i

AAA

ppprrrooobbbllleeemmm---pppooosssiiinnnggg

iiinnnttteeerrrvvveeennntttiiiooonnn iiinnn ttthhheee dddeeevvveeelllooopppmmmeeennnttt

ooofff ppprrrooobbbllleeemmm---sssooolllvvviiinnnggg

cccooommmpppeeettteeennnccceee ooofff

uuunnndddeeerrraaaccchhhiiieeevvviiinnnggg,,,

mmmiiiddddddllleee---yyyeeeaaarrr

ssstttuuudddeeennntttsss...

Deborah Jean Priest

M. Ed. (QUT), BSc Ed. (Melb.)

Faculty of Education

Queensland University of Technology

Kelvin Grove Campus, Brisbane.

A Thesis submitted in fulfilment of the requirements leading to the award of

the degree of Doctor of Philosophy

May 2009

ii

CERTIFICATE RECOMMENDING ACCEPTANCE

iii

DEFINITION OF ACRONYMS

ACER Australian Council for Educational Research

ANTA Australian National Training Authority

DEST Department of Education, Science and Training

GSA Graduate Skills Assessment Test

IQ Intelligence Quotient

MCEETYA Ministerial Council on Education, Employment, Training and

Youth Affairs

MYAT Middle Years Ability Test

NAPLAN National Assessment Program: Literacy and Numeracy

NCB National Curriculum Board

NCTM National Council of Teachers of Mathematics

(United States of America)

NNR National Numeracy Review

NMAP National Mathematics Advisory Panel

(United States of America)

NRC National Research Council (United States of America)

POPS Profiles of Problem Solving Test

iv

KEYW ORDS

assessment

cognition

developmental learning

education

engagement

intervention

mathematics

middle year

multiple intelligences

pedagogy

problem solving

problem posing

self-regulation

teaching experiment

underachievement

iv

ABSTRACT

This study reported on the issues surrounding the acquisition of problem-

solving competence of middle-year students who had been ascertained as

above average in intelligence, but underachieving in problem-solving

competence. In particular, it looked at the possible links between problem-

posing skills development and improvements in problem-solving

competence.

A cohort of Year 7 students at a private, non-denominational, co-educational

school was chosen as participants for the study, as they undertook a series

of problem-posing sessions each week throughout a school term. The

lessons were facilitated by the researcher in the students’ school setting.

Two criteria were chosen to identify participants for this study. Firstly, each

participant scored above the 60th percentile in the standardized Middle Years

Ability Test (MYAT) (Australian Council for Educational Research, 2005) and

secondly, the participants all scored below the cohort average for Criterion B

(Problem-solving Criterion) in their school mathematics tests during the first

semester of Year 7.

Two mutually exclusive groups of participants were investigated with one

constituting the Comparison Group and the other constituting the Intervention

Group. The Comparison Group was chosen from a Year 7 cohort for whom

no problem-posing intervention had occurred, while the Intervention Group

was chosen from the Year 7 cohort of the following year. This second group

received the problem-posing intervention in the form of a teaching

experiment. That is, the Comparison Group were only pre-tested and post-

tested, while the Intervention Group was involved in the teaching experiment

and received the pre-testing and post-testing at the same time of the year,

but in the following year, when the Comparison Group have moved on to the

secondary part of the school. The groups were chosen from consecutive

Year 7 cohorts to avoid cross-contamination of the data.

v

A constructionist framework was adopted for this study that allowed the

researcher to gain an “authentic understanding” of the changes that occurred

in the development of problem-solving competence of the participants in the

context of a classroom setting (Richardson, 1999). Qualitative and

quantitative data were collected through a combination of methods including

researcher observation and journal writing, video taping, student workbooks,

informal student interviews, student surveys, and pre-testing and post-

testing. This combination of methods was required to increase the validity of

the study’s findings through triangulation of the data.

The study findings showed that participation in problem-posing activities can

facilitate the re-engagement of disengaged, middle-year mathematics

students. In addition, participation in these activities can result in improved

problem-solving competence and associated developmental learning

changes. Some of the changes that were evident as a result of this study

included improvements in self-regulation, increased integration of prior

knowledge with new knowledge and increased and contextualised

socialisation.

vi

TABLE OF CONTENTS

Certificate ____________________________________________________ ii

Definition of Acronyms __________________________________________ iii

Keywords ____________________________________________________ iv

Abstract ______________________________________________________ v

List of Tables__________________________________________________ x

List of Figures ________________________________________________ xii

List of Appendices ____________________________________________ xiii

Statement of Authenticity _______________________________________ xiv

Acknowledgments _____________________________________________ xv

Chapter 1 - Introduction to the Research Study

1.1 INTRODUCTION ___________________________________________ 16

1.2 DEFINITION OF TERMS _____________________________________ 16

1.3 RATIONALE FOR THE STUDY _________________________________ 18

1.3.1 Summary ____________________________________________ 25

1.4 BACKGROUND TO THE STUDY ________________________________ 26

1.4.1 The Value of Problem Solving in Today’s Society _____________ 27

1.4.2 The Place of Problem Posing in a Responsive Curriculum ______ 28

1.4.3 Disparity in Student Mathematical Performance ______________ 31

1.5 PURPOSE OF THIS STUDY ___________________________________ 32

1.6 SIGNIFICANCE OF THE RESEARCH _____________________________ 32

1.7 THESIS OVERVIEW ________________________________________ 34

Chapter 2 - Theoretical Perspectives

2.1 CHAPTER OVERVIEW ______________________________________ 36

2.2 UNDERSTANDING DEVELOPMENTAL LEARNING ____________________ 36

2.2.1 Information Processing Theory ___________________________ 39

2.2.2 Psychometric Theory ___________________________________ 42

2.2.3 Multiple Intelligences Theory _____________________________ 50

2.2.4 Summary ____________________________________________ 51

2.3 PROBLEM-SOLVING PERSPECTIVES ____________________________ 53

2.3.1 Introduction ___________________________________________ 53

2.3.2 The Power of Teaching through Problem Solving _____________ 56

2.3.3 Can Problem Solving Drive Mathematical Reform? ___________ 57

vii

2.3.4 Issues Related to the Assessment of PSC __________________ 58

2.3.5 Should Specific Problem-solving Strategies be Taught? ________ 59

2.3.6 Student's Understandings of Problem Structures _____________ 60

2.3.7 Summary ____________________________________________ 63

2.4 PROBLEM-POSING PERSPECTIVES ____________________________ 65

2.4.1 Introduction ___________________________________________ 66

2.4.2 Problem Posing as a Tool for Mathematical Reform ___________ 66

2.4.3 Problem-posing Skills for Lifelong Learning _________________ 68

2.4.4 Fostering a Problem-posing Environment ___________________ 70

2.4.5 Connections between Problem Solving and Problem Posing ____ 71

2.4.6 Summary ____________________________________________ 75

2.5 STUDENT UNDERACHIEVEMENT PERSPECTIVES ___________________ 76

2.6 CONSTRUCTIONIST PERSPECTIVES ____________________________ 80

2.7 CONCLUSION ____________________________________________ 83

Chapter 3 - Research Design

3.1 CHAPTER OVERVIEW ______________________________________ 87

3.2 INTRODUCTION ___________________________________________ 87

3.3 RESEARCH QUESTIONS ____________________________________ 92

3.4 RESEARCH DESIGN _______________________________________ 93

3.4.1 Research Design Rationale and Structure __________________ 93

3.4.2 Participants ___________________________________________ 96

3.5 METHODS _____________________________________________ 102

3.5.1 Data Collection _______________________________________ 105

3.5.2 Instruments __________________________________________ 111

3.5.2.1 The Middle Years Ability Test (MYAT) __________________ 111

3.5.2.2 The Profiles of Problem Solving (POPS) Test ____________ 113

3.5.2.3 The Student Survey ________________________________ 115

3.5.2.4 The Problem Criteria Sheet __________________________ 116

3.5.3 Data Analysis ________________________________________ 117

3.5.3.1 Researcher Journal ________________________________ 118

3.5.3.2 Student Surveys ___________________________________ 118

3.5.3.3 Student Workbooks ________________________________ 120

3.5.3.4 Researcher Observations ___________________________ 120

3.5.3.5 Informal Interviews _________________________________ 121

3.5.3.6 The Profiles of Problem Solving (POPS) Test ____________ 122

3.5.4 Reliability and Validity Issues ____________________________ 123

viii

3.5.5 Ethical Issues ________________________________________ 125

3.6 CONCLUSION ___________________________________________ 127

Chapter 4 - The Teaching Experiment

4.1 CHAPTER OVERVIEW _____________________________________ 129

4.2 THE PHILOSOPHICAL UNDERPINNINGS AND STRUCTURE OF THE

TEACHING EXPERIMENT____________________________________ 130

4.2.1 The Philosophical Underpinnings of the Teaching Experiment 130

4.2.2 The Structure of the Teaching Experiment _________________ 133

4.3 THE PRE-TEST AND POST-TEST LESSONS ______________________ 134

4.3.1 Introduction __________________________________________ 134

4.3.2 First Lesson - Pre-test and Initial Survey ___________________ 136

4.3.3 Last Lesson - Post-test and Final Survey __________________ 137

4.4 THE SEVEN TEACHING EPISODES (LESSONS 2-8) ________________ 137

4.4.1 The First Teaching Episode - Lesson 2 ____________________ 138

4.4.2 The Second Teaching Episode - Lesson 3 _________________ 139

4.4.3 The Third Teaching Episode - Lesson 4 ___________________ 143

4.4.4 The Fourth Teaching Episode - Lesson 5 __________________ 143

4.4.5 The Fifth Teaching Episode - Lesson 6 ____________________ 144

4.4.6 The Sixth Teaching Episode - Lesson 7 ___________________ 146

4.4.7 The Seventh Teaching Episode - Lesson 8 _________________ 148

4.5 CONCLUSION ___________________________________________ 148

Chapter 5 - Reporting and Analysis of the Data

5.1 CHAPTER OVERVIEW _____________________________________ 150

5.2 OBSERVATIONS AND INTERVIEWS WITH THREE CASE STUDY STUDENTS 150

5.2.1 Paul _______________________________________________ 152

5.2.2 Andrew _____________________________________________ 161

5.2.3 Nicole ______________________________________________ 170

5.3 STUDENT SURVEYS ______________________________________ 176

5.3.1 Question One ________________________________________ 176

5.3.2 Question Two ________________________________________ 179

5.3.3 Question Three_______________________________________ 182

5.3.4 Question Four ________________________________________ 184

5.4 PROFILES OF PROBLEM SOLVING TEST - THE PRE-TEST AND THE

POST-TEST _____________________________________________ 186

5.4.1 Descriptive Analysis of the POPS Test Results ______________ 190

5.4.2 Paired Samples T-Test Results __________________________ 195

ix

5.4.3 Analysis of Improvement of Scores from the Pre-test and the

Post- test ___________________________________________ 198

5.5 CONCLUSION ___________________________________________ 201

Chapter 6 - Responses to the Research Questions

6.1 CHAPTER OVERVIEW _____________________________________ 204

6.2 RESEARCH QUESTION 1 ___________________________________ 204

6.3 RESEARCH QUESTION 2 ___________________________________ 207

6.4 RESEARCH QUESTION 3 ___________________________________ 209

6.5 CONCLUSION ___________________________________________ 213

Chapter 7 - Limitations and Implications for Future Research

7.1 CHAPTER OVERVIEW _____________________________________ 215

7.2 LIMITATIONS OF THE STUDY ________________________________ 215

7.2.1 Limitations in the Selection of Students ____________________ 216

7.2.2 Limitations in the Timing of the Research __________________ 217

7.2.3 Limitations of the Size of the Control and Intervention Groups __ 218

7.2.4 Limitations of the Withdrawal of Students from their Usual

Classroom Environment ________________________________ 219

7.2.5 Limitations in the Length of the Problem-posing Intervention ___ 220

7.2.6 Limitations of Question Three of the Student Survey _________ 222

7.3 IMPLICATIONS OF THE RESEACH _____________________________ 222

7.4 CONCLUDING COMMENTS __________________________________ 223

REFERENCES _____________________________________________ 225

APPENDICES ______________________________________________ 252

x

L IST OF TABLES

Table 1.1 Percentage Comparison of How Time is Allocated in

Year Eight mathematics Classrooms in Germany, the

United States and Japan

23

Table 1.2 Comparative Problem-solving Scale Scores from the

2003 PISA Test

24

Table 1.3 Overall Combined mathematical Literacy Scores from

the 2003 PISA Test

25

Table 1.4 The Eight Skill Groupings of the Employability Skills

Framework

29

Table 2.1 Comparison of Stage Development in Cognitive

Development Theories

39

Table 2.2 Spearman’s Correlations of Student Scores Between

Subjects

45

Table 3.1 Data Used to Respond to the Three Research

Questions of the Study

107

Table 4.1 Variations to Pre-arranged Lesson Times in 2007 135

Table 5.1 Paul’s Profiles of Problem Solving Pre-test and Post-

test Results

153

Table 5.2 Andrew’s Profiles of Problem Solving Pre-test and

Post-test Results

163

Table 5.3 Nicole’s Profiles of Problem Solving Pre-test and Post-

test Results

171

Table 5.4 Do you enjoy solving problems? 177

Table 5.5 What type of problems do you prefer to solve? 180

Table 5.6 Do you think learning to solve problems is a useful

thing to do?

183

Table 5.7 What things could teachers do to assist you to become

better at solving problems?

185

Table 5.8 Comparison Group Pre-test and Post-test results 188

Table 5.9 Intervention Group Pre-test and Post-test results 189

xi

Table 5.10 Mean Score and Standard Deviation Statistics for each

Aspect of the Profiles of Problem Solving Test for

Students in the Comparison and Intervention Groups

191

Table 5.11 Paired Samples Test for each Subscale of the Profiles

of Problem Solving Test for Students in the

Comparison and Intervention Groups

197

Table 5.12 Numbers of Improvements in Individual Aspect Scores

of Comparison and Intervention Group Students, from

the Pre-test to the Post-test

198

xii

List of Figures

Figure 2.1 A Schematic Diagram of Sternberg’s Triarchic Theory

of Intelligence

42

Figure 3.1 Research Study Framework 95

xiii

L IST OF APPENDICES

Appendix A Project Information Sheet and Parent Consent Form

for Comparison Group

253

Appendix B Project Information Sheet and Parent Consent Form

for Intervention Group

258

Appendix C Student Survey Sheet 263

Appendix D Teaching Experiment Lesson One 266

Appendix E Teaching Experiment Lesson Two 270

Appendix F Teaching Experiment Lesson Three 275

Appendix G Teaching Experiment Lesson Four 280

Appendix H Teaching Experiment Lesson Five 288

Appendix I Teaching Experiment Lesson Six 295

Appendix J Teaching Experiment Lesson Seven 302

Appendix K Teaching Experiment Lesson Eight 309

Appendix L Teaching Experiment Lesson Nine 314

Appendix M Profiles of Problem Solving Assessment Instrument

(Stacey, Groves, Bourke, & Doig, 1993)

317

Appendix N Problem Criteria Sheet 327

Appendix O Participant Pseudonym Code to Psuedonym Name

Conversion for Comparison Group

329

Appendix P Participant Pseudonym Code to Psuedonym Name

Conversion for Intervention Group

331

Appendix Q Marking Scheme for Profiles of Problem Solving Test 333

xiv

STATEMENT OF AUTHENTICITY

The work contained in this document has not previously been submitted for a

degree or diploma at any other higher education institution. To the best of my

knowledge and belief, the document contains no material previously

published or written by another person except where due reference is made

in the document itself.

Deborah Jean Priest

May 2009

15

ACKNOW LEDGEMENTS

I wish to acknowledge the valuable and ongoing support I have received from

Professor Lyn English in the first instance, and also Dr Mal Shield and Associate

Professor Rod Nason. In particular, I wish to thank Professor Lyn English and

Dr Mal Shield for their substantial guidance and encouragement that has been

instrumental in the completion of my PhD journey. In addition, I would like to

thank Dr Mark Bahr for his assistance in becoming familiar with the Statistical

Package for Social Sciences software (SPSS Inc., 2007).

I would also like to thank the Year 7 teachers, the Deputy Principal in charge of

the Year 7 students at the research school, and the Principal for allowing me to

work with their students. I would like to acknowledge you all by name but am

unable to do so as the participants in this study may be more readily identified as

a result. Please accept my deepest appreciation for your cooperation and

assistance.

As those who have previously completed their PhD journeys will fully understand,

there are many activities that must be set aside in order to find the necessary

time to undertake and complete detailed research such as that reported in this

document. My journey to completion would not have been possible without the

understanding of my husband, John and my two daughters Megan and Ashley.

Their patience has been greatly appreciated, cannot be understated and will be

rewarded in the future.

16

Chapter 1

Introduction to the

Research Study

1.1 Chapter Overview

Seven main sections comprise this chapter. The first section is a definition of

terms used frequently throughout this report (see Section 1.2), while the second

section introduces the rationale that led to the overarching question for this study

(see Section 1.3). The third section provides some preliminary background to

the research study including discussion about the value of problem solving and

problem posing in a contemporary mathematics curriculum and introduces the

concept of disparity between a student’s actual mathematical performance and

their potential performance (see Section 1.4). Section four of this chapter

introduces the three research questions investigated in this study (see Section

1.5) while the fifth section considers the significance of this research (see Section

1.6). The final section presents an overview of the chapters in this report (see

Section 1.7).

1.2 Definition of Terms

The following terms, with their associated meaning, are used frequently

throughout this report:

Cognition “refers to the processes or faculties by which knowledge is acquired

and manipulated” (Bjorklund, 2000, p. 3) and “includes conscious, effortful

processes such as those involved in making important decisions and

17

unconscious, automatic processes, such as those involved in recognizing a

familiar face, word or object” (pp.19, 20).

Developmental learning changes refer to cognitive (e.g., Goswami, 2002) and

behavioural (e.g., Lesh & Doerr, 2003) changes that can be attributed to an

intervention or experiences that occurred over a period of time.

Engagement refers to the willing participation of students in activities (Ryan &

Patrick, 2001).

Middle years refer to Years 5 - 9 in Australian schools. Students enrolled in

these year levels are most commonly aged between 10 and 14 years.

Problem posing is the act of creating a new problem for oneself or for peers to

solve. The problem may be presented in an oral, written or other visual format

(English, Fox, & Watters, 2005; Lowrie, 2002).

Problem solving occurs when a specific goal exists that cannot be solved

immediately due to the presence of one or more obstacles (DeLoache, Miller, &

Pierroutsakos, 1998). Problem solving is “getting from givens to goals when a

solution path is not readily apparent” and requires the problem solver to recall

information, draw upon previously learned skills, choose appropriate solution

strategies, and express information in a meaningful way. It involves the

acquisition and utilisation of knowledge, metacognition and socio-cultural

contexts (Lesh & Zawojewski, 2007).

Self-regulation refers to a student’s ability to be actively and productively

involved in an activity that does not intentionally distract or interfere with the

learning of other students (Schunk, 2001).

Underachievement occurs when there is a “distance between the actual

developmental level [of a child] as determined by independent problem solving

and the level of potential development as determined through problem solving

18

under adult guidance or in collaboration with more capable peers” (Vygotsky,

1978, p. 86). In this study, underachieving students will be defined to be those

students who achieve above average results in the MYAT test (Australian

Council for Educational Research, 2005) while also achieving lower than the

average results in the problem-solving criterion of their mathematics tests,

compared to their cohort.

1.3 Rationale for the Study

It could be said that today, children resemble their times more than they

resemble their parents. This is not surprising when we consider that our current

times are typified by dynamic advances in technology and a resultant, ever-

changing job market that requires the workforce to embrace flexibility and

creativity. The responsibility to prepare our students to be effective and

productive citizens in such a world is mandated, in part, to the education system

of the day. In response to this mandate, rigorous reviews of the State-based

education systems in Australia have lead the Australian Federal Government to

move towards a national, futures-focussed curriculum that recognises “that

society will be complex, with workers competing in a global market, needing to

know how to learn, adapt, create, communicate, and interpret and use

information critically” (National Curriculum Board, 2008, p.5).

Two reports the Australian National Numeracy Review Report (National

Numeracy Review, 2008) and Foundations for Success: The final report of the

National Mathematics Advisory Panel (National Mathematics Advisory Panel,

2008) from the United States of America, have provided a foundation for

discussion papers leading to the development of an Australian, national

mathematics curriculum. The establishment of this national mathematics

curriculum is a unique opportunity to redefine not only the appropriate curriculum

content, but also to reconsider and redefine the most appropriate pedagogy to

achieve the desired student outcomes.

19

With mathematics education having a long history of marginalising and

disengaging students through traditional teaching practices, one could argue that

a review of teaching practices is timely (English, 2002; Lesh & Zawojewski, 2007;

Skovsmose & Valero, 2002). Currently not all students are being presented with

mathematics curriculums that allow them to draw on their knowledge to solve

meaningful problems that are relevant to them and to society. Indeed, “an

unintended effect of current classroom practice is to exclude some students from

future mathematics study” therefore creating a need to engage more students in

mathematical activities that are connected meaningfully to real-life contexts

(National Curriculum Board, 2008).

Education departments and national curriculum organisations across Australia

have continued to develop policies to promote contemporary teaching practices

to address this concern, with mixed success. The New Basics Framework is an

example of a recent four-year project in Queensland schools that created new

opportunities to connect the curriculum to real-life contexts (Department of

Education Training and the Arts, 2007). The Framework provided an alternative

organisational and conceptual framework for the curriculum and was intended to

reflect the new demands placed on students, and hence on curriculums,

assessment and pedagogy, by the “new times”.

The New Basic’s trial curriculum was organised around four “clusters”; Life

Pathways and Social Futures; Active Citizenship, Multiliteracies and

Communication Media, and Environments and Technologies. Assessment was

adapted from assessing and reporting against students’ learning outcomes,

through traditional pen and paper tests, to student demonstrations of learning

throughout the transdisciplinary “Rich Tasks”. While some Queensland State

schools have continued, in part, to pursue and support this new direction in

curriculum ideology, broad-scale implementation of the Rich Tasks has not

subsequently occurred across all Queensland schools. Reasons given for the

lack of broad-scale implementation included insufficient professional

20

development for teachers and reduced class time available for the development

of basic student literacy and numeracy skills that will now be measured and

compared between the States of Australia (Department of Education Training

and the Arts, 2007). The first national comparison of literacy and numeracy skills

between students in different States took place in May 2008 as part of the

National Assessment Program Literacy and Numeracy (NAPLAN) (Ministerial

Council on Education, Employment, Training, & Affairs, 2008a).

Despite the Queensland Government’s decision not to proceed with the full

implementation of the New Basics Framework, curriculum organisations are still

calling for meaningful connections to be made between school-based

curriculums and real-life contexts (Department of Education Training and the

Arts, 2007). In 2008, the National Curriculum Board of Australia opened public

discussion about what is important in the teaching of mathematics, by publishing

for public comment, papers about a nationally administered mathematics

curriculum. One such paper, The National Mathematics Curriculum: Framing

Paper argued that “mathematics is important for all citizens” and that “some

students are currently excluded from effective mathematics study” (National

Curriculum Board, 2008, p. 1). The paper stated that equity of opportunity is a

central goal in the construction of a national mathematics curriculum and

included discussion about how specific groups have been excluded and how to

re-engage more students in the study of mathematics. According to the paper,

the students at most risk of disengagement are students in their middle years of

schooling. The paper suggested the alienation and disengagement of these

students is largely attributed to “irrelevant curriculums”, unconnected to real-life

contexts, and “ineffectual learning and teaching processes” (National Curriculum

Board, 2008, p. 5). The report went on to state that it is “imperative that we

reverse this trend” (p. 5).

The concept of irrelevant curriculums is not new (Secada & Berman, 1999) with

Hollingsworth, Lokan and McCrae (2003) reporting that, in Year 8 mathematics

21

lessons, more than seventy-five percent of the problems provided to students

were low in complexity, emphasised procedural fluency, rather than higher-order

critical thinking, and only twenty-five percent of the problems were connected to

real-life contexts. When looking a little further into the senior years of schooling,

Barrington (2006) reported a drop in student participation rates in Year 12

mathematics classes and the National Numeracy Review (NNR) reported a

decline in tertiary students undertaking substantial studies in mathematics which,

in part, has lead to a national shortage of secondary mathematics teachers

(National Numeracy Review, 2008). These reports have major implications for

educators and in particular, teachers of middle-year mathematics; for it is in the

middle years of schooling that students appear to be forming enduring

dispositions and perceptions about the personal relevance of the study of

mathematics that can lead to underachievement, disengagement or both

(National Curriculum Board, 2008).

According to the National Declaration on Education Goals for Young Australians

draft report, Australia has no “inherent advantage – except through the quality of

education” to prepare students for the “radically evolving and uncertain context”

of future life in a global society (Ministerial Council on Education, Employment,

Training, & Affairs, 2008b, p. 4). This is supported in the National Mathematics

Curriculum: Framing paper where the authors stated that “a fundamental goal of

the mathematics curriculum is to educate students to be active, thinking citizens,

interpreting the world mathematically, and using mathematics to help form their

predictions and decisions about personal and financial priorities” (National

Curriculum Board, 2008, p. 3).

The paper defined, as goals of a national mathematics curriculum, four

proficiency strands;

1. understanding (conceptual understanding);

2. fluency (procedural fluency);

22

3. problem solving (strategic competence) and,

4. reasoning (adaptive reasoning).

It stated that problem-solving competence, including “the ability to make choices,

interpret, formulate, model and investigate problem situations, and communicate

solutions effectively”, is central to ensuring a futures orientation to a national

curriculum (National Curriculum Board, 2008, p. 3). The importance of

developing problem-solving competence was previously discussed by Cai (2003)

during his investigation of Singaporean students’ mathematical thinking in

problem solving and problem posing. Cai stated, following his exploratory study,

that problem solving was the most purposeful activity in the study of

mathematics. Later researchers such as Brown and Walter (2005) suggested

that it was the formulation or posing of problems, more so than the solving of

problems, that was fundamental in the development of mathematical skills.

Previous researchers such as Lowrie (2002) had already undertaken some

research into the usefulness of problem posing and had discovered that, when

used as a regular strategy in the study of mathematics, problem posing had the

potential to increase the engagement of underachieving students.

Shimizu (2002) also considered the engagement of students when he

investigated how the structured problem-solving approach to teaching

mathematics in Japanese schools and its associated impact on how Japanese

students perceive their lessons, compared with the pedagogy used by German

and American mathematics teachers and the perceptions of their students. One

of the differences he noted was that fostering mathematical thinking was the

main goal of mathematics lessons for the majority of Japanese teachers whereas

61 percent of American teachers and 55 percent of German teachers had the

development of mathematical skills as their main goal. A second difference he

discussed was the time spent by Japanese, German and American students on

the practice of routine procedures compared to time spent thinking about

23

mathematical problems and inventing new solutions. His data were taken from a

Third International Mathematics and Science Study (TIMSS) video classroom

study (Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1999) and can be found in

the Table 1.1.

Table 1.1

Percentage Comparison of How Time is Allocated in Year Eight Mathematics

Classrooms in Germany, the United States and Japan

OECD Country Practising routine

procedures

Thinking about

mathematical problems and

inventing new solutions

Japan 40.8 44.1

Germany 89.4 4.3

United States 95.8 0.7

Note. Adapted from " The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States” by J.W. Stigler, P. Gonzales, T. Kawanaka, S. Knoll & A. Serrano, 1999, Washington, D.C..

According to the study (Stigler et al., 1999), students in Japan spend less than

half the amount of time practising routine procedures and more than ten times

the amount of time working with problems than do their German and American

counterparts. These statistics become more notable when we consider the

performance of Japanese students compared to American and German students

in The Program for International Student Assessment (PISA) test undertaken by

students in twenty-nine member countries of the Organization for Economic

24

Cooperation and Development (OECD) in 2003 (Lemke et al., 2004). The

average country scores for fifteen-year-old students from Japan, Germany,

Australia and the United States on the problem-solving scale are reported in

Table 1.2 while the overall combined mathematical literacy scores can be found

in Table 1.3. Statistics about Australian students have been included for

comparative purposes.

Table 1.2

Comparative Problem-solving Scale Scores from the 2003 PISA Test

OECD Country Average student score

(average = 500, S.D.=100)

OECD Ranking

N=29

Japan 547 3rd

Australia 530 5th

Germany 513 13th

United States 477 24th

Note. Adapted from "International outcomes of learning in mathematics, literacy and problem solving: PISA 2003 results from the U.S. perspective - highlights” by M. Lemke, A. Sen, E. Pahlke, L. Partelow, D. Miller & T. Williams, 2004, Washington, D.C.: National Center for Educational Statistics.

It could be deduced from the results in Tables 1.1, 1.2 and 1.3 that a

mathematics classroom rich in problem-solving opportunities can not only lead to

enhanced performance on international problem-solving testing instruments, it

can also support the development of mathematical literacy and is therefore

worthy of further research in an Australian school context.

25

Table 1.3

Overall Combined Mathematical Literacy Scores from the 2003 PISA Test

OECD Country Average student score

(average = 500, S.D.=100)

OECD Ranking

N=29

Japan 534 4th

Australia 524 8th

Germany 503 16th

United States 483 24th

Note. Adapted from "International outcomes of learning in mathematics, literacy and problem solving: PISA 2003 results from the U.S. perspective - highlights” by M. Lemke, A. Sen, E. Pahlke, L. Partelow, D. Miller & T. Williams, 2004, Washington, D.C.: National Center for Educational Statistics.

1.3.1 Summary

Four foci arose from the preliminary review of the literature surrounding

mathematics education in Australia and internationally:

1. problem solving

2. problem posing

3. middle years and,

4. underachievement

It has been suggested that posing problems can re-engage underachieving

students (Lowrie, 2002) and that middle-year students are at most risk of being

26

disengaged and underachieving in the study of mathematics (National

Curriculum Board, 2008). Problem posing has been attributed as being an

important skill in the development of problem-solving competence (e.g., Cai,

2003; English, 2003; Silver & Cai, 1993a), which is one of the four proficiency

strands that make up the structure of the new national mathematics curriculum

(National Curriculum Board, 2008). International research has reinforced the

value of a mathematics curriculum, rich in problem-solving, to the student

development of mathematical literacy skills and problem-solving competence

(Lemke et al., 2004; Stigler et al., 1999). In light of these observations and to

progress the reform of mathematical curriculums, the following overarching

question was investigated in this present research study:

How might a problem-posing intervention impact upon the development of

problem-solving competence of underachieving, middle-year students?

The decision to investigate this overarching research question was consistent

with international curriculum documents such as those written by the American

National Research Council (e.g., NRC, 2004), and the National Council of

Teachers of Mathematics (NCTM, 2000) that recommended that teachers

provide regular opportunities for students to pose and solve problems within

meaningful contexts. The results of this present study provide education policy

makers, syllabus writers, and teachers with insights into how underachieving,

middle-year, mathematics students may be assisted to develop problem-solving

competence through a problem-posing intervention (e.g., Bjorklund, 2000; Jones

& Myhill, 2004; Kanevsky & Keighley, 2003).

1.4 Background to the Study

This section considers further background information on problem solving,

problem posing and disparity between student results and their potential that was

used to develop the three Research Questions for this study.

27

1.4.1 The Value of Problem Solving in Today’s Society

Problem solving is widely argued as the most purposeful activity in a

mathematics curriculum (Cai, 2003; Cai & Hwang, 2002; Costa, 2005; NCTM,

2000). It is not surprising then to find the States of Australia have been collecting

data about students’ problem-solving performance from all students in Years 3, 5

and 7 for almost ten years (e.g., Queensland Studies Authority, 2005). Despite

these and similar efforts at collecting data, it seems that little of the data have

been converted into reform of the teaching and learning of mathematics (e.g.,

Lowrie, 2002). As international researchers (e.g., Brown & Walter, 2005; Lester,

2003) have indicated, a review of current practices was needed, as was a “fresh

perspective of problem solving … that goes beyond current school curricula and

state standards” (Lesh & Zawojewski, 2007, p. 52). Of equal concern is Lesh

and Zawojewski’s recent review of literature that reported there is a “growing

recognition that a serious mismatch exists (and is growing) between the low-level

skills emphasized in test-driven curriculum materials and the kind of

understanding and abilities that are needed for success beyond school”

(Gainsburg 2003a in Lesh & Zawojewski, 2007 pp. 5-6). In fact, they went so far

as to say that the challenging and novel problems encountered outside of the

school environment, requiring extensive use of mathematics, are frequently

inconsistent with the underlying assumptions of conventional approaches to

solving mathematical problems in schools (Lesh & Zawojewski, 2007).

Indeed, the extent to which our education system is successful in developing

these skills has broad implications for students as they leave the school system.

Universities and employers throughout Australia and overseas are looking to

organisations like the Australian Council for Educational Research (ACER) to

screen prospective students and employees for their problem-solving

intelligence. Testing instruments such as the Commonwealth Government

funded Graduate Skills Assessment (GSA) (ACER, 2003), can now be used by

28

employers and universities to assist in the determination of university placements

and employment suitability.

Further evidence for the value of problem solving in Australian society can be

found in a more recent government initiative, which saw the Department of

Education, Science and Training (DEST) and the Australian National Training

Authority (ANTA) contract a project to establish the Employability Skills

Framework (DEST & ANTA, 2004). The purpose of this project was to inform

educators about employer perspectives on the personal attributes and skills of

desirable employees. The framework specified eight skill groupings that defined

and described employability skills (see Table 1.4).

There is a need to “continue building Australia’s capacity to effectively operate in

the global knowledge-based economy” and “education and training providers will

have a key role in equipping the community for this challenge” (Australian

Chamber of Commerce and Industry, 2002, p. 1). Reports, such as the

Employability Skills Framework (DEST & ANTA, 2004) attempt to address this

need and provide implications for researchers of educational pedagogy. Not only

is the acquisition of problem-solving competence fundamental in acquiring

important mathematical concepts (e.g., Adams, Brower, Hill, & Marshall, 2000;

Bobis, Mulligan, & Lowrie, 2004), it can also impact on the employability of

graduates entering the work place.

1.4.2 The Place of Problem Posing in a Responsive Curriculum

Problem-posing skills are a fundamental building block in the development of

mathematical skills (Brown & Walter, 2005; Lowrie, 2002; NCTM, 2000).

Problem-posing activities are a means to demystify problems and to empower

students to connect with mathematics in a more personal and meaningful way.

However, despite the clear benefits of problem-posing activities, students are not

often given the opportunity to pose their own mathematics problems publicly

(Silver, 1997).

29

Table 1.4

The Eight Skill Groupings of the Employability Skills Framework

Skill Description

Communication Skills that contribute to productive and harmonious relationships between employees and customers

Team Work Skills that contribute to productive working relationships and outcomes

Problem-solving Skills that contribute to productive outcomes

Initiative and enterprise Skills that contribute to innovative outcomes

Planning and organisation Skills that contribute to long-term and short-term strategic planning

Self-management Skills that contribute to employee satisfaction and growth

Learning Skills that contribute to ongoing improvement and expansion in employee and company operations and outcomes

Technology Skills that contribute to effective execution of tasks

Note. From “Employability skills final report: Development of a strategy to support universal recognition and recording of employability skills - A skills portfolio approach.” by Department of Education, Science and Technology and Australian National Training Authority. 2004. Canberra, ACT.

The virtues and benefits to students of posing problems have been known for

some time. Hart (1981) marvelled at how the activity of allowing students to pose

30

their own problems afforded her the opportunity to “open a window” through

which to view students’ thinking. Van Den Brink (1987) expressed a similar view

when he said problem posing provided him with a “mirror” that reflected the

content and character of a student’s mathematical experience. However, Silver

and Cai (1993b) suggested more profound reasons for including problem posing

as a learning activity, as it presents the opportunity to consider students’ views

on issues of morality, justice and human relationships. These virtues and

benefits are as valid today as they were twenty years ago.

Research has been undertaken in recent years that also espouses the benefits of

mathematical problem posing and solving in a balanced mathematics curriculum

(Bjorklund, 2000; Bobis et al., 2004; Brown & Walter, 2005; Cai, 2003; Daniel,

2003; English et al., 2005; Knuth & Peterson, 2002; Stoyanova, 2003). While

problem posing and problem solving feature highly in most Australian and

American policy documents on Mathematics education, in some American

mathematics classrooms the learning of knowledge and processes received over

one hundred times the attention afforded to the development of problem solving

(Stigler et al., 1999).

