and likely to be beyond that of other well-educated adults) includes the ability to

analyse students errors and evaluate their alternative ideas, give mathematical explanations, and

TMME, vol6, nos.1&2, p.5

use mathematical representations. Ball et al. (2005) also subdivide the category of pedagogical

content knowledge into two components, namely knowledge of content and students, and

knowledge of content and teaching. These two parts of teacher knowledge bring together aspects

of content knowledge that are specifically linked to the work of the teacher, but are different

from specialised content knowledge. Knowledge of content and students includes the ability to

anticipate student errors and common misconceptions, interpret students incomplete thinking,

and predict what students are likely to do with specific tasks and what they will find interesting

or challenging. Knowledge of content and teaching deals with the teacher s ability to sequence

the content for instruction, recognise the instructional advantages and disadvantages of different

representations, and weigh up the mathematical issues in responding to students novel

approaches.

Although statistics is considered to be part of school mathematics, there are some significant

differences that have implications for the teaching and learning of statistics. In mathematics,

students learn that mathematical reasoning provides a logical approach to solve problems, and

that answers can be determined to be valid if the assumptions and reasoning are correct (Pereira-

Mendoza, 2002), that the world can be viewed deterministically (Moore, 1990), and that

mathematics uses numbers where context can obscure the structure of the subject (Cobb &

Moore, 1997). In contrast, statistics involves reasoning under uncertainty; the conclusions that

one draws, even if the assumptions and processes are correct, are uncertain (Pereira-Mendoza,

2002); and statistics is reliant on context (delMas, 2004; Greer, 2000), where data are considered

to be numbers with a context that is essential for providing a meaning to the analysis of the data.

It becomes necessary when teaching statistics, to encourage students to not merely think of

statistics as doing things with numbers but to come to understand that the data are being used to

address a particular issue or question (Cobb, 1999; Gal & Garfield, 1997).

Statistical literacy, reasoning, and thinking have featured in the statistics education literature in

recent years. Ben-Zvi and Garfield (2004) provide some clarity for these terms, although with

regard to statistical thinking, Wild and Pfannkuch s (1999) paper provided a model for statistical

thinking. Wild and Pfannkuch describe five fundamental types of statistical thinking: (1) a

recognition of the need for data (rather than relying on anecdotal evidence); (2) transnumeration

being able to capture appropriate data that represents the real situation, and change

representations of the data in order to gain further meaning from the data; (3) consideration of

variation this influences the making of judgments from data, and involves looking for and

describing patterns in the variation and trying to understand these in relation to the context; (4)

reasoning with models from the simple (such as graphs or tables) to the complex, as they

enable the finding of patterns, and the summarising of data in multiple ways; and (5) the

integrating of the statistical and contextual making the link between the two is an essential

component of statistical thinking. Along with these fundamental types of thinking are more

general types that could be considered part of problem solving (but not exclusively to statistical

problem solving). Wild and Pfannkuch s dimension of types of thinking is one of four

dimensions that explain statistical thinking in empirical enquiry. The other three dimensions are:

the investigative cycle (problem, plan, data, analysis, and conclusions these are the procedures

that a statistician works through and what the statistician thinks about in order to learn more from

the context sphere (Pfannkuch & Wild, 2004, p. 41)); the interrogative cycle (generate, seek,

interpret, criticise, and judge) this is a generic thinking process that is in constant use by

Burgess

statisticians as they carry out a constant dialogue with the problem, the data, and themselves

(Pfannkuch & Wild, 2004, p. 41); and dispositions (including scepticism, imagination, curiosity

and awareness, openness, a propensity to seek deeper meaning, being logical, engagement, and

perseverance), which affect or propel the statistician into the other dimensions. All these

dimensions constitute a model that encompasses the dynamic nature of thinking during statistical

problem solving, and is non-hierarchical and non-linear.

This model for statistical thinking was developed through reference to the literature following

interviews with statisticians and tertiary statistics students as they performed statistical tasks

(Wild & Pfannkuch, 1999). Although it was developed as a model applicable to the statistical

problem solving of statisticians and tertiary students, it has subsequently been used in a variety

of other studies, such as an examination of the thinking of primary students (Pfannkuch &

Rubick, 2002) and pre-service primary teacher education students (Burgess, 2001), through a

professional development workshop with secondary teachers (Pfannkuch, Budgett, Parsonage, &

Horring, 2004), and an investigation into how statistical thinking of learners can be encouraged

through a teaching activity (Shaughnessy & Pfannkuch, 2002).

The Framework

Teacher knowledge frameworks from the mathematics education domain are inadequate for

examining teacher knowledge for statistics because of the differences between statistics and

mathematics, as discussed earlier. The development of a teacher knowledge framework that takes

into account the particular needs of statistics teaching and learning is therefore required. Such a

framework must be specific to statistics, since teacher knowledge is organised in content-specific

ways (Hill et al., 2004). Consequently the framework on which this study is based draws heavily

on the statistical thinking model of Wild and Pfannkuch (1999). The categories of teacher

knowledge that are described by Hill, Schilling, and Ball (2004) and Ball, Thames, and Phelps

(2005), namely mathematical content knowledge and pedagogical content knowledge, and each

of these with two sub-categories, provide a good starting point for examining statistics content

knowledge as enacted in classroom teaching.

A matrix for a conceptual framework, against which statistical knowledge for teaching can be

examined, is shown in Table 1.

TMME, vol6, nos.1&2, p.7

Table 1: The framework for teacher knowledge in relation to

statistical thinking and investigating.

Statistical knowledge for teaching

Content knowledge Pedagogical content

knowledge

Common Specialised Knowledge Knowledge

knowledge knowledge of content of content

of content of content and and

(ckc) (skc) students teaching

(kcs) (kct)

Need for data

Transnumeration

Variation

Thinking

Reasoning with

models

Integration of

statistical and

contextual

Investigative

cycle

Interrogative

cycle

Dispositions

The columns of the matrix refer to the types of knowledge that are important in teaching. These

four types are: common knowledge of content (ckc); specialised knowledge of content (skc);

knowledge of content and students (kcs); and knowledge of content and teaching (kct). Hill,

Schilling and Ball (2004) and Ball, Thames, and Phelps (2005) describe the features of these four

categories of teacher knowledge in relation to number and algebra. These descriptions arise from

a consideration of the question, What are the tasks that teachers engage in during their work in

the classroom, and how does the teachers mathematical knowledge impact on these tasks?

From those researchers close examination of teachers work, it is apparent that much of what

teachers do throughout their teaching is essentially mathematical.

Just as Ball et al. (2001) claim that many of the everyday tasks of the teacher of mathematics are

essentially mathematical, it is suggested that much of what a teacher engages in during the

teaching of statistical investigations essentially involves statistical thinking and reasoning.

Consequently, the four teacher knowledge categories are examined in relation to statistical

thinking. The main feature that sets this framework apart from those offered for the mathematics

Burgess

domain is the inclusion of the elements of statistical thinking and empirical enquiry (Wild &

Pfannkuch, 1999), which are listed as the rows of the matrix.

THE STUDY

Since teacher knowledge is acknowledged to be important in relation to what and how students

learn and is dependent on the context in which it is used (Ball & Bass, 2000; Barnett & Hodson,

2001; Borko, Peressini, Romagnano, Knuth, Willis-Yorker, Wooley et al., 2000; Cobb, 2000;

Cobb & McClain, 2001; Fennema & Franke, 1992; Foss & Kleinsasser, 1996; Friel & Bright,

1998; Marks, 1990; Sorto, 2004; Vacc & Bright, 1999), it is argued that research should

therefore take place in the classroom. Also, research on teacher knowledge must acknowledge

and accommodate the dynamic aspects of teacher knowledge (Manouchehri, 1997), and be based

on an understanding of how knowledge evolves. A post-positivist realist paradigm (Popper,

1979, 1985) was chosen because of the explanations about where knowledge comes from and

how it grows in a dynamic fashion. Popper argued that knowledge develops through trial and

elimination of error, and the logic of learning model (Burgess, 1977) was proposed as being

appropriate for examining learning in classroom settings (Swann, 1999).

Using this post-positivist realist paradigm, case study research was undertaken with four

inexperienced primary teachers (all in their second year of teaching), Linda, John, Rob, and

Louise (all pseudonyms). The four classes were in the Year 5 (about 9-10 years old) to Year 8

(about 12-13 years old) level of primary school. The teachers were given a teaching unit that

required students to investigate some multivariate data sets. The teachers developed their

teaching based on this unit. The data sets generally consisted of 24 cases, each with four

variables (or attributes). The first set used by each teacher included four category variables,

while the other sets included at least two numeric variables along with the category variable(s).