To address the research that suggests traditional practices in the teaching of

mathematics can contribute to the disengagement of students (e.g., English,

2002; Lesh & Zawojewski, 2007), research into problem posing has continued

(e.g., Brown & Walter, 2005; English et al., 2005). Of particular interest are the

reports by researchers of increased engagement of underachieving students in

the study of mathematics when problem posing was used as a regular teaching

strategy (English, 1997a, 1997b; Lowrie, 2002). However, despite these

findings, connections between a problem-posing intervention and increased

problem-solving competence of students, who achieve above average results in

standardised intelligence tests and who underachieve on problem-solving tests,

are yet to be made. This study has attempted to fill this void in the research.

31

1.4.3 Disparity in Student Mathematical Performance

It is widely accepted that the mathematical abilities of students of different ages

vary enormously; but so do the intellectual abilities of same-aged students (Case,

1998). These differences have been the study of many research projects

investigating intelligence and the means to measure intelligence (e.g., Sternberg,

2002; Vernon, Wickett, Bazana, & Stelmack, 2000). A number of standardised

intelligence tests have been devised over the past one hundred years and have

been used to benchmark cognitive development (e.g., Spearman, 1904;

Wechsler, 1991). These tests distinguish between the mental age of a child and

the chronological age of a child. The power of the message sent to students

when their performance on such tests is alluded to, or even articulated to the

child, cannot be underestimated. What students believe about their intelligence

and mathematical performance has been shown to be a powerful indicator of

achievement outcomes (Stipeck & Gralinski, 1996).

While we can readily accept that mathematical abilities of students vary from

student to student, it is perplexing when intelligence tests suggest a strong

potential for mathematical ability, yet results from classroom tests do not support

this prediction. In particular, the scenario becomes more perplexing when a

student achieves a high predictive score in an intelligence test, scores highly in

routine procedural questions in class tests, yet continues to perform below

average in questions that require significant problem-solving capabilities. This is

an area of research that has received little attention in the corpus of knowledge

connecting students and their problem-solving capabilities, and precipitated one

of the foci of this study.

1.5 The Purpose of this Present Study

The purpose of this present study was to investigate how a problem-posing

intervention might impact on the development of students’ problem-solving

competence, with a particular focus on the engagement of under-achieving,

middle-year students. This present study provided opportunities for selected

32

students, from four different Year 7 classes in the one school, to pose and

explore their own problems over a seven-lesson teaching experiment. Eighteen

participants met the selection process (see Section 3.3.2) and were withdrawn

from their customary Monday morning assembly each week. They met together

as a group in a multi-purpose classroom in their School library. Data from three

of these students was disregarded, due to the multiple absences of these

students from the teaching episodes, leaving data from fifteen students to be

analysed. From the remaining students, three case-study students were chosen

for a detailed investigation of the changes that occurred for them as a result of

the problem-posing intervention (see Section 5.2 for the selection process of the

three case-study students).

To address the purpose of this present study, three research questions were

investigated during the teaching experiment.

Research Question 1

Can, and if so, how can participation in problem-posing activities facilitate the re-

engagement of middle-year mathematics students?

Research Question 2

Can, and if so, how can participation in problem-posing activities facilitate

improved problem-solving competence of middle-year, mathematics students?

Research Question 3

In terms of problem-solving competence, what developmental learning changes

occur during the course of a problem-posing intervention?

1.6 Significance of the Research

Ceci (1996) argued that it is not possible to deduce the intelligence of a person

from their performance on a set of standardised questions such as those found

33

on commonly used Intelligence Quotient (IQ) tests. Indeed, he argued that

cognition occurs within the framework defined by parents, teachers, peers, and

the culture of the time. It follows then that it may not be possible to accurately

deduce students’ mathematical potential from a set of questions presented to

them in a standardised test or examination, as is the current status quo in many

schools across Australia. It has been mooted by several authors that alternative

activities, such as problem posing, may provide teachers with more authentic and

accurate insights into their students’ understandings of mathematical processes

and concepts. Performance at problem-posing tasks may therefore be a more

accurate indicator of student’s mathematical potential (Anderson, 1997; Bobis et

al., 2004; Brown & Walter, 2005).

Siegler (1996) maintained that teachers can influence their students’ cognitive

development in three significant ways. Firstly, they can influence what their

students think about. Secondly, they can influence how their students will

acquire and construct their information and, thirdly, they can influence why their

students engage in the learning process. This view is supported by Tate and

Rousseau (2002) who found that mathematics was the favourite subject of most

Year 1 and 2 students, yet was one of the least favourite by the time they

reached the middle years of schooling. They attributed this phenomenon to

either the students removing themselves from the challenging programs in

mathematics or the teachers removing the challenging programs from them. In

either situation, mathematics teachers clearly have an important role to play in

constructing effective learning opportunities for their students.

The use of a problem-posing intervention has been investigated by many

researchers. For example, Bandura (1997) discussed the impact of problem-

posing opportunities on students’ self-efficacy, while Knuth (2002) considered its

impact on the development of students’ intrinsic motivation to engage in the

learning of mathematics. Graham, Harris and Larsen (2001) looked at how

problem posing could be used in the prevention of writing problems for students

34

with learning difficulties, while Lowrie (2002) focussed on the influence of the

teacher on the types of problems students pose. Contreras (2003) and Lavy and

Bershadsky (2003) investigated a problem-posing approach to solving geometry

problems, while Stoyanova (2003) considered the impact of problem posing on

gifted and talented students. Despite this apparent breadth of problem-posing

research, there appears to be little research into the role of a problem-posing

intervention in assisting underachieving mathematics students who have above-

average performance in standardised intelligence tests. This study has

addressed this shortcoming.

1.7 Thesis Overview

This thesis comprises seven chapters. The first chapter provides an introduction

to the research study while the second chapter provides a report on the relevant

literature pertaining to problem-solving, problem-posing and underachievement

of students in their middle years of schooling. This review highlights where the

shortcomings in the research exist. A discussion about the design and

theoretical foundations of the research study and a detailed description of the

instruments used to collect data, can be found in Chapter Three. This chapter

also includes a section outlining the selection process for participants of this

study and a more detailed description of how three case-study students came to

be chosen from the participant group.

Issues pertaining to reliability and validity of the data collected and the

associated ethical considerations arising from this study are discussed towards

the end of Chapter Three. Chapter Four introduces the structure of the teaching

experiment and discusses each teaching episode in detail. These discussions

are particularly useful in highlighting the situational challenges, and associated

implications for data collection and analysis, that arose throughout the

experiment. The fifth chapter reports on the data collected during the teaching

experiment and contains an in-depth review of the impact of the problem-posing

intervention on three case-study students; Paul, Andrew and Nicole. Chapter Six

35

provides an analysis and synthesis of the data collected throughout the teaching

experiment that enabled the three Research Questions to be answered. The

limitations of this study and the implications of the study’s findings for future

research are discussed in Chapter Seven.

36

Chapter 2

Theoretical Perspectives

2.1 Chapter Overview

This chapter contains a critical review of current literature pertaining to this

present study. The review begins in Section 2.2 with the literature pertaining to

the developmental learning of students. It starts with a brief introduction to the

main theories discussed by education researchers and then focuses on the three

theories that are particularly relevant to educational research related to the

learning of mathematics. The literature surrounding the development of problem-

solving competence and its relevance and role in developing mathematical skills

is reviewed in Section 2.3, while literature about the use of problem-posing as an

intervention to promote student learning is reviewed in Section 2.4. This latter

section concludes with a review of the literature surrounding the relationship

between the development of problem-solving competence and student

opportunities to pose their own problems. Literature related to the possible

causes of underachievement of middle-year students is reviewed in Section 2.5.

The literature surrounding the theoretical framework that underpins this present

study and the investigation of the Research Questions is presented in Section

2.6, while a conclusion for the chapter can be found in Section 2.7.

2.2 Understanding Developmental Learning

“Developing an understanding of the developmental status of students’ thinking

and learning is fundamental to improving that learning” (Cai & Hwang, 2002, p.

401). As student development of problem-solving competence was a goal of this

present study, this section presents an overview of the literature surrounding

developmental learning of students. Links between developmental learning and

37

problem-solving competence are established, as are the areas in the research

where disagreement between authors exists and uncertainty occurs.

Researchers have provided many methods and concepts that increase our ability

to observe, explain and describe the process of student’s developmental

learning. For example, Siegler (1991) said,

all types of thinking involve both products and processes. The products

of thinking are the observable end states – what children know at

different points in development. The processes of thinking are the

initial and intermediate steps, often accomplished entirely inside

people’s heads that produce the products. (p. 3)

He compared children to scientists because they both ask innumerable

elementary questions about the nature of the universe, which seem entirely trivial

to everyone else, and are both given the time by society to pursue their

ruminations. This inquisitive nature of children is the very attribute that lends

itself to the development of problem-solving competence and problem-posing

expertise from a very early age. Siegler (1991) exemplified this view when he

talked about it not being uncommon to see a toddler in a high chair deliberately

drop food from their tray onto the floor to see what happened to the food.

Together with investigations on intelligence and developmental learning,

researchers are gaining a clearer picture of how to assist students to narrow their

“zone of proximal development” (Vygotsky, 1978) in problem-solving

competence. However, it is not clear, from the current research, whether

problem posing is an appropriate teaching strategy for the particular group of

middle-year students who underachieve in problem solving, yet who appear to

have above average intelligence compared to their peers. Whether intelligence

and developmental learning are a function of nature or nurture has been actively

38

debated for many years. In fact, many researchers have published a plethora of

theories, to understand differences in children’s cognition and developmental

learning, that are worthy of review (e.g., Bjorklund, 2000). Despite some

researchers supporting conceptual frameworks of more than one theory, for

example, Sternberg (1999a; 2002) supporting the multiple intelligences and

information processing theories, and Case (1998) supporting the stage and

information processing theories, in general, most researchers’ work aligns with

one of five theories, which are highlighted in Table 2.1. This present study draws

most heavily from the Information Processing Theory, with some reference made

to the Multiple Intelligences Theory, and the Psychometric Testing Theory, where

relevant.

Regardless of which theory a researcher supports, it is helpful to acknowledge

three basic characteristics of developmental learning. Firstly, we can

acknowledge that the brain is capable of finite information storage and

information processing capacity. Secondly, the human brain is constantly

adapting to a changing environment and thirdly, Goswami (2002) would have us

believe that “cognitive skills almost always can be increased, at least to some

degree” (p. 619). These three characteristics will be discussed further, within the

context of the Information Processing Theory, the Multiple Intelligences Theory

and the Psychometric Testing Theory in the next three sections.

39

Table 2.1

Comparison of Stage Development in Cognitive Development Theories

Theory Underpinning Beliefs Leading Researchers

Stage Learning occurs in stages and a child needs to pass through one stage completely before entering the next stage.

(Piaget & Inhelder, 1969); (Case, 1998)

Information Processing

Mental representations, processes, strategies, and knowledge develop over time.

(Sternberg, 2002); (Halford, 2002); (Klahr, 1992); (Deary, 2000); (Lohman, 2000); (Siegler, 1991, 1996)

Psychometric testing

Intelligence can be described in terms of mental factors and psychometric testing instruments can be constructed to reveal such factors.

(Spearman, 1904); (Brand, 1996); (Hernstein & Murray, 1994) ; (Jensen, 1998); (Wechsler, 1991)

Multiple Intelligences

Intelligence is not a unitary concept, but more a multiple one, where intelligence may be domain specific or domain general.

(Gardner, 1999a); (Sternberg, 1997a); (Thelan & Smith, 1998);

Biological, Environmental and Social Factors

Intelligence characteristics are acquired partly through heredity. Cognitive development occurs through the internalisation of concepts experienced through environmental and social contact.

(Vygotsky, 1981); (Feuerstein, 1979); (Rogoff, 1998); (Ceci, 1990); (Grigorenko, 2000); (Vernon et al., 2000)

2.2.1 Information Processing Theory

Information processing theorists argue that thinking is like processing

information. The quality of the thinking is dependent on the processing capability

and memory limitations of the child. In other words, what information the child

40

chooses to use in a particular situation, how the child processes the information

to achieve their desired outcome, and how much of the information they can

retain in memory at anyone time, will be decisive factors in their overall success

at solving problems. Siegler (1996) spoke about an “essential tension” (p. 58)

that exists for children between their limitations to retain and process information

and their automatic striving to find ways to overcome these limitations. He

discussed a variety of strategies commonly used by children in this pursuit which

included:

1. practice and rehearsal to overcome limited memory capacity,

2. increased use of resources such as books or the internet to overcome

limited knowledge, and

3. the use of problem-solving strategies, such as breaking a problem into

smaller sub-problems, to overcome an inability to deal with long

sequences of tasks.

.

According to Siegler (1991) “it is no accident … that the two main theoretical

approaches to cognitive development – the Piagetian and the information

processing approaches – both place great emphasis on problem solving” (p.

252). He said that when children regularly solve problems they are in fact

contributing to their own cognitive development as problem solving requires them

to create solutions for themselves, rather than relying on procedures and

practised routines they have learnt. This active involvement by a child in their

own developmental learning, by engaging in continuous self-modification

(Siegler, 1996), was also supported by Bjorklund (2000) who said “cognitive

development is a constructive process, with children playing an active role in the

construction of their own minds” (p. 481).

Researchers who support an information-processing theory, discuss four change

mechanisms that they believe play a significant role in childhood cognitive

41

development: automatisation (the increasingly efficient execution of mental

processes), encoding (the selection and prioritising of important aspects of

situations), generalisation (the use of prior knowledge of numerous familiar

situations), and strategy construction (the synthesis of change processes to

produce cognitive growth) (see Sternberg, 2000). In previous research

Sternberg (1985) referred to only three information processing components of

general intelligence in his Triarchic Theory of Intelligence (see Figure 2.2), these

being knowledge acquisition components (discrimination between relevant

and irrelevant data), metacomponents (selection and planning of appropriate

strategies) and performance components (combination of the selected data

and appropriate strategy to solve the problem). However, none of these

components explicitly acknowledge the efficiency with which a student solves a

problem as a significant factor of intelligence. The efficiency of execution in the

solution of a problem warrants further investigations where the time allowed for

an assessment of skills is a controlled factor.

While Sternberg’s earlier work is over twenty years old, and has been

superseded by the four change mechanisms to a large extent, researchers (e.g.,

Goswami, 2002; Thomas & Karmiloff-Smith, 2002) still refer to Sternberg’s

Triarchic Theory of Intelligence when discussing childhood cognitive

development. According to Goswami (2002), “individual differences in cognition

derive largely from individual differences in the execution of these three kinds of

components. The components are highly interdependent.” (p. 608)

42

Selective Selective Strategy Strategy Encoding Application Encoding Combination Construction Selection Selective Strategy Inference Comparison Coordination

Figure 2.1. A Schematic Diagram of Sternberg’s Triarchic Theory of Intelligence

(in Siegler, 1991 p. 69).

2.2.2 Psychometric Theory

If we assume that infants come into the world poorly endowed, the

question becomes how they are able to develop as rapidly as they do. But

if we assume that infants come into the world richly endowed, the question

becomes why development takes so long. (Siegler, 1991, p. 3)

This section explores the issues surrounding this nature versus nurture debate

that begun in the late 1800s by researchers such as Sir Francis Galton (1883),

Charles Darwin’s cousin, who popularised the now famous Bell Curve and its

associated normal (Gaussian) distribution. A review of the history of

psychometric theory is relevant to current research as views held by

contemporary proponents of the psychometric theories have not changed

Intelligence

Metacomponents Knowledge Acquisition Components

Performance Components

43

significantly to those of the founding researchers in this field. The Bell Curve,

first introduced by Galton, is still used as a standard tool for comparing students

and prospective employees as well as being used by researchers for interpreting

data in social science research projects.

Galton’s (1883) interest in comparing individuals stemmed from his advocacy for

eugenics, the inter-breeding of intelligent people in order to strengthen the gene

pool of the human species. While Galton initially investigated the distribution of

physical measurements such as weight and height, he later theorised that since

psychological characteristics were based on physiological characteristics, then

human intelligence could also be represented by the Bell Curve. While Galton

had begun founding research into human intelligence, he did not construct broad-

scale instruments to measure intelligence levels of children or adults. This work

was taken up a few years later in France when universal education was

introduced in the late 1890s, as a result of the Industrial Revolution, with

psychologists Alfred Binet, Director of the Sorbonne in France, and Theophile

Simon being engaged to develop a testing instrument to determine which

children needed “special education” (Binet & Simon, 1905).

The first Binet-Simon test (Binet & Simon, 1905) was used in 1905 and included

thirty questions on reasoning, memory, language and problem solving, ordered

by difficulty, and was used to identify children who may experience difficulty with

a common curriculum. The test was based on data from 50 subjects, therefore

lacking validity, and was criticised because it relied heavily on the reading and

language ability of the children. Almost one hundred years later, this same

criticism is leveled at authors of psychometric tests in current use (Bjorklund,

2000; Gardner, 1999b). The Binet-Simon test was revised in 1908 following

further research with 203 subjects and had test items grouped according to age

level rather than increasing difficulty. It was at this stage that Binet and Simon

introduced the concept of mental age (MA), as compared to chronological age

44

(CA), which later resulted in the establishment of the ‘intelligence quotient’ (IQ)

by German psychologist William Stern (1912) and Terman (1916) that is still

used today to define, label and categorise students.

At around the same time as Binet and Simon (1905) established their first test,

research into human intelligence and developmental learning took a different

direction in England with Spearman (1904) investigating the existence of a

general intelligence factor that he abbreviated to a more commonly used

expression, a g factor. According to Spearman, all individual differences in

cognitive ability were due to a general factor that is present at birth and that he

believed was as a result of differences in mental energy. This g factor impacted

upon performance in all cognitive tests, whereas a specific factor, (commonly

called an s factor) could impact upon an individual’s performance in a specific

type of test. To support his proposition, he examined correlations between

student scores on different school subjects (see Table 2.2) and offered the high

positive correlations as evidence of the existence of a single common general

intelligence factor. This suggestion of individuals having specific s factors

maintained momentum and, 85 years later, was paralleled by Ceci’s (1990) view

that the context in which a test occurs is a decisive and determining agent in an

individual’s performance on the test. This position has important implications for

current research where researchers are interested in the participant’s

developmental learning changes as opposed to their connectedness to the

context of the questions used in the assessment instrument or the style of the

questions.

If the position of specific and general factors of human intelligence was to be

accepted, a new testing instrument was needed to measure and compare the

intelligence of individuals. Spearman (1904) in association with Cyril Burt,

another British psychologist, were some of the earliest researchers to develop a

range of intelligence tests, to measure the mental abilities of British school

45

children, that took general and specific intelligence factors into consideration.

They pioneered the concept of factor analysis that other researchers, such as

Thurstone (1938) and Wechsler (1991), further developed many years later.

These tests allowed gender differences to be considered. For example, Halpern

(1997) reported that, on average, boys score higher on tasks that involve visual

and spatial awareness than do girls, while girls perform better than boys at tasks

that require access to long-term memory, fine motor skills, perceptual speed, and

writing and comprehension of complex prose. These findings require current

researchers to consider whether assessment instruments favour the natural

differences of either gender. Without these consderations, the validity of data

could be challenged.

Table 2.2.

Spearman’s Correlations of Student Scores between Subjects

Subject Classics French English Math Pitch Music

Classics - .83 .78 .70 .66 .63

French .83 - .67 .67 .65 .57

English .78 .67 - .64 .54 .51

Math .70 .67 .64 - .45 .51

Pitch .66 .65 .54 .45 - .40

Music .63 .57 .51 .51 .40 -

Note. From "General intelligence, objectively determined and measured” by C. Spearman, 1904, American Journal of Psychology, 15(2), pp. 201-293.

46

By the early 1920s, the use of psychometric testing had expanded to the United

States and was being used as a means to determine which immigrants were

suitable for residency and which should be deported, and later in the 1930s to

determine intelligence levels of American school children. To achieve this goal,

Lewis Terman (1916), a Stanford Professor, revised the French Binet-Simon

(1905) test calling it the Stanford-Binet test. Results from this test were

compared to a standardised sample of 3184 mainly white, urban children from

eleven states in America, chosen by father’s occupation. This revised test was

administered under the assumption that not all children of a particular age think

and reason in the same way or at the same level. Terman’s results were more

reliable for older children aged between twelve and sixteen years than for

younger children, or children in the lower IQ ranges, but he found standard

deviations for children in different age groupings made the interpretation of data

difficult.

The use of intelligence tests continued to grow throughout American schools

over the next eighty years with the most common uses being for the identification

of children with special needs and children with special gifts and talents (Piirto,

2007). The use of IQ tests to investigate differences in intelligence levels of

different ethnic groups became widely provocative with the publishing of

Hernstein and Murray’s (1994) book entitled The Bell Curve: Intelligence and

Class Structure in American Life. The researchers stated in their book that they

had proven that people from minority ethnic backgrounds had lower IQs than

white Americans. Researchers in education, such as Kincheloe, Steinberg and

Gresson (1996), were quick to refute the allegations in their book Measured Lies:

The Bell Curve Examined. They wrote “The Bell Curve … emerges from a

crumbling paradigm often deemed inadequate for the study of human

intelligence” (Kincheloe et al., 1996, p. 28). While this latter book sold widely, it

did not impact on the growth of intelligence testing in American schools. In fact,

47

according to Piirto (2007, p. 14) “the increase in the use of aptitude,

achievement, and personality tests has been marked.”

Psychometric testing began to emerge in Australian schools in the early 1920s

and is now a well-established and accepted part of testing for Australian school

children (Hughes, 2002). The establishment of the Australian Council for

Educational Research in 1930 provided standardised resources for psychometric

testing to be undertaken in schools. By 1936, all Year 6 students in New South

Wales (NSW) were administered intelligence tests to determine which form of

secondary education best suited them. However twelve years later, a United

Nations Educational, Scientific and Cultural Organization (UNESCO) report on

educational psychology services across 41 countries in 1948, estimated that only

20 psychologists were employed across all Australian school systems and were

mostly based in NSW (Korniszewski & Mallet, 1948). Therefore, while a wealth

of data was being collected, the analysis of the data was generally limited to

superficial interpretation by school administrators.

The dominance of psychometric testing in Australian and international schools

has been driven by a widespread need of modern society to quantify individuals’

intellectual capacity. This is evident when one considers that most students in

Australia, and particularly those attending private schools, will not leave formal

schooling without having undertaken at least one Intelligence Quotient (IQ) test

and a dozen more specific intelligence tests (Bjorklund, 2000). The testing is

sometimes undertaken internally by educators using standardised instruments

such as the Middle Years Ability Test (Australian Council for Educational

Research, 2005) or administered privately by organizations such as The Sydney

Child Assessment and Testing Service (SCATS) which provides a private testing

environment for children aged between 3 and 16 years using predominantly the

Wechsler (1991) testing instruments.

48

Despite the widespread use and acceptance of psychometric testing in Australia

and internationally, researchers such as Naglieri and Kaufman (2001) raised

concerns about weaknesses in traditional IQ tests when they are used as a tool

to determine giftedness. They said these tests were theoretically old, they were

weak in theory and they were achievement driven. In addition, Jensen (as

reported in Jensen & Miele, 2004) raised concerns about the blatant lack of

understanding among users of IQ tests that has lead to the common misuse of

data generated from such testing instruments. Richhart (2002, p. 16) alerted us

to the impact of test practice on test scores when he said that some critics

“contend that test scores are highly influenced by one’s test-taking competence

and familiarity”. The existence of Core Skills Test (Queensland Studies

Authority, 2009) preparation courses in many Queensland schools, where

students in Year 12 practise tests from previous years to ensure they are familiar

with the test format and timing, could be considered as evidence of the widely

held acceptance of this viewpoint.

While IQ results correlate positively with academic success and employability

(Brody, 1997) and have been strongly supported by researchers such as Jensen

(1998), other researchers argued that IQ tests are limited in what they can

measure and that it is misleading to use an IQ score as a sole indicator of a

child’s overall intelligence. Gardner (1999b) was one of these researchers and

suggested that IQ tests provided at best a distorted view of an individual’s

potential, as they clearly advantaged individuals with strengths in the linguistic

and mathematical intelligences. Individuals with strengths in other intelligence

areas, such as the bodily-kinaesthetic intelligence, are often neglected and

hence do not receive an education sympathetic towards their unique form of

intelligence. Surprisingly, Gardner is not opposed to the use of intelligence tests

for determining intelligence of individuals. He would however, prefer that testing

instruments were constructed to measure and evaluate all of the multiple

intelligences. In additional to these concerns, other critics of IQ tests say they

49

are culturally biased, that is, they are based on knowledge and skills of middle-

class individuals from majority cultures rather than being inclusive of the

traditions, values, predominant language or experiences of minority cultures

(Bjorklund, 2000). The concerns mentioned here are relevant to new research

when IQ testing is used as an instrument to collect data or determine participants

for research studies. The literature would seem to suggest that, at best, IQ test

data can be used as an indication, rather than a definitive measure, of an

individual’s intelligence.

For as long as the existence of general and specific factors has been mooted,

there have been researchers who support the existence of only a single general

factor, or only specific factors, or both. A number of researchers supported the

existence of specific factors but challenged the existence of a general factor.

Debate continues about the existence of a higher-order, general intelligence

factor that oversees and orchestrates these other cognitive factors. For example,

Jensen (1998) is still seeking to demonstrate the factor’s existence, while others

like Ceci (1996) proclaiming the search is “fruitless”. On the other hand, other

researchers, such as Guilford (1988) and Sternberg (2002), have repeatedly

attempted to disprove, without success, the pivotal influence of a g factor in

determining intelligence of individuals however, according to Piirto (2007, p. 15),

“general intelligence is pervasive, even in tests that purport not to measure g-

factor intelligence.”

Brody (2003, p. 319) adds his support to the existence of a g factor when he says

that the “g theory is required to understand the relationships obtained by

Sternberg and his colleagues” who were proponents of information processing

theories of intelligence. That being said, it is now generally accepted that there is

more to intelligence than the general intelligence factor alone (Gottfredson,

2003). The challenge for future research is to develop a theoretical framework

and appropriate testing tools that incorporate the notion of a g factor in

50

combination with the widely accepted multiplicity in intellectual functioning that is

reported by researchers such as Gardner (1999b).

Despite disagreement between researchers about the accuracy of psychometric

testing as an accurate measure of an individual’s intelligence, or whether a

psychometric test is a reliable instrument to measure intelligence for all

individuals, the widespread use of IQ testing remains a feature of our present

education systems both internationally and in Australia. Gottfredson (2003) and

Piirto (2007) supported the use of IQ testing for the purposes of indentifying

individuals for suitable interventions to address their particular developmental

learning needs, however, to measure an individual’s ability within a specific

context and in a specific area of learning, more specific testing instruments are

required (Gardner, 1999b; Sternberg, 2000).

2.2.3 Multiple Intelligences Theory

In contrast to how Sternberg (1999a) emphasised the connectedness of the three

aspects of his Triarchic Theory of Intelligence, Gardner (1999b) emphasised the

separateness of his multiple intelligences. For him, there were up to ten unique

intelligences that represented a modular, brain-based capacity, some of which

were linguistic intelligence, logical-mathematical intelligence, and intrapersonal

intelligence. His was the first theory to account for the diverse range of important

capacities of individuals by considering a diverse range of competences and

based his theory on a diverse range of evidence. His evidence included the

selective damage of specific cognitive abilities following brain trauma and the

existence of individuals who present as low-achieving in IQ testing, yet who

display extraordinary abilities within one intelligence domain, such as Mozart who

was a musical genius but particularly ordinary in his performance at school tests

(Gardner, 1999b).

51

Like Gardner, Bjorklund (2000), when discussing the multiplicity of intelligence,

acknowledged the existence of both domain specific and domain general

intelligences. He said,

Intelligence, I believe, is multifaceted, with intellectual functioning varying

considerably as a function of a person’s knowledge and the context in

which the cognitive operations were acquired and are assessed…..The

most tenable position from my point of view is that some aspects of human

intelligence are domain general in nature, whereas others are domain

specific in nature. (Bjorklund, 2000, p.435)

Like Bjorklund (2000), Gardner (1999b) stated that although each of his multiple

intelligences had its roots in biology, each intelligence was flexible and could be

improved with education given an appropriate facilitation of learning experiences.

The implications of this statement are noteworthy for research that involves

teaching experiments with student participants in classrooms as facilitating

developmental learning is a key goal for many educators. Ritchhart (2002, p. 13)

extended the notion of appropriate learning experiences to beyond the classroom

and encouraged educators to consider intelligence in the context of the real world

where it will ultimately be applied rather than “the artificial world of school and

testing”. While such variables are difficult to control or accommodate in an

intervention with children, mention should be made in the limitations of such

research reports to ensure appropriate interpretation of the findings can be

made.

2.2.4 Summary

In section 2.2 of this chapter, the five major cognitive development theories that

researchers refer to today were identified. The three theories that are particularly

relevant to educational research, and that relate to problem solving and the

learning of mathematics, were then discussed in more detail. While the theories

52

were presented and discussed separately, there are some common threads that

lend support for the design of this present study. For example, Sternberg’s

(1988) Triarchic Theory sits well within the information processing theory due to

his definition of its three distinct information processing components; knowledge

acquisition components, metacomponents, and performance components.

However, the Triachic Model had its foundations in biological and social factors,

thus making an alignment with the biological, social and environmental theories,

and could be said to have links with the Psychometric Theory due to Sternberg’s

discussion of an overarching, general intelligence factor. Similarly, there is

agreement between the Psychometric theorists (e.g., Hutton, Wilding, & Hudson,

1997) and the Information Processing theorists (e.g., Siegler, 1996) that

intelligence is dependent on cognitive factors related to memory capacity and

function, and cognitive processing speed. A discussion of these shared ideas

and beliefs between the theories serves to inform what underpins our

understanding of childhood developmental learning with respect to the

development of problem-solving competence in a mathematical context.

Information processing theorists (e.g., Halford, 2002; Sternberg, 2002) suggested

that it is a student’s ability to select the correct information with which to work,

their ability to process that information, and how much of the information they can

retain at any one time, that will be the significant factors in developing problem-

solving competence. In addition, the amount of time a student is given to

undertake a task can determine the level of success they are able to

demonstrate in specific skills. These are important factors to consider when

designing a study that measures problem-solving competence and

developmental learning changes. Further, (Siegler, 1991 p. 59) suggested that

cognitive change occurred as a result of “self-modification” and (Bjorklund, 2000)

said it is a process where students best construct their own learning by taking an

active role. It follows therefore, that a traditional approach to teaching, where the

teacher instructs and the student’s feedback what has been instructed to them,

53

may not be the most beneficial if long term cognitive change is a goal of the

learning experience. The National Council of Teachers of Mathematics (2000)

and Ritchhart (2002) supported the need for learning opportunities for students

that involve experiential learning of real world, novel problems where they have

opportunities to be risk takers in the advancement of their own developmental

learning. More research in this area is being called for with problem posing and

problem modelling being two forms of intervention that have been identified as

worthy of further investigation as a means to promote the active engagement of

students in the development of mathematical skills (e.g., Lesh & Zawojewski,

2007).

A review of developmental learning theories highlighted the contribution of

problem-solving activities to the development of general, cognitive growth, and

as such, a review of the research surrounding the development of problem-

solving competence can be found in the following section.

2.3 Problem-solving Perspectives

This section reports on the current literature connecting problem solving to the

developmental learning of students, with particular emphasis on how it can be

used as a tool to develop mathematical learning and why the ability to be a

competent problem solver is so important. The benefits to students of teaching

mathematics from a problem-solving perspective are considered as are the ways

that problem-solving can drive mathematical reform. Issues surrounding the

development and assessment of problem-solving competence and student’s

understandings of problem structures are also reviewed.

2.3.1 Introduction

There is an increasing need for students to be able to solve real world, novel

problems (e.g., NCTM, 2000) and while substantive literature can be found which

expounds the virtues of providing students with opportunities to solve problems,

54

there is inadequate research into practical teaching strategies for teachers in

classrooms to use. Despite this, it is heartening to see significant research has

taken place in recent years surrounding the use of problem posing and problem

modelling as a means to facilitate increased problem-solving competence (e.g.,

Brown & Walter, 2005; English, 2003; English et al., 2005; Stoyanova, 2003).

However, little research has focussed on the specific group of students that

teachers would be expecting to succeed in problem solving, that is, the students

who perform above average, compared to their peers, in academic intelligence

tests. The reasons for underachievement of these students are still an

unexplained phenomenon (Baker, Bridger, & Evans, 1998; Lesh & Zawojewski,

2007).

As socio-cultural contexts change Lesh and Zawojewski (2007) suggested that it

follows then that teaching and learning strategies need to change to

accommodate them. Students are now looking towards an ever-changing job

market characterised by dynamic advances in technology, and one in which they

will need higher-order thinking skills and increased problem-solving ability to

cater for the inevitable ambiguities they will encounter. Researchers, who

question the ability of our curricula to prepare students for such a world, have

asked, “What if education were less about acquiring skills and more about

cultivating the dispositions and habits of mind that students will need for a lifetime

of learning, problem solving, and decision making?” (Ritchhart, 2002 p. xxii) This

question appears to have gone unanswered in the literature.

In 2003, Lester and Kehle undertook a review of the literature relating to problem

solving over the previous ten years. They found for example, that some

researchers had reported on how students use specific strategies for different

types of problems (e.g., Bjorklund, 2000; Ceci, 1996), while others had reported

on children’s use of talking with peers to solve problems (e.g., Teasley, 1995)

and the effect of metacognitiion in the development of problem-solving

55

competence (e.g., Kramarski, Mevarech, & Arami, 2002). As a result of their

meta-review, Lester and Kehle reported that minimal progress had been made,

and the literature undertaken in that time offered little to the pedagogical reform

of mathematics teaching (Lester & Kehle, 2003). Similar findings were reported

by Lesh and Zawojewski (2007) who reported there was still a pressing need to

improve the way students view problem solving and mathematics, while others

reported the need to increase students’ abilities to solve unfamiliar, novel

problems (English & Larson, 2005; English et al., 2005; Accreditation Board for

Engineering and Technology 2004 in Lesh & Zawojewski, 2007).

According to Lesh and Zawojewski (2007), this lack of significant reform in the

teaching of mathematical problem solving is not surprising as many developed

countries, such as the United States of America, have experienced a strong

swing back to curricula that emphasise basic skills. They reported this swing as

a result of a world-wide emphasis on high-stakes, standardised testing (Lipman,

2004; Stecher, 2002) which, according to the American National Research

Council (as cited in Kilpatrick, Swafford, & Findell, 2001, p. 4), has resulted in

mathematics curricula that are “shallow, undemanding, and diffuse in content

coverage”, with students becoming “notably deficient in their ability to apply

mathematical skills to solve even simple problems”. It is also worth noting that a

compromising feature of these standardised tests is that the results of the tests

are sometimes used to determine teacher pay rates, ongoing employment and

funding for schools in the United States of America and in some States of

Australia. According to Stecher (2002) this focus on basic computational skills,

leading to high-stakes test results, has resulted in the systematic marginalisation

of innovative and experimental teaching practices and less focus on conceptual

understanding. In addition, it has lead to a decline in the amount of problem-

solving research being undertaken (Lester & Kehle, 2003; Stein, Boaler, & Silver,

2003).

56

When reviewing the historical literature on the development of mathematical

understanding and knowledge, it becomes apparent that the bulk of this literature

presupposes that the development of mathematical understanding occurs along

a one-dimensional line (e.g., Piaget & Beth, 1966; Vygotsky, 1978). More recent

mathematical research however, particularly research surrounding models and

modelling perspectives of learning (e.g., Hamilton, Lesh, Lester, & Yoon, 2006;

Lesh & Zawojewski, 2007; Lester & Kehle, 2003), assumes that mathematical

understanding develops from a multi-dimensional perspective and that students’

final understandings of mathematics are a direct result of many influences within

the learning process. Van de Walle (2004) refers to this deeper understanding

as “relational” knowledge and attributes it with higher levels of intrinsic

motivation, improved attitudes and improved problem-solving competence of

students. According to Lesh and Zawojewski (2007), this more integrated and

multi-dimensional view of mathematical learning requires new methodologies of

teaching and warrants further research.