Each case was presented on a data card (see examples below from three different data sets), so

that the students could easily manipulate and sort the cards in order to discover interesting things

in the data.

Each lesson was videotaped, then edited by the researcher in order to focus on interesting

episodes from the lesson. The edited videotape was shown to the teacher, and the discussion

between the teacher and the researcher was audiotaped. The videotapes and the audiotapes from

the post-lesson discussions were analysed in relation to the cells of the framework. Segments

from the lessons or the discussions were identified in relation to the categories of teacher

knowledge and the components of statistical thinking that were in evidence.

TMME, vol6, nos.1&2, p.9

This paper reports on the results pertinent to the following research question:

What are the features of teacher knowledge in relation to aspects of statistical thinking that are

used in the classroom?

DESCRIPTIONS OF THE FRAMEWORK

An understanding of the need for data on which to base sound statistical reasoning, instead of

relying on and being satisfied with anecdotal evidence, is important in the development of

statistical thinking. This corresponds to the first row of the framework. Classroom investigations

can be conducted through two different approaches. First, an investigation can start with a

question or problem to be solved and move onto data collection, which requires an understanding

that data needs to be collected in order to solve the question or problem. The second approach is

to start with a data set and generate questions for investigation from that data. By adopting this

second approach for this study, teachers and students were not faced with the issues pertinent to

establishing the need for data to help solve their questions. Consequently the need for data did

not feature in this research. As such, the need for data is not described in relation to the four

categories of teacher statistical knowledge for the framework.

Dispositions (corresponding to the final row of the framework), as another component of

statistical thinking, did not emerge specifically in relation to the individual components of

teacher knowledge but in a more general way. Teachers statistical dispositions were apparent in

the classroom. For example, inquisitiveness and readiness to think in relation to data along with

an anticipation of what was to come was evident when Linda asked the students what they had

started to notice when filling in their own data cards. She justified this question in the subsequent

interview by saying that it was to give them a hint of what was to come to see if the students

had the inclination to start making their own conclusions already.

Common knowledge of content

As described by Ball, Thames, and Phelps (2005), common knowledge of content refers to what

the educated person knows and can do; it is not specific to the teacher. They describe it as

including the ability to recognise wrong answers, spot inaccurate definitions in textbooks, use

mathematical notation correctly, and do the work assigned to students.

Wild and Pfannkuch (1999) describe transnumeration as the ability to: sort data appropriately;

create tables or graphs of the data; and find measures to represent the data set (such as a mean,

median, mode, and range). In general, transnumeration involves changing the representation of

data in order to make more sense of it.

For teaching, common knowledge of content: transnumeration includes the knowledge and skills

described above, along with the ability to recognise whether, for instance, a student gave the

correct process or rule for finding a measure, had created a table correctly, or had sorted the data

cards appropriately. Evidence of this category (as well as others involving common knowledge

of content) was not often observed because the teachers generally used other types of teacher

knowledge in relation to transnumeration. However if, for example, a teacher asked questions

that led the students towards sorting the data in a particular way, it was assumed that the teacher

Burgess

also had the common knowledge of content of how to do this for him or herself. There were

instances where the researcher verified that this was indeed the case by asking the teacher during

the interview to sort the cards, calculate a measure, or something similar. Consequently, common

knowledge of content: transnumeration was subsumed within other categories of knowledge.

Consideration of variation in data is an important aspect of statistical thinking (Wild &

Pfannkuch, 1999). It affects the making of judgments based on data, as without an understanding

that data varies in spite of patterns and trends that may exist, people are likely to express

generalisations based on a particular data set as certainties rather than possibilities.

The knowledge category of common knowledge of content: variation manifests itself in the

classroom when the teacher gives examples of statements about data that acknowledge variation

through the language used. Some of the more common situations that were observed related to

inferential statements. Such statements were either about the actual data set and based on it, or

generalisations about a larger group (population) from the smaller data set (sample). Such

language included words and phrases such as maybe, it is quite likely that, and there

is a high probability that . In addition, when the teacher talked about another sample being

similar, but not identical, to the first sample, common knowledge of content: variation was

evidenced.