2.3.2 The Power of Teaching through Problem Solving

From the literature it can be seen that many researchers (e.g., Lesh &

Zawojewski, 2007) believe that problem solving is a powerful way to learn

mathematics, as opposed to simply a means to challenge talented mathematics

students or apply mathematical knowledge and procedures that have already

been taught in a traditional drill and practice fashion. English and her colleagues

(2005) supported this position and extended it to encompass problem posing and

solving using mathematical modelling when they said:

57

As children work interactively in solving the modelling problems they share

ideas, question one another’s claims, ask ‘what-if’ questions, contemplate

numerous decisions, and consider alternate courses of action. In so

doing, children generate important mathematical ideas and processes and

use a variety of representations to display their findings. The multifaceted

nature of these problems makes them ideal vehicles for advancing

children’s learning in numerous directions. (p. 11)

According to the National Council of Teachers of Mathematics (2000) the majority

of mathematics teachers remain supporters and practitioners of traditional

teaching methodologies. However, researchers have repeatedly called for

innovation and reform in mathematics education for over four decades (e.g., Ben-

Chaim, Fey, Fitzgerald, Benedetto, & Miller, April, 1997; Brown & Walter, 1983;

Bruner, 1965; Cattell, 1971; Collins, 1986; Ritchhart, 2002). Nearly forty years

ago, Freire (1970, p. 45) argued that education was suffering from “narration

sickness” whereby the teacher “narrates” the subject content to passive students

who are expected to memorise and regurgitate it. Little had changed by 1996

with Roberts (1996) referring to this pedagogy as the “banking” model where

teachers “deposit” ideas into willing receptacles (students). He lamented that it is

still widespread in modern education where “students are treated as acquiescent

automatons to be controlled in both thought and action” (Roberts, 1996, p. 2).

2.3.3 Can Problem Solving Drive Mathematical Reform?

Despite the limited success of reformists, optimists like Costa (2005) continued to

promote problem solving as a valid teaching strategy and encouraged teachers to

work toward producing “effective problem solvers [who] know how to ask

questions to fill in the gaps between what they know and what they don’t know”

(p. 5). One of the most ardent supporters of a “liberating” education, Costa, is still

“marketing” this form of education today:

58

We hope that students will learn how to take intellectual as well as

physical risks. Students who are capable of being different, going

against the grain of the common, thinking of new ideas and testing

them with their peers as well as teachers are more likely to be

successful in an era of innovation and uncertainty. (p. 8)

One of the apparent obstacles in the broad scale use of problem solving as a

teaching strategy appears to be the view by teachers that problem solving is not

a pedagogically sound or motivating approach for the introduction, development,

and application of mathematical concepts and skills. Rather, it is more commonly

seen as a tool to consolidate and reinforce mathematical knowledge and

procedures (Adams et al., 2000; Bobis et al., 2004). Earlier researchers such as

Pejouhy (1990), recognised that the destiny of students’ problem-solving

successes, and the success of any mathematics curriculum reforms, ultimately

lay in the hands of classroom teachers. To effect reform teachers must be aware

that understanding mathematics and doing mathematics are different (Canobi,

Reeve, & Pattison, 2003; Cassel & Reid, 1996) and researchers need to

construct new tools to measure the constructs required for understanding

mathematics (Lesh & Zawojewski, 2007).

2.3.4 Issues Related to the Assessment of Problem-Solving Competence

While the main emphasis in developing problem-solving competence has

surrounded the acquisition of problem-solving strategies (Bjorklund, 2000), there

are many students who are very good at numeration and computation but who do

not demonstrate a commensurate ability in problem-solving skills (English, 2003;

Sternberg, 2002; Thomas & Karmiloff-Smith, 2002). Students with these

particular characteristics are worthy of further research studies to determine the

nature of this phenomenon.

59

Recent research has shown that using the wrong strategy to solve a problem can

either slow the solution process or inhibit a student’s opportunity to solve

problems (e.g., Bjorklund, 2000; Cai, 2000; Ceci, 1996). An awareness and

understanding of strategies and when to use them can make the difference

between a student being a successful problem solver or not (e.g., Goswami,

2002; Thomas & Karmiloff-Smith, 2002). A number of researchers (e.g.,

Kramarski et al., 2002) have tried different interventions to assist students to

improve their problem-solving competence. Kramarski and his colleagues

investigated the notion of teaching students metacognitive strategies to assist

them to self-regulate and monitor their problem-solving processes. They

believed this form of instruction might have the additional benefit of improving

students’ self-efficacy with regard to mathematical problem solving. Bentley

(1996) had previously examined the benefits of teaching metacognitive practices,

and while this research focussed on middle-year students in Years 6 and 7, he

did not consider students of differing intelligence levels within his study. This is a

shortcoming in the research on problem solving and worthy of further

investigations.

2.3.5 Should Specific Problem-solving Strategies be Taught?

While the specific instruction of problem-solving strategies might therefore seem

prudent, we are reminded by Bobis, Mulligan, and Lowrie (2004) that:

the teaching of specific strategies may increase the likelihood of success

in problem solving but unfortunately teachers tend to model specific

strategies in whole-class situations. This can lead to the perception that

you need to solve a problem the way the teacher has completed it. (p.

47)

This is consistent with Lesh and Zawojewski’s (2007) findings. They reported that

there is insufficient evidence to suggest conclusively that prescribing the problem-

60

solving strategies of “experts” to “novices” is effective in improving the novices’

problem-solving performances. This was also reported earlier by Schoenfeld

(1992) and Lester (1994) in their literature reviews when they summarised their

findings by saying that teaching students to use general problem-solving

strategies was found to be unsuccessful and has done very little to improve the

ability of students to solve problems. In fact, Lester and Kehle (2003) suggested

that researchers have made much rumination, but few practical suggestions, to

improve school practice in this area and there is a need for further research to

address the unanswered questions on this issue. Two primary concerns reported

by several researchers were the “relative ineffectiveness of instruction to improve

students’ ability to solve problems” (Lester & Kehle, 2003, p. 510) and the decline

of research focussing on problem solving (Lester & Kehle, 2003; Stein et al.,

2003). Lesh and Zawojewski (2007) added their additional concern about the

complete separation of problem-solving activities from the learning of “substantive

mathematical concepts”. They alternatively emphasised a “synergistic

relationship” between the developmental learning of mathematics and problem

solving (Lesh & Zawojewski, 2007).

2.3.6 Students’ Understandings of Problem Structures

The research into the development of students’ awareness and understanding of

how problems are constructed began almost thirty years ago. Rumelhart (1980)

wrote about how students encoded problems using a rich, embedded schema.

He said that once the problem is encoded, understanding it and solving it, are

nearly the same thing. Van Essen and Hamaker (1990) also followed this line of

research and reported that representing the problem in a meaningful format is the

most important step in the problem-solving process, since it is during this time

that the student determines the actual structure of the problem. English and

Halford (1995) referred to the important role of “reasoning by analogy” in

identifying these structures. Such reasoning occurs when a student discovers the

“source” or “base” problem that has all the “base” elements that corresponds to

61

the stated problem, but without any of the superfluous, distracting information.

Some years later, Lester and Kehle (2003) summarised similar findings and said

that “good problem solvers know more than poor problem solvers and what they

know, they know differently – their knowledge is well connected and composed of

rich schemas” (p. 507).

Whether this form of analogical reasoning is available to students has been

debated over the years (e.g., Bernardo, 2001; DeLoache et al., 1998; Goswami,

1996). Followers of the Piagetian model would suggest that it is not available to

adolescents (Inhelder & Piaget, 1958) but more recent research suggests it may

be present from birth (Goswami, 1996) and is indeed the basis for problem

solving and other reasoning tasks (Bernardo, 2001). The conflict in findings from

this present review of literature suggests that more research is needed to

establish whether analogical reasoning is available to middle-year students as a

tool for cognitive development. This position is supported by Cai and Hwang

(2002) who said that, “the more information teachers can obtain about what their

students know and how they think, the more opportunities they can create for

student success” (p. 401). They believe that research into the teaching of

mathematics through problem solving can fill some of the gaps in the current body

of research.

There is “general agreement that new perspectives are needed regarding the

nature of problem solving and its role in school mathematics” (Lester & Kehle,

2003, p. 509). Lesh and Zawojewski (2007) and English, Fox and Watters

(2005) would like to see researchers focus their problem-solving investigations

on the bridge that links the problem-solving instruction in schools with the real-life

problem solving that occurs beyond the classroom. The ability of students to be

able to transfer their mathematical and problem-solving knowledge to real-life

situations is a goal of many researchers (e.g., English et al., 2005) and was a

particular focus of a report by the President of the NCTM, which proposed

62

“building bridges of mathematical understanding” (as cited in Price, 1996, p. 3).

Lesh and Zawojewski (2007) discussed this goal in their recent research and

wrote:

The notion of transfer is important when considering situated knowledge,

because one might argue that the knowledge and routines are developed

for only the situation at hand. However, in everyday contexts, people

quickly move from creating mathematical procedures to dealing with not

only the original context, but also isomorphic situations routinely ... (p. 36)

The findings of The Connected Mathematics Project, conducted at Michigan State

University and focussing on middle-year students, also valued the methods of

instruction that assist students to construct meaning and build effective networks

of skills and understanding (Ben-Chaim et al., April, 1997). The findings from the

two-year study indicated that the more time students spend working in the project,

the more their results in the state-run standardised mathematics tests improved,

and indeed the gap between their results and those students who were not

participating in the project widened. These findings provide evidence to refute the

claims of those who suggest that for students to succeed in broad-scale

standardised testing, teachers are required to devote the majority of class time to

the development of basic skills and less time to activities such as problem solving

(see Lipman, 2004; McNeil & Valenzuela, 2000).

The context in which students explored and solved problems was investigated by

Gravemeijer (1994) in the Netherlands during his longitudinal study entitled The

Realistic Mathematics Education Study. This study used everyday contexts to

pose problems to students that caused “internal conflict” which in turn lead to

students creating “abstract conceptualisations” of the problems. The findings

from the study recommended that familiar contexts when setting problems for

students to solve as it was more likely to result in higher intrinsic interest levels

63

and engagement by the participants. Following similar lines, but taking a more

conservative approach to the research undertaken in the Netherlands, the major

research projects in this area in the United States of America have focussed on

the more traditional methods for developing problem-solving competence that

included using concrete materials to conceptualise the problems (e.g., Ball &

Cohen, 1996; Cobb, 1997). One such study in Texas was developed following

the publication of the Third International Mathematics and Science Study (TIMSS)

results (Adams et al., 2000). The researchers in the study worked with 350

teachers from over 100 schools to determine which practices should be used to

promote mathematics and science reform in middle-school education. Not

surprisingly, the researchers reported that reform would require increased

emphasis on developing student’s problem solving and reasoning abilities, but fell

short of providing a detailed report of practical suggestions for teachers. These

findings suggest there is still a need for investigations into intervention strategies,

such as a problem-posing teaching experiment, to fill this void in the research of

the development of problem-solving competence.

In Australia, research projects such as the Literacy and Numeracy Development

in the Middle Years of Schooling Project (Luke et al., 2003) have investigated a

variety of issues surrounding the learning of mathematics. The aim of this project

was on the development of literacy and numeracy skills amongst middle-year

students. However, the report suggested in its findings that students in

classrooms where higher-order thinking tasks, such as problem-solving tasks,

were commonplace, and where the content was linked to real-life situations with

which the students were familiar, demonstrated superior numeracy skills to those

students in classrooms where lower order, more abstract tasks were prevalent.

2.3.7 Summary

There are repeated calls in the literature for the need for mathematics curriculum

reform (e.g., Adams et al., 2000; Groves, Mousley, & Forgasz, 2006; NCTM,

64

2000). The findings of these studies suggested that problem solving should

receive more intensive focus in the mathematics curriculum since it facilitates in

students a deeper understanding and connectedness of mathematical concepts

(e.g., Ben-Chaim et al., April, 1997). Problem solving is, after all, an everyday

activity, at some level, and a necessary skill for individuals to be able to adapt to

our dynamic society where change and innovation are considered to be routine.

The ability to reason by analogy, and recognise and transfer similar structures

between classes of problems, has been demonstrated as particularly effective in

improving students’ performances at solving problems (English & Halford, 1995).

Teaching students which strategies to use for problems of a particular structure,

has however, produced conflicting reports about its effectiveness and long-term

benefits to improve students’ problem-solving competence (e.g., Bobis et al.,

2004).

Problem solving is a skill that can be learnt by students of all achievement levels

in mathematics (e.g., Groves et al., 2006). Underachieving students have been

shown to develop quite sophisticated and complex strategies for solving problems

following intervention programs. The review of literature has highlighted multiple

directions for future studies. However, there is significant evidence to suggest

that further study in the use of intervention strategies with underachieving middle-

year students is particularly warranted. More importantly, there is significant

indication in the current research findings to suggest that not only can

underachieving students improve their problem-solving competence but they can

also “construct important mathematical ideas through solving novel problems”

(Groves et al., 2006, p. 200). To fill the void in the research, a future study with

underachieving, middle-year students who have been ascertained with above

average intelligence would provide evidence to move this area of research

forward.

65

Research has been conducted into student attitudes to solving novel problems

based on real-life and abstract scenarios (e.g., Bjorklund, 2000; English & Larson,

2005; English et al., 2005), and the findings of these reports suggest that making

connections between classroom mathematics and real-life mathematics has a

positive impact on student outcomes. In addition, discussion surrounding the

facilitation of such teaching practices indicates that some changes to classroom

teacher goals and teaching methodologies may be required in order to integrate

meaningful problem-solving activities into the current curriculum (e.g., Costa,

2005; Lester, 2003; Ritchhart, 2002). It became apparent when reviewing the

literature for this section, that there were a number of reports into the merits of

problem posing by researchers who were investigating ways to develop students’

problem-solving competence. A review of the literature pertaining to problem-

posing perspectives can therefore be found in the following section.

2.4 Problem-posing Perspectives

The formulation of a problem is often more essential than its solution,

which may be merely a matter of mathematical or experimental skill. To

raise new questions, a new possibility, to regard old problems from a new

angle, requires creative imagination and marks real advances. (Albert

Einstein, n.d.)

These words from an eminent and past mathematician encapsulate the contents

of this following section. Einstein’s words have held meaning for recent

researchers who also believe that providing students with opportunities to pose

problems can provide them with meaningful learning opportunities that can result

in improved problem-solving competence (e.g., Brown & Walter, 2005; Lesh &

Doerr, 2003; Lesh & Zawojewski, 2007). This section reports on the literature

surrounding problem posing with particular emphasis on problem posing as a life-

long learning tool and for mathematical reform. Literature reporting on how to

66

foster a problem-posing environment in the classroom is then reviewed followed

by a review of the connections that have been reported in the literature between

problem solving and problem posing.

2.4.1 Introduction

According to Silver (1994), problem posing refers to one of three distinct

mathematical activities:

1. Pre-solution posing, whereby a student poses a question from given or

perceived stimuli;

2. Within-solution posing, whereby a student reformulates a given problem

while it is being solved; and

3. Post-solution posing, whereby a student changes the conditions or goal

of a problem at the end of the solving process in order to generate more

problems.

In other words, problem posing can happen before, during or after problem

solving (Silver, Mamona-Downs, Leung, & Kenney, 1996). However, problem

posing involves far more than simply creating new problems. For example, it

provides students with opportunities to develop creativity (Silver, 1997), improve

self-efficacy, (Bandura, 1997; Marat, 2005) and to share problems with peers,

have them critiqued and hence refine problems using peer feedback (English,

Cudmore, & Tilley, 1998).

2.4.2 Problem Posing as a Tool for Mathematical Reform

“Problem posing … has the potential to create a totally new orientation toward the

issue of who is in charge [of the learning process] and what has to be learned”

(Brown & Walter, 2005, p. 5). Brown and Walter (2005) believed that problem

posing can change the orientation of mathematics lessons from a goal of finding

the correct answer and recording an appropriate method, to finding the correct

67

question from a infinite array of imaginative possibilities. In fact, Lesh and

Zawojewski (2007) reported that the development of systems for interpreting

problems is equally as important, if not more so, than simply developing problem-

solving processes, while Lesh and Doerr (2003) reported the ability to “see” a

problem is as important as the ability to “do” a problem. With such strong support

from researchers, one would imagine that a revolution in mathematics education

might have already taken place. However, Ritchhart (2002, p. xxi) lamented that:

The fact is that most schools today do not try to teach for intelligence.

Rather than working to change who students are as thinkers and

learners, schools for the most part work merely to fill them up with

knowledge ….This form of schooling serves to bind rather than free the

mind.

Brown and Walter (2005) say that rather than students looking inward at

problems, through a lens with which they have become accustomed to use, and

which narrows their focus, students should be concentrating on developing

divergent forms of thinking which may present unexpected discoveries and new

and deeper opportunities for learning. This would create a major shift for the

students from a state of being to a state of becoming. As an example, Brown and

Walter (2005) cite the following problem:

Given two equilateral triangles, find a third one whose area is equal to the

sum of the areas of the other two. (p. 112)

This problem stimulates a plethora of problem-posing opportunities in order to find

a solution. Students need to ask themselves questions about the dimensions of

the triangle’s sides, the ratio between the side lengths of each of the triangles and

what variables they may use to reference the triangles and their dimensions.

Along the way, attributes of equilateral triangles will need to be revisited by the

68

students, as will algebraic theory. This is a perfect example of the type of

learning that encourages reflective practice whereby students continually cycle

through iterative steps of thinking when they detect weaknesses in their previous

solution strategies and thinking (Hamilton, Lesh, Lester, & Yoon as cited in Lesh

& Zawojewski, 2007).

It can be seen from this example that much discussion and questioning would

need to occur within and between students before a satisfactory method of

solution and justification is determined. These forms of learning experiences

have been attributed with combating mathematics anxiety since posing and

solving your own problems is far less threatening than simply solving someone

else’s problem (e.g., Brown & Walter, 2005).

2.4.3 Problem-posing Skills for Lifelong Learning

Patton (2002) believed that problem posing was not merely an activity to be

undertaken in a classroom by students. He saw far reaching applications of

problem posing for adults and indeed for the advancement of our society. He

referred to problem posers as “problem pioneers” and defined them as

“individuals with a problem that is not only new, but that is destined to be

important to an entire field or the community at large some time in the future” (p.

111). He saw problem posing as an essential aspect of planning for the future

and he maintained that “those who concern themselves ahead of time with the

effects of a potential calamity such as war, famine, earthquakes, landslides, or

hurricanes are in effect problem pioneers” (p. 123).

From less dramatic settings, Patton (2002) cited several examples of problem

pioneers who have made substantial contributions to society. He talked about the

world-renowned paediatrician and author, Benjamin Spock, who posed many

questions, and solutions to questions, about how parents cope with breakdowns

in family ties as a result of raising a family in today’s society. Patton also

69

discussed Freeman Dyson who posed significant questions about the research

being undertaken by two independent radiation scientists. The questions Dyson

posed allowed each radiation scientist to see the merit in the other researcher’s

work and inspired them both to move forward together to eventually discover a

new theory of radiation, for which the two scientists then received a Nobel Prize.

These examples, and others in Patton’s paper, serve as significant motivation to

educators and the broader community to value the activity of problem posing and

to ensure that students are provided with ample opportunities to practice and

hone these skills while they are still at school.

Problem posing is a natural inclination for all people; however, some people are

more naturally inquisitive and inclined to pose more problems than others (Costa,

2005; Gonzales, 1998). Inquisitive people might pose questions like; “Why can’t

we see wind?” or “When will humans first inhabit a planet other than the Earth?”

or “What makes the waves roll up onto the beach each day?”. In fact, Costa

(2005, p. 7) believed that, “all human beings have the capacity to generate novel,

original, clever or ingenious products, solutions, and techniques – if the capacity

is developed”.

There are many examples in history of famous, inquisitive people who posed

very important questions that were later investigated by many researchers, and

for many of which we now have the answers. One such example is that of Albert

Einstein who in 1911 posed the question; What if light does not travel in straight

lines but is indeed deflected around objects of huge mass such as the sun? (as

cited, in Kline, 1996) It was not until eleven years later, in 1922, that his original

question received its first independent, experimental verification and Einstein was

finally heralded worldwide as the brilliant mathematician and physicist that he

was. Had he not posed the question all those years earlier, it is unclear where

our current theories on radiation might be today. Einstein and Infeld (1938)

believed that posing problems was one of the key components of exploring

70

mathematics. They said that in scientific enquiry, posing a problem well was

often a more important task than solving the problem.

2.4.4 Fostering a Problem-posing Environment

Stoyanova (2003) suggested that it was not only Einstein that supported the

notion of problem posing from as far back as the early twentieth century. She

said that “prominent scientists like Einstein, Darwin, Wertheimer, and many others

have placed far greater emphasis on the importance of posing significant

questions than on attempts to solve them” (p.32). Despite the historic discussion

of the significance of problem posing, it has only been recently that problem

posing has begun to receive broad based and renewed attention by researchers

(e.g., Brown & Walter, 2005; Cai, 2003; Contreras, 2003; Crespo, 2003; English

et al., 2005).

Although these reports, and others such as the those published by the NCTM

(2000), have championed the cause for an increased focus on problem-posing

activities for mathematics students, sadly it seems that problem-posing activities

still receive limited if any regular attention (Bobis et al., 2004). Despite this,

researchers like English (1997b) and Lowrie (2002) continue to propose that

greater emphasis be given to problem posing, particularly within the framework of

realistic contexts. On discussing the promotion of a problem-posing classroom,

English (1997b, p. 173) said “one of the main strengths of such an environment is

that it can empower all students to explore problem situations and to pursue lines

of inquiry that are personally satisfying”. Knuth and Peterson (2002, p. 579)

added their support to these comments when they said that “providing students

with ample opportunities to engage in problem posing strengthens their problem-

solving abilities and, perhaps more important, promotes the development of their

mathematical thinking”.

71

2.4.5 Connections between Problem Solving and Problem Posing

It is widely accepted that problem solving is a worthwhile activity in which

students should be actively engaged (e.g., English & Larson, 2005; Lester &

Kehle, 2003; NCTM, 2000). The level of success at solving problems varies

dramatically between students (e.g., Lesh & Zawojewski, 2007), and a variety of

strategies have been trialled to address this issue (e.g., Brown & Walter, 2005;

English et al., 2005). While problem posing has been investigated in recent

years by a number of authors, those investigations have focussed more on

students from a specific year level (e.g., Lowrie, 2002), a particular strand of

mathematics (e.g., Contreras, 2003; Lavy & Bershadsky, 2003), or on the

process of problem posing in general, without consideration of the particular

cognitive attributes of the students within the study groups (e.g., Brown & Walter,

2005; Cai, 2003). In addition, the majority of the research conducted on middle-

year students was conducted in the 1990s (e.g., Gonzales, 1994; Silver & Cai,

1996), thus creating a void of recent research into the development of problem-

solving competence of middle-year students.

Over the past thirty years, significant research progress has been made in many

aspects of problem solving. However, according to Stoyanova (2003) “in

mathematics education research, problem posing has long been under the

shadow of problem solving” (p. 32). Silver and Cai (1996) concurred with

Stoyanova and added that “far less is known about the cognitive processes

involved when solvers generate their own problems or about the instructional

strategies that can effectively promote productive problem posing” (p. 522). In

their research study, Silver and Cai worked with 509 middle-year students, who

represented the range of student intelligences found within a mainstream school

setting, and investigated the correlation between the students’ problem-solving

and problem-posing performances. Their findings were convincing as they

reported a high correlation between the students’ problem-solving and problem-

posing performances throughout the eight open-ended tasks that were used to

72

determine the students’ mathematical problem-solving abilities. This is consistent

with an earlier report by Silver and Jinfa (1996) who reported that middle-year

students, who were regarded as good problem solvers, were the ones who

generated the more complex and challenging problems, compared to their weak

problem-solving peers.

Rather than looking at differences in problem-solving ability, other authors such

as Cai and Hwang (2002) and Becker, Sawada and Shimizu (1999) have

considered cultural differences that impact upon a student’s ability to pose and

solve problems, with particular emphasis on differences between Asian and

American students. Interestingly, these studies showed that Asian students are

far more likely to use symbolic representations in their methods of solutions, while

American students are far more likely to seek out concrete representations to

assist them with their problem posing and solving. In addition, Cai and Hwang

found a high correlation between problem-solving and problem-posing ability of

Chinese students, but did not find a similarly strong correlation amongst the

American students in the study. These findings are in direct contrast to those

found by Silver and Cai (1996).

Researchers, such as Mestre (2000), have suggested that problem-posing

activities provide far more opportunity for students to demonstrate their

mathematical understandings than do solving problems. When solving problems

he reported that:

73

the subject is provided with a problem that not only has been formulated in

advance by an expert, but also is well defined and solvable, [whereas]

problem-posing requires that the subject perform the job of the expert in

constructing a suitable problem, a job that entails combining a viable story

line with appropriate surface features in ways that embody specific

concepts. Hence, to do well at posing problems, the subject must be well

versed in how concepts apply across a wide range of problem contexts,

whereas it is possible for the solver to avoid the meaning of concepts

altogether by solving problems by means-ends analysis. (p. 160)

He goes on to recommend that teachers should not only provide problems for

students to solve, they should also provide situations or contexts within which to

construct a variety of problems to solve.

Problem-posing research has also included a focus on the pre-service teachers’

perspectives, with the premise that if pre-service, mathematics teachers have

been introduced to the underlying theory and supporting literature of problem

posing during their internships, then they may facilitate this style of teaching in

their classrooms when they become fulltime teachers (Crespo, 2003; Gozales,

1994). Crespo (2003) reported success in this endeavour when she found that

the pre-service teachers who had adopted the new teaching practices

incorporating problem-posing strategies, viewed mathematical problems as far

more useful in providing challenges for students and for probing their

mathematical thinking.

Fosnot and Dolk (2001) were also interested in what mathematics teachers did in

the classroom. They undertook a meta-study of recently completed research to

investigate what had been discovered about effective practice in the modern

mathematics classroom. As a result of their study, they developed the notion of

74

‘situations for mathematizing’ as being a useful focus for teachers. In a middle-

school environment where teachers are able to teach across syllabuses, they

recommended teachers be on the constant lookout for these situations as they

encourage students to ask mathematical questions, look for and see patterns,

and become inquisitive. In particular, they said that by students posing their own

smaller problems, in the process of solving bigger problems, the bigger problem

was “owned” and it became alive and meaningful to the students.

This transdisciplinary approach to problem posing may serve to build the

connections between “school maths”, intuitive knowledge and everyday, familiar

contexts. These were noted by English (1996) as lacking from students

participating in her study. One of the recommendations from her report

suggested “getting children into the habit of recognising mathematical situations

wherever they might be” (p. 238). Bernardo (2001) followed a similar research

path and reported that “an important objective of education is to increase a

student’s capability to competently address varied problems in a changing

environment” (p.137). This was also discussed in Section 2.3. He said that one

of the challenges for students in meeting this goal is the ability to transfer

knowledge and skills from one problem in one particular setting, to another

analogous problem in another type of setting, thus creating a mapping between a

familiar problem and a problem that is being constructed (Bernardo, 2001).

Bernardo (2001), and other authors (Cai, 1997; Silver & Cai, 1996), have cited

students’ lack of understanding of problem structures as the inhibiting factor in

developing these transfer skills. He investigated a problem-posing intervention as

a means to assist students to engage more readily with problem structures. This

experience, although limited to word problems associated with probability,

allowed the high school students in his study to recognise common characteristics

in the structures of various analogous problems and hence assisted them to

become more competent at solving problems based on probability.

75

2.4.6 Summary

Within this review of problem-posing literature, it has become apparent that

problem-posing activities should be seen an essential part of a mathematics

curriculum and as an important activity in the community at large. In this way,

teachers can assist their students to become “mathematicians in a mathematics

community” (Fosnot & Dolk, 2001) and make links between theory and the

practice (Mestre, 2000). Mestre (2000, p. 167) supported his position by saying

that “in traditional physics instruction, we train students to solve many problems,

believing that in solving problems correctly, the student also understands the

underlying concepts. … Instructors often delude themselves that problem-solving

proficiency implies conceptual understanding.”

From a student’s perspective, reports from the research suggested that a

problem-posing intervention has not only the ability to impact positively on

students’ problem-solving outcomes, but it can also have a positive effect on

students’ self-efficacy, creativity, sense of ownership of the learning process, and

divergent thinking. Despite the persuasive nature of this research, Silver’s (1994)

findings drew our attention to the fact that there was insufficient research on how

students respond to problem-posing intervention programs. This present study

addresses this issue.

Problem-posing research has been conducted with students, pre-service

teachers, and teachers. Authors such as Gonzales (1998) have told us that

problem-posing skills develop naturally for students when they are investigating

mathematical scenarios. Research foci on problem-posing studies have included

cultural difference, context, strategy preference, mathematics anxiety, teaching

style and student enjoyment of the learning process. While some smaller

research projects have been undertaken looking at differences between

participant students of higher and lower problem-solving ability, there appears to

be little research focussed on students who have been ascertained as high-

76

achieving in standardised intelligence tests, yet underachieving in the problem-

solving criterion of school-based mathematics tests. In addition, there is some

disagreement throughout the research about whether a consistent, positive

correlation exists between a student’s problem-solving and problem-posing

abilities in middle-year students (e.g., Silver & Cai, 1993b). This study contributes

to the shortcomings in the body of knowledge regarding both of these issues.

2.5 Student Underachievement Perspectives

While problem-posing intervention, as a suitable strategy to assist in the

development of problem-solving competence, was a primary focus of this

research, a secondary focus was the group of middle-year students who

underachieve in problem solving yet who demonstrate superior scores to their

peers in mathematical intelligence tests. Therefore, the themes emerging from

the literature surrounding underachievement in middle-year students are reported

in this section.

Underachievement of students is all about potential and little, if anything, to do

with a lack of ability (Jones & Myhill, 2004). The view adopted throughout this

present study was that underachieving students have latent, unrealised talents

that may be impacted upon by appropriate intervention strategies. Delisle (1992)

made a salient distinction between “gifted non-producers” and “gifted

underachievers” that is worthy of mention before continuing with this review. He

noted that “gifted non-producers” are academically, but not psychologically, at

risk. They are typically self-assured and have independently chosen to withdraw

from the set work due to its irrelevance to them or due to boredom (see also

Gentry, Gable, & Springer, 2000). They are frustrated individuals who have

choices to make concerning how they react (Schultz, 2000). Schultz (2000, p.

42) asked “Do they change their habits to fit in with the crowd, or flex their

independence and stand up for their specific needs?”. He says teachers rarely

cater adequately for the “under-challenged” student and that teachers generally

77

respond negatively to a student who says they are not learning anything new in

their class. This results in “a power struggle that leads ultimately to the

emotional and social downfall of the under-challenged student” (p. 43). By

contrast, “gifted underachievers” often do not complete work because of low self-

esteem or because they are dependent learners requiring ongoing instruction

and reinforcement from a teacher. In a group of students who have been

ascertained as above average in intelligence, yet who appear to have

underachieved in aspects of their mathematical testing, it is worth noting that

both types of underachievers may exist within the group.

Gender differences in students have also been reported upon in recent studies

on underachievement. Jones and Myhill (2004, p. 531) reported that:

the identity of the underachiever has become synonymous with the

stereotypical identity of boys. Teachers know what underachievement

looks like: it looks like a boy who is bright, but bored. … [whereas]

underachievement in girls is often overlooked or rendered invisible. … It

becomes a matter of concern if teachers perceive boys as the vessel of

potential and of latent ability, while the high achievement of girls is seen to

be about performance, not ability.

Reis and Siegle (2006) also undertook a study that focussed on gender

differences and made a contrasting comparison. They reported that girls who

underachieved almost always had serious boyfriends who came first, before

school work, whereas boys who underachieved often got into trouble and spent

hours playing videogames or watching television. These findings were followed

a year later by an international study conducted on 1,700 German, Canadian and

Israeli middle-year students. Boehnke (2007), investigated whether

underachievement was caused as a consequence of high peer pressure. With

the assumption that students underachieve to avoid social exclusion, Boehnke

78

reported that the primary victims of peer pressure that resulted in

underachievement in mathematics were girls, while the perpetrators of the peer

pressure were more likely to be boys. These gender differences are useful to

consider when undertaking future studies about underachievement of students in

a co-educational setting.

Whether we are considering a male or a female student, according to Reis and

Siegle (2006), there are three degrees of underachievement:

1. Pervasive and devastating – which results in student drop-outs and “life

failures”;

2. Moderate – which results in failing grades in some areas; and

3. Minimal – which results in lower grades than expected across some or all

subjects.

The literature suggests that it is members of these last two categories that are

most able to be positively influenced by the adaptation and modification of

situational variables in and out of the school setting (e.g., Cummins & Sayers,

1995; Gootman, 2001; Simons-Morton, Crump, Haynie, & Saylor, 1999).

Dispositional factors such as gender (e.g., Reis & Siegle, 2006; Younger &

Warrington, 1996), boredom (e.g., Kanevsky & Keighley, 2003), and

computational proficiency (Johnson, 2000) require individual intervention

strategies for maximum improvement of student performance and as such, are

not part of this present study. However, since attention to the common

situational factors, such as pedagogy and classroom structure, have been shown

to have a positive influence on students affected by dispositional factors (e.g.,

Gootman, 2001; Kanevsky & Keighley, 2003; Simons-Morton et al., 1999), a

group-intervention process may be an appropriate strategy for use in further

research.

79

Family, school, and the unique attributes of preadolescents impact upon the

development of underachievement (Baker et al., 1998). Researchers such as

Cummins and Sayers (1995) suggested that underachievement in mathematics

is commonly the result of traditional teaching pedagogy whereby the subject is

“taught sequentially through drill and memorisation” (p. 145) with a classroom

structure adopting an emphasis on straight rows of desks and chairs, equanimity,

and acquiescent behaviour. Cummins and Sayers considered this form of

learning to “encourage passivity, and to be ineffective and xenophobic” (p. 145),

and is particularly unhelpful to primary and middle-year boys (Jones & Myhill,

2004; Younger & Warrington, 1996). Kanevsky and Keighley (2003, p. 1)

supported this position and indicated that some students develop a “growing

sense of moral indignation” toward this style of pedagogy and respond by

disengaging from the learning process.

This review on recent literature has uncovered a varying range of views

regarding the underlying causes of underachievement in middle-year students.

Some researchers (e.g., Boehnke, 2007; Jones & Myhill, 2004; Reis & Siegle,

2006) have focussed on the gender differences while others (e.g., Cummins &

Sayers, 1995; Gootman, 2001; Kanevsky & Keighley, 2003) have categorised the

differences into situational and dispositional factor categories. With general

agreement from researchers that middle-year students are particularly vulnerable

to underachievement in mathematics, it could be expected that a large body of

research may have already reported specific teaching strategies and

interventions to assist these students to reach their potential. Some guidance for

future research may be found from researchers of developmental learning, such

as Lincoln and Denzin (2000) who have reported that empowering students to be

in control of their own learning may address issues of underachievement and

disengagement of learners, or from Bjorklund (2000) who emphasised the

importance of students playing an active role in their own education. If this line of

80

advice is to be followed a constructionist (as opposed to constructivist)

theoretical perspective would be helpful as a foundation for future research.

2.6 Constructionist Perspectives

According to Crotty (1998), the predominant paradigm of research in the western

world has it roots in ancient Greek philosophy and has previously focussed

around a belief that “objective truth” combined with suitable methods of inquiry,

can lead us to “accurate and certain knowledge of that truth” (p. 42). The

assumption then is that truth and meaning are contained within objects and

artefacts and are independent of human consciousness, interaction and thought.

This stance is rejected by followers of constructionism (e.g., Schwandt, 2001)

who prefer to view truth and reality as being constructed through the interactions

between human beings, and conveyed within a generally social context, as is

clearly the case with students in a classroom. This is particularly relevant in the

learning of mathematical problem solving as the development of mathematical

concepts and problem-solving competence are more interdependent,

contextually situated and socially constructed than traditional theorists have lead

us to believe (Lesh & Zawojewski, 2007). Indeed, recent research has reverted

to a focus on the social perspectives of group learning of mathematics as a

means to investigate how students develop cognitive models and ways of

thinking (e.g., Greeno, 2003; Zawojewski, Lesh, & English, 2003). However,

rather than the main learning of mathematical concepts arising from student-

student interactions, with the teacher as a facilitator to ensure the intended

learning occurs, it is still the case that most learning in mathematics classrooms

is dependent solely on teacher-student interactions (Lesh & Zawojewski, 2007).

Constructionism began to emerge as a theoretical perspective in the 1980s as a

result of a series of computing technology projects undertaken by Seymour

Papert of the Epistemology and Learning Group at the MIT Lab (see Papert,

1980). He used the foundation of constructivism that supports learning as

“building knowledge structures” and added the concept that learning occurs most

81

felicitously when the learner is consciously involved in constructing a “public

entity” (Papert, 1991). Constructivism, as Piaget and Inhelder (1969) saw it,

required knowledge to be passed from one person to the next as a whole,

whereas Papert’s view of constructionism held that knowledge was constructed

by individuals in the process of their practice and was not contained solely within

the mind of an individual (Papert, 1991). A fundamental premise of

constructionist theory is therefore that best learning occurs when students are

actively engaged in creating objects that are meaningful to them and that they

can share with their peers (Crotty, 1998). According to the NRC (as cited in

Kilpatrick et al., 2001, p. 135) “to become proficient, they [students] need to

spend sustained periods of time doing mathematics – solving problems,

reasoning, developing understanding, practicing skills – and building connections

between their previous knowledge and their new knowledge”. However, in

mathematics classrooms, it is clear that students do not often explore and create

novel problems of their own, about topics of interest to them, or share them with

their peers (Lesh & Zawojewski, 2007).

As mentioned earlier, the embedded theory of knowledge in constructionism is

founded on the principle that all knowledge “is contingent on human practices,

being constructed in and out of interaction with human beings” (Crotty, 1998, p.

42) and brings subjectivity and objectivity together in a way that no other single

theory does. Therefore, constructionists believe that understanding and

interpretation come from construction rather than discovery alone. This

construction centres on the “collective generation of meaning” facilitated by the

experiences of the learner (Schwandt, 2001 p. 42). For a constructionist study to

investigate developmental and affective changes in middle-year students, two

theoretical frameworks can provide a clear structure around which to develop

further research projects; the post-modern theoretical framework and the critical

theoretical framework. Critical postmodernists hold the view that as researchers

we do not simply seek to investigate our world; we seek to create change as a

result of our investigations and findings (Tierney, 1997). Indeed, as Tierney

82

wrote, it is the role of the critical postmodernist researcher “to assemble new

practices, languages and ways of seeing and hence acting in the world so that

individuals and groups will not of necessity need to subsume their identities into a

homogenous mass” (p. 24). A teaching experiment, incorporating a problem-

posing intervention, could provide such an opportunity for students to create their

own learning, through posing and publicly sharing their own problems, in a forum

that supported, encouraged and celebrated a student’s individuality.

Throughout a teaching experiment, a constant awareness remains that

conducting research with child participants is inevitably based on a perceived

power differential between the participants and the adult researcher and that the

identities of both the researcher and the participants frame the social situation

created by the intervention (Broido, 2002). This awareness is important when

conducting an experiment and analysing the data, as the research is not

independent of the researcher’s and participant’s assumptions about the

research process and anticipated findings of the research study. Consistent with

the framework of critical theory, the researcher must attempt to empower the

participants to take control of their learning experiences and hence have a sense

of “emancipation” from the normal constraints of the teacher/student relationship

(Lincoln & Denzin, 2000).

Issues of social justice, power and identity are at the base of critical theory. As

Lincoln and Denzin (2000) write, “critical theorists claim that society, in its current

form, is oppressive” (p. 1056). While the word oppressive may seem overly

dramatic when discussing a middle-year mathematics classroom, it is clear that

the traditional teaching methodologies, employed by practising mathematics

teachers, have the “power” in the classroom held predominantly by the teacher

(e.g., Roberts, 1996; Stein et al., 2003). From a management and safety

perspective, it is important that the adult in charge is in clear control of what is

occurring in the classroom. However, this does not necessarily preclude the

students from sharing in the decision making process of how learning can occur

83

and indeed in playing a significant role in customising the learning experience for

themselves. A problem-posing intervention, where students choose how they

want to pose and share problems, could contribute to addressing this issue and

could be used as a means to partly restructure a power base in the classroom. A

teaching experiment could therefore promote a liberating and transformative

learning environment (Broido, 2002) that dismantles traditional oppressions, thus

paving the way for “jointly constructed knowledge” and “respectful participation”

(Alvesson & Skolberg, 2000).

Together, the underlying premises and truths, held in the critical and postmodern

theories, define the relationship between the researcher and the participants as

being interactive, subjective, and interdependent. They acknowledge the reality

of the mathematics classroom as being pluralistic, complex, and difficult to

quantify. The theories acknowledge that research is context specific and that

any findings and interpretations of the data are most appropriately viewed in such

a context.

2.7 Conclusion

This review of the literature has considered five fields of research; cognitive

developmental theories, problem solving, problem posing, student

underachievement, and constructionist perspectives. There are shortcomings in

all five fields of research that leave open a possible focus on underachieving,

high-ability, middle-year students, and the correlation between a problem-posing

intervention and the development of the students’ problem-solving competence.

The review of research into cognitive developmental theories demonstrated that

students learn best when they are actively engaged in their learning (e.g.,

Bjorklund, 2000). For the learning of mathematics, Kelly and Lesh (2000)

argued that:

84

Doing mathematics involves (more than anything else) interpreting

situations mathematically, that is, it involves mathematizing. When this

mathematization takes place, it is done using constructs [conceptual

models]. …These constructs must be developed by the students

themselves; they cannot be delivered to them through their teachers’

presentations. (p. 215)

The need for curriculum reform in mathematics teaching has been heralded for

many years, with recent advocates such as Lester (2003) and Groves and her

colleagues (2006) supporting the position stated in the reports by the NCTM

(2000) and the NRC (2005) that both call for problem solving to receive far more

attention than it currently does in mathematics curriculums. The review provides

evidence to suggest that problem-solving activities are a rich form of instructional

strategy for the development of skills in mathematics (e.g., English, 2002). While

there is little difference of opinion by researchers to say that problem solving

should be an integral part of a mathematics curriculum, the most appropriate way

to develop problem-solving competence has created conflict amongst

researchers. Some suggest that the most traditional method of teaching problem

solving, through teaching strategies alone, does not necessarily result in

improved problem-solving competence (e.g., Bobis et al., 2004), and the NRC

(2004) warns us that significant classroom time can be lost if student-focussed

activities are allowed to become unfocussed. Despite this warning, other

researchers suggest that teaching problem solving through problem-posing and

modelling activities is worthwhile (e.g., English et al., 2005), and the NRC (2005)

acknowledges that issues, such as losing valuable class time, are not

insurmountable.

Silver (1997) was one of the researchers who investigated possible correlations

between problem-posing activities and the development of problem-solving

competence, calling for further research into a possible connection. There is

85

disagreement among the researchers about the strength of the correlation

between problem posing and problem solving. Earlier researchers such as Silver

and Cai (1993a) were not convinced about the link whereas more recent

researchers reported that the correlation was evident through their research (e.g.,

Lavy & Bershadsky, 2003; Lowrie, 2002). This review of the literature has

demonstrated that further research into this possible correlation is warranted

(e.g., Gonzales, 1998).

Consideration has also been given to the research surrounding the possible

reasons for underachievement of middle-year students. Groves and her

colleagues (2006) looked at the attributes of the students they were investigating

and reported that underachieving students are capable of developing

sophisticated strategies for solving problems. Cummins and Sayers (1995)

reported that underachievement is commonly the result of a traditional teaching

pedagogy, while Jones and Myhill (2004) found the physical arrangement of the

classroom played a significant role in the underachievement of middle-year boys.

Reis and Siegle (2006) investigated the varying degrees of underachievement

and how they may individually be addressed, while Gootman (2001) and others

(e.g., Simons-Morton et al., 1999) investigated the impact of situational and

dispositional variables in underachievement. Kanevsky and Keighley (2003)

continued the research into situational and dispositional variables, finding that

improvements in underachievement for students demonstrating dispositional

variables, were less likely to occur through group intervention than individual

attention. They did however, support group intervention as a beneficial strategy

for addressing situational variables. Despite this consideration of research into

why students underachieve, there was little research to be found on practical

strategies to assist underachieving students with high ability in mathematics. This

is an area in need of further research.

86

The review of constructionist perspectives established a need for research into

teaching practices that allow students to become more active and responsible for

their own learning. A teaching experiment, involving problem posing with

students as authors, could provide an opportunity for them to become more

engaged in the learning process. In addition it could provide students with an

opportunity to be acknowledged as individuals in terms of how they prefer to

construct their own learning (Gardner, 1999b). The traditional power base in the

classroom would become shared between the facilitator and the participants.

This sharing of power is a strong tenant of critical theorists (e.g., Broido, 2002)

who suggest it is a requirement for improved performance of students who may

currently be underachieving due to their disenfranchisement from the learning

process.

In summary, the need for students to be competent at solving problems is a

recurring message in the literature, suggesting that teaching practices that

facilitate problem-solving opportunities are to be encouraged. There is a need

for further research to be conducted into ways students can improve their

problem-solving competence and into the links between a problem-posing

intervention and the development of problem-solving competence, which are

currently unclear. In addition, little consideration has been given to students who

underachieve but are of high ability, in the studies of problem solving. This

particular group of students could therefore become a focus group of future

studies.

87

Chapter 3

Research Design

3.1 Chapter Overview

The purpose of this present study was to investigate and explain the links

between a problem-posing intervention and the development of problem-solving

competence of middle-year students who have been ascertained as above

average in standardised intelligence tests, yet below average in the problem-

solving criterion of their school mathematics tests. This chapter outlines how this

study was structured to achieve this aim.

Five main sections comprise this chapter. The first section provides an

introduction to the design of the present study (see Section 3.2) while the second

section provides the reasoning that lead to the establishment of the three specific

research questions (see Section 3.3). The third section contains the rationale

and structure of the research design and the selection process of participants

(see Section 3.4), while the fourth section of this chapter describes the methods

for the collection and analysis of the data and the associated considerations that

were given to reliability, validity and ethical issues (see Section 3.5). The final

section presents a conclusion of this chapter (see Section 3.6).

3.2 Introduction

A compelling theme arising from the literature was that students appear to

experience better outcomes when they are actively engaged in the learning

process (e.g., Bjorklund, 2000). The philosophical foundations of

constructionism are centred on this premise and extend to a student’s

construction of objects, such as problems, that can be shared with others

88

(Papert, 1991). This philosophy of learning provided an ideal framework within

which to construct the design of this present study.

The review of the literature in Chapter 2 considered five fields of research;

developmental learning theories, problem solving, problem posing, student

underachievement, and constructionist perspectives. The review uncovered a

number of shortcomings in these fields of research. It was discussed in Chapter

1 that traditional practices of teaching mathematics have been held responsible

for the marginalisation of some students and, as a result, there have been

repeated calls for a review of mathematics pedagogy (e.g., English, 2002; Lesh &

Zawojewski, 2007; NRC, 2004). Research that can inform new teaching practices

is an area needing further attention by researchers, while teaching practices, that

actively engage students in directing their own learning, have been suggested as

a useful focus for future research studies (e.g., Bjorklund, 2000). A student-

centred approach to learning, underpinned by a ‘Critical’ theoretical perspective,

was therefore incorporated into the design of this present study and will be

discussed in more detail later in this chapter.

The literature review provided evidence to suggest that problem-solving activities

are a rich form of instructional strategy for the development of skills in

mathematics (e.g., English, 2002), yet there is a decline in research focussing on

problem solving (Lester & Kehle, 2003; Stein et al., 2003). Lesh and Zawojewski

(2007) have however, pursued their investigations on this topic and expressed

concerns about “substantive mathematical concepts” being taught in isolation of

problem-solving activities. They emphasised a synergistic relationship between

problem solving and learning. While the focus of this present study is on

developing problem-solving competence in apparent isolation of the student’s

mathematics curriculum, it is through the focus of developing problem-solving

competence that ultimately students may become more successful at learning the

“substantive mathematical concepts” that Lesh and Zawojewski discussed in their

report. In this way, it can be justified to withdraw students from their normal

89

mathematics classes to become involved in the problem-posing intervention

incorporated into this present study.

While there is little difference of opinion by researchers to say that problem

solving should be an integral part of a mathematics curriculum, the most

appropriate way to develop problem-solving competence has created conflict

amongst researchers and warrants further investigation (see Chapter 2). Despite

some researchers suggesting that significant classroom time can be lost if a less-

traditional approach to teaching mathematics is pursued and student focus on the

task at hand is lost (e.g., Bobis et al., 2004; NRC, 2004), other researchers

suggested that teaching problem solving through problem-posing activities is

worthwhile (e.g., English et al., 2005; Lavy & Bershadsky, 2003; Lowrie, 2002). A

consistent viewpoint of whether or not a problem-posing intervention can assist all

students to develop their problem-solving competence, and as a result their

mathematical skills, cannot be found in the literature. This study has therefore

investigated the link between a problem-posing intervention and the development

of problem-solving competence.

To determine a focus group for the present study, a review of the literature into

underachievement was conducted. The desire to provide a learning environment

that allows students to be actively engaged in learning and to “assemble new

practices” (Tierney, 1997) that have measureable and positive outcomes, is

consistent with a post-modernist theoretical perspective and is a goal of this

present study. Some researchers reported that underachievement is commonly

the result of a traditional teaching pedagogy (e.g., Cummins & Sayers, 1995),

while others discussed that middle-year students were particularly at risk of

underachievement as a result of disengagement, for a number of situational and

dispositional reasons (e.g., Gootman, 2001; Jones & Myhill, 2004; Kanevsky &

Keighley, 2003). While these views are generally accepted, there was little

research found that provided practical ways for educators to address

underachievement in middle-year students who have demonstrated a high level of

90

mathematical ability in routine knowledge and procedural aspects of a

mathematics curriculum, or indeed those students who have demonstrated

above-average intelligence in intelligence tests. This group of students therefore

formed the focus group for this present study, that is, middle-year students who

have performed below the average for their cohort in problem-solving

competence but who have also demonstrated above average results in

intelligence testing.

A research study that empowered students to become more active and

responsible for their own learning (critical theory), that expected positive learning

outcomes to be achieved for the participants (post-modern theory), and that

incorporated a problem-posing intervention with middle-year students as a means

to develop problem-solving competence and re-engagement of these students in

the learning process, could be facilitated through a teaching experiment. This

present study incorporated these features into a seven-episode teaching

experiment with Year 7 students that took place in the final term of their school

year.

The constructionist approach to social science research supports opportunities to

collect both qualitative and quantitative data. Gergen (1996) supported this

triangulation approach to data collection when he raised concerns about the sole

use of quantitative approaches to test changes in learning development. He

wrote, “there is nothing about a social constructionist psychology that rules out

empirical research” (p. 5) however, “from a constructionist perspective, the

traditional attempt [using only quantitative approaches] to test hypotheses about

universal processes of the mind …. seems at a minimum misguided, and more

tragically, an enormous waste of resources” (p. 5). A number of researchers

have supported Gergen in this position. For example, Lesh and Zawojewski

(2007 p. 51) said that:

91

general theories of learning are useful for guiding the design of worthwhile

learning activities and theoretical models, but useful educational models

for learning and teaching in specific contexts, often need to draw on and

build from more than a single practical or theoretical perspective.

Further, Bell (2004, p. 3) reported that “learning is too complex a phenomena to

be the sole province of any one discipline, theoretical perspective, or research

method”. Support for multiple approaches to data collection in an educational

setting with students was also offered by a number of other researchers (e.g.,

Brown & Campione, 1998; Bruner, 1996; Goldman-Segall, 1998; Love, 2002).

The repeated recommendation to collect both qualitative and quantitative data

precipitated the use of two theoretical frameworks for this present study that are

discussed in the following section.

The review of the literature pertaining to the development of problem-solving

competence highlighted that the bulk of research undertaken in this area

presupposes that problem solving is a skill acquired after a particular cycle of

learning has been facilitated. The cycle usually consists of:

1. Learning mathematical knowledge

2. Learning problem solving strategies

3. Acquiring metacognitive strategies

4. Applying correct strategy for solution, and

5. Unlearning ineffective beliefs and dispositions while learning effective

ones (Lesh & Zawojewski, 2007).

The research questions that have arisen from the literature will challenge the

validity of this learning cycle and address the shortcomings of the assumptions

underpinning the cycle. As Lesh and Zawojewski reported, “research based on

these preceding assumptions have been unimpressive” (2007, p. 88).

92

3.3 Research Questions

The following research questions related to the focus group of students for this

present study, who scored highly on a widely used intelligence test but who were

considered by their teachers to be underachieving in problem-solving

competence, as evidenced by the results of their Year 7 mathematics tests.

The overarching question that was responded to in this study was:

How might a problem-posing intervention impact upon the development of

problem-solving competence of underachieving, middle-year students?

The review of literature related to this overarching question generated the

following more specific research questions.

Research Question 1

Can, and if so, how can participation in problem-posing activities facilitate the re-

engagement of middle-year mathematics students?

Research Question 2

Can, and if so, how can participation in problem-posing activities facilitate

improved problem-solving competence of middle-year, mathematics students?

Research Question 3

In terms of problem-solving competence, what developmental learning changes

occur during the course of a problem-posing intervention?

Research Question 1 is at the heart of this study. The response to this question

provides further evidence to determine whether a problem-posing intervention

can have a beneficial impact on the engagement of middle-year students in the

learning process. As the link between engagement and underachieving has

been established in the literature (e.g., English, 2002; Lesh & Zawojewski, 2007),

93

if the engagement of these students can be increased through a problem-posing

intervention, then an opportunity exists to address their underachievement.

Directions for curriculum reform are provided by the response to Research

Question 2. The study findings associated with this response also contributed

information to the debate in the literature between those who believe there is a

strong correlation between a problem-posing intervention and problem-solving

competence (e.g., Lavy & Bershadsky, 2003; Lowrie, 2002) and those who are

yet to be convinced (e.g., Silver, 1997). It was a goal of this study to investigate

a teaching experiment that may result in increases in problem-solving

competence. However, other developmental learning changes that resulted from

the teaching experiment may also contribute to the effectiveness of mathematical

teaching programs. The investigation of Research Question 3 allows data related

to developmental learning changes that were collected throughout this study, to

be analysed.

3.4 Research Design

3.4.1 Research Design Rationale and Structure

As writers like Crotty (1998) point out, there are four main elements to consider

when designing a research proposal. The study should be informed by a

philosophy of beliefs (epistemology) that are underpinned by theoretical

perspectives leading to a suitable methodology for appropriate methods of

data collection.

In order to determine these four elements, Crotty suggests researchers consider

the following four questions:

94

1. What methods do we propose to use?

2. What methodology governs our choice and use of methods?

3. What theoretical framework lies behind the methodology in question?

4. What epistemology informs the theoretical perspective?

Crotty (1998, p. 2)

These four questions are answered in the following section. The design of this

present study incorporated two mutually exclusive groups of students to enable

comparisons between the cohorts, identifying developmental learning changes

and other changes that resulted from the problem-posing intervention. The 31

participants, 16 in the Comparison Group and 15 in the Intervention Group, were

drawn from consecutive year cohorts at the one school. While Comparison and

Intervention students from the same year cohort may have been considered

more ideal in terms of the control of experimental variables, the possible cross-

contamination of students who were in the Intervention Group, sharing

information with the Comparison Group students, would have been counter

productive to this research. This compromise was supported by Lesh and his

colleagues who said that, “planning research often involves trade-offs similar to

those that occur when an automobile is designed to meet conflicting goals (such

as optimising speed, safety, and economy)” (2000, p. 19). Students in the

Comparison Group were pre-tested and post-tested in 2006, while students in

the Intervention Group were involved in the teaching experiment and tested in

the following year (see Figure 3.1). Three case study students were selected

from the Intervention Group (see Section 3.4.2 for further information on how

these students were chosen).

The NCTM-2000 Standards (2000, p. 18) highlighted the multifarious nature of

teaching mathematics and reported that the teaching of mathematics “must

balance purposeful, planned classroom lessons with the ongoing decision-

making that inevitably occurs as teacher and students encounter unanticipated

95

discoveries or difficulties that lead them to unchartered territory”. This being the

case, it was clear that the methodology chosen for this study needed the

flexibility contained within design-based research (Brown, 1992) and also the

rich, contextualised and flexible structure afforded by a teaching experiment

(Malara, 2002). Both methods rely on the cyclical nature of classroom-based

research that incorporates the refinement of the instruction process throughout

the intervention.

Comparison Group

Term 4 2006

Intervention Group

Term 4 2007

Pre-test

Post-test

Teaching

Experiment

Researcher observation

Researcher journal

Student workbooks

Informal interviews

Pre-test & Survey

Post-test & Survey

Figure 3.1. Research Study Framework

Teaching is a fluid and dynamic undertaking that occurs in response not only to

work programs and lesson plans, but also to the characteristics of the students in

the class, the weather conditions and the activities that are occurring within the

96

school environment on any one day. For example, a teacher may become aware

at the start of the year that there is a student in the class who has performed

particularly poorly in testing for mathematics in the previous year. This may

influence how the teacher talks to that particular student (to be mindful of the

self-esteem of the student), where the teacher sits the student in the classroom,

(to assist the teacher to closely monitor and assist the student) and to the

modification of tasks for the student to undertake (individualising the work

program). From a whole-class perspective, on any given day, the “mood” of the

class may be impacted by the weather, resulting in students being particularly

unsettled and unable to self-regulate their behaviour. In response, a teacher

may provide opportunities for students to do more structured tasks. These

examples are typical factors that need to be accommodated by a teacher through

the refinement of lesson plans both within and between lessons. This flexibility

was built into the design of this present study and is further discussed in Chapter

4.

As the purpose of this present study was to gain an “authentic understanding”

(Richardson, 1999) of the changes that occurred as a result of a problem-posing

intervention, without a loss of scientific rigour, the design of this study blended

elements of both qualitative and quantitative methodologies. The significant

strengths of combining both qualitative and quantitative methodologies have

been reported upon by a number of researchers including Bruner (1996) and

Brown and Campione (1998). Discussion related to the instruments and

methods used for data collection can be found later in this chapter.

3.4.2 Participants

As mentioned previously, the participants for this present study, typically aged

between 11 and 12 years of age, were drawn from two consecutive Year 7

cohorts. Year 7 is the first year of secondary studies in New South Wales,

Victoria, Tasmania and the Australian Capital Territory but is the final year of

primary school studies for students in Queensland, South Australia, Western

97

Australia and the Northern Territory. While the middle-years of schooling

generally refer to students in Years 5 to 9, students in Year 7 were chosen as

this was the final year of primary schooling at the research school. After Year 7,

students move into the senior part of the school with different lesson times,

separate classrooms and recreation areas. The new Year 8 students do not

commonly come into regular contact with students in the current Year 7 cohort

and are less likely to have communication with them. Had students from the

same Year 7 cohort be chosen to be participants in both the Comparison and

Intervention Groups, there would be a chance of contamination of the data due to

conversations held between Intervention Group students returning from a

teaching episode with the researcher and Comparison Group students who

remained in their usual classes. In addition, if the number of students who

satisfied the selection criteria in any one year were to be divided equally to create

the Comparison and Intervention Groups, then the numbers available to each

group would be substantially reduced.

Two specific criteria were used to select the Year 7 participants for this study.

The criteria were chosen in consultation with the Head of Department –

Curriculum and the Coordinator of Year 7 Mathematics at the research school,

who were both wanting to seek answers to the apparent underachievement of

students who seemed, from intelligence testing, to have the potential to have a

greater competence in problem solving than that which they had demonstrated in

their mathematics tests. The Head of Department – Curriculum, the Coordinator

of Year 7 Mathematics and the researcher felt that this particular group of

students could gain most immediate and significant benefit from the findings of

this present research study (see also Lesh & Zawojewski, 2007).

The first selection criterion required the participants to have scored above the

60th percentile in the standardized Middle Years Ability Test (MYAT) (Australian

Council for Educational Research, 2005), which is routinely administered to all

students at the research school, in their first term of Year 7. The MYAT is a

98

general ability test that provides an overall, norm-referenced, ability score

combined with a profile of the student’s performance. Further details on this test

can be found in Section 3.5.1.1.

The second criterion required the students to have scored below the Year 7

cohort average for Criterion B (Problem Solving Criterion) in their school

mathematics tests during the first semester of Year 7. Meeting with the students

who satisfied the two criteria in 2006 and 2007 and the researcher were arranged

by the Coordinator of Year 7 Mathematics in July of both years. That is, students

who were eligible to participate in the Comparison Group had their meeting in

July of 2006 and students who were eligible to participate in the Intervention

Group had their meeting in July of 2007. It was at this stage of the year that the

Semester 1 mathematics results for these consecutive cohorts had been

finalised. The students who attended the meetings were given a brief overview

of the purpose of the study and the voluntary nature of their participation should

they wish to become involved in the study.

At the meetings, students were given an information letter and permission slip

(see Appendices A and B) to take home to their parents. Students of each group

were asked to return the signed permission slips within two weeks of the

meetings for them to be included in the study. This time frame was chosen to

provide the students with time to remember to hand the sheets to their parents

and to provide parents with time to contact the researcher should they have any

questions about their son’s or daughter’s involvement. No attempt was made to

choose equal numbers of male and female participants or participants that were

representative of all ethnic groups at the school. These considerations were

outside the boundaries of this present study. However, students who met the

two criteria for the Intervention Group but who had a sibling or cohabitant that

that had been a participant in the Comparison Group, were not chosen for

participation in this present study. Their inclusion would have opened up

opportunities for cross-contamination of the data.

99

As mentioned previously, an aim of this study was to investigate students who

had been ascertained as above average in the MYAT test (Australian Council for

Educational Research, 2005) but who had performed below the average of their

cohort in the problem-solving aspect of their mathematics test. From a

mathematician’s point of view this would suggest that any student scoring in the

51st percentile or higher would be considered above the average and therefore a

potential participant. However, testing instruments such as the MYAT test are

useful in providing a strong indication of intelligence only, rather than an inflexible

and unquestionable representation of a student’s ability (e.g., Bjorklund, 2000;

Gardner, 1999b).

Reliability and validity factors, beyond the control of a test administrator such as

the pre-disposition of a candidate at the time of taking the test can influence test

results (see Section 3.5.3). Therefore, rather than choosing students who were

in the 51st percentile or higher, the researcher allowed for a 10% margin in

testing results and chose students scoring in the 61st percentile and above. The

size of the margin was chosen to be as large as possible without reducing the

potential number of candidates for each of the Comparison and Intervention

Groups to below 24 students, the size of a usual class of students in the research

school. As it happened, the size of the two groups was further reduced by the

number of students who did not return their permission forms, those who were

absent on one or both of the pre-testing and post-testing days, or by Intervention

Group students who were absent for more than two of the teaching episodes. As

a result of the attrition, the final Comparison Group was comprised of 16 students

while the final Intervention Group was comprised of 15 students.

To undertake a deeper investigation of the research questions associated with

this study, three students were identified early in the teaching experiment as

case study students. Initially, all participants were potential case study students.

However, the three case study students were selected for their potential to

provide data to assist in responding to the research questions of this present

100

study. For example, students who began the teaching experiment as particularly

disengaged were considered potential case study students to assist with the

response to Research Question 1. In addition to being initially disengaged, other

qualities such as their score in the MYAT test (Australian Council for Educational

Research, 2005) or their behaviour during their disengagement, would provide

some useful comparisons between case study students. The first two students

were identified within the first teaching episode due to their specific

characteristics.

The first case study student, Andrew, had the lowest MYAT (Australian Council

for Educational Research, 2005) score of the Intervention Group students

(scoring in the 76th percentile) and was noticeably disengaged from the class

activities in the first teaching episode. In fact, he was far more interested in

talking to his neighbours than posing or solving problems. The second case

study student, Nicole, had the highest MYAT score of the Intervention Group

students (scoring in the 94th percentile) and was noticeably less inclined than her

peers to participate in posing problems. She was however, very quiet and did

not disturb other students, preferring to watch what other students were doing.

The characteristics displayed by these two students were consistent with the

findings of Jones and Myhill (2004) who reported that underachieving boys are

often typified by rowdy behaviour and disengagement while underachieving girls

are often “rendered invisible” because they quietly disengage from activities.

The study of these two students produced useful data to respond to Research

Questions 1 and 3.

The third case study student, Paul, was identified in the first teaching episode but

only confirmed as a case study student by the fourth teaching episode. His

initial enthusiasm, yet inability to organise his thoughts, drew attention at the start

of the teaching experiment. While his levels of enthusiasm did not waver or

diminish throughout the teaching experiment, his ability to become more

101

organised in his thoughts and coherent in his explanations became apparent

within a few teaching episodes. These observations were consistent with

Sternberg’s (2000) proposition that four change mechanisms; automatisation,

encoding, generalisation and strategy construction play a substantial role in

developmental learning and discussion about how Paul demonstrated each of

these mechanisms can be found in Chapter 4. Paul had the median MYAT

(Australian Council for Educational Research, 2005) score of the Intervention

Group students (scoring in the 85th percentile) and was the most willing and

enthusiastic student to participate in the problem-posing activities and to share

his problems with his peers. Data collected about Paul were particularly useful in

responding to Research Questions 2 and 3.

Throughout the entire teaching experiment, observational data primarily about

the three case study students were collected using three audio-video tape

recorders. The cameras were labelled Camera One, Camera Two and Camera

Three. This enabled accurate transcripts to be made of conversations between

students and the researchers, and between the students themselves. All

extracts of the transcripts used in this document are described using the number

of the camera, the teaching session number from which the recording was made

and the time into the recording. The time is displayed as mm:ss which refer to

the number of minutes (mm) and seconds (ss) into the recording of a particular

session. As soon as the case study students were identified, the two fixed-

position cameras were positioned to ensure that footage of these students was

captured as well as the activities of surrounding students. Paul and Andrew sat

at adjacent tables which enabled a single fixed camera to make recordings of

both students simultaneously. The second camera was focussed on the table

where Nicole sat. The third camera was mobile and was used to capture

informal interviews with the three case study students as they occurred as well

as recording details of general class activity. This arrangement of recorders

ensured that accurate transcripts could be made of the conversations the three

102

case study students had with their peers and with the researcher. In addition, it

also allowed the researcher to repeatedly review each teaching episode from

different perspectives in order to respond to the research questions.

3.5 Methods

An introduction to the teaching experiment, the methods of data collection and

the testing instruments used in this present study are discussed in this next

section. This is followed by a detailed description of how the data were analysed

and the reliability and validity issues were taken into account throughout the data

collection process. A more detailed description of each teaching episode within

the teaching experiment can be found in Chapter 4.

The teaching experiment took place in the library classroom of the research

school. The experiment was not able to be conducted in the students’ usual

classrooms as participants were drawn from four different Year 7 classes. The

sourcing of participants from across the four classes was necessary to obtain

enough students to proceed with the study. The withdrawal of students from

their usual classrooms, classmates and teachers was therefore, a necessary

construct of the research design. The undertaking of research with the students

out of their everyday classroom setting and environment does have implications

for the validity of the data if it is to be used to make statements about student

learning in their usual classroom environment. However, the library classroom in

their school library was a room with which the students were very familiar and in

which they often undertook activities such as language or research lessons. The

students were familiar with working in groups other than their ‘homeroom’

groups, as alternative Year 7 groupings occurred for language classes and sport.

Therefore, the use of the library classroom with students from four different Year

7 classes was deemed to be an acceptable compromise in attempts to create an

authentic and familiar classroom setting for the students in both groups.

103

While the withdrawal of students from their usual class groupings could be seen

as a limitation of this present study, it is also a feature supported by the

philosophical underpinnings of this study. The participating students in the

Comparison and Intervention Groups have all been ascertained as

underachieving in their usual classroom environments. This present study gave

the students in the Intervention Group an opportunity to work in a different

environment where they could feel safer to explore the concept of mathematical

problem solving from a unique and different perspective. Problem posing offers

students an opportunity to incorporate “ingenuity” and “playfulness” into learning

that is not available in a traditional mathematics classroom (Brown & Walter,

2005). Brown and Walter said that it was through posing problems that students

can discover that “a slight turn of phrase, or recontextualising the situation, … will

transform it [a problem] from one that appears dull to one that ‘glitters’” (Brown &

Walter, 2005, p. 5). They add that a problem-posing environment is “the

beginnings of a mechanism for confronting the rather widespread feelings of

mathematical anxiety” that can lead to underachievement (Brown & Walter, 2005,

p. 5).

The research methods chosen for this study were designed to be consistent with

both the theoretical frameworks of postmodernism and critical theories, whereby

postmodernists are looking to effect positive change as a result of research and

intervention, and critical theorists are concerned with issues related to equity,

identity and power sharing (e.g., Alvesson & Skolberg, 2000; Broido, 2002). In

the design of the teaching experiment the freedom of the students to choose how

to engage with the learning activities was a focus of the study. However, it was

framed by the resources available in the library classroom, the customs of the

School and the ‘duty of care’ expected of all registered teachers in the State of

Queensland. For example, students were required to remain quiet at the start of

each teaching episode while a roll was taken and they were expected to respect

each other and listen while students were sharing their problems with the group.

For reasons of student safety, the researcher was required to remain in the

104

classroom and to supervise the students at all times. It was inevitable then that

the researcher would need to direct the students in some activities such as the

whole-class sharing of problems, when to pack up, when to hand in workbooks

and when the students should return to their next class. Student direction by the

researcher was however kept to a minimum.

While for practical purposes, such as those mentioned in the previous paragraph,

the researcher facilitated each teaching episode; the students were given many

options on how to engage with the learning in order to encourage a student-

focussed approach to the teaching experiment (see Chapter 4). As the

mathematical teaching environment with which these students were familiar was

traditional and one where the teachers provided limited, if any, choices about

how to work within the classroom, the level of freedom provided in this study was

deemed to be significant and consistent with a critical theorist approach.

Students were able to choose where to sit, how to pose their problems, whether

to pose problems individually, in pairs or in small groups, whether to pose their

problems to the students sitting near to them or to the whole class or not at all, or

whether to become involved in providing feedback to the problems posed by their

peers. This was a genuine attempt to emancipate the students from their usual

structured way of learning where the teacher held all of the power in the

classroom (Lincoln & Denzin, 2000).

The teaching experiment consisted of a preliminary week for pre-testing and

conducting the first student survey (see Appendix C for the Survey Sheet and

Appendix D for the lesson plan related to the preliminary week lesson), followed

by seven, weekly teaching episodes lasting one hour each (see Appendices E to

K for the lesson plans of each consecutive teaching episode), finalised by a

session for post-testing and the repeated administration of the student survey

(see Appendix L for the final lesson plan). A lesson plan was prepared in

advance for each teaching episode. Each subsequent lesson plan was refined

both prior to the lesson occurring, as a result of student feedback and researcher

105

observations of the previous teaching episode, and during the lesson depending

on the activities that were happening on the day in the research school that could

not have been predicted. For example, during a teaching episode, some

students may have arrived late due to heavy rain and road closures. This would

necessitate the researcher speaking to the individual students as they arrived to

provide them with an explanation of what the other students were doing (The

rationale for the construction of each lesson plan can be found in Chapter 4 and

in the lesson plans found in Appendices D to L.)

3.5.1 Data Collection

Both qualitative and quantitative data, in a variety of forms, were collected

throughout this study (see Table 3.1). The different forms of data were collected

for all students in the Intervention Group and were used in various ways to

respond to the three research questions. That is, all students in the Intervention

Group undertook the surveys (see Appendix C), the pre-test and the post-test

(see Appendix M), had their work books reviewed weekly, were videotaped and

were equally encouraged to pose their problems and share them with their peers.

However, for practical purposes, transcripts of the videotaped conversations and

informal interviews were only completed for the three case study students. Since

the researcher was also the facilitator of the teaching episodes, the recording of

observational data could not be undertaken in great detail during each teaching

episode. Therefore, detailed observations of the students in the Intervention

Group were undertaken primarily through a weekly review of videotapes from the

three audio-video recorders. Lesh and Clarke (2000, p. 137) supported the

collection of observational data when they said that “teacher’s observations can

lead to rich portrayals (or models) of their students”. According to Burns (1995,

p. 260) “observation serves to elicit from people their definitions of reality and the

organizing constructs of their world”. The data collected from observations

throughout the teaching experiment were invaluable in determining any

developmental learning changes that occurred in students.

106

The qualitative data collected from the observations, student surveys, workbooks

and the informal interviews provided evidence of transformation in student

engagement and changes to students’ attitudes towards solving problems. For

example, the length of time taken by a student to begin posing problems, their

ability to remain focussed or the quality of the problems posed in their

workbooks, could all indicate whether a student was engaged in learning or not.

In addition, a student’s response to question in an informal interview could

indicate if a student had been actively engaged in the posing of problems.

The review of the tapes and transcripts provided a rich source of data on

individual students that in turn, provided the opportunity to make detailed

investigations of Paul’s, Andrew’s and Nicole’s learning development throughout

the teaching experiment. In addition, it provided supporting data to be used in

the refinement of subsequent teaching episodes. The data collected from the

student workbooks, and the observations of students, provided ongoing insights

into the students’ development learning and engagement during the teaching

episodes in the experiment (Confrey & Lachance, 2000). The level of

mathematical content and level of sophistication of the problems posed by the

students, provided evidence of how successfully the students’ could

“mathematize” situations (Fosnot & Dolk, 2001). This ability to recognise

mathematical situations was noted by English (1996) as being important and may

result in greater connections being made by students between their “school

maths”, intuitive knowledge and everyday, familiar contexts.