For people to be able to make sense of data, statistical thinking requires the use of models. At the

school level, appropriate models with which students could reason include graphs, tables,

summary measures (such as median, mean, and range), and as used in this research, sorted data

cards. If teachers demonstrated evidence of common knowledge of content: reasoning with

models, it would be through making valid statements for the data, based on an appropriate use of

a model.

Wild and Pfannkuch (1999) describe the importance of continually linking contextual knowledge

of a situation under investigation with statistical knowledge related to the data of that situation.

The interplay between these two enables a greater level of data sense and a deeper understanding

of the data, and is therefore indicative of a higher level of statistical thinking.

The component of common knowledge of content: integration of statistical and contextual is

characterised by the ability to make sense of graphs or measures, and by an acknowledgement of

the relevance and interpretation of these statistical tools to the real world from which the data

was derived. For example, John gave some possible reasons to support the finding that all the

youngest students could whistle. He suggested that the older siblings could have taught the

younger ones to whistle. This shows thinking of the real-life context in association with what the

statistical investigation had revealed; such integration of the two aspects can sometimes enable

the answering of why might this be so that is being illustrated by the data.

One of the four dimensions of statistical thinking, as defined by Wild and Pfannkuch (1999), is

the investigative cycle. This cycle, characterised by the phases of problem, plan, data, analysis,

and conclusions, is what someone works through and thinks about when immersed in problem

solving using data. If a teacher can fully undertake and engage with an investigation, then that

teacher would be demonstrating common knowledge of content: investigative cycle. The teacher

would be able to: pose an appropriate question or hypothesis, or set a problem to solve; plan for

TMME, vol6, nos.1&2, p.11

and gather data; analyse that data; and use the analysis to answer the question, prove the

hypothesis, or solve the problem.

For example, Linda discussed how data might be handled with an open-response type of question

in a survey or census. Linda had considered, at the problem-posing phase of the investigation,

how the responses from such an open-response type question would present a challenge at the

analysis stage. This clearly indicated that Linda had some knowledge of the phases of the

investigative cycle. She was able to maintain an awareness of a later stage of the cycle (analysis)

while dealing with an early stage (planning data collection), and consider how decisions at that

early stage could impact on the later stages.

A teacher would have common knowledge of content: interrogative cycle if it was evident that

possibilities in relation to the data were considered and weighed up, with some possibilities

being subsequently discarded but others accepted as useful. Engaging with data and being

involved in debating with it would be evidence of such knowledge. Likewise, developing

questions that the data may potentially be able to answer is an aspect of common knowledge of

content: interrogative cycle. Teachers who had immersed themselves with a data set prior to

using it in teaching, so that they were aware of some of the things that might be found from the

data, would be showing common knowledge of content: interrogative cycle. Such teachers would

be prepared for knowing what their students might find in the data and what conclusions might

be drawn from that data.

Specialised knowledge of content

A teacher requires specialised knowledge of content: transnumeration to analyse whether a

student s sorting, measure, or representation was valid and correct for the data, particularly if the

student has done something in a non-standard and unexpected way. It includes the ability to

justify a choice of which measure is more appropriate for a given data set, or to explain when

and why a particular measure, table, or graph would be more appropriate than another. Some of

these skills, although considered part of statistical literacy (Ben-Zvi & Garfield, 2004), are still

currently beyond what many educated adults can undertake. As such they are considered to be

part of specialised knowledge of content: transnumeration rather than common knowledge of

content:transnumeration.

Specialised knowledge of content: transnumeration was identified for all the teachers in the

study. For example, Linda attempted to follow a student s description of how she had sorted the

data and converted it into an unconventional table involving all four variables. The table

consisted of: four columns labelled G, B, G, B; four rows with labels on the left to account for

two more variables; labels on the right for three rows to account for the fourth variable; but no

numbers or tally marks in the cells of the table to represent the sorted data. To determine the

statistical appropriateness of that particular representation, Linda had to call on her specialised

knowledge of content: transnumeration as she tried to make sense of the table. In another

example in relation to some students deciding which measure or measures they should calculate

for the data set (out of the mode, median and mean), Rob recognised that the mode would not be

the most appropriate measure to use for the numerical data in question, and was able to give

some justification regarding the inappropriateness of the mode.

Burgess

Making sense of and evaluating students explanations around whether it is possible to generalise

from the data at hand to a larger group involves specialised knowledge of content: variation. For

instance, when Linda asked whether there would be many boys who watched a particular

programme on TV based on the class data that showed only a small proportion of such boys, a

student answered, Don t know; she hasn t asked all the classes yet. The teacher had to

evaluate whether that was a reasonable response in relation to understanding of variation; Linda

explained that there are factors that might affect the validity of this generalisation, but that the

student s justification (about not having the data from the population so therefore it was not

possible to make such a generalisation) was not a good reason for not generalising from the class

data.

Specialised knowledge of content: reasoning with models is needed to interpret students

statements to determine the validity or otherwise of those statements. Students often struggled

with making sensible and valid statements about the data based on a particular model they were

using, and as a consequence it was not always straightforward for the teachers to make sense of

the students statements. Consequently, this category is seen as being quite distinct from common

knowledge of content: reasoning with models.

Specialised knowledge of content: reasoning with models was a very commonly occurring

component of teacher knowledge, especially as the focus of the unit was on finding interesting

things in multivariate data sets, and making statements about these data sets. In many cases,

students justified their statements through reference back to the model and as such, the teachers

needed specialised knowledge of content: reasoning with models to help check the veracity of the

students statements. For example, the following interaction, initially between Linda and one

student but later extended to the whole class, exemplifies the challenge for teachers to listen to

and make sense of students statements:

Student: That most girls can write with their right hand, most girls write with their right hand ...

[inaudible].

Teacher: Sorry, I didn t catch what you said. Can you say that again for me? Slower this time.

Student: Most girls can write with their right hand are the youngest in

Teacher: Hang on. Most what are you saying? Most girls who produce their neatest handwriting

with their right hand can whistle. [pause]. Okay [pause]. How many girls who produce

their neatest handwriting with their right hand can whistle? [pause] Is that what you have

got in front of you? [pointing at the cards on the desk] ... How many is that? [Student can be

seen nodding as he counts cards] Is that these ones?

Teacher: So there are 5? These ones can whistle as well? But are they right handed? Okay. So

what are you comparing that with? You said most. So most compared with what? [No

response from student.] In comparison with the right handed boys or in comparison with the

left handed girls?

Student: Left handed girls.

Teacher: Okay [pause] So R and J have taken that a step further and they have got [teacher

moves to the whiteboard and starts drawing a type of two-way table see Figure 1] here

right-handed girls and right-handed boys and they have taken just this square [lower right]

and sorted those people [the right handed girls] into different piles, into whistlers and non-

whistlers. And they have found that there are more whistlers who are girls who are right

handed than non-whistlers who are girls who are right handed. I think that is what they are

trying to say.

TMME, vol6, nos.1&2, p.13

Figure 1: Diagram drawn by Linda to help students make sense of the

statement from a student.

The interaction indicates the use of specialised knowledge of content: reasoning with models by

the teacher, involving initially the model of sorted data cards on the student s desk, followed by

the model on the board that she created from transnumeration of the data cards.

Being able to evaluate a student s explanation based on both statistical data and a knowledge of

the context under investigation is one aspect of the category of specialised knowledge of content:

integration of statistical and contextual knowledge. There were a number of situations in which

the teacher prepared the students to gather data. Data collection questions had been suggested,

such as, What position are you in the family, youngest, middle or eldest? When the students

were considering the question prior to the actual data gathering, Linda was asked:

Does it count if you have half brothers or sisters?

What if your sister or brother has died?

What if your brother or sister is not living at home?

What would you put if you were an only child?

Each of these questions, and others involving the definition of family, were unexpected by

Linda. She had to decide on the spot how to respond to each question from students. She was

required to weigh up the statistical issues related to answering such a data gathering question

with the contextual issue of interpretation of family . Her answers indicated that she was able to

do so satisfactorily and therefore were evidence of her having specialised knowledge of content:

integration of statistical and contextual.

A teacher needs specialised knowledge of content: investigative cycle when dealing with

students questions or answers in relation to phases of the investigative cycle, or when discussing

or explaining various phases of the cycle and how they might interact. When thinking about

suggestions for what could be investigated in a data set, the teacher needs to be able to evaluate

the suitability of the problem/question, and whether it needs to be refined to be usable and

suitable, in relation to the subsequent analysis.