107

Table 3.1

Data Used to Respond to the Three Research Questions of the Study

Research Question Qualitative/Quantitative Data Instruments

1. Can, and if so, how can

participation in problem-posing

activities facilitate the re-

engagement of middle-year

mathematics students?

Qualitative Informal interviews

Observations

Workbook analyses

Student survey

2. Can, and if so, how can

participation in problem-posing

activities facilitate improved

problem-solving competence of

middle-year mathematics

students?

Quantitative and

Qualitative

*Profiles of Problem Solving test results

Informal interviews

Observations

Workbook analyses

3. In terms of problem-solving

competence, what developmental

learning changes occur during the

course of a problem-posing

intervention?

Qualitative Informal interviews

Observations

Workbook analyses

Note. * From “Profiles of Problem Solving” by K. Stacey, S. Groves, S. Bourke, and B. Doig, 1993, Australian Council of Educational Research, Hawthorne, Vic.

The review of student workbooks on a weekly basis provided opportunities to

identify emergent themes in the problems posed by the students. An emergent

theme was considered to occur when a substantial number of students were

demonstrating a similar characteristic in their problems. For example, the first

108

problems posed by the students were typically simplistic and minimalist. That is,

they required at most two steps to solve and were posed at a level that a student

of average ability in Year 4 may have been able to solve. It was at this point that

a refinement to the subsequent teaching episodes was planned as it seemed

apparent that students had little concept about what constituted a “good” problem

(see definition below). The workbooks from the 15 students were collected at

the end of each teaching episode and provided data about the development of

student learning and the levels of engagement that students demonstrated

throughout the teaching experiment. They also provided invaluable evidence to

respond to Research Questions 1 and 3. In addition, the lesson by lesson review

of these workbooks allowed the subsequent activities in the teaching experiment

to be adjusted where necessary, as “even researchers experienced in teaching

may not know well enough what progress students will make or know well

enough their mathematical thinking and power of abstraction to formulate

learning environments prior to teaching” (Steffe, Thompson, & von Glasersfeld,

2000, p. 279).

For the students, a “good” problem was defined to be one that was high in

interest factor, was appropriately challenging for the intended audience and had

sufficient data embedded, within it, to be solved. This concept of a “good”

problem was revisited in two ways during each teaching episode to provide a

scaffold around which students could pose problems that their peers would want

to solve. In the first instance, students were given opportunities each week to

self-rate their own problem by completing a self-rating sheet that described the

three qualities (see Appendix N for a copy of the self-rating sheet). In the second

instance, students were given opportunities to rate their peers’ problems using a

simple card system. Each student was provided with three cards upon which

were written the numbers 1 on one card, 2 on another card and 3 on the third

card. After a student had posed a problem to the class, other students were

asked to provide feedback to that student by holding up the card that, in their

opinion, had the number of qualities of a good problem that were demonstrated

109

by the problem they had just heard. The use of these cards was further refined

in subsequent teaching episodes as the students began asking for more specific

feedback from their peers on each individual quality of a good problem, rather

than the problem as a whole.

Informal interviews with students were conducted during the teaching episodes,

throughout the teaching experiment. It was important to conduct the interviews in

an informal manner, while the students were working, to ensure they were

immersed in the problem-posing tasks and therefore responding about their

immediate thought processes, rather than from what they thought they might

have been thinking at the time. Students were asked to discuss what they were

thinking as they worked through a particular problem-posing task. An example of

some of the questions asked of students are: “What has been the biggest

challenge for you in constructing this problem today?” or “How did you go about

constructing your problem today?”. The responses were recorded for later

analysis.

The data from these interviews with case study students supplemented the

researcher’s observations and provided data to demonstrate change processes

occurring in the learning development and attitudes towards problem solving of

the students throughout the teaching experiment. Mestre (2000, p. 167)

commented on this form of data collection saying that “although these

methodologies are extremely time consuming to administer and the data are

difficult to analyse, they can provide detailed information of a subject’s

conceptual knowledge and its links both to contexts in which it can be applied

and to conditions of applicability”. He further said that multiple methods, such as

those used in this present study, provide researchers with different data, and it is

common that more than one method is required to satisfactorily respond to all of

the research questions in a particular study (Mestre, 2000).

110

At the completion of each lesson, general observations and notes about the

lesson were recorded in a researcher journal. The journal observations

contextualised and recorded any events that impacted upon the behaviour and

concentration of students during individual teaching episodes. These events

included any incidents that took place before a teaching episode, or distractions

that occurred nearby the library classroom on a particular day, during a particular

episode. They were recorded to monitor the “mood” of the classroom throughout

the experiment and thereby contextualise student behaviours. The data

collected from the observations were taken into consideration when

determinations were being drawn about students’ behaviours during each

teaching episode.

The quantitative data collected in the pre-testing and post-testing were necessary

to provide empirical evidence of transformation in problem-solving competence of

the students. The triangulation of data obtained from both the qualitative and

quantitative methods used in this experiment is consistent with Gardner’s

(1999b) call for a variety of assessment instruments to provide multiple and

different opportunities for individuals to demonstrate their understanding of

concepts. For example, one student may be able to successfully demonstrate

the understanding of a concept by undertaking an oral presentation to peers

whereas another student may prefer to quietly respond to some questions in an

essay or short response format. In the context of this study, accommodation of

different learning styles and intelligence types is not possible with pre-testing and

post-testing alone. Kelly and Lesh (2000, p. 229) agreed and said that “pretest-

posttest designs … tend to presuppose that the best way to get complex systems

to evolve is to get them to conform to a single one-dimensional conception of

excellence”, which in this study would have prohibited discussion about

observations of developmental learning demonstrated throughout the teaching

experiment.

111

3.5.2 Instruments

The instruments used to collect the data in this present study are discussed in

this section. The first instrument, the Middle Years Ability Test (Australian

Council for Educational Research, 2005), was used to assist in the participant

selection process. The second instrument, the Profiles of Problem Solving

assessment instrument (Stacey et al., 1993), was used as both the pre-test and

the post-test for this study. The third instrument, the Student Survey Sheet, was

created by the researcher to collect attitudinal data from the students that may

assist in responding to Research Question 1. These instruments are described

in greater detail I the following sub-sections.

3.5.2.1 The Middle Years Ability Test (MYAT)

The Middle Years Ability Test (2005) is an intelligence testing instrument

developed by the Australian Council for Education Research (2005) and was

used to identify participants for this study. The normative data, provided to

administrators of the MYAT test by the Australian Council for Education

Research, to be used for the assessment of students came from the

achievement scores of over 2000 Australian middle-year students tested in July

and August of 2004. This test was chosen as it was specially designed for

Australian middle-year students and was routinely administered to students in the

research school at the beginning of Year 7, thus negating the need for the Year 7

cohort to undertake additional testing as a whole group. In addition, unlike other

IQ tests, the MYAT test contains 25 out of the 75 questions contained in the test

that assess non-verbal reasoning skills. As mentioned earlier, Gardner (1999b)

has suggested that intelligence tests can distort the view about an individual’s

potential but, he was not against them if they made attempts to address

intelligences other than linguistic and mathematical intelligences. The MYAT test

goes some way to addressing this issue.

112

The MYAT test (Australian Council for Educational Research, 2005) contains a

pair of parallel general ability tests that were designed to assist teachers to

assess the general abilities and skills of their middle-year students. Each of the

two tests contains 75 items based on literacy, numeracy and non-verbal

reasoning skills that are arranged in order of difficulty from simplest to most

difficult. Students respond to each question by marking their choice of five

possible answers for each question on an answer sheet. Only one answer of

each set of five possible answers is correct. Consideration was given to the

debate in the literature about the validity of administering intelligence tests that

have time as a component of the testing conditions, such as is the case with the

MYAT test (Australian Council for Educational Research, 2005). Opponents,

such as Goswami (2002), suggest the reliability of results from such timed

instruments is questionable, whereas supporters, such as Hutton and his

colleagues (1997), believe that speed in processing information, does indeed

correlate highly with intelligence. To address this reliability issue, a 10% margin

of consideration in the student results was made before considering potential

participants for this present study.

The Head of Department – Curriculum and the Coordinator of Year 7

Mathematics at the research school reported that the MYAT test had provided a

sound indication of student intelligence based on comparisons with the

classroom performances of students generally across all subjects. However,

they noted that there seemed to be some inconsistencies with some students

who had high predictive scores on the MYAT test but whose performance on the

problem-solving criterion of their mathematics tests was below the average score

of their peers. The specific nature of why this situation occurs is beyond the

scope of this present study, however, strategies to overcome this potential

underachievement of students have been addressed in the findings of this

present study (see Section 3.4.2).

113

3.5.2.2 The Profiles of Problem Solving (POPS) Test

A number of criteria were considered before choosing the instrument to be used

for the pre-testing and post-test of participants in this study. Firstly, it was

important that the test was specifically designed and written for Australia middle-

year students. While Australia is not typified by a mono-culture, it is reasonable

to at least ensure that the language and contexts used in the test are the

language and contexts that the students would commonly experience in their

day-to-day activities. It is worth noting that students who had English as a

second language were not considered as potential participants for this study.

Their inclusion could have created additional variables and lead to results that

were more about a student’s ability to understand and read the English language

than they were about their ability to pose or solve problems.

The second consideration, when choosing a testing instrument, was that the

instrument should measure five different aspects of problem-solving competence:

1. Correctness of answer – a fundamental criterion for developing a high

level of problem-solving competence

2. Method used – the use of a systematic plan that realistically can lead to

the solution of a problem is integral in developing problem-solving

competence

3. Accuracy – without the ability to complete accurate calculations a

problem cannot be solved

4. Extracting information – whether a student can identify and filter useful

information from extraneous information provided in a problem, and

5. Quality of explanation – a student’s ability to effectively communicate

their mathematical ideas, processes and calculations as well as the final

answer

114

These five aspects of problem solving are consistent with Polya’s (1957) four

stages of solving a problem that are still commonly referred to in schools as the

‘See, Plan, Do, Check’ scaffold for solving problems. (The ‘Accuracy’ and

‘Extracting Information’ are combined in the ‘Do’ aspect.) While the majority of

test instruments written in Australia focussed on a student’s ability to solve

problems across the five strands of mathematics; number, algebra, space,

measurement, and chance and data, the Profiles of Problem Solving (POPS) test

(Stacey et al., 1993) was the only test that specifically addressed the five aspects

of problem solving.

The third consideration when choosing a testing instrument was that it needed to

be a reliable and valid instrument to use in the context of this study. The POPS

(Stacey et al., 1993) test incorporated everyday contexts that were within the

typical experiences of an Australian middle-year student, for example, money

and birthday candles. This instrument had pre-established validity and reliability

(Impara & Plake, 1998; McLellan, 1998; Medina-Diaz, 1998) and was a useful

tool in determining changes in problem-solving competence of the students

between the start and end of the teaching experiment. The only testing

instrument that satisfied all the three considerations of assessing the five

problem-solving criteria, being designed in Australian for middle-year students

and having verified validity and reliability, was the POPS test and hence it was

used as the pre-testing and post-testing instrument for this study.

The POPS test took approximately 40 minutes to administer, with 32 minutes

being specifically set aside for the students to answer the questions. While

authors, such as Ridgeway and colleagues (2000), suggested that

understandings related to problem-solving competence are difficult to measure,

and that standardised tests can fall short of capturing the true nature of students’

problem-solving competence, the use of multiple data collection methods,

including both qualitative and quantitative methods used in this present study,

115

provided for triangulation of the data and hence, increased faith in the results of

this study (Mestre, 2000).

3.5.2.3 The Student Survey

The purpose of the student survey was to determine if there had been a shift in

student attitudes towards solving novel problems as a result of the problem-

posing intervention. The four questions in the survey were designed by the

researcher to be simple to understand and answer by the students. The four

questions in the survey are listed below.

Question One: Do you enjoy solving problems?

This question was included in the survey as it has direct links with the

dispositional state of students and the possible reasons for their

underachievement in this criterion of their routine mathematics tests (e.g.,

Gootman, 2001; Kanevsky & Keighley, 2003).

Question Two: What type of problems do you prefer to solve?

This question was included in the survey to determine if students were discerning

about the types of problem they like to solve. The student responses to this

question had the potential to provide some data to explain possible

disengagement of students in the Intervention Group. In addition, possible

correlations were made possible between the similarity or difference of

responses to this question in the initial survey compared to the final survey, and

the observations of student engagement in the classroom throughout the

teaching experiment.

Question Three: Do you think learning to solve problems is a useful thing to do?

Students’ attitudes to problem solving have been demonstrated by Kanevsky and

Keighley (2003) and Gootman (2001) as pivotal is their willingness to solve them.

The responses to this question in the initial survey, when compared to responses

116

to this question in the final survey, had the potential to provide an insight into

whether involvement in a problem-posing intervention could facilitate a change in

student’s attitude and possible their levels of engagement in classroom activities.

Question Four: What things could teachers do to assist you to become better at

solving problems?

This question was included in the survey out of respect for, and interests of, the

opinions of the participants in this study. It is commonplace for teachers to make

decisions for students on the premise that they are more wise and

knowledgeable about what and how students should learn (e.g., Roberts, 1996;

Stein et al., 2003). However, the philosophical framework, upon which this study

was founded, supported a liberating and transformative learning environment

(Broido, 2002) that respected the views of the students (Alvesson & Skolberg,

2000). Student responses to this question had the potential to direct refinement

in the lesson plans for each teaching episode and to provide direction for future

research into problem solving and problem posing.

3.5.2.4 The Problem Criteria Sheet

The problem criteria sheet (see Appendix N) was designed to allow students to

become reflective and mindful about the qualities of the problems they were

posing for their peers and to provide the researcher with a regular opportunity to

provide feedback to the students on their problems. The sheet was made up of

two sections; one for the student to self-rate their problems and a second

identical section below for the ‘teacher’ to provide feedback. The word ‘teacher’

was used on the sheet as opposed to ‘researcher’ as a genuine attempt to make

the students feel that this experience was as similar to a normal teaching

environment as possible.

Each week, the researcher would glue a new criteria sheet into the students’

workbooks so that it would be in place when the students completed posing their

new problem during the next teaching episode. Each lesson the students would

117

be given time to pose problems and then to share them. They received

immediate peer feedback during each lesson from the 1, 2, 3 card system

described in Section 3.5.1. Students then self-rated their posed problems by

ticking one of three boxes in each of the three criteria, interest factor, challenge

level and do-ability. In addition, students were given an opportunity to make any

comments they wanted to about their problem in a section alongside the boxes

before handing their workbooks to the researcher. This data contained in the

‘Comment’ section, provided clarification for why a student rated their problem as

they did and provided data that could be correlated with other data sources to

respond to Research Question 1.

The researcher reviewed the posed problems and the self-ratings recorded in the

problem criteria sheets of the student workbooks each week. The researcher

then completed the ‘teacher’ section of the criteria sheets, rating each student’s

posed problem against the three criteria, and provided each student with an

individualised comment that included affirmations like “Your problem is very

interesting this week, well done!” and constructive feedback like “When you are

posing your next problem, remember to include all of the information that a

person would need to solve the problem. By making the time to do the problem

yourself, before you pose it to your peers, you will be able to ensure it is do-able.”

3.5.3 Data Analysis

Data analysis is a “dynamic and creative process” through which researchers

attempt to make sense of what they are investigating (Taylor & Bogdan, 1998, p.

141). The qualitative data collected from this teaching experiment were analysed

from a number of perspectives to identify emergent themes. The quantitative

data collected from students in the Comparison Group and the Intervention

Group were analysed according to accepted statistical methods both within and

between groups. This section will look at how the data from all sources were

analysed singularly and in combination and, where applicable, how emergent

themes were identified.

118

3.5.3.1 Researcher Journal

The purpose of the researcher journal was to record details about the “mood” of

the class and to identify external factors that impacted upon the teaching

experiment. Trends across the class were recorded in the researcher’s journal at

the conclusion of each teaching episode and were used to determine if the

students were displaying group behaviours that may be as a result of external

influences on the day, such as an upcoming test or an extremely hot day.

Possible correlations between atypical behaviours of students and situational

factors were reviewed on a lesson by lesson basis. For example, if the

engagement of some students was noted in the tapes and video recordings as

lower than in previous weeks, the researcher could consider possible correlations

with entries for that teaching episode in the researcher journal. A possible

scenario may have been that this observation correlated to an entry in the

researcher journal mentioning these same students arrived late to the teaching

episode due to a vaccination program being conducted at the research school on

that morning. It would therefore be reasonable to suggest that some of the

disengagement of these students may have been due to the factors external to

the teaching experiment. This triangulation of data was of particular importance

in identifying the apparent trends of engagement, attributable to the problem-

posing intervention, throughout the teaching experiment and provided information

to respond to Research Question 1.

3.5.3.2 Student Surveys

There were four questions in the student survey (see Appendix C and Section

3.5.2.3). The students’ responses to Question One were categorised as a

positive, a negative or a neutral response (see Table 5.4). The ‘closed’ nature of

the wording of this question made these categories obvious and appropriate.

The administration of the survey sheets did not pre-empt possible student

responses to Questions Two, Three and Four and students were not prompted in

119

any way while they were responding to the questions. The categories shown in

Tables 5.5, 5.6 and 5.7 were determined after the students completed both the

initial and final surveys, as natural groupings became evident.

However, as is often the case when categorising data, a conflict can arise

between the number of categories finally chosen and the ‘fit’ of the data into the

categories. The categories were recorded in the tables as they arose in the

responses. For this present study it was decided that unless a student response

naturally belonged to a category that had previously been recorded (as a result of

another student making similar comments), then a new category would be

created in the table. This process ensured that all students’ views were

represented and not ‘lost’ in the compromises that are sometimes made when

categories are made known to students in advance of a survey, and is consistent

with the critical theorist framework that underpins this present study.

Students were asked to explain their answers to Questions One, Two and Three.

These responses were used to interpret if the students may have changed their

view from the start to the end of the teaching experiment as a result of the

problem-posing intervention. These responses could be correlated with data

collected from student workbooks, observations and the pre-test and post-test.

For example, a student may have written in their first survey that they disliked

solving all problems because they were too hard. This comment may have

contrasted to the comment they wrote in their final survey which could have been

“I now like solving problems because I understand how they are made and I am

better at knowing where to start to solve them”. It would be reasonable to

deduce from this change in attitude that something that occurred in the problem-

posing intervention had affected the change of view. The opportunity then

existed to correlate this attitudinal change with any changes in the quality of

problems progressively written in this student’s workbook, their self-ratings,

researcher observations of student engagement and results from the pre-test and

post-test.

120

3.5.3.3 Student Workbooks

The student workbooks were collected each week and the problems posed by

the students and their self-ratings on the problem criteria sheets were reviewed

by the researcher. Weekly feedback was provided to the students through the

researcher entries in the problem criteria sheets. At the conclusion of the

teaching experiment, the student workbooks of the three case study students

were analysed from a number of perspectives. In response to Research

Question One, the comments that students wrote, and any changes in the quality

of the problems posed by the students, could be correlated with other data

sources. For example, if a student had increased their score in the post-test as

compared to their score in the pre-test, the workbook may or may not show a

corresponding improvement in the quality of problems posed by the student

throughout the teaching experiment. This information would assist in responding

to Research Question Two. Analysis of workbook responses was also very

helpful when comparing the students’ answers in the initial and final surveys with

their comments and self-ratings in their problem criteria sheets. It was, for

example, useful to determine if a change in answer to Question One on the

student survey correlated or contrasted with any changes in students’ comments

in the problem criteria sheets of their workbook.

The themes that students choose to write about when they posed their problems

was not a focus of this present study, however, the reading of the problems on a

weekly basis inevitably provided guidance to the researcher about what themes

interested students and therefore, informed refinements in subsequent teaching

episodes. An awareness of which themes engage students emerged from this

study as a useful focus for further research.

3.5.3.4 Researcher Observations

The audio-video tape recordings of each teaching episode allowed detailed and

unobtrusive observations to be made of the three case study students as well as

providing secondary data about the “mood” of the class. Following each teaching

121

episode, the three video cassettes from the audio-video recorders were

electronically transferred to digital video disks (DVDs) and the cassettes were re-

used in the following teaching episode.

Each DVD was viewed and any conversations between a case study student and

peers, or informal interviews between the researcher and a case study student,

were transcribed. These data were then correlated with data from other sources,

such as the workbooks and the student surveys, to respond to Research

Questions 1 and 3. For example, the length of time a student took to begin

writing a problem in each teaching episode could be determined by reviewing the

tapes. This data, in combination with other data, such as the quality of the

problems posed in the student’s workbook or the comments they wrote in their

problem criteria sheet, could provide evidence about the student’s levels of

engagement throughout the teaching experiment.

The DVDs were also used to collect data on the case study students’ body

language and behaviour throughout the teaching episodes. Developmental

learning changes involve changes in internal processes that essentially are only

identifiable by the ‘products’ produced by an individual (e.g., Bjorklund, 2000;

Siegler, 1991). These products can be physical artefacts such as posed

problems written in workbooks and test results, or they can manifest as changes

in social behaviour such as a student’s body language or their ability to

communicate effectively or willingly with other individuals or groups of individuals.

As the DVDs were viewed, notes were made about each case study student’s

behaviours and types of interaction with peers and the researcher. These data

were used to respond to Research Question 3.

3.5.3.5 Informal Interviews

As mentioned in the previous section, informal interviews were conducted with

the case study students throughout the teaching experiment and were recorded

by one of three audio-video recorders positioned around the room. The

122

questions were designed to uncover some of the conscious and unconscious

thought processes that had been experienced by the students throughout the

teaching experiment. The questions asked in the informal interviews were not

scripted to ensure they were in context with each individual’s circumstances at

the time of the interviews. Analysis of this type of data is often difficult (Mestre,

2000) but it does provide insights into an individual’s developmental learning

(Confrey & Lachance, 2000). As mentioned in the previous section, the

interviews were transcribed and extracts from these transcripts were used in

conjunction with the pre-test and post-test data, to respond to Research Question

3.

3.5.3.6 The Profiles of Problem Solving (POPS) Test

The quantitative data collected from the five problem-solving aspects of the pre-

test and the post-test were analysed using the Statistical Package for Social

Sciences (SPSS) Version 15 (SPSS Inc., 2007). The data for both the

Comparison and Intervention Groups were entered using the SPSS software and

a paired t-test was performed to establish the means and standard deviations of

the paired sets of data. The results of the paired t-test provided opportunities to

compare the means and standard deviations of each problem-solving aspect

between the two groups. To assist with the analysis of the paired t-test results,

the mean scores for the Comparison and Intervention Groups, within each

problem-solving aspect, were graphed on single sets of axes.

The data from the POPs test (Stacey et al., 1993) were further analysed by

considering the number of students, in each of the Comparison and Intervention

Groups, who improved their score in the individual problem-solving aspects from

the pre-test to the post-test. This provided a further opportunity to compare and

contrast the data sets from the two groups and to explore the impact that the

problem-posing intervention may have had on students in the Intervention Group.

123

3.5.4 Reliability and Validity Issues

Issues of validity and reliability were considered in order to “trust” the results of

this educational research (Merriam, 1998) and to determine the quality of the

research study design. That is, trustworthiness was a major consideration

throughout the design and implementation of the study, including the collection,

analysis, interpretation and presentation of data and results. The qualitative

analysis required the establishment of a “strong chain of evidence” to be

collected in the survey sheets, audio-video tape recordings, student workbooks,

and the researcher’s journal, and by the laying of an “audit trail” that provided a

complete documentation of the research process for potential replication by other

researchers (Borg, Gall, & Gall, 1999).

The quantitative analysis was undertaken using accurate mathematical

techniques and commonly used statistical tests for the purpose of analysing data

in social science studies. Issues of instrument validity were addressed in

multiple ways; by ensuring the testing instrument was designed for Australian

middle-year students; the pre-test and post-test questions were of a format with

which the students were familiar, yet contained questions that were still novel in

nature; and by ensuring the testing instrument assessed the subject of this study,

that is, aspects of problem-solving competence. This was possible because the

testing instrument, the Profiles of Problem Solving (POPS) test (Stacey et al.,

1993), provided information on all five aspects of problem solving rather than

providing information about a student’s strength within a particular strand of the

mathematics curriculum in Queensland.

Validity of the design of the study was addressed predominantly through the

choice of a teaching experiment for the study’s intervention. The teaching

experiment provided the opportunity for students to become participants within a

familiar environment and with familiar peers. The students were not required to

attend school outside of their usual school hours thus minimising any disruption

to the students’ usual weekday routines. In addition, the teaching experiment

124

provided an opportunity to collect multiple sources of data and to observe the

students over an extended period of time.

Reliability of the study’s findings was maximised by appropriate and consistent

participant selection, whereby the 16 Comparison Group students and the 15

Intervention Group students came from cohorts who attended the same school,

in the same year level (albeit consecutive years), with the same mathematics

teacher and the same mathematics curriculum. In addition, the same selection

process was used for both groups and the Profiles of Problem Solving (POPS)

testing instrument (Stacey et al., 1993) was administered to the students in the

two groups at the same time of the school year to ensure similar cognitive

development and maturation of the students in the two groups. Students from

both groups undertook their pre-testing and post-testing during the morning of

the testing days to ensure that students were not tired from other testing or

activities that may have naturally occurred on the school day. The effect of

students guessing correct solutions to the test did not have a major impact on the

reliability of the data as it was the process of solution that was analysed more so

than the solutions themselves.

Practising of test questions was not allowed with either the Comparison or

Intervention Group students, with the exception of students having undertaken

the pre-test eight weeks prior to undertaking the post-test. McLellan highlighted

her concerns about the “practice effect” (Burns, 1995) of individuals undertaking

the same pre-test and post-test. She wrote, “a limitation of the POPS test is the

consumable nature of the product” (1998, p. 2). In this study, the administration

of the POPS test ensured that neither group received feedback on their pre-test

results or was made aware that the post-test contained the same questions as

the pre-test.

The reliability of the POPS test (Stacey et al., 1993) has been demonstrated by

the authors and widely accepted by researchers (e.g., Booker & Bond, 2001;

125

Impara & Plake, 1998; McLellan, 1998; Medina-Diaz, 1998). The test has been

widely used in Australia and internationally as a pre-testing and post-testing

instrument for measuring the problem-solving competence of middle-year

students. It was originally trialled with over 200 students from a selection of

schools that represented students from a diverse range of socio-economic

backgrounds in Australia and is accepted as a reliable testing instrument.

Subsequent to the original trial, a number of authors had considered the

instrument for its internal validity and reliability and determined the instrument to

be sound in both aspects (e.g., Booker & Bond, 2001; Impara & Plake, 1998;

McLellan, 1998; Medina-Diaz, 1998). Indeed, McLellan commended the

development methodology of the instrument when she wrote, “the number of

children used in the development of this assessment tool is impressive,

considering the individual involvement required from each child and the depth of

analysis applied to each child’s performance” (1998, p. 2).

The reliability of the testing instrument is further strengthened by the prescriptive

method of marking provided by the authors. To demonstrate the reliability of

their instrument, Stacey, Groves, Bourke and Doig, authors of The Profiles of

Problem Solving (1993), provided a random sample of fifty test responses to five

independent markers consisting of three of the authors, one primary teacher and

one secondary teacher, three of whom marked the tests twice. Using a Pearson

Product Moment correlation (Burns, 1995, pp. 185,193) between the various

pairs of markers, the calculated correlation coefficients were all above 0.95

(Stacey et al., 1993). According to Stacey et al. (1993, p. 57)., “these high

correlations indicate that the marking scheme is sufficiently explicit to enable

markers to make consistent judgements, even in relatively subjective areas”.

3.5.5 Ethical Issues

This study received ethics approval number 4193H from the QUT University

Human Research Ethics Committee.

126

Conducting ethical research requires a researcher to be aware of ethical issues

such as; researcher/participant relationships, confidentiality, informed consent,

storage, analysis and reporting of data, and privacy. As Glesne and Peshkin

said, “by their nature, ethical dilemmas defy easy solutions” (1992, p. 124). This

being said, every attempt was made to minimise the impact of these issues on

the participants and the research outcomes of this study. Ethical considerations,

that were specific to the research site or subjects, were carefully considered. While

this study involved participants who were unable to give their own consent to the

research, consent was sought from the parents or guardians of the participants.

All testing and permission sheets, and study results, were stored in a secure

office. Thus, this research was of Low Risk in terms of Ethics Clearance.

The participants in this present study were provided with full anonymity through

the use of pseudonyms in the reporting process and at no time were the students

required to participate involuntarily. The choice of a pseudonym for each

participant was influenced by a pseudonym code allocated in a particular way by

the researcher. This process allowed the researcher to back-track to a data

source, while at the same time protecting the participants from identification by

readers of the research report.

Students in the Comparison Group received their pseudonym code in the order

of receipt of their signed permission notes. The first student received the code of

a06 with the next being b06, c06, d06, and so on, with the first character of the

code following the alphabetic order and the “06” representing 2006, the year of

testing. Similarly, participants in the Intervention Group also received their

pseudonym code in the order of receipt of their permission notes. The first

participant received the code A07, with the next being B07, C07, D07, and so

on, following the alphabetic order and the “07” representing 2007, the year of the

teaching experiment with the Intervention Group.

127

In recognition of the participants of this study being people and not objects, the

pseudonym codes were then converted to given names using the letter of the

pseudonym code to be the starting letter of the names that were chosen at

random (see Appendices E and F). For example, the Comparison Group male

student with the code a06 became known as Adam, while the Intervention

Group, female student with the code G07 became known as Georgia. This use

of names, to refer to the students, in the reporting of this study was more

appropriate than the use of an alphanumeric code that could be perceived as de-

humanising or disrespectful to the students. This process is in keeping with the

theoretical framework underpinning this study and is more user-friendly to the

reader. While gender was not a factor being controlled or investigated in this

present study, the use of pseudonym names was kept gender consistent.

3.6 Conclusion

Three specific research questions arose from the literature review. Research

questions 2 and 3 provided an opportunity to investigate developmental learning

issues that relate to the development of problem-solving competence. The first

question however, provided an opportunity to investigate affective factors that,

through research, have been linked to underachievement in problem-solving

competence. The importance of considering affective factors was reported by

Leder and Forgasz (2003, p. 95) who said that “ it is now widely accepted that

cognitive as well as affective factors – such as attitudes, beliefs, feelings, and

moods – must be explored if our understanding of the nature of mathematics

learning is to be enhanced.”

A single method of data collection would have provided insufficient evidence to

respond to the three research questions and hence, a triangulation approach

(Gergen, 1996) to data collection was adopted to collect both qualitative and

quantitative data. This approach was consistent with Gardner’s (1999b) view of

the existence of multiple intelligences whereby he suggested that multiple forms

of data should be collected before making determinations about changes in the

128

developmental learning status of students. The collection of multiple forms of

data was also consistent with a constructionist approach to educational research

(Schwandt, 2001) whereby learning occurs progressively, as opposed to

summatively, through active student participation in the construction of

knowledge and artefacts for sharing.

The data in this study was collected during an eight-week, teaching experiment

that occurred in the participants’ school, in a familiar classroom, albeit not their

‘homeroom’. With much of the previous research on problem posing and

problem solving occurring in non-school settings, findings were open to criticism

of sustainability in everyday classroom situations (Lester & Kehle, 2003). The

design of this study addressed the need “for classroom research that is more

speculative and in which some of the constraints of typical classrooms are

relaxed while others remain in force” (Confrey & Lachance, 2000, p. 231).

The design of this study was informed by two theoretical perspectives that work

closely together; postmodern theory and critical theory (Lincoln & Denzin, 2000).

These two perspectives underpinned the teaching experiment that aimed to

effect positive change for its participants as a result of the intervention (post-

modernist theory) (see Tierney, 1997), and at the same time allowed individuals

to experience liberation and recognition throughout the learning process (critical

theory) (see Broido, 2002; Stein et al., 2003).

Issues of reliability and validity of data were addressed by the collection of

multiple sources of data, and the choice of testing instrument for the pre-test and

post-test. The selection of methods provided opportunities to correlate findings

and thus created a higher level of confidence in the data obtained from the study.

Ethical considerations about the choice of participants and their anonymity, and

the collection and storage of data have been discussed and the study was

acknowledged by the QUT University Human Research Ethics Committee as

being of low risk and therefore received ethics clearance.

129

Chapter 4

The Teaching Experiment

4.1 Chapter Overview

This chapter describes in detail the structure and evolution of the teaching

experiment, and the unforeseen events and circumstances that impacted upon

the individual teaching episodes and which resulted in refinements of subsequent

teaching episodes. While this chapter was designed to complement the

information provided in Chapter 3, about the methodology used in this present

study, it was inevitable that some references to what occurred during the

teaching experiment, and what ultimately influenced some refinements to

subsequent teaching episodes, were discussed in this chapter. However,

detailed discussion about results is reported in Chapter 5.

Chapter 4 comprises four main sections. The first section introduces the general

overall structure of the experiment and briefly re-visits the philosophical

underpinnings that were considered in its design (see Section 4.2). The second

section looks specifically at the first and last lessons in which the students of the

Intervention Group undertook their pre-test and initial survey, and post-test and

subsequent survey respectively (see Section 4.3), while the third section looks at

the development of the seven individual teaching episodes and how experiences

from each episode influenced the refinement of subsequent teaching episodes

(see Section 4.4). The final section presents a conclusion of this chapter (see

Section 4.5).

130

4.2 The Philosophical Underpinnings and Structure of the Teaching

Experiment

4.2.1 The Philosophical Underpinnings of the Teaching Experiment

Chapter 3 provided a rationale for the choice of a teaching experiment as the

appropriate intervention for investigating the three research questions presented

in Section 3.2. The teaching experiment methodology provided the ideal

framework to support constructionist beliefs that students learn best when they

are creating artefacts to share with peers, and when they are actively engaged in

their own learning (e.g., Crotty, 1998; Papert, 1991). Students were able to be

observed and interviewed over the eight-week timeframe of the intervention

process to investigate changes in their problem-solving competence,

engagement and developmental learning. During the design of the teaching

experiment, the following five student learning goals were established to enable

the research questions to be responded to and to direct the development of each

teaching episode.

1. Pose problems

2. Become increasingly engaged and interested in posing problems

3. Increase problem-solving competence

4. Choose to pose problems individually, in pairs, or in small groups, and

5. Choose to share problems independently, via a chosen peer, or via the

researcher

The two major foci of this study were a problem-posing intervention and problem-

solving competence, hence the development of Goals 1 and 3. The group of

students being investigated in this present study were ascertained as

underachieving, therefore, Goal 2 was developed to respond to the connections

established in the literature review between underachievement and engagement

131

(e.g., Jones & Myhill, 2004; Reis & Siegle, 2006). The constructionist

perspective underpinning this present study supported the fundamental premise

that best learning occurs through human interaction and the creation of

meaningful artefacts to share within a human-interaction framework (Crotty,

1998). Goals 4 and 5 were developed to ensure this focus was present

throughout the teaching experiment. These five goals provided the impetus for

the development of each of the teaching episodes (see lesson plans for each

teaching episode in Appendices D to L) and will now be further discussed in turn.

Goal 1: Pose problems

This goal was at the heart of the teaching experiment, as a problem-posing

intervention was being investigated as an appropriate strategy to improve the

problem-solving competence of the participants. There has been considerable

debate over the use of problem-posing instruction to improve students’ problem-

solving competence (e.g., Brown & Walter, 2005; Silver, 1997) and this present

study has contributed to the body of knowledge surrounding this debate.

Students were given opportunities in each of the seven teaching episodes to

pose problems from a variety of stimuli sympathetic to their interests,

experiences, and maturation. The choices of each stimulus are discussed in

Section 4.3. While students worked individually or collaboratively to pose their

own problems they created an opportunity for all three of the research questions

to be investigated.

Goal 2: Become increasingly engaged and interested in posing problems.

This study was underpinned by two theoretical frameworks; the post-modern and

the critical frameworks. A significant premise of post-modernist research, such

as this present study, is the desire to create change as a result of research

investigations (e.g., Tierney, 1997) as without this perceived opportunity of

change, there seems little point in pursuing the line of research. To fully

132

investigate Research Question 1, the increased engagement and interest of

students in posing problems was seen to be an important goal of the teaching

experiment.