So what does specialised knowledge of content: interrogative cycle look like, as distinguished

from common knowledge of content: interrogative cycle? When a teacher has to consider

Burgess

whether a suggestion from a student is viable for investigating within that data, the teacher

requires specialised knowledge of content: interrogative cycle. Also, it involves determining

whether a student s suggested way of handling and sorting the data would be useful to enable the

later interpretation of results in relation to the question at hand.

Knowledge of content and students

The knowledge of content and students: transnumeration component includes: knowledge of the

common errors and misconceptions that students develop in relation to the skills of

transnumeration (including sorting data, changing data representations such as into tables or

graphs, and finding measures to summarise the data); the ability to interpret students incomplete

or jumbled descriptions of how they sorted, represented, and used measures to summarise the

data; an understanding of how well students would handle the tasks of transnumeration; and an

awareness of what students views may be regarding the challenge, difficulty, or interest in the

tasks of transnumeration.

There were situations in which students, when handling the data cards and sorting them, tried to

consider too many variables at once and could not manage the complexity in the sorting of the

cards and in making sense of what the cards showed. Linda was aware of this difficulty and

guided the students to sort the cards more slowly . She suggested sorting by one variable, and

then splitting the groups by a second variable; she knew how many groups of data there would

be from sorting by three variables and therefore that it needed to be simplified for the students. In

general, the teachers did not realise how much the students would struggle with sorting the data

cards, especially when the students were looking at numeric data such as arm spans, heights, and

so forth. The teachers were surprised that the students did not naturally order the numeric data

but simply grouped the data cards into piles. Furthermore, sorting data cards to check for and

show relationships between two data sets was difficult for students, and most of the teachers

underestimated the level of challenge that students would therefore face with sorting to show

relationships in the data.

Knowledge of content and students: variation includes knowing what students may struggle with

in relation to understanding variation, and to predict how students will handle tasks linked to

variation. Whether students can appreciate and think about variation in data while looking for

patterns and trends in the data is something that a teacher needs to listen for in students

explanations and generalisations. Although all the teachers posed questions as to whether it was

possible to generalise from the class data to a wider group, there was no significant evidence of

knowledge of content and students: variation being used by the teachers. It may be that for the

investigations being conducted, such teacher knowledge of variation was not called on because

the students were not ready for this inferential-type thinking. Since it was something new for the

teachers to teach, they had not considered the statistical implications relevant to the students

readiness for thinking in relation to variation.

If a teacher can anticipate the difficulties that students might have with reasoning using models,

or can make some sense of students incomplete descriptions, then the teacher would be showing

evidence of knowledge of content and students: reasoning with models. In one example of such

knowledge, Rob described how he worked with a group of students who had made a statement

TMME, vol6, nos.1&2, p.15

from the data cards comparing the number of boys with the number of girls who were right or

left handed. Rob knew that the students were capable of proportional thinking so he encouraged

them to consider proportions. He did so because the numbers of boys and girls in the data cards

were different, and therefore using proportions for the comparison would be more appropriate

than using frequencies. Rob knew these students sufficiently to encourage them to reason with a

proportional model, which two of the students handled particularly well.

Can a teacher anticipate that students may have difficulty with linking contextual knowledge

with statistical knowledge? Are students, through focusing on statistical knowledge and skills,

likely to ignore knowledge of the real world, that is, contextual knowledge, or vice versa? Such

aspects would give an indication of a teacher s knowledge of content and students: integration of

statistical and contextual.

Whereas Linda s students questions which related to the data question of position in the family

(as discussed above) were unexpected, John anticipated such possible difficulties for his students

and pre-empted their questions by asking the class how each child from a four-child family

might answer the question, Are you youngest, middle, or eldest in the family? John s question

encouraged the students to think about the data question (the statistical) in association with their

knowledge of particular families (the contextual). This helped the students understand that

statistics is not performed in a vacuum, removed from real issues, but deals with numbers that

have a context (delMas, 2004).

Knowledge of where students might encounter problems or particular challenges in an

investigation, and whether students will find an investigation interesting or difficult, are aspects

of knowledge of content and students: investigative cycle.

One teacher predicted that students could have a problem with knowing how to interpret a data

collection question so had to consider how he would deal with t

Contact this candidate