Goal 3: Increase problem-solving competence.

Similarly, as for Goal 2, the problem-posing intervention in this teaching

experiment was designed to provide opportunities for developmental learning

changes in the students, as well as affective changes. The establishment of this

goal ensured the focus of the teaching experiment extended to developmental

learning changes and provided data that informed the response to Research

Question 3.

Goal 4: Choose to pose problems individually, in pairs, or in small groups.

The second theoretical framework that underpinned this study was the critical

framework (e.g., Broido, 2002). Critical theorists support research that

celebrates, encourages, and supports individuality. In this way, students can feel

valued and safe and a transformative learning environment is created (Lincoln &

Denzin, 2000). In this teaching experiment, individuality was extended to both

how the students preferred to work with their peers (Goal 4) and how they

wanted to express themselves in front of their peers (Goal 5). By focussing on

both of these goals all students were given an opportunity to work within their

preferred learning style using their dominant intelligence(s) (Gardner, 1999b).

Goal 5: Choose to share problems independently, through a chosen peer or

through the teacher.

Constructionism, as a belief system, underpins this teaching experiment by

students being actively engaged in posing problems for peers to solve. For

students to be able to share their problems (artefacts) with their peers,

consideration had to be given to ways in which the students felt most comfortable

in doing so. Therefore, students were given a variety of suggestions on how

133

they might share their problems and were empowered to suggest alternative

methods of sharing. For example, if the problem they had posed was a written

one, they could choose to read the problem themselves, or have a friend or the

researcher read it for them. Both Goal 4 and Goal 5 ensured that the

epistemology and underpinning theoretical perspectives of this study were

supported throughout the teaching experiment.

In summary, five goals were developed for this teaching experiment to ensure

that:

1) the data collected would provide evidence to respond to the research

questions, and

2) the theoretical perspectives that informed the design of this study, were

supported throughout the teaching experiment.

4.2.2 The Structure of the Teaching Experiment

In 2006, a one-hour session in week one, and a one hour session in week eight

of Term 4 were used to pre-test and post-test the Year 7 students at the research

school who met the participant criteria for the Comparison Group. Students were

excused from their weekly assembly in both weeks to undertake the testing. Pre-

testing and post-testing was undertaken in the same way for students in the

Intervention Group in the first and last lesson of the teaching experiment in 2007,

with the additional aspect of the students in the Intervention Group completing

both initial and final survey sheets.

In Term 4 of 2007, the researcher was initially provided with one hour per week

access to the Year 7 students at the research school who met the participant

criteria for the Intervention Group (same criteria as for the Comparison Group).

Term 4 consisted of eight teaching weeks. With one hour required for each of

the pre-testing and post-testing sessions, only six weeks would have remained

between the first and last week of Term 4, in which to conduct the teaching

134

experiment with the Intervention Group. Following negotiation with the Head of

Department – Curriculum and the Coordinator of Year 7 Mathematics at the

research school, permission was given to have an additional hour with the

Intervention Group students in the final week of the term since all school-based

assessment would be completed and students would be engaged in end-of-year

activities. Thus, the researcher had nine hours access to the Intervention Group

over the eight weeks of Term 4 in 2007. Lesson plans for all nine lessons can be

found in Appendices D to L.

The one-hour teaching episodes began at 8.30 a.m., fifteen minutes before

school started, and continued through the forty-five minute assembly. This timing

was agreed to by the senior staff at the research school to ensure that no

students were absent from any of their regular teaching classes during the

teaching experiment. Situating this study in a working school environment, as

required with the experimental approach, was an important aspect of this present

study but required some accommodation of atypical circumstances due to

unforeseen events (see Section 4.4). In addition, some variations to the timing of

lessons occurred for a variety of normal, school-based reasons. A summary of

the variations appears in Table 4.1.

4.3 The Pre-test and Post-test Lessons

4.3.1 Introduction

The design of this study included both qualitative and quantitative data collection

and analysis. The same testing instrument and survey sheet were used in the

first and last lessons, however, the students of both the Comparison and

Intervention Groups were unaware, until the final lesson, that the pre-test and

post-test were identical, or that the two surveys contained identical questions.

Discussion about the validity of using the same tests and surveys, and the

corresponding reliability issues of data collected from these methods, has

previously been discussed in Section 3.4.3.

135

Table 4.1

Variations to Pre-arranged Lesson Times in 2007

Lesson Number Proposed date

Resultant date

Reason for change

1.Pre-test and

first survey

8th

October

8th

October

No change

2. First teaching

episode

15th

October

15th

October

No change

3. Second

teaching episode

22nd

October

29th

October

Pupil-free day at the research

school on 22nd October

4. Third teaching

episode

29th

October

30th

October

Catch-up day for 29th October

5. Fourth

teaching episode

5th

November

5th

November

No change

6. Fifth teaching

episode

12th

November

12th

November

No change

7. Sixth teaching

episode

19th

November

19th

November

No change to date/ room change

due to a library stock take

8. Seventh

teaching episode

26th

November

26th

November

No change

9. Post-test and 26th 26th No change to date / lesson time

136

final survey November November was from 9.30 a.m. to 10.30 a.m.

4.3.2 First Lesson – Pre-test and Initial Survey (see Appendix D for lesson

plan)

The research room was set up with video cameras in place and the desks

arranged separately to ensure that students were unable to easily see a

neighbour’s work. Each desk had a survey sheet and a pen placed upon it.

Students were asked not to bring their pencil cases to the research room as they

would likely contain erasers, pencils and correction fluid that would provide a

student with an opportunity to erase working when he or she felt it was not

leading to a correct solution. Marking of the standardised Profiles of Problem

Solving test (Stacey et al., 1993) required the marker to consider all written work

to investigate the method used by the student, the accuracy of calculations, the

ability of the student to extract information, and their quality of explanation, as

well as the correctness of their final answer (see Appendix Q for the Profiles of

Problem Solving marking scheme).

All students were present for the pre-test and the initial survey. The first ten

minutes of the session were spent introducing the research to the students and

their role in the research, together with the pseudonym system that was used to

protect their identity. The provision of pseudonyms to the students was

deliberately held over until the following week to avoid distractions during this

testing session. Students were given ten minutes to complete the survey sheet.

This time appeared to allow all students to write a response to each of the four

questions. No students asked for additional time. The pre-tests were then

handed out to the students face down until all students had a copy. The test was

then completed in the required thirty-two minutes.

137

4.3.3 Last Lesson – Post-test and Final Survey (see Appendix L for lesson

plan)

This lesson followed on directly from the final teaching episode and therefore

occurred an hour later in the day than the pre-test and initial survey had been

conducted. The same process of administration was followed for this lesson as

occurred with the first lesson. Students made comment about the survey sheet

questions and the Profiles of Problem Solving test (Stacey et al., 1993) questions

being the same as they had seen eight weeks earlier, but no students seemed to

be adversely concerned.

4.4 The Seven Teaching Episodes (Lessons 2-8)

The teaching episodes began with an opportunity for the students to adapt a

given novel problem to create a new problem. This experience served to

introduce the concept of students being authors of problems and gave them their

first opportunity to seek feedback from their peers by reading their problems to

the class.

Each subsequent teaching episode gave students the opportunity to pose a

variety of novel problems to their peers and to receive feedback from both their

peers and the researcher. In some of the teaching episodes unforeseen issues

arose that necessitated the refinement of subsequent teaching episodes. As

seen in Table 4.1 some adjustments were made to the location or timing of a

teaching episode, however, other changes were made to subsequent lesson

plans as a result of students misinterpreting tasks. For example, the first

problems posed by students were superficial and involved only simple

computations to be solved. In response, the stimuli provided in the subsequent

teaching episode was more abstract thus encouraging students to be more

divergent in their thinking and creative in their problem posing. In addition,

factors that made a problem interesting, such as increased detail and a challenge

factor, were discussed. Further examples and explanations of lesson plan

138

refinements can be found in Sections 4.4.1 to 4.4.7. Details of how particular

students engaged with the learning activities in the teaching episodes can be

found in Chapter 5.

4.4.1 The First Teaching Episode - Lesson 2

The detailed lesson plan for this teaching episode can be found in Appendix E.

On this morning, rather than the students being dressed in their usual school

uniform, they were dressed in their sports uniform. It quickly became apparent

that this day was allocated by the school for the taking of sporting group

photographs; for example, the girls’ hockey team photograph, or the boys’ rugby

team photograph. The usual practice was for students in the different teams to

leave their classes, as they were called for by the photographer. Therefore,

throughout this first teaching session, different students left the research room for

up to five minutes to have their photographs taken in the adjacent hall. The

students appeared to be familiar with this practice and, on returning to the

research room, they immediately resumed their work. During this teaching

session, only three students were called for photographs and hence, the process

had a minimal impact on the Intervention Group as a whole.

The purpose of this first teaching session was twofold:

1) to encourage the students to consider more than one solution to a given

novel problem, and

2) to begin the process of posing their own problems by modifying a given

novel problem.

Students were provided with a novel problem and then given opportunities to

consider each other’s answers, as well as an opportunity to challenge and/or

support them. This part of the lesson laid the ground work to validate and ‘give

permission’ to the students to think independently from both their peers and the

139

teacher, and prepared them for their first attempts at posing problems for their

peers to solve.

Three fixed video cameras were used in this lesson as a means to capture

discussions between peers, informal interviews with students, and to monitor

student engagement. While the cameras were focussed on the case study

students, vision and conversations of some neighbouring students were also

captured by the cameras. Following the lesson, the video tapes were reviewed

and it became apparent that the sound quality from the fixed locations was poor

due to the level of background noise in the room. It was therefore decided to use

a research assistant, for each subsequent teaching episode, to enable one of the

video cameras to be mobile and hence capture significant conversations and

informal interviews from a closer proximity. This proved to be a successful

solution to the issue.

4.4.2 The Second Teaching Episode - Lesson 3

The detailed lesson plan for this teaching episode can be found in Appendix F.

This teaching episode occurred two weeks after the first one, due to the

scheduled session in the previous week falling on a “pupil-free day” when

students were not required to attend school. To add to the challenge of the two-

week break between sessions, the heavy rain throughout the previous evening

and the morning had caused localised flooding and hence many school buses

were running late. This resulted in three students being absent, and five

students being late for the session. The students who arrived more than five

minutes late were clearly unsettled however, following individual briefing on what

they were required to do, all late students settled into posing their problems. It is

worth noting that two of the students who arrived particularly late were either

unable to complete their problems and solutions in the remaining time, or wrote

very superficial problems during the remainder of the lesson.

140

Written below, are two examples of superficial problems written by students who

arrived late to this session.

Ethan, who was 30 minutes late to this session, wrote (without corrections to

Ethan’s spelling or grammatical mistakes):

If there are 40 skittles inside one pack and Liam ate 6, Kip ate 20, Elliot ate 4,

how many are left for us to eat?

Andrew, who was also 30 minutes late to this session, wrote (without corrections

to Andrew’s spelling or grammatical mistakes):

If there are 50 skittles and there are 10 different colours and 5 of each what is

the percentage of green?

These two problems contrasted with one written by Felicia and transcribed

below, who was on time for the start of this session. She wrote (without

corrections to her spelling or grammatical mistakes):

Maddie has two piles of scittles. In one pile, for every red scittle, there are 3

green scittles. In the other pile, for every green scittle, there are 3 red skittles.

Both piles have 40 scittles. How many green scittles are there?

Both Andrew’s and Ethan’s problems required only one or two steps to solve and

involved basic arithmetic, although Ethan did incorporate the concept of

percentage. The problems written by the two boys, contrasted with the question

written by Felicia. Her question incorporated the concepts of ratio and or

fractions, and involved several multiplications and additions to solve the problem,

once it had been determined how to solve the problem.

The main purpose of this teaching episode was to introduce the concept that a

“good” problem satisfies the following three criteria:

1) the problem contains an interest factor,

141

2) the problem is appropriately challenging for its intended audience, and

3) the problem has enough detail to be “do-able”.

In addition, students were introduced to reflective practice through a weekly

rating system of their problems and through immediate peer feedback (see

Appendix N for the Problem Criteria Sheet).

As students each shared their posed problems with the group, peers were

encouraged to ‘rate’ each problem by holding up one of three cards provided to

them at the start of the lesson. One card had the number one printed on it, while

the other two cards had the numbers two and three respectively printed on them.

It was explained to the students that a problem that satisfied all three of the

accepted qualities of a ‘good’ problem should receive a ‘three-rating’, whereas a

problem that satisfied only two or one of the criteria should receive a ‘two-rating’

or ‘one-rating’ respectively.

After each problem had been read to the group, and when students had rated the

problem by holding their chosen card up, selected students who held differing

numbers were asked to provide a brief justification of their choice of rating.

Students were constructive with their feedback and did not use the opportunity to

diminish a peer’s self-esteem.

For example, Paul’s problem, which was confusing to other students and not do-

able in its present form, is recorded below, followed some of the feedback from

his peers. Paul’s problem is written as it appears in his workbook without

corrections made to his spelling or grammar.

The skittles company wanted to create a new couler (pink) although the

maximum amount of skittles is 40 and there are already 37. the minimum

amount of skittles in one couler is 4.

142

Courtney rated Paul’s problem with a “2” card and when asked to justify her

answer she replied, “it was interesting <pause> and confusing and kind of do-

able, I think”. Ethan rated Paul’s problem with a “1” card and said “it was like,

<pause> hard <pause> and I didn’t get what the question was”. Paul smiled at

this latter comment, despite it being associated with a “1” rating. He appeared,

from his body language, to be proud of being acknowledged for creating a

challenging problem.

This mutual respect, such as was developed between the students during the

teaching experiment, was a phenomenon reported by Crotty (1998, p.45) who

said about humans engaging in learning with other humans, “it is in and out of

this interplay that meaning is born”. The method of immediate peer feedback,

incorporated into the teaching experiment, was consistent with the constructionist

beliefs that underpin this form of intervention. The students’ general willingness

and involvement in the problem posing and the peer rating was testament to how

“safe” and valued they felt during the session.

Towards the end of this and subsequent teaching episodes, students were asked

to self-rate their posed problems against the three criteria of a ‘good’ problem by

completing a problem criteria sheet (see Appendix N). At the conclusion of each

teaching episode, the student workbooks were collected and the researcher then

rated the students’ problems on the same problem criteria sheet. Workbooks

were returned to students at the start of each subsequent teaching episode and

time was allocated for students to read the teacher ratings and comments before

beginning the new teaching episode.

This was the first teaching episode that students were provided with a variety of

objects as stimuli about which to pose their problems. These objects were

chosen to facilitate problem posing across some or all of the strands of the

mathematics syllabus including; number, algebra, measurement, chance and

data, and space and were chosen as objects with which they would be familiar.

143

In subsequent teaching episodes the stimuli were chosen to respond to issues

that arose in a previous teaching episode. For example, students may not have

been inspired to pose problems about everyday objects found in the classroom,

such as bulldog clips, as they held little interest for them. However, if a student

was asked to pose a problem about an object that was meaningful to them, such

as an iPod, they were more likely to be engaged in posing a problem.

4.4.3 The Third Teaching Episode - Lesson 4

The detailed lesson plan for this teaching episode can be found in Appendix G.

This lesson occurred one day after the previous lesson, since the scheduled

Monday session had been used to catch-up the session missed on the ‘pupil-free

day’. Problems posed in the previous lesson had been superficial and focussed

on very basic counting strategies, so a goal of this lesson was to challenge the

students to pose more complex problems using a seemingly, unquantifiable

stimulus, thereby creating an opportunity for the students to think more deeply.

Therefore, a close-up, colour photograph of a patch of grass was provided to

each student as the stimuli for their problem posing for this teaching episode.

Counting the blades of grass was an unlikely option for students; hence they

were encouraged to think differently about the type of problem they would be

able to pose.

4.4.4 The Fourth Teaching Episode - Lesson 5

The detailed lesson plan for this teaching episode can be found in Appendix H.

After the posed problems from the previous week were read, it was noted that

the setting out of student’s solutions lacked detail. Therefore, as a prelude to the

problem posing of this lesson, the researcher posed a problem to the students

about the patch of grass photograph (stimuli from the previous lesson) and asked

for two volunteers to solve it on a double-sided whiteboard (one student either

side). This created the opportunity to focus on the quality of a solution, in

addition to the quality of a problem, and thereby assisted the students to refine

their posing ability. By considering the steps involved in a quality solution, the

144

students could then focus on posing problems that required a more detailed

mathematical solution, rather than a one or two-step superficial solution that

lacked challenge or interest.

The stimuli for this lesson were three familiar shapes; a triangle, a circle and a

square, all of the same proportion, but with no recorded dimensions. These were

chosen because they were familiar to the students, yet an immediate challenge

was present because of the lack of dimensions. Once again, the aim of this

choice of stimuli was to encourage the students to think broadly and deeply,

rather than superficially, about the type of problem they may be able to pose.

To encourage the students to focus on the three individual attributes of a good

problem, students were asked to hold up their 1, 2, 3 rating cards three times for

each problem; once for interest factor, once for challenge level and once for ‘do-

ability’, thus providing more immediate and specific feedback to the problem

posers. For example, students could now determine if their peers thought the

problem they had posed was very interesting, moderately interesting or not

interesting at all. This refinement was consistent with the problem criteria sheet

(see Appendix N) on which the students self-rated, and the researcher rated,

each problem posed by a student on a 3-point scale for each of the qualities of a

‘good’ problem.

4.4.5 The Fifth Teaching Episode - Lesson 6

The detailed lesson plan for this teaching episode can be found in Appendix I.

This teaching episode occurred on another rain-soaked morning which meant

school buses ran late again. Five students arrived between five and thirty

minutes late to the classroom, in varying states of concentration.

In the previous session, as students read their problems to their peers, it was

apparent that the length of the problems had increased substantially on

previously shared problems. This may have been as a result of students

becoming more engaged with writing their problems, or as a result of the

145

students wanting their peers to be more interested in their problems. It may also

have been as a result of a student perception that a long problem was a “good”

problem.

An example of one of these longer problems was written by Georgia and is

recorded below. Georgia wrote (without corrections made to spelling or

grammatical mistakes):

Bob usually travels to school on his skateboard. There is a new shopping

centre that is known to be a shortcut to his school. It just so happens that

today he is late for school. He gets ready quickly and decides to go through

the shopping centre. However, as he arrives to the shopping centre he can

not skateboard through it. If he were to walk it would take him twice as long if

he skateboarded. This is equivilent to his normal trip (not going through the

shop). If it takes him twenty minutes on his normal trip what would it be if he

could travel through the shopping centre on his skateboard, it would take him

twice as long if he did that. What would be the ratio on a skateboard through

the shopping centre, to his normal trip, to the WALKING through the shopping

centre.

After Georgia read her problem there seemed to be a slight pause before anyone

held up a rating card. This may have been because they were still trying to

determine what the problem was about or because they were trying to assimilate

the information provided within the problem. This is what Sternberg (2000)

referred to as the “encoding” phase of solving a problem. When students were

asked to provide feedback after the problem was shared with the group, students

commented on how hard it was to remember all the details of the problem when

they could only ‘hear’ it rather than ‘read’ it for themselves.

The feedback, provided by the students, suggested that they needed some re-

direction on how to make a problem interesting, challenging and do-able, without

it being a very long problem. Without this re-direction it would have been difficult

146

for all students to share their problems with their peers in the time provided for

each session. In addition, the refinement of the lesson plan for this session

allowed students to think differently by focussing on more concise problems.

They were now challenged to make a problem interesting, using as few words as

possible.

To accommodate this refinement, students were introduced to an example

problem consisting of one short sentence (see Appendix I) and asked to solve it

and then rate it against the three criteria for a ‘good’ problem established earlier

in the teaching experiment. The purpose of this activity was to show the students

that a problem could be interesting, challenging and do-able, despite it only

requiring one sentence to write.

The stimuli chosen for this lesson, a skateboard and an iPod, were both visual

and very meaningful to the students and were objects with which all of them were

familiar. Students were asked to try to pose a problem that could be written in

one or two short sentences.

At the conclusion of this teaching episode, students were reminded that the

library was having its stock-take during the following week so the venue for the

next session would be changed to an alternative classroom with which the

students were familiar.

4.4.6 The Sixth Teaching Episode - Lesson 7

The detailed lesson plan for this teaching episode can be found in Appendix J.

When the students were collected from their classrooms for this teaching

episode, it became apparent that a number of them were missing. Georgia,

Joanne and Leah, who all held school leadership positions, had been asked to

guide parents of prospective new students around the school campus. Quick

negotiation with the Year 7 teachers enabled other students, not participating in

this present study, to be sent to takeover from the leaders and Georgia, Joanne

147

and Leah were able to re-join the Intervention Group within ten minutes of the

start of the session.

In previous teaching episodes, despite students being free to move around the

room, all students chose a seating position at the start of each lesson and

remained seated for the duration of the lesson. In an effort to maintain student

interest, the methodology for this teaching episode incorporated visual stimuli

and required the students to move around the room and interact differently than

they had in previous sessions. (Modifying the structure and design of

consecutive lessons is a strategy commonly used by experienced teachers to

ensure students remain engaged throughout a ‘unit’ of work.) The furniture in the

alternative classroom was re-arranged to allow students to initially sit in a semi-

circle of chairs at the front of the room, with all of the desks in two long rows

behind them. This allowed some visual problems to be laid out on the tables

without the students initially being able to see them directly.

The initial consideration of short problems in the previous session had not

successfully resulted in the students posing more concise problems that were

any easier to rate than long ‘wordy’ problems, when presented orally. The

methodology adopted for this session addressed the issue by taking advantage

of student preferences to work with visual aids (e.g., Rose, 2007). Students

began by attempting to solve a selection of visual problems presented to them by

the researcher on the board (see Appendix J for examples). They were then

asked to stand up and move to the desks at the back of the room where twelve

visual problems were placed in a row. (These three-dimensional problems had

been constructed by Year 7 students in previous years during a problem-posing

unit of work.) After attempting to solve, and ultimately discussing, all of the three-

dimensional, visual problems, students were asked to construct their own visual

problems at home and share them with their peers in the following session.

148

4.4.7 The Seventh Teaching Episode - Lesson 8

The detailed lesson plan for this teaching episode can be found in Appendix K.

This session was the first of two consecutive lessons with the Intervention Group

on this day and it occurred on the final Monday of the school year. The purpose

of this session was to provide the students with an opportunity to showcase and

share their visually-posed problems. The method of peer feedback was varied

this week to add a level of competition and culmination to the teaching

experiment. Students were consulted about, and gave approval for, the changed

method of feedback. After all students had assembled their problems along the

row of tables in the classroom, they vacated the room and stood quietly in the

corridor. Only one student at a time was permitted to re-enter the room to place

a 1, 2 or 3 card alongside the problems they felt were the best; three points being

allocated for the best, two points for the second best and one point for the third

best. The word ‘best’ was defined to be a problem that most addressed the three

criteria of a ‘good’ problem (see Section 4.4.2). When all students had

completed this process, all students re-entered the room and the scores were

tallied and shared.

4.5 Conclusion

Research undertaken with students in a school setting is often beset by

unexpected dispositional factors (see for example Reis & Siegle, 2006) and

situational factors (see for example Gootman, 2001) that cannot be pre-empted

during the design of the research methodology. Factors such as rain, late school

buses, school photographs and final week activities all impacted to varying

degrees on the implementation of the teaching experiment. The design of the

teaching experiment, and the flexibility of the staff at the teaching site, ensured

that any negative impact caused by these factors was minimised. As is the

normal case for school teachers and researchers in school settings (Steffe et al.,

2000), refinements were made to subsequent teaching episodes as factors

arose. For example, when the library chose a Monday to have their stock-take,

an alternative room was chosen as the research room.

149

During the problem-posing intervention, students were encouraged to choose

how they wished to work; individually, in pairs or in small groups. The research

intervention was contingent on human interaction between the students and with

the researcher and occurred in a way consistent with constructionist beliefs (e.g.,

Broido, 2002). The teaching experiment provided the students with varied

opportunities to assemble new practices and ways of seeing problems (Tierney,

1997) and provided them with choice and control over the direction of their

learning.

Time to accommodate pre-testing and post-testing is acknowledged as

problematic, as was finding a regular intervention time to ensure that students

would not miss any of their normal school classes. A student arriving late to a

session was a regular occurrence for a variety of valid reasons, but the reasons

were always beyond the control of the students and not as a result of their

reluctance to participate in the teaching experiment. Given the context of the

teaching experiment, and the time of the year the research was conducted, the

setting for this intervention was suitable to investigate the impact of a problem-

posing intervention on the development of problem-solving competence.

150

Chapter 5

Reporting and Analysis of

the Data

5.1 Chapter Overview

This chapter reports on the qualitative and quantitative data collected during the

teaching experiment and comprises four sections that follow the Chapter

Overview. Further discussion and interpretation of the data, reported in this

chapter, can be found in Chapter 6 where responses to the three research

questions are presented. The first section of this chapter reports the data

collected from observations of, and informal interviews with, three case-study

students during the teaching experiment (see Section 5.2), while the second

section reports on student surveys data collected from all of the students in the

Intervention Group (see Section 5.3). This is followed in Section 5.4 by a report

on the quantitative data, collected from the Profiles of Problem Solving

assessment instrument (Stacey et al., 1993), collected from students in the

Comparison and the Intervention Groups. The final section presents a

conclusion of this chapter (see Section 5.5).

5.2 Observations and Interviews with the Three Case Study Students

This section will begin with a brief re-introduction to the three case study

students, Paul, Andrew and Nicole, followed by an in-depth review of their

individual progress throughout the teaching experiment. As did their peers, these

three students completed the Profiles of Problem Solving (POPS) (Stacey et al.,

1993) pre-test and post-test that was scored out of a possible 53 marks. Paul

151

scored 31 and 40 respectively, while Andrew scored 42 and 38 respectively and

Nicole scored 37 on both the pre-test and the post-test.

Paul had a median result in the Middle Years Ability Test (MYAT) (Australian

Council for Educational Research, 2005), compared to the other students in

the Intervention Group, yet he demonstrated one of the highest increases in

problem-solving competence as measured by the POPS (Stacey et al., 1993)

test at the end of the teaching experiment. The POPS test allowed data to be

collected on five aspects of problem-solving competence; correctness of

answer, method used to obtain an answer, accuracy of calculations, ability to

extract useful information and the quality of the student explanation of their

answer (see Section 3.4.3 and Section 5.4). Paul was of particular interest in

the investigation of Research Questions 2 and 3.

Andrew was chosen as a case study student for his increased engagement

in learning activities throughout the teaching experiment. He was of particular

interest in the investigation of Research Questions 1 and 3. His engagement

behaviours changed significantly from being off-task, to a student who was

interested to listen to his peer’s problems and one who became focussed

while writing increasingly mathematical problems of his own. These latter

problems, posed by Andrew, required more than two mathematical

calculations to be completed, in order to find a solution.

Nicole began the teaching experiment as the participant with the highest

MYAT (Australian Council for Educational Research, 2005) score of all

participants in the Intervention Group. While her POPS (Stacey et al., 1993)

test results did not indicate improvement in problem-solving competence, the

problems Nicole wrote in her workbook and posed to the group, consistently

became more mathematically sophisticated throughout the teaching

experiment. That is, the problems Nicole progressively posed, required an

increasing number of mathematical calculations to be undertaken in order to

152

solve her problems. In addition, Nicole did not actively participate in posing

problems in her workbook or sharing them with her peers in the first few

teaching episodes. These behaviours steadily changed throughout the

teaching experiment. Nicole was of particular interest in the investigation of

Research Questions 1 and 3.

5.2.1 Paul

As mentioned in Section 3.4.2, one of the selection criteria for inclusion in this

study was for the participants to have scored above the 60th percentile in the

MYAT (Australian Council for Educational Research, 2005). Paul had a MYAT

score equal to the median result for all students in the Intervention Group. He

scored in the 85th percentile compared to his peers with the following group

statistics: N=15; range: 76 to 94; mean = 85 and median = 85. His pre-test and

post-test results from the POPS test (Stacey et al., 1993) recorded in Table 5.1.,

show that Paul improved his overall score from 31 to 40 marks out of a possible

53 marks, improving in four of the five criteria.

In the first criterion, ‘Correctness of answer’, Paul’s high score remained

unchanged with 11 marks out of a possible 13 marks in both the pre-test and the

post-test. However, his score for the ‘Method used’ aspect increased from 6

marks to 10 marks, out of a possible 14 marks (see Table 5.1). Information

Processing theorists, such as Sternberg (2002) and Halford (2002), discussed an

individual’s chosen methods of solution to a problem in terms of change

mechanisms that they said were largely responsible for improvements in

developmental learning. In Paul’s case, improvements in the ‘Method used’

criterion would align with an increasingly efficient execution of mental processes

(automatisation) and an improved ability to select and prioritise important aspects

of a problem (encoding) (see Section 2.2.1). Sternberg (2000) said it was the

synthesis of these change processes that resulted in cognitive growth and

developmental learning. The improved results in the Paul’s ‘Method used’ score,

153

suggested that Paul may have experienced some developmental learning as a

result of the problem-posing intervention.

Table 5.1

Paul’s Profiles of Problem Solving Pre-test and Post-test Results

Pre-test result Post-test result

Correctness of answer (maximum 13 marks)

11 11

Method used (maximum 14 marks)

6 10

Accuracy (maximum 10 marks)

6 8

Information extraction (maximum 8 marks)

6 8

Quality of explanation (maximum 8 marks)

2 3

Total test score (maximum 53 marks)

31 40

Paul increased in his score for ‘Accuracy’ and ‘Information extraction’ by 2 marks.

In this latter criterion, Paul achieved the maximum marks possible by the end of

the teaching experiment. In the fifth criterion, he demonstrated a modest

improvement in his ‘Quality of explanation’. While Paul could determine the

correct answer for most questions in the POPS test (Stacey et al., 1993), his

increased ability, to demonstrate skills in these latter three criterion, following the

problem-posing intervention, needed further analysis. During Paul’s schooling,

154

he may have only been encouraged or expected to achieve the correct answer.

That is, the product, rather than the process, may have always been the primary

focus of his problem solving. Lokan and McCrae (2003) reported that only 25%

of problems provided to students in Year 8 classes in Australia were rich in

complexity emphasising procedural fluency. It is reasonable to suggest then,

that this may also be the experience for Paul. This more traditional approach to

teaching mathematics has been identified and criticised by the Australian

National Curriculum Board (2008) which referred to such teaching as “ineffectual”

and “irrelevant” in the pursuit of mathematical learning that is connected and

meaningful to a futures-focussed society.

An alternative viewpoint, to explain Paul’s improved scores in the individual

criterion of the POPS (Stacey et al., 1993) test, may come from consideration of

the amount of time dedicated to solving problems in Pauls’ normal mathematics

classes. In particular, within the amount of time spent solving problems, it would

be interesting to compare the amount of time that was focussed on finding the

correct answer as compared to ‘thinking’ about problems and on how they are

constructed and solved. This fundamental distinction was discussed in Shimizu’s

(2002) findings and in the report of data collected in the Third International

Mathematics and Science Study (TIMSS) (Stigler et al., 1999). Shimizu reported

that, for Japanese students, it was the time spent thinking about mathematical

problems and finding alternative solutions that most influenced a student’s ability

to become a more competent problem solver.

The TIMSS study presented some very pertinent data for comparing

mathematical pedagogy that supports the mathematical teaching practices in

Japanese schools. It reported that Japanese students were ranked 3rd in the

comparative problem-solving scale scores of the test, compared to American

students who ranked 24th. In addition, it reported that Japanese students spent

45 times more class time thinking about mathematical problems and inventing

new solutions than did their American peers (see Tables 1.1 and 1.2). While

155

these findings are reported about Japanese and American students, and not

Australian students, it could be argued that it may have been the opportunity to

focus on the process, rather than the product in the teaching experiment, that

allowed Paul to improve his ability to articulate his methods of problem solution.

Paul was an enthusiastic participant from the very first teaching episode. He was

full of self-confidence and very willing to share his posed problems with his

peers. He enjoyed working in pairs or by himself and once he began posing his

problems he focussed on the task at hand and ignored any distractions around

him. An early example of this was captured on Camera Three (time 06:11) in the

third teaching episode when a neighbouring student made several attempts to

tease him for his industriousness. It was clear that Paul found problem posing to

be a liberating activity and he felt empowered to demonstrate his individuality.

This may have been as a result of the critical, post-modernist framework

underpinning this study (Broido, 2002; Lincoln & Denzin, 2000), or it may have

been as a result of Paul looking for public notoriety or reinforcement from his

peers and the researcher, as discussed by Schultz (2000).

In Paul’s case there is more evidence from this present study to support the latter

explanation. When students were asked for volunteers to pose their problems to

the class, Paul would always volunteer with enthusiasm despite not always

knowing exactly what he would be required to do. This often resulted in him not

being able to communicate his thoughts coherently. For example, during the

fourth teaching episode, the lesson began with students being given a problem

posed by the researcher. Students were asked to deconstruct it and then to try

to solve it. Camera Three captured the following scene:

05:17 Researcher: “Now, I’m going to ask for two brave people.” (Paul

immediately waved his hand in the air before he knew what he was going to be

asked to do.)

156

06:00 Paul and another student were chosen to record their full solution on

either side of a mobile whiteboard in front of the class.

<Both students recorded separate solutions on individual sides of the

whiteboard. The other student was first to discuss his solution with the class.>

16:13 Paul attempted to describe his thinking to the class as he went through

the steps of his solution. He stumbled over his words and scratched his head

twice while pausing.

Paul’s solution was discussed by the group and subsequently all students were

provided a researcher solution for comparison (see Appendix H for researcher

solution). The researcher solution provided some guidance to the students about

the level of mathematical detail required to effectively communicate a solution to

a reader. Paul immediately appeared to understand the difference between how

he was accustomed to writing a solution and the researcher solution. This was

evident by Paul’s ensuing discussions with his peers and his regular reference to

the researcher’s solution in subsequent teaching episodes, and was

demonstrated in his improved scores for the ‘Method Used’ and ‘Quality of

Explanation’ criterion of his post-test.

From the fourth teaching episode, Paul began to seek preliminary feedback from

the researcher before he posed his problems to the class. An example of such a

conversation was captured by Camera Three in the second teaching episode

when students were provided with a small packet of colourful lollies (Skittles), as

well as several other objects to use as stimuli about which to pose a problem:

13:57 Paul to Researcher: “Mine will be hard if it’s just er <pause> like spoken,

but, if like, <pause> ‘cos, no-one will understand it ‘cos it’s got heaps of

colours.”

14:03 Researcher read the following problem, as written in Paul’s workbook

(without corrections to Paul’s spelling or grammatical mistakes):

157

If a new couler was created (pink) and the maximum amount of skittles is 40.

The creators wanted pink to have the same amount as the red even if they had

to take some skittles from the yellow and change them to pink. How many

skittles were in green, red, pink, orange, purple, yellow?

14:32 Researcher to Paul: “Have another read of it Paul because I’m thinking

about the third criteria. Is it do-able? In your mind you can say ‘I know what I

mean?’ but other people have to. If we publish that in a Maths book other

people have got to read it and understand it.”

14:48 Paul nodded and began to refine his problem immediately.

Paul did not complete his refined problem due to lack of time but he had begun to

consider the feedback he had received from the researcher. His partly

completed, refined problem, as written in his workbook was as follows (without

corrections to Paul’s spelling or grammatical mistakes):

The skittles company wanted to create a new couler (pink) although the

maximum amount of skittles is 40 and there are already 37. the minimum

amount of skittles in one couler is 4.

From early in the teaching experiment, students were encouraged to ‘rate’ each

other’s problems and their own. This process was made simple by including only

three criteria for deliberation, all of which were fully explained to the students to

ensure their understanding. Students were expected to consider how interesting

the problems were, whether they were appropriately challenging to their peers,

not too hard and not too easy, and whether there was sufficient information

contained within the problem for a solution to be determined (see Section 3.5.1

for discussion about peer and self-rating and see Appendix N for the Problem

Criteria Sheet template). Paul self-rated this problem as two out of three in the

three criteria of a ‘good’ problem; interest factor, challenge level and do-ability.

He wrote in the General Comments section of the Criteria Sheet, “Make it much

158

less confusing”. This demonstrated that he had been reflective about the

feedback he received from the other students and it supported Crotty’s (1998)

findings which suggested that best learning occurs when students are actively

engaged in creating meaningful artefacts to share with their peers.

Paul’s problems were always high in ‘interest factor’. In the first teaching

episode, students were presented a problem about some people going on a

holiday and they had asked their neighbour’s children to care for their dog and

cat. Students were asked to re-write the problem to pose a new problem (see

Appendix E). Most students simply changed the amount of money being offered

to care for the pets, or changed the frequency of the care needed by each pet.

Paul posed the following problem, as written in his workbook (without corrections

to Paul’s spelling or grammatical mistakes):

My neighbours are going to Chad, they have a pet lion and a pet tiger, the tiger

needs to eat 5 females every fortnight and the lion needs to eat 2 females

every second day. If we had to look after them for 5 weeks …….

The gruesome theme drew smiles and laughter from the students sitting at Paul’s

desk when he read the first line to them. This problem, and later problems

written by Paul and other male students, often contained more gruesome themes

compared to those written by the female students in the group. This is consistent

with the findings of Jones and Myhill (2004) who said that boys typically like to

explore more aggressive games and activities while female students are more

interested in collaborative and nurturing themes and activities.

Despite Paul’s problem not being completed in the time provided, he self-rated

the problem as two out of three for interest factor and challenge, and three out of

three for do-ability which was interesting considering it was incomplete. He did

not write any general comments in the space provided on the criteria sheet. It

was during this teaching episode that it became clear that Paul had some

difficulty in getting his stream of ideas into a logical sequence and order to create

159

a complete, meaningful problem in the time provided. In the first few teaching

sessions he often used more time to pose an imaginative problem than he did to

consider if it were actually do-able.

Paul’s subsequent problems were no less entertaining. In the fourth teaching

episode students were provided with a triangle, a square and a circle as stimuli

for posing a problem. Providing stimuli each session assisted the students to

focus on the posing of problems rather than the thinking about a topic about

which to pose a problem. Most students wrote about blocks of land or animal

enclosures that referred directly to the shapes provided. However, on his second

attempt, Paul wrote the following problem (transcribed without corrections to

Paul’s spelling or grammatical mistakes):

Every day the cadbury factory in Tasmania sells 50032 boxes of chocolate.

They get all there cocoa beans from a factory called cocoa bean Hevean. CbH

has had some problems with the bank lately and havn’t been able to sell their

cocoa beans. For every 100 boxes they sell Cadbury factory earns $200.

Because they havn’t been getting cocoa beans from CbH they havn’t been able

to sell any chocolates. How much money would the cadbury factory have lost if

CBh didn’t sell them coca beans for a fortnight?

When asked how the stimuli took his thoughts to chocolates, Paul said the

square reminded him of a box of chocolates and the circle and the triangle

reminded him of chocolates in the box. This was the first problem where Paul

clearly demonstrated development in his mathematical thinking and analogical

reasoning. Sternberg (2002) discussed developmental learning, such as that

typified by Paul’s newly demonstrated skills, as an increasingly efficient

execution of mental processes (automatisation), an increased ability to select

and prioritise relevant data from a situation being mathematised (encoding), and

an increased ability to draw on prior knowledge to inform current thinking

(generalisation). Paul had incorporated mathematical information into an

160

elaborate storyline in a meaningful way. After he read his problem to the class,

Paul was able to clearly describe the steps that would be needed to solve the

problem. He had successfully transferred his knowledge gained from peer and

researcher feedback of his previous problems to create a new and meaningful

novel problem. This finding supports Mestre’s (2000) research that suggested

posed problems provide evidence of whether the author has become ‘well-

versed’ in how mathematical concepts can apply across a range of problem

contexts.

Paul continued to demonstrate the development of his mathematical thinking

throughout the teaching experiment. Each successive problem posed by Paul,

albeit containing grammatical and spelling errors, was increasingly mathematical

in content while remaining entertaining to his peers. Brown and Walter (2005)

reported that problem posing has the potential to encourage students to think

divergently, which in turn has the potential to promote developmental learning

opportunities. Their finding is supported by Paul’s progress in posing meaningful

and detailed problems as he moved further into the teaching experiment.

In the fifth teaching episode students were provided with a cartoon about

skateboarding and an iPod as stimuli for this session (see Appendix I). Paul

posed the following highly-entertaining problem (transcribed without corrections

to his spelling or grammatical mistakes):

At school kids were getting really angry and went on a strike, because the

teachers didn’t want any kids leaving the school they decided to let the kids run

the school for 1 Day. The kids created a poster to show how mean the

teachers had been the teachers didn’t care. The kids were soo angry about

how rude the teachers were being they decided to destroy the school. In the

school there was 100 bricks 30 lights and 30 desks. To Destroy 1 brick it takes

1 minute and a half to destroy 7 desks it takes 9 minutes and to destroy 13

lights it takes 17 minutes. How long does it take to Destroy the whole school?

161

This problem showed that Paul was now able to incorporate some increasingly

sophisticated mathematical challenges into his problems. Not only did he

challenge his peers to extract information and do some multiplications to solve

his problem, he also challenged them to do additions and conversions of units.

Paul had begun to demonstrate what Sternberg (2002) referred to as the

“synthesis of change processes” that results in cognitive growth. Previously Paul

had not been able to coherently explain his thoughts, whether they were in

describing how he would solve a peer’s problem or how he was going to pose

one of his own problems. However, Paul was now able to demonstrate

sequencing of logical thoughts to achieve a meaningful product, that is, an

interesting, challenging and ‘do-able’ problem. Despite the problem being

implausible, Paul recorded an accurate solution to the problem in his workbook.

According to studies undertaken by Bernardo (2001), analogical reasoning is

available to all students to different degrees, however, it can be further

developed through appropriate intervention strategies. Paul’s increasing

understanding of problem structures had enabled him to ‘map’ between his

previous problems and his new problems, thus suggesting that his experiences in

the problem-posing intervention has facilitated some developmental learning

changes for him. His problems had become ‘do-able’ as well as highly

interesting to his peers.

5.2.2 Andrew

In the selection process, Andrew had the lowest MYAT score of all participants in

the Intervention Group, scoring in the 76th percentile. He was by far the most

lacking in confidence and most easily distracted student at the start of the

teaching experiment, thus making him a suitable candidate for a case study

student. During the first few teaching episodes, Andrew did not pose complete

problems or work earnestly at the task. Shultz (2000), when talking about “gifted

underachievers”, said that students, such as Andrew, typically do not fully

engage with work for a number of reasons including possible low self-esteem,

162

while Boehnke (2007) suggested disengagement may be due to the individual’s

fear of social exclusion. While Andrew seemed quite confident in his interactions

and was liked by his peers, he initially seemed self-doubting and reticent about

his ability to pose or solve problems.

In the first few teaching episodes, when volunteers were asked to share their

problems, Andrew would not raise his hand. When he was asked about what he

thought of his problems, he replied that they were not very good and not worth

sharing with his peers. However, as the teaching experiment unfolded, Andrew’s

self-esteem and confidence developed and, in the third teaching episode, he first

offered to share with his peers a problem he had posed. When his peers were

asked to ‘rate’ his problem using the 1, 2, 3 card system, Andrew initially looked

anxious. As the cards were raised and Andrew saw that there were some 2’s

and 3’s amongst them, he smiled then immediately dropped his head as if he did

not want to show his peers that he was so pleased.

As the teaching experiment continued to unfold, Andrew not only engaged with

the problem-posing activities, he also gained in self-confidence and actively

ignored students who tried to distract him. As the teaching episodes progressed,

his problems became far more interesting, including detailed story lines about

topics that were familiar to himself and his peers. In addition, he began to enjoy

the opportunity to pose and share his problems and joined Paul as one of the first

students to offer to share his problems with the group when given an opportunity.

At the completion of the fourth teaching episode, students were asked if they

were enjoying the sessions. Andrew responded by saying that it was good to

learn how to solve problems by learning how to “write” them. This was an

interesting response because Andrew had himself made the connection between

posing and solving problems despite no direct mention being made of this

connection in the question. The ability for problem-posing activities to impact on

a student’s self-efficacy was reported by Bandura (1997) as significant, while

163

Knuth (2002) reported its positive impact on the development of students’

intrinsic motivation to engage in learning activities. Both authors associate

problem-posing activities with these two positive changes that were increasingly

demonstrated by Andrew throughout the teaching experiment. It would seem

reasonable to suggest then that we may be able to attribute Andrew’s increased

engagement to the problem-posing activities undertaken as part of this current

study.

Andrew’s pre-test and post-test results are shown in Table 5.2. While the test

results would appear to indicate that he made no gains in problem-solving

competence as a result of the teaching experiment, the observations made of

Andrew by the researcher and the developing quality of problems posed in his

workbook, present a different picture.

The session in which the pre-test occurred was the first session the students

came together as a group and met the researcher. All students, including

Andrew, quietly and conscientiously completed the student survey and the pre-

test. As a group, they were far more reserved in this session than they were

throughout the teaching experiment or in the final session (the post-test session)

by which time they had become familiar with the researcher, the teaching

experiment, posing problems and being withdrawn from their usual class

activities. On the day of the post-test, Andrew appeared particularly distracted.

His distraction may have been due to the final session taking place in the final

school week of the year, or to the lack of problem-posing opportunities in this

session. Jones and Myhill (2004) and Baker (1998) reported that perceived

underachievement, such as Andrew’s, has little to do with a lack of ability and is

more likely to do with situational factors such as the timing of a test. While a

number of the students were clearly excited and more animated at the start of the

post-test lesson than they were at the start of the pre-test lesson, most students

were able to re-focus and apply themselves to complete the post-test to the best

of their ability. Andrew did not have the same ability to moderate his physical

164

and emotional responses to the situational factors of the day. He seemed

distracted and fidgeted throughout the session.

Table 5.2

Andrew’s Profiles of Problem Solving Pre-test and Post-test Results

Pre-test result Post-test result

Correctness of answer (maximum 13 marks)

11 10

Method used (maximum 14 marks)

11 8

Accuracy (maximum 10 marks)

8 8

Information extraction (maximum 8 marks)

8 8

Quality of explanation (maximum 8 marks)

4 4

Total test score (maximum 53 marks)

42 38

From the earliest teaching sessions Andrew had demonstrated how easily he

was impacted by situational factors. His engagement in the first teaching

episode was characterised by fiddling with pens, being easily distracted by any

form of movement or noise around him, and with minimal writing. He required re-

direction from the researcher on three occasions during this session. An

example of a three minute time segment, captured on Camera One in the first

session, can be seen below:

165

22:30 laughing with neighbours – off task

23:11 “I have an idea” – finally started writing

23:48 put hand up for assistance but researcher was with another student

24:06 put hand down and started to write

24:29 looked up and chatted to neighbour

24:32 began writing again

24:38 neighbour wanted to swap pens but Andrew playfully put his pen

behind his back

24:53 pen situation resolved – started writing again

25:46 Andrew finished modifying the researcher-provided problem and looked

up (see Appendix I for original problem provided to students)

Andrew’s problem, written below without grammatical corrections, was

superficial, unclear and lacked detail. He did not provide a worked solution,

choosing instead to simply record how he would divide the money.

Tom and Sue have asked to look after the neighbours dogs they have left for a

holiday 5 days and are giving the $100 to share between them Tom has to look

after fifi for 50 mins Jane has to look after Pluto for 25 mins everyday.

Tom = $75

Jane = $25

As students completed writing their problems, volunteers were sought to read

their problems to the class. Andrew did not offer to read his problem. At the end

of the session he chose not to complete his first self-rating criteria sheet despite

other students around him taking the time to fill in their sheets.

166

During the second teaching episode Andrew’s engagement in the sessions

began to change noticeably. He started to respond to questions from the

researcher without shrugging his shoulders or looking away. An example of such

a response was recorded on Camera One:

07:40 Researcher: “What was the first thing you thought to do Andrew?”

07:43 Andrew: “Um, well, condense five squared to find out how much the

perimeter <pause> ‘cos the whole thing is 72 you have to find out how much

more than five squared it is.”

This was followed later in the lesson by another example of engagement

captured on Camera Two:

13:03 Researcher: “Andrew, tell me what were you thinking?” (Students had

been asked to determine a strategy to calculate how many dots were in a piece

of aboriginal art as a warm-up exercise for their subsequent problem posing.

See Appendix H for details.)

13:05 Andrew: “If you really want to, you can count up the outside ones of the

circle and times it by how many rings are in the circle.”

Later in the lesson, when students were given an opportunity to pose their first

novel problem, using a photograph of some grass as a stimulus (see Appendix

H), Andrew began to work immediately. He continued to work uninterrupted until

the researcher asked for volunteers to read their problems aloud. The following

dialogue was captured on Camera Two:

48:18 Researcher: “Andrew, have a go at yours.” (Andrew smiles)

48:21 Andrew: “In a grass patch there is a quarter green grass and the

remainder is dead. If the grass patch is a square and one side is 18cm, what is

the area of dead grass?”

167

48:36 Researcher: “How many steps do you think there are to mathematically

solve that?”

48:41 Andrew: “Three” (Andrew smiles with confidence)

This problem was quite different from Andrew’s previous problem because it

made sense and the grammar and spelling were correct. Andrew also took the

time to record a fully worked solution in his workbook. When asked to self-rate

his problem Andrew gave himself a two out of three for each of the three criteria;

interest factor, challenge level and do-ability. In addition, he wrote the following

sentence in the General Comments section of the criteria sheet, “I think my

problem is better than last because it does actually have steps to take to work it

out.” This comment and Andrew’s increased engagement demonstrated that he

had responded to feedback and had become engaged with the process of posing

meaningful novel problems. Andrew had demonstrated what Brown and Walter

(2005) reported when they said that posing problems puts the poser in charge of

the learning process which in turn has the potential to develop more divergent

forms of thinking. From the problems Andrew had written in his workbook, it

could be concluded that Andrew had either chosen to apply himself to the task at

hand or that he had begun to think about problems differently to the ways he had

done in the past.

Andrew continued to be fully engaged and on-task over the remaining teaching

episodes. From the third teaching episode he began to seek feedback from his

neighbours about his posed problems, before the class had been asked to share

their problems. By the fifth teaching episode he was also seeking preliminary

feedback from the researcher before posing his problem to his peers. An

example of this was captured on Camera One at 35:16 and resulted in Andrew

writing the following problem, with a full solution, in his workbook:

If Andrew is riding his skateboard at 17 km/hr and 50-50 grinds down the

handrail and increases his speed by 5% how fast is he going? If Andrew then

168

lands and his speed drops by 2% what is his speed after the trick and while he

is doing it?

This problem, posed by Andrew, could be considered to be the “observable end

state” in the process of thinking as discussed by Siegler (1991). Seigler reported

about processes of thinking as being the initial and intermediate steps to

developmental learning accomplished inside an individual’s mind and that are

unseen by an observer except through the end product, which in this case was

Andrew’s problems. If we accept Siegler’s position, then the change in the

‘quality’ of Andrew’s problems could be considered as evidence of developmental

learning. Andrew had consciously and unconsciously become involved in the

cognitive processes that resulted in him posing increasingly sophisticated

problems for his peers to solve. He became increasingly efficient in the

execution of his mental processes (automisation), he was clearly able to select

appropriate data and prioritise how to present it in the form of a coherent problem

(encoding), he responded to prior feedback from his peers and the researcher in

the posing of subsequent problems (generalisation) and was able to coherently

discuss solutions for his problems and his strategies for constructing them

(strategy construction).

The increased demonstration of these four skills, automisation, encoding,

generalisation and strategy construction, was identified by Sternberg (2000) as

playing a significant role in developmental learning. It is therefore reasonable to

suggest that we may be able to attribute some developmental learning changes

demonstrated by Andrew to the problem-posing intervention in this study.

Andrew’s ability to remain on-task was repeatedly challenged by distractions

around the room during the teaching episodes. As time progressed he moved

from a position of ignoring these distractions to actively discouraging them. For

example, in the fifth teaching episode, Camera Three captured the following

incident:

169

41:25 Blair, who had finished posing his problem, started to move irritably from

side to side on his seat in an attempt to distract Andrew who was still writing.

41:28 Andrew said to Blair, without looking up from his writing, “Don’t pay me

out!”

41:30 Blair smiled and replied, “Sorry, I’m not paying you out, I’m just rocking.”

In the sixth teaching episode students were investigating and posing visual

problems (see Appendix J). Andrew was particularly engaged during this

teaching episode and was clearly not prepared to be interrupted by other

students. Towards the end of the lesson the researcher asked all students to sit

quietly for an evaluation of the lesson, but some students continued to chat to

each other about their problems. Camera One captured the following scene:

45:15 Andrew puts his finger to his lips and said “Ssh!” to the students around

him. Students responded and the researcher continued to talk without any

further student interruption.

While involved in the problem-posing intervention, Andrew had become a fully-

engaged student with an ability to self-moderate his behaviour and remain on

task without re-direction. This outcome would not be unexpected by English

(1997b) who reported that a problem-posing classroom can empower and

encourage students to pursue “lines of inquiry” that are personally satisfying to

them, as was clearly the case with Andrew. He had even begun to ‘take some

risk’ with regard to how he was perceived by his peers transforming from a

‘follower’ in the first teaching episode to a ‘leader’ in the final teaching episodes

by encouraging others to engage in the problem posing when they were off-task.

The problem-posing intervention had served to re-engage Andrew to work in a

focussed way with novel problems. His increased focus and change of

behaviour was noticeable and increasingly consistent while he was engaged in

problem posing. In the absence of an opportunity to pose problems, Andrew’s

170

behaviour returned to being affected by situational factors and he disengaged

with set tasks. This was evident in the final session of the teaching experiment

when Andrew was required to solve problems in the POPS test (Stacey et al.,

1993). For Andrew, it seemed that it was the opportunity to pose problems and

receive feedback on those problems from his peers that allowed him to engage in

learning throughout the teaching experiment.

5.2.3 Nicole

In the selection process, Nicole had the highest MYAT score of all participants in

the Intervention Group. She scored in the 94th percentile compared to her peers.

During the teaching episodes, students had four opportunities for the researcher

to ‘rate’ their problems against the three qualities of a ‘good’ problem; interest

factor, appropriate level of difficulty and do-ability (see section 3.5.2.4). Nicole’s

researcher-rating range was 5.5 to 7.5 out of a possible 9 marks, compared to

the group average researcher-rating range of 4.6 to 6.7. Interestingly, while

Nicole posed consistently sophisticated problems during the intervention, her

pre-test and post-test results from the POPS test did not vary significantly. Her

pre-test and post-test results are shown in Table 5.3.

Nicole’s ‘Quality of explanation’ mark decreased from 4 to 2 marks out of a

possible 8 marks. As she had previously demonstrated an increased capacity in

this criterion and the pre-test and post-test were identical test papers, it can be

assumed that either dispositional or situational factors had influenced Nicole

during the post-test. The ability of such factors to have a substantial influence on

an individual’s preparedness or ability to demonstrate skills, was reported by

Gootman (2001, p. 5) who said that “many children haul the baggage of

dysfunction straight into the classroom” where it can manifest in

underachievement or disengagement.

The students in this present study were not immune from day-to-day challenges

that can arise at any given time on any given day, and the exact nature of

171

situational factors impacting Nicole on this particular day were not known.

However, dispositional factors such as Nicole being bored or simply disinterested

in completing a test that she had already seen and completed seven weeks

earlier, cannot be dismissed as contributing to Nicole’s decreased score on the

post-test in this criterion. Gentry, Gable and Springer (2000) reported that

boredom is a major contributor for the withdrawal from activities by middle-year

students and can therefore not be discounted as contributing to Nicole’s

disengagement from the activities in the final lesson of the teaching experiment.

Table 5.3

Nicole’s Profiles of Problem Solving Pre-test and Post-test Results

Pre-test result Post-test result

Correctness of answer (maximum 13 marks)

9 9

Method used (maximum 14 marks)

10 10

Accuracy (maximum 10 marks)

6 8

Information extraction (maximum 8 marks)

8 8

Quality of explanation (maximum 8 marks)

4 2

Total test score (maximum 53 marks)

37 37

172

At the start of the teaching experiment, Nicole was an acquiescent student who

preferred to pose problems by herself. While she was willing to pose problems,

when students were first given the opportunity to rate each other’s problems

using the 1, 2, 3 cards, she did not participate in the rating process. By the

second teaching episode she was scanning the students sitting at her table to

see what ratings they were giving. Only then would she lift her hand from the

table with her chosen rating card. As she lifted a card she kept her wrist on the

table which meant the cards were never raised high enough for them to be

clearly seen. By the third teaching episode, Nicole had begun to lift both her

hand and her wrist from the table to raise a rating card and she did not always

wait to see which cards her peers were raising. It appeared that she had begun

to feel ‘safe’ in the teaching environment.

Nicole continued to become more engaged and enthusiastic about rating other

student’s problems as the teaching experiment progressed. The behaviours,

demonstrated by Nicole, are consistent with those reported by Alvesson and

Skolberg (2000) who found that students would participate more readily in

activities if they felt ‘respected’ and if the learning environment was ‘supportive’

and involved ‘jointly constructed knowledge’, as was the case in these problem-

posing sessions. In the first two teaching episodes Nicole did not volunteer to

share her problems with her peers and once she had posed her problems, she

would not refer back to them. That is, she would quietly look around the room, or

would ask to read a neighbour’s problem. By the third teaching episode, Nicole

had begun to seek ideas from her neighbour before posing a problem and had

willingly offered to read her problem to her neighbour. She had still not

volunteered to read her problems to the class. The following scene was captured

during the third teaching episode by Camera One. The students had just been

given a photograph of some grass as a stimulus for their problem posing (see

Appendix G).

173

20:53 Nicole to her neighbour while she looked at the photograph of the grass:

“I don’t know what to do.”

22:43 started to pose a problem in her workbook.

23:04 looked up and mouthed silently to herself, “I don’t know what I am doing.”

24:09 stopped writing and doodled on her page.

24:29 started writing again

24:29 Nicole lifted her head with a form of non-focussed, rapid-eye movement.

25:12 started writing again

25:33 looked up and scratched her head with a quizzical look on her face

25:35 started writing again

26:23 stopped writing, looked up and began to chew the top of her pen

26:26 started writing again

26:36 stopped writing, looked up and flicked her pen through her hair

27:03 started writing again

28:03 read her problem to herself

28:48 read her problem to her neighbour and laughed happily when she has

finished

29:10 made some changes to her problem

Nicole appeared to gain confidence from her neighbour’s positive feedback. This

spurred her to seek further feedback from the researcher that was captured

again by Camera One.

174

32:27 Nicole put her hand up to attract the researcher’s attention

32:39 Nicole to researcher while pointing to her problem: “Can you see what

you think of that?”

<researcher read problem and offered some feedback>

34:45 Nicole nodded and smiled and, when the researcher had moved away,

tore the page out of her workbook. She then re-wrote the problem on a fresh

page.

Nicole continued to work on her problem and its solution despite the class being

asked to stop work so volunteers could pose their problems to their peers. She

did not look up to listen to her peers or to rate their problems with her rating

cards. Her completed problem is recorded below:

My friend Oscar pulls out grass for a job. Yesterday, he had a patch of grass

32m x 21m and he was pulling the grass out at a rate of half a square metre per

20 mins. If he got paid $1.25 for each hour he did, how much money did he

earn?

On her self-rating criteria sheet, Nicole gave herself a three out of three rating for

challenge level and do-ability, and a two out of three for interest factor. In the

General Comments section she wrote, “I liked it. I think it’s my best one so far.”

Gardiner (1999b), who emphasised the ‘separateness’ of intelligences, reported

on students such as Nicole. If the multiple intelligences viewpoint is accepted, it

could be suggested that Nicole’s perceived disposition towards problem posing

may have been more related to her naturally dominant intelligence of being

‘intrapersonal’ rather than her response to the activity itself. Despite Nicole

having a more introverted personality than that of her peers, she continued to

become more engaged with posing problems over the remaining teaching

episodes. She took less time to start posing her problems and the problems

were always interesting and challenging. Rather than sitting quietly and looking

175

around the room, Nicole became proactive in seeking help from the researcher to

start a problem. The following scene was captured by Camera Three in the fifth

teaching episode after the students had been provided with a skateboard cartoon

and an iPod as stimuli (see Appendix I):

35:35 Nicole to researcher: “I kind of know what I want to do. I just don’t know

how to do it.”

<researcher read what Nicole had written so far>

35:49 Researcher: “Okay, tell me what it is you want the reader to find out?”

35:52 Nicole: I was going to say she has this CD collection and she has to

download all of them but she doesn’t know if she’s got enough room <on her

iPod>.”

After considering the advice provided, Nicole posed the following problem in her

workbook:

Leanne has just bought a new iPod that can hold 8MB. She has already

downloaded 204 songs, and she still has 5.7MB left. She has 27 albums that

she wants to download to the iPod. If the average number of songs on each

album is 18 – 22, will she have enough room?

Nicole rated her problem on her self-rating Criteria Sheet as two and a half out of

three for interest, one and a half out of three for challenge and three out of three

for do-ability. In the General Comments section she wrote, “I liked this one

because it involved music.” It was during this teaching episode that Nicole first

volunteered to pose her problem to the class. In addition, she became more

actively engaged in listening and rating her peer’s problems as recorded on

Camera 2 starting at 38:49. Problem posing had created an opportunity for

Nicole to develop not only her mathematical skills, but also to become more

personally involved with the process of her learning and to improve her self-

176

efficacy. This ability of problem-posing activities to positively influence an

individual’s self-efficacy has previously been reported by Bandura (1997) and

Marat (2005). While Nicole was initially an introverted student, her involvement

in the problem-posing intervention in this present study, had encouraged her to

become a more outwardly focussed and socially interactive student.

5.3 Student Surveys

There were four questions in the survey that the students in the Intervention

Group responded to prior to the pre-test and then again prior to the post-test.

(see Appendix C). While the student responses were narrative in nature, it was

possible to identify common emergent themes for which a tally was recorded to

indicate the level of student support for each similar type of response (see

Section 3.5.2.3 for the methodology of identifying these emergent themes). All

student responses were accommodated in the categories chosen for each

question as seen in Tables 5.4 to 5.7. Most students responded to each

question with two or three sentences, hence their responses were simple to

interpret and to categorise. The data from each question are reported in turn,

beginning with Question One and responses from the case study students as

well as others students in the Intervention Group are reported here.

5.3.1 Question One

The students’ responses to Question One, as summarised in Table 5.4, were

enlightening and supported the existence of a dispositional shift towards

engagement with problem solving as a result of the problem-posing intervention.

This position is supported by the dramatic increase in affirmative responses to

‘Do you enjoy solving problems?’ from 2 before the intervention, to 11 at the

conclusion of the teaching experiment.

177

Table 5.4

Do you enjoy solving problems?

Pre-intervention Post-intervention

Indicated affirmative response 2 11

Indicated negative response 8 2

Neutral or undecided 5 2

Note. The number of students in the Intervention Group was 15.

A typical set of responses to Question One included those from Blair and Felicia

who wrote:

Blair prior to the teaching experiment:

It really depends on the difficulty. If a problem is easy and interactive most

people will enjoy it and accept they can do it. But if it is hard, and requires

special practice, then I usually end up simply becoming frustrated and lose

patience for the problem…

Blair at the completion of the teaching experiment:

Depends on their do-ability, interest factor and challenge. Problems that are

too challenging and are not very interesting quickly lose their charm, while

fun, interesting and challenging problems are great.

178

Felicia prior to the teaching experiment:

I hate solving problems. I hate it because it takes so long to do and I hate

having to set out all of my working…because it always makes me confused

with my answer.

Felicia at the completion of the teaching experiment:

Now doing these sessions I feel more confident solving problems and I enjoy

them more.

Blair’s and Felicia’s responses indicated that, as a result of the problem-posing

intervention, they had adopted new attitudes to solving problems. Blair’s view

had changed from, a good problem being an “easy” problem, to a good problem

being a “challenging” problem, while Felicia’s view had changed from hating

solving problems, to enjoying solving them more. Brown and Walter (2005, p. 5)

reported similar findings on the potential of a problem-posing intervention to

“create a totally new orientation” of students’ thinking towards mathematical

problem solving. This position was further supported by many of the other

students whose responses in the post-intervention survey included (presented

without spelling or grammatical corrections):

Paul at the completion of the teaching experiment:

I enjoy problem solving much more now that I know how to figure them out.

Hayley at the completion of the teaching experiment:

The problems were fun to solve as we got to make our own.

Rodney at the completion of the teaching experiment:

I do enjoy solving problems now because they are fun and if you do more of

them it is funner and you become smarter.

179

Georgia at the completion of the teaching experiment:

I found after these sessions that I really, really like working out problems and

creating them.

The affirmative responses to Question One on the survey had increased from

13% of the Group before the problem-posing intervention, to 73% of the Group at

the completion of the teaching experiment. This result, combined with the

observations of students throughout the teaching experiment, could suggest that

problem-posing activities have the potential to re-engage middle-year

mathematics students.

5.3.2 Question Two

The responses to this question in the initial survey, compared to those in the final

survey, indicated that the way students viewed mathematical problems had

changed (see Table 5.5). For example, in the first survey, a problem with few

words and requiring only a few calculations to solve was the most highly-rated

category by students. However, by the end of the teaching experiment, the most

highly-rated category was the visual problems category. This change in student

preferences cannot easily be interpreted without looking further into the

supporting comments that students wrote in their responses to this question on

the survey sheets. However, it would seem that, as a result of the problem-

posing intervention, the students may now have been looking at the structure of

the problems from an informed, discriminating position as problem posers rather

than simply as problem solvers. They were now able to see the problems more

clearly for how they were constructed rather than for simply what the problems

were asking them to find. The responses suggested that the students were now

looking at problems through a different ‘lens’. Lesh and Doerr (2003) reported

this ability to “see” a problem as being equally important to being able to “do” a

problem.

180

Table 5.5

What type of problems do you prefer to solve?

Pre-intervention Post-intervention

Number problems – problems with few words

that require several mathematical calculations 5 1

Word problems – problems that require

substantial reading to extract data for solving 2 2

Visual problems – problems that can be

solved without the use of calculations or reading more than a few words

2 6

Short problems – problems that contain no

more than two short sentences and that require only one or two calculations

3 3

Real-life problems – problems that are

written about familiar real-life contexts 2 1

No preference 1 2

Note. The number of students in the Intervention Group was 15.

In the first survey, conducted before the students had begun the intervention, the

students’ responses to Question Two, focussed on the level of difficulty and

length of problems. Some examples of responses written by the students are

recorded below:

Joanne prior to the teaching experiment:

Easy problems.

Nicole prior to the teaching experiment:

I like solving problems that are kind of hard, but not impossible.

181

Rodney prior to the teaching experiment:

I like to solve easy but not too easy [problems] and if I finish the easy ones I

like doing hard ones…

Kelly prior to the teaching experiment:

I prefer to solve problems that are shorter because I figure it out faster….

Andrew prior to the teaching experiment:

I prefer to answer just questions that get straight to the point because they

are simple.

Following the problem-posing intervention, the responses to this question had

changed notably. They now focussed on the structures and content of the

problems. Some examples of responses written by the students are recorded

below:

Blair at the completion of the teaching experiment: (without spelling

corrections)

Problems in my field of interest are the best for me. Problems with soccer,

snowboarding, archetecture and others with pictures and diagrams are

fantastic.

Paul at the completion of the teaching experiment: (without spelling

corrections)

I like to solve problems that have twists in them because it’s a challenge and

once I figure out what it is I feel forfilled.

182

Georgia at the completion of the teaching experiment:

We have learnt different types of problems…. I really like looking at a

problem and thinking outside the box.

Brown and Walter (2005, p. 165) discuss the potential of a problem-posing

intervention to reorientate students’ thinking from ‘narrow’ and ‘inward’ to

‘divergent’ thus creating new learning opportunities. They said that problem

posing “...is also a handmaiden of other aspects of mathematical activity – from

problem solving to greater personal understanding”. The responses to Question

Two in the final survey supported this finding and suggested that students had

consciously and unconsciously made new personal discoveries about their own

problem-solving identities. Not only were these discoveries about problem

structures, they were also about their own feelings towards different types of

problems and their solutions. Responses to this survey question suggested that

involvement in a problem-posing intervention has the potential to change the way

that students feel about problems, which can then impact upon the engagement

of these students in solving problems.

5.3.3 Question Three

The responses to this question, as summarised in Table 5.6 above, did not vary

substantially between the initial and final surveys. In the initial survey, 13 out of

the 15 students wrote that problem solving is a skill used in ‘real-life’ situations

that is needed for life outside of school. The students’ responses were largely

unchanged in the final survey. The one student who seemingly changed her

view gave the following responses to this question:

Hayley prior to the teaching experiment: (without grammatical corrections)

Yes, because you learn a new skill and you are faced with problems every

single day of our live. They help you with your estimation and trial and error.

183

Hayley at the completion of the teaching experiment: (without

grammatical corrections)

No, not really, because you wouldn’t use it unless you wanted to be a

crossword person or something. You would use it but it’s not really a must

like maths and english.

Further, more specific questioning may have allowed an informed interpretation

of the apparent change in Hayley’s position on this question. Her first response

appeared considered while her second response seemed flippant and conceived

without due attention to the question or the importance of her response.

Table 5.6

Do you think learning to solve problems is a useful thing to do?

Pre-intervention Post-intervention

Indicated affirmative response 15 14

Indicated negative response 0 1

Neutral or undecided 0 0

Note. The number of students in the Intervention Group was 15.

Like Nicole and Andrew, Hayley may simply have been distracted by the

situational factors occurring in the final week of the year, thus influencing her

readiness to be meaningfully engaged in this final survey. Schultz (2000) offered

an alternative viewpoint to inconsistent responses, reporting that gifted

underachieving students make conscious and sub-conscious choices about how

184

they react in situations and are far more likely to choose to “withdraw” from

engagement in an activity if they feel “under-challenged”. In Hayley’s case, it

may be that she simply felt disinterested in a task that she had seen before and

that was not, in her view, as important as other activities in which she may

otherwise be engaged. Indeed, it may have been a combination of both factors

that resulted in her quite different responses to the same question.

5.3.4 Question Four

The responses to this question were varied within each survey and between

surveys. (A summary of the results is presented in Table 5.7.) However, in the

first survey it seemed that students were of the view that there was a lock-step

process to learning how to solve problems. Two typical responses are recorded

over page:

Rodney prior to the teaching experiment: (without grammatical and

spelling corrections)

They [teachers] could have a couple of problems then do the first one with us

then the students could do them by them selfs that way the children know

how to do it and it would be easier.

Oliver prior to the teaching experiment:

They [teachers] teach us the different ways to solve problems. They also

teach us about the different types of problems.

Oliver’s view had changed by the time he responded to the same question at the

end of the problem-posing intervention. His second response is recorded over

the page:

185

Oliver at the conclusion of the teaching experiment: (without spelling

corrections)

…They [teachers] make us write our own problems so if we no how to write

the problems we know how to solve them.

This new response indicated that Oliver had adopted a positive view on the

potential of problem-posing to assist him to become a more competent problem

solver.

Table 5.7

What things could teachers do to assist you to become better at solving problems?

Pre-intervention Post-intervention

Make problems about real life situations 1 1

Teach us the steps to solve problems 4 3

Start with simple problems first 2 0

Help students understand the problems 3 4

Provide a positive class environment 1 0

Do problem solving regularly in class 3 5

Let students pose problems 0 1

Allow students to use visual aids 0 1

No help required 1 0

Note. The number of students in the Intervention Group was 15.

186

Paul, who made one of the largest improvements in the POPS test (Stacey et al.,

1993) following his participation in the problem-posing intervention, wrote a

similar response to Oliver when he answered Question Four in the final survey.

His response was:

Paul at the conclusion of the teaching experiment: (without grammatical

corrections)

Well they [teachers] could … let us figure out how to create a problem so that

you know how to fix it

Silver (1997) had reported the potential that problem-posing activities have to

demystify problems, over a decade ago. The responses to this question, from

both Oliver and Paul, add support to Silver’s findings and remind us of the need

to consider problem-posing activities as worthwhile for inclusion in mathematics

teaching practice. In addition, as mentioned earlier in Paul’s case, we can go

further to add that significant developmental learning changes can occur during

the course of a problem-posing intervention.

5.4 Profiles of Problem Solving Test – The Pre-test and the Post-test

The Profiles of Problem Solving test (POPS) (Stacey et al., 1993) is designed to

assess mathematical problem solving skills and to monitor the effectiveness of

intervention that has been specifically designed to improve student’s problem-

solving competence. In the test, students were presented, at the start and the

end of the seven-week teaching experiment, with a variety of novel problems that

simulated real-life situations. The design of the test allowed data to be collected

on five aspects of problem-solving competence:

1. Correctness of answer

2. Method used to obtain an answer

3. Accuracy of calculations

187

4. Ability to extract useful information, and

5. Quality of the student explanation of their answer.

The data collected in the pre-testing and post-testing of students in the

Comparison and Intervention Groups are recorded in Table 5.8 and Table 5.9

respectively with the post-test data being ‘bolded’ for easy discrimination from the

pre-test data. The data contained in the tables provides an opportunity to

consider individuals’ subscale score changes that occurred between the pre-test

and the post-test. Each table shows both pre-test and post-test results, in each

of the five problem-solving aspects, for each student in the respective groups. In

addition, the total score possible for each individual aspect is recorded at the top

of each column. According to the authors of the POPS test (Stacey et al., 1993),

the final two sub-columns, under the heading ‘Total’, can be used to determine if

a student has developed an increased overall competence in solving problems.

188

Table 5.8

Comparison Group Pre-test and Post-test results

Correctness /13

Method /14

Accuracy /10

Extraction /8

Explanation /8

Total /53

Pre-test

Post-test

Pre-test

Post-test

Pre-test

Post-test

Pre-test

Post-test

Pre-test

Post-test

Pre-test

Post-test

Laura 10 9 8 6 9 6 8 6 4 2 39 29

Jack 5 6 6 4 4 3 7 3 2 1 24 17

Fay 9 10 9 10 8 8 8 8 6 5 40 41

Gayle 8 9 6 9 8 6 8 7 4 3 34 34

Penny 6 11 7 11 6 8 8 8 4 2 31 40

Diane 7 10 5 5 4 8 5 7 3 2 24 32

Matt 11 10 12 8 8 6 8 8 5 3 44 35

Nola 11 10 11 10 8 7 7 8 4 2 41 37

Ben 9 10 6 4 6 6 8 8 3 2 32 30

Adam 8 8 6 6 6 6 7 7 2 2 29 29

Kyle 7 10 7 8 6 6 8 8 5 4 33 36

Imogen 10 9 8 8 7 6 8 8 1 1 34 32

Clare 8 7 7 5 5 6 8 7 2 1 30 26

Oliver 9 8 8 7 8 6 8 8 3 1 36 30

Ellen 11 8 8 8 8 8 8 6 4 2 39 32

Helen 6 9 3 4 7 8 5 3 0 0 21 24

Note. The number of students in the Comparison Group was 16.

189

Table 5.9

Intervention Group Pre-test and Post-test results

Correctness /13

Method /14

Accuracy /10

Extraction /8

Explanation /8

Total /53

Pre-test

Post-test

Pre-test

Post-test

Pre-test

Post-test

Pre-test

Post-test

Pre-test

Post-test

Pre-test

Post-test

Leah 13 11 11 10 9 8 8 8 6 6 47 43

Blair 8 9 6 8 7 9 7 8 2 5 30 39

Joanne 8 10 10 8 8 6 8 6 4 4 38 34

Paul 11 11 6 10 6 8 6 8 2 3 31 40

Courtney 10 11 10 11 7 8 8 8 6 6 41 44

Hayley 11 9 11 8 8 8 7 8 5 4 42 37

Nicole 9 9 10 10 6 8 8 8 4 2 37 37

Rodney 9 8 8 9 7 8 8 8 4 5 36 38

Oliver 9 9 8 9 5 8 8 8 5 6 35 40

Georgia 6 7 6 9 6 6 7 8 4 5 29 35

Ethan 11 13 9 11 8 8 7 8 2 5 37 45

Kelly 8 10 6 7 8 6 8 6 4 3 34 32

Andrew 11 10 11 8 8 8 8 8 4 4 42 38

Felicia 11 11 11 11 8 10 8 8 7 8 45 48

Danielle 10 10 10 10 8 8 8 8 5 3 41 39

Note. The number of students in the Intervention Group was 15.

190

5.4.1 Descriptive Analysis of the POPS Test Results

The mean and standard deviation scores of students for each aspect in the pre-

test and post-test were calculated using the student version of the Statistical

Package for Social Sciences (SPSS Inc., 2007) and compared between the

Comparison Group and the Intervention Group. These scores can be found in

Table 5.10. The mean score data in Table 5.10 indicate a trend of improvements

in each of the five subscales comprising the POPS (Stacey et al., 1993) test for

Intervention Group students that was not apparent consistently for students in the

Comparison Group. This trend suggested that the problem-posing intervention

may be responsible for some aspects of the development of problem-solving

competence amongst the Intervention Group students.

The ‘Quality of Explanation’ aspect of the test, the fifth subscale, produced some

of the most interesting comparisons between students in the Comparison Group

and students in the Intervention Group. While the increase in scores from the

pre-test to the post-test is only modest for students in the Intervention Group, the

decline in the scores from the pre-test to the post-test, for students in the

Comparison Group, is noteworthy. In their test administration book, the authors

of the POPS (Stacey et al., 1993) test stated that, from their trials, the ‘Quality of

Explanation’ aspect had, by far, the lowest score of the five problem-solving

aspects. They attributed this to its perceived “non-essential nature that results in

it receiving little attention in classroom mathematics” (Stacey et al., 1993 p. 62).

That is, students generally perceive the goal of problem solving to be achieving

the correct answer as opposed to focussing on the method or quality of the

explanation that leads to the correct answer. This focus on the ‘destination’, as

opposed to the ‘journey’, limits opportunities for students to explore the nature of

problems and generate important mathematical concepts and ideas (English et

al., 2005).

191

Table 5.10

Mean Score and Standard Deviation Statistics for each Aspect of the *Profiles of

Problem Solving Test for Students in the Comparison and Intervention Groups

Comparison Intervention Comparison Intervention

Problem-solving aspects Pre- test

M

Post-test M

Pre- test M

Post-test M

Pre- test

SD

Post-test SD

Pre- test

SD

Post-test SD

Correctness of answer 8.60 9.00 9.67 9.87 1.84 1.36 1.76 1.46

Method used 7.60 7.27 8.87 9.27 1.92 2.22 2.03 1.28

Accuracy of calculations 6.73 6.40 7.27 7.80 1.58 1.30 1.10 1.08

Extraction of information 7.60 7.13 7.60 7.73 0.83 1.36 0.63 0.70

Quality of explanation 3.47 2.20 4.27 4.60 1.36 1.15 1.49 1.55

Note. The number of students in the Intervention Group was 15 and the number of students in the

Comparison Group was 16. All results in this table are correct to two decimal places and were calculated using the Statistical Package for Social Sciences (Student Version, Version 15) by SPSS Inc., 2007, Prentice Hall, Melbourne, Victoria. * From “Profiles of Problem Solving” by K. Stacey, S. Groves, S. Bourke, and B. Doig, 1993, Australian Council of Educational Research, Hawthorne, Victoria.

The reflective practice of focussing on the ‘journey’ with peers, as experienced by

the Intervention Group students throughout the teaching experiment required

them to think critically, to detect their weaknesses and to self-regulate improved

practices. It may be reasonable to suggest that this opportunity resulted in these

students having an enhanced ability to explain their solutions. Students in the

Comparison Group did not have this same opportunity. According to Hamilton

and his colleagues (2006), opportunities for reflective practice, such as those

found in the problem-posing intervention, can lead to other significant gains such

as reduced mathematics anxiety and improved performance in solving problems.

The responses to the Question One of the Student Survey supported this

192

statement showing a substantial improvement in student attitudes towards

problem solving following the teaching experiment (see Section 5.3.1).

The difference in improvements between the Comparison Group and the

Intervention Group, in the ‘Quality of Explanation’ aspect of problem solving, is

not surprising. Throughout the teaching experiment, students in the Intervention

Group were repeatedly encouraged and given opportunities to discuss their

reasoning and to explain their methods of solutions with their peers. Brown and

Walter (2005) discussed the benefits of changing the “orientation” of learning

mathematics from finding a correct solution to finding a good question as being

very useful in developing divergent ways of thinking and deeper opportunities for

learning.

On their own, the standard deviations of the subscale scores for both the

Comparison and Intervention Groups portray a limited interpretation of the data.

However, when combined with the mean score data, they are able to contribute

to a deeper understanding of the differences between the Comparison and

Intervention Groups that may be attributed to the problem-posing intervention.

Earlier in this section, it was noted that the mean score of students in the

Comparison Group, for the ‘Method used’ aspect of the POPS (Stacey et al.,

1993) test, had decreased from 7.60 to 7.27, while the mean score of students in

the Intervention Group had increased from 8.87 to 9.27. In themselves, these

results are only mildly persuasive in suggesting that the problem-posing

intervention may have contributed to the improvement in this aspect for the

students in the Intervention Group. However, these results become more

convincing when we consider that, not only had the students in the Intervention

Group improved their mean score in this aspect, they had also produced more

consistent results following the problem-posing intervention than they had in the

pre-test. This can be seen from the reduction in the standard deviation of their

scores from 2.03 to 1.28. This contrasts to the reduced mean score of students

193

in the Comparison Group combined with an increase in standard deviation of

their scores, between the pre-test and the post-test, from 1.92 to 2.22.

The more consistent and improved scores of the Intervention Group may have

been as a result of the students’ improved ability to understand problem

structures. Bernardo (2001) mooted this potential of problem-posing activities, to

influence problem-solving competence, when he reported on students’ abilities to

transfer knowledge and skills from one problem in one setting, to another

problem in another setting. He and other researchers, such as Cai (1997),

suggested that the experience students gained from posing problems allowed

students to recognise common characteristics in the structures of analogous

problems.

During the teaching experiment, students were required to focus on the

structures of problems as they posed them. This was facilitated through the

posing of the problems and the recording of solutions in their workbooks, as well

as by the ‘rating’ of their own and their peers’ problems. In addition, students

were regularly challenged to determine how many steps would be required to

solve their problems. This focus on the steps required to solve their problems, in

combination with the peer rating of ‘do-ability’ and ‘level of challenge’, required

the students to look back at the structure of their problems. Stoyanova (2003, p.

39) reported on this opportunity when she said “problem-posing activities provide

environments that seem to engage students in a natural way in reflective

mathematical abstraction”. In the absence of similar findings with the

Comparison Group, it would seem reasonable to suggest that the problem-

posing intervention may have contributed to a focus on problem structures that

had in some way influenced the Intervention Group’s improved and more

consistent scores in the ‘Method used’ aspect of the POPS (Stacey et al., 1993)

test.

194

It has been noted earlier in this section that it may have been the focus on the

process (method of construction and method of solution) of problem solving as

opposed to the product (the solution) that allowed the students in the Intervention

Group to improve their scores in the ‘Quality of Explanation’ aspect of the POPS

(Stacey et al., 1993) test. In comparison to the students in the Intervention

Group, the mean score for the students in the Comparison Group fell from 3.47

to 2.20. Not only did their mean score fall, the standard deviation of the

individual scores for this problem-solving aspect also fell which indicates a

reduced spread of the scores. The trend of reduced scores is therefore more

consistent between the students in the Comparison Group and the reduced

mean score cannot be attributed to a few ‘outlier’ scores that may have reduced

the mean score for the entire group. It is worth noting, that by contrast, the

standard deviation for students in the Intervention Group varied only slightly

between the pre-test and the post-test.

In summary, both the measure of central tendency, the mean scores, and the

measure of spread, the standard deviation of scores, suggested that the students

in the Intervention Group had undergone some form of change, perhaps

developmental learning and perhaps affective, that had influenced their scores in

the post-test. This same pattern of change was not apparent for students in the

Comparison Group. It is reasonable to suggest then that it was the involvement

of the Intervention Group students in the problem-posing intervention that was

responsible for the apparent trend of improvement in problem-solving

competence.

195

5.4.2 Paired Samples T-Test Results

To further investigate the impact of the intervention in this study on students’

scores of the POPS (Stacey et al., 1993) test a paired samples t-test was

conducted using the student version of the Statistical Package for Social

Sciences (SPSS Inc., 2007). This test provided an opportunity to consider

another parameter of interest, the difference between the means within the

Comparison and Intervention Groups and adds further weight to the discussions

in Section 5.4.1. Results of the paired samples t-test can be seen in Table 5.11.

While no significant differences were found in students’ scores on the POPS test

for the Intervention Group, a paired samples t-test did demonstrate however, that

for the Comparison Group there was a statistically significant decrease in scores

on the Quality of Explanation subscale from the pre-testing (M=3.47, S.D.=1.36)

to the post-testing (M=2.20, S.D.=1.15, t=6.97, p<.05). The Eta Squared Statistic

(.78) indicated a large effect size.

As mentioned in Section 5.4.1 the decrease in scores on the Quality of

Explanation subscale for the Comparison Group may be attributed to a lack of

involvement in the teaching intervention and the study by the members of this

group. That is, the only involvement of the participants in this group in the study

was to complete the pre-test and the post-test which were separated by eight

weeks in which time there was no discussion or further involvement with the

study. In comparison, participants in the Intervention Group spent time together

as a unique group during the eight-week teaching experiment leading to

increased opportunity to develop an interest in and a sense of commitment to the

study. This interest and commitment may in turn have resulted in participants in

the Intervention Group striving to achieve their potential in the post-test whereas

students in the Comparison Group may have had limited interest in completing

the post-test as it may have appeared irrelevant or meaningless to them.

196

The transformative nature of teaching experiments was reported by Confrey and

Lachance (2000) who commented on the benefits of facilitating student-centred

learning environments that were free of the usual restraints in traditional

classroom settings. Unlike students in the Comparison Group, students in the

Intervention Group were provided with the student-centred learning opportunities

throughout the teaching experiment in which they could share, with their peers,

explanations of how they solved problems, thus raising their awareness of this

important aspect of problem-solving competence. The responses to the Student

Surveys from students in the Intervention Group supported the increased interest

and commitment to solving problems students (see Section 5.3).

Apart from the statistically significant decrease in Comparison Group scores on

the Quality of Explanation subscale of the POPS (Stacey et al., 1993) test from

the pre-test to the post-test no other significant differences were found.

197

Table 5.11

Paired Samples Test for each Aspect of the *Profiles of Problem Solving Test for

Students in the Comparison and Intervention Groups

Paired Differences

95% Confidence Interval of the Difference

M S.D. Std. Error

Mean

Lower Upper t df Sig.

(2-tailed)

Comparison pre to post

Correctness of answer

-.40 2.06 .53 -1.54 .74 -.75 14 .47

Method used

.33 2.06 .53 -.81 1.47 .63 14 .54

Accuracy of calculations

.33 1.76 .45 -.64 1.31 .73 14 .48

Extraction of information

.47 1.41 .36 -.31 1.25 1.28 14 .22

Quality of explanation

1.27 0.70 .18 .88 1.66 6.97 14 .00

Intervention pre to post

Correctness of answer

-.20 1.32 .34 -.93 .53 -.59 14 .57

Method used

-.40 2.03 .52 -1.52 .72 -.76 14 .46

Accuracy of calculations

-.53 1.51 .39 -1.37 .30 -1.37 14 .19

Extraction of information

-.13 1.06 .27 -.72 .45 -.49 14 .63

Quality of explanation

-.33 1.50 .39 -1.16 .50 -.86 14 .40

Note. The number of students in the Intervention Group was 15 and the number of students in the Comparison Group was 16 although subjects with unchanged scores were excluded from these statistics.

All results in this table are correct to two decimal places and were calculated using the Statistical Package for Social Sciences (Student Version, Version 15) by SPSS Inc., 2007, Prentice Hall, Melbourne, Victoria. * From “Profiles of Problem Solving” by K. Stacey, S. Groves, S. Bourke, and B. Doig, 1993, Australian Council of Educational Research, Hawthorne, Victoria.

198

5.4.3 Analysis of Improvement Scores from the Pre-test to the Post-test

When we compare the number of improvements in individual problem-solving

aspect scores of students from the pre-test to the post-test (see Tables 5.8 and

5.9), we can detect changes that may be consistent with developmental learning

occurring as a result of participation in the problem-posing intervention. A

summary of the number of improvements can be found in Table 5.12.

Table 5.12

Numbers of Improvements in Individual Sub-scale Scores of Comparison and

Intervention Group Students, from the Pre-test to the Post-test

Problem-solving aspects Comparison Group Intervention Group

Correctness of answer 8 6

Method used 5 8

Accuracy of calculations 4 7

Extraction of information 2 5

Quality of explanation 0 7

Note. The number of students in the Intervention Group was 15 and the number of students in the

Comparison Group was 16.

From considering only the number of students with improved scores in the

‘Correctness’ aspect of the test, it would seem that the Comparison Group’s

results demonstrated they had achieved greater improvements than had the

students in the Intervention Group. However, the pre-test mean score in this

aspect for the Comparison Group was 8.60 compared to a higher pre-test mean

199

score of 9.67 achieved by students in the Intervention Group (see Table 5.10). In

addition, twice as many students in the Intervention Group, compared to students

in the Comparison Group, had pre-test scores of 11 or more out of a possible 13

marks (see Tables 5.8 and 5.9) thus reducing the range of values to which they

could improve.

In the ‘Method Used’ aspect of the test, a similar situation to that described for

the ‘Correctness’ aspect of the test can be seen. The pre-test mean score in this

aspect for the Comparison Group was 7.60 compared to a pre-test mean score

of 8.87 achieved by students in the Intervention Group (see Table 5.10). As with

the ‘Correctness’ aspect of the test, twice as many students in the Intervention

Group, compared to students in the Comparison Group, had pre-test scores of

11 or more (see Tables 5.8 and 5.9). It is reasonable to suggest that both of

these factors would have made it more difficult for students in the Intervention

Group to achieve high scores than students in the Comparison Group for both of

these aspects. Despite this possible impedance to improved scores, 8 students

in the Intervention Group improved their score in the ‘Method used’ aspect

compared to only 5 students in the Comparison Group.

Improvements between the groups in the ‘Accuracy of Calculations’ aspect of the

test are consistent with those for the ‘Method Used’ aspect. The Intervention

Group began with a higher mean score of 7.27 compared to the Comparison

Group’s mean score of 6.73, but still managed to have more students improve

their score in this aspect of the test. Seven students made improvements in the

Intervention Group compared to only four students making improvements in the

Comparison Group. This trend was continued when we looked at the increased

ability of students to extract useful and relevant information from data presented

in the problems contained in the test. Interestingly, the mean score of 7.60 out of

a possible 8 marks for the ‘Extraction’ aspect of the test was the same for both

groups in the pre-test. However the Intervention Group were able to improve

200

their mean score to 7.73 in the post-test while the Comparison Group’s mean

score fell to 7.13.

Initially it may seem unusual that a mean score in an identical pre-test and post-

test, undertaken by the same students in a group, could fall. However, the

students in the Comparison Group may not have developed a sense of

ownership or respect for the teaching experiment as they had not committed the

same amount of time as had students in the Intervention Group, thus possibly

impacting the students’ willingness to wholeheartedly complete the post-test to

the best of their abilities. On the other hand, the students in the Intervention

Group had experienced a routine of coming together as a group over an eight-

week period, thus having a sustained opportunity to investigate the nature of

problems and ultimately to pose their own problems to their peers. This coming

together to share the learning experience is discussed by Crotty (1998) who

attributes it with the development of interpretation and understanding of

knowledge in a way that traditional forms of teaching are unable to compete.

Crotty’s (1998) constructionist (as opposed to constructivist) proposition, that

best learning occurs when students are actively engaged in activities that allow

them to create artefacts [problems] to share with peers in a group situation, is

consistent with findings from the NRC (2001) that stated that students need to

spend “sustained periods of time” involved with problem solving to become more

competent at solving problems and to make connections between their prior

knowledge and their new knowledge.

The final problem-solving aspect of the POPS (Stacey et al., 1993) test

measured the students’ ability to explain their solutions. As with the first three

problem-solving aspects, students in the Intervention Group had a higher mean

score in the pre-test compared to the students in the Comparison Group. While

students in the Intervention Group improved their mean score from 4.27 to 4.60

out of a possible 8 marks, the mean score for the Comparison Group fell from

3.47 to 2.20. As was discussed for the ‘Extraction of Information’ aspect, it is not

201

easy to explain why a mean score can fall substantially from an identical pre-test

to a post-test when both tests were taken under the same conditions by two

groups of students selected with the same selection criteria. On the basis that

the only difference between the Comparison and Intervention Groups was the

problem-posing intervention, it is reasonable to suggest that the intervention may

have contributed to the improvement in mean scores for the Intervention Group

and therefore to the overall problem-solving competence of the students in that

group.

5.5 Conclusion

A report on both the qualitative and quantitative data, collected throughout the

teaching experiment, was presented in this chapter. Specific responses to the

three research questions are addressed in the following chapter. The data

collected from the video-tapes, student interviews and the student workbooks,

combined with the data from the surveys, pre-tests and post-tests has provided

evidence to determine the extent to which a problem-posing intervention can

assist in the development of problem-solving competence in underachieving

middle-year students.

This chapter began with an introduction to the three case-study students; Paul,

Andrew and Nicole, who were chosen for the unique developmental learning and

behaviour changes that occurred for them as a result of the problem-posing

intervention. Paul was a student who began the teaching episode as

disorganised in his thinking, as it related to problem solving. He demonstrated

the most improvement in his problem-solving competence from the start to the

end of the teaching experiment. Andrew, the student with the lowest Middle

Years Ability Test (MYAT) score (Australian Council for Educational Research,

2005) and Nicole, the student with the highest MYAT score, both began the

teaching experiment as disengaged in the problem-posing activities. Despite

Andrew being easily distracted by his peers and Nicole being a passive non-

participant at the start of the teaching experiment, they both responded well to

202

the problem-posing intervention and as a result, became engaged with learning

and enthusiastic about posing problems to their peers.

A report on the data obtained from the student surveys, undertaken at the start of

the teaching experiment and then again at the completion of the teaching

experiment, is found in Section 5.3. The analysis of data from Question One

provided some interesting findings including a substantial dispositional shift of

students from negative to positive feelings about problem solving as a result of

the problem-posing intervention. Student responses to Question Two suggested

that, as a result of their involvement in the teaching experiment, students had

begun to look at problem structures more knowledgeably. It could be suggested

that the students’ foci had shifted from considerations of the difficulty of problems

to the characteristics of the problem, as the discerning factor in enjoyment. Data

from Question Three of the initial and final surveys confirmed that students

remained constant in their view that solving problems was a worthwhile activity.

An investigation of the data from Question Four of the surveys showed students

were more divergent in their views about how a teacher could assist them to

become a better problem solver as a result of the problem-posing intervention.

Section 5.4 began with a brief reintroduction to the problem-solving testing

instrument used to determine changes in problem-solving competence from the

start to the end of the teaching experiment. The descriptive statistics of the data

were discussed in Section 5.4.1 and then further explored using a paired

samples t-test in Section 5.4.2 to determine if the problem-posing intervention

could be attributed with improved problem-solving competence of the students in

the Intervention Group. The changes in mean scores of the five problem-solving

aspects tested in the POPS test (Stacey et al., 1993) were compared within and

between the Comparison and Intervention Groups.

Students in the Intervention Group had improved mean scores in each of the five

problem-solving aspects between the pre-test and the post-test although these

203

improvements were not statistically significant. Therefore, analysis of the

quantitative data revealed that the problem-posing intervention did not

significantly improve students’ test scores on the POPS (Stacey et al., 1993) test

for the Intervention Group. However, the qualitative data suggested that the

intervention did generally enhance students’ attitudes towards problem solving

thus paving the way for students to become more engaged in problem-solving

activities.

For the Comparison Group, the quantitative data analysis revealed that there

were no significant differences in students’ scores on the POPS (Stacey et al.,

1993) test except for the subscale Quality of Explanation in which scores

significantly decreased from the pre-test (M=3.47) to the post-test (M=2.20).

Possible reasons for this finding might be attributed to teachers’ lack of attention

given to this aspect when teaching mathematics. For the Intervention Group, the

problem-posing intervention focused on students’ quality of explanations when

undertaking problem-solving activities and this may be reflected in the students

maintaining consistent scores on this subscale in both the pre-test (M=4.27) and

the post-test (M=4.60).

The analysis of both the qualitative and quantitative data, presented in this

chapter, uncovered interesting findings and valuable evidence that lead to the

research questions being responded to with confidence in Chapter 6.

204

Chapter 6

Responses to the

Research Questions

6.1 Chapter Overview

In this chapter the research questions are responded to using the data presented

and discussed in Chapter 5 and reflect findings about students who have English

as their first language. Section 6.2 provides some discussion and a response to

Research Question 1, while Section 6.3 and Section 6.4 provide discussion and

responses to Research Questions 2 and 3 respectively. The final section,

Section 6.5, addresses the overarching question to be responded to as a result

of this study; How might a problem-posing intervention impact upon the

development of problem-solving competence of underachieving, middle-year

students?.

6.2 Research Question 1

The first research question was:

Can, and if so, how can participation in problem-posing activities facilitate the

engagement of middle-year mathematics students?

An investigation of the data collected in this study suggests an affirmative

response to this question. As mentioned in Chapter 3, a number of data sources

provided an opportunity for triangulation of the data which added an increased

level of certainty when responding to this research question. Accurate details of

informal interviews between the three case study students and the researcher,

205

and between three case study students and their peers, were captured by three

audio-visual recorders used throughout the teaching experiment. In addition, the

problems posed by these students in their workbooks, and their associated self-

ratings and comments, added a more personal depth to the data collected from

the students. This qualitative data, in conjunction with the student responses

recorded in the student surveys, provided convincing evidence of the potential for

problem-posing activities to re-engage previously disengaged students.

In Chapter 5, Paul, Andrew and Nicole were presented as three case-study

students (see Sections 5.2.1, 5.2.2 and 5.2.3). While Paul was consistently

engaged in learning throughout the teaching experiment, Andrew and Nicole

began the teaching experiment as disinterested in learning. In Andrew’s case,

he was easily distracted from learning and lacking in self-efficacy. His increased

engagement and increased levels of public participation in sharing his posed

problems with his peers, supported Bandura’s (1997) findings that problem-

posing activities have the potential to improve self-efficacy. Similarly, Van de

Walle (2004) investigated middle-year students, like Andrew, and attributed

improved attitudes and higher levels of intrinsic motivation to multi-dimensional

pedagogies found in classrooms where students are given options about how

they want to learn, such as those incorporated into the teaching experiment in

this present study.

The transcripts from the videotapes of the three cameras used in the experiment,

showed significant changes in the sentiment of the conversations held by Andrew

and Nicole with the researcher and with their peers. Both of these students

seemed to increasingly enjoy the opportunity to pose and share their problems

throughout the problem-posing intervention. From the data collected in this

study, it could be concluded that the non-traditional methodologies, underpinned

by constructionist beliefs and built into the teaching experiment, provided an

environment in which Andrew and Nicole felt increasingly ‘safe’ and willing to

‘expose’ their thinking to their peers. This conclusion was also supported

206

through the research reported in Chapter 2. Broido (2002) reported on the

potential of “liberating” and “transformative” learning environments, such as was

offered to the Intervention Group in this teaching experiment, to assist students

to re-engage with learning. Similarly, Alvesson and Skolberg (2000) suggested

that a learning environment that supported “respectful participation” and “jointly

constructed knowledge” had the potential to re-engage students. In Andrew’s

case, he not only became engaged in learning, he also became active in

encouraging others to be more engaged so that he could focus more fully on the

problem-posing activities provided in the teaching experiment (see Section

5.2.2). In Nicole’s case, she was able to show her re-engagement in a physical

way through the use of the ‘1,2,3 rating cards’ as well as through the improved

quality of the problems she posed in her workbook and her willingness to share

her problems with her peers (see Section 5.2.3).

An analysis of the answer to the first question in the initial and final student

surveys also provided supporting evidence for a response to Research Question

1. In the first survey, only two out of fifteen students, compared to eleven out of

the fifteen students at the end of the teaching experiment, said they enjoyed

solving problems (see Table 5.4). While Question One in the survey asked

students about their enjoyment of solving problems and therefore was not a

direct measure of student engagement, it did however provide an opportunity to

consider changes in how students viewed their experiences of solving problems,

and it has been argued that there is a direct relationship between enjoyment of

activities and student engagement in activities (e.g., Adams et al., 2000;

Bjorklund, 2000).

The data collected from the first survey question that demonstrated a positive

dispositional shift to enjoying problem solving, was consistent with findings by

Brown and Walter (2005), Lesh and Doerr (2003) and Lesh and Zawojewski

(2007) who all reported that opportunities to pose problems can provide students

with meaningful learning environments in which they are more likely to engage.

207

In summary, the survey responses add confidence to the response that

participation in problem-posing activities can facilitate the engagement of

disengaged, middle-year mathematics students.

6.3 Research Question 2

The second research question was:

Can, and if so, how can participation in problem-posing activities facilitate

improved problem-solving competence of middle-year, mathematics

students?

The qualitative and quantitative data collected in this present study, presented

contrasting evidence to respond to this research question. There were two

primary data sources, the pre-test and post-test results, as well as three

secondary sources, informal interviews, observations and student workbooks that

were used to support this statement.

Paul, one of the three case-study students in this teaching experiment,

demonstrated one of the largest improvements in problem-solving competence,

as measured by the difference between his pre-test and post-test scores. Paul

improved in four of the five problem-solving aspects being tested in the Profiles

of Problem Solving (POPS) test (Stacey et al., 1993) and scored the same result

in the fifth aspect in both the pre-test and the post-test. His overall score

increased from 31 marks at the start of the problem-posing intervention, to 40

marks out of a possible 53 marks at the end (see Table 5.1). Knuth and

Peterson (2002) reported similar improvements for students in their study when

they said that, as a result of providing students with many opportunities to pose

problems, students improved their problem-solving competence as well as

developed their mathematical thinking.

An improvement by one student in this present study is insufficient to suggest

that participation in this problem-posing intervention resulted in improved

208

problem-solving competence for all participants. Indeed, Nicole, another case

study student, made no overall improvement in her total score (see Table 5.3),

while Andrew’s score fell from 42 marks to 38 marks out of a possible 53 marks

(see table 5.2). Further investigations of the data, collected from the Comparison

and Intervention Groups, did however suggest that a trend of improvement was

evident for the Intervention Group that was not apparent for students in the

Comparison Group (see Tables 5.10 and 5.11). The graphical presentations of

the mean scores in each of the five problem-solving aspects of the POPS test,

for both the Comparison and Intervention Groups, showed a trend of improved

scores for students in the Intervention Group that were not apparent for the

students in the Comparison Group. The visual representation of the mean-score

data were persuasive and suggested that the problem-posing intervention could

be responsible for the increases in problem-solving competence of students in

the Intervention Group. In addition, when a comparison of the number of

students who improved their scores in each problem-solving aspect is made

between the two groups, evidence can be found to suggest that the problem-

posing intervention has contributed to the improvements of students in the

Intervention Group (see Table 5.10).

Throughout the teaching experiment, students were given a number of

opportunities to solve researcher and peer problems. Some students, like Paul,

took every opportunity to tell his peers how he would solve a particular problem.

Costa (2005, p. 8) acknowledged students, like Paul, as risk takers who are

prepared to go “against the grain, … thinking of new ideas and testing them with

their peers” as being more successful at solving problems than those who don’t.

Early in the teaching experiment, Paul attempted to solve a researcher-provided

problem on a whiteboard at the front of the classroom (see Section 5.2.1). He

was unable to fully explain either his thinking or his incorrect solution. His

perceived confusion of his ideas typified his early replies to similar researcher

questions about how he would solve other problems. However, towards the end

of the teaching experiment, Paul demonstrated that some developmental

209

learning changes had taken place (discussed further in Section 6.4).

Gravemeijer (1994) discussed opportunities to pose, explore and solve problems,

as opportunities to create “internal conflict” that ultimately can lead to a student

developing more abstract and detailed conceptualisations of problems. In Paul’s

case, it appeared that as a result of his experiences in the problem-posing

intervention, he had learnt how to order his thoughts in a coherent and

appropriate order to pose meaningful, challenging and interesting problems and

generate correct solutions. This was demonstrated in the problems he

progressively posed in his workbook and in his responses to questions in the

POPS (Stacey et al., 1993) test that resulted in his improved overall score.

In summary, the qualitative and quantitative data collected to determine if, and if

so, how can a problem-posing intervention have an impact on student’s problem-

solving competence, provided contrasting findings. The numbers of students in

the Intervention Group who improved their overall POPS (Stacey et al., 1993)

test scores was, in itself, insufficient to demonstrate with certainty that the

problem-posing intervention was solely responsible for the Intervention Group’s

improvements in problem-solving competence. However, when the individual

problem-solving aspect scores were compared between students in the

Comparison Group and the Intervention Group, the suggestion that the

intervention may have been responsible for improved problem-solving

competence began to emerge. When considering all the evidence presented in

this study, support can be given to Silver and Cai’s (1996) findings which state

that participation in problem-posing activities can result in the improved problem-

solving competence of some middle-year, mathematics students.

6.4 Research Question 3

The third research question was:

In terms of problem-solving competence, what developmental learning

changes occur during the course of PPI?

210

A number of student developmental learning changes were identified throughout

the teaching experiment. These included changes in self-regulation for some

students. For example, Andrew increasingly demonstrated an improved ability to

self-regulate as the experiment proceeded (see Section 5.2.2). Self-regulated

learning is a complex interactive process involving both cognitive and

motivational functions (Boekaerts, 1997) and can be influenced by instructional

conditions and task difficulty (Paris & Newman, 1990), such as those presented

throughout the teaching experiment. It is noteworthy that while Andrew was able

to self-regulate throughout the teaching sessions that involved problem-posing

activities, he was unable to demonstrate the same degree of self-regulation in the

post-test session. Possible explanations for this apparent anomaly can be found

in Section 5.2.2.

When students are actively engaged in activities that provide opportunities to

integrate prior knowledge with new knowledge, they are then able to construct

new meaning that results in cognitive change (e.g., Goswami, 2002; Siegler,

1996). At the start of the teaching experiment, Paul appeared to be full of

enthusiasm but lacking in specific cognitive tools that allowed him to order his

thoughts logically (see Section 5.2.1). Sternberg (2000) referred to these tools

as the metacomponents and performance components of general intelligence

that, when lacking in development, can impede an individual’s progress in solving

problems.

Evidence of Paul’s development in these components can be seen by comparing

his solutions in the pre-test questions to those in the post-test, and by comparing

his early entries to his later entries in his workbook. New and deeper

opportunities for developmental learning, as a result of involvement with problem-

posing activities, were reported by Brown and Walter (2005) who attributed

problem posing with presenting unexpected discoveries about an individual’s

learning potential. Throughout the problem-posing intervention, Paul had

discovered unexpected abilities that neither he nor his teachers had previously

211

identified. Since Paul was always a willing participant in all of the teaching

episodes, we can reasonably suggest that the problem-posing intervention was

responsible for the development of learning skills that resulted in his increased

problem-solving competence.

The opportunity to pose problems provided Paul with the knowledge of problem

structures that allowed him to make the necessary connections with prior

knowledge, resulting in cognitive change. This opportunity to pose problems,

that results in cognitive change, is discussed at great length by Siegler (1991)

and is generally supported by both Piagetian and Information Processing

theorists (e.g., Piaget & Inhelder, 1969; Sternberg, 2002). However,

Psychometric theorists would say that if performance in tests, such as the POPS

test (Stacey et al., 1993), is to be used as a measure of intelligence, then Paul’s

improved overall score from 31 to 40 out of a possible 53 marks is more to do

with improved memory capacity and cognitive speed than it is to do with

developmental learning changes (Hutton et al., 1997). This is an unlikely

explanation for Paul’s improved score and it is more likely that the problem-

posing intervention activities have provided the new knowledge that Paul needed

to create new meaning in the process of solving problems.

Lesh and Doerr (2003) reminded us that conceptual and developmental learning

changes include social perspectives as well as psychological perspectives. This

study was underpinned by contructionism theory (Schwandt, 2001) that stressed

the importance of the social interaction in the production of shared artefacts,

which in this case were the problems posed by the students. This opportunity

for social interaction, throughout the teaching experiment, had a notable impact

on a number of students and their levels of involvement in the activities. Nicole

was a student who became increasingly engaged in the learning process as a

result of the problem-posing intervention. Her Middle Years Ability Test (MYAT)

(Australian Council for Educational Research, 2005) results that lead to her

selection as a participant in this study, put her in the 94th percentile of students.

212

However, her perceived talent did not manifest itself in a correspondingly high

score in the pre-test, nor was she more enthusiastic or engaged than other

students at the start of teaching experiment.

Evidence from researcher interviews and observations recorded by the cameras

in the research room, provided evidence that Nicole had progressively developed

socially-oriented learning skills as a result of her participation in the problem-

posing activities (see Section 5.2.3). This potential for problem-posing activities

to encourage students “to explore problem situations and to pursue lines of

enquiry that are personally satisfying” was reported by English (1997b). At the

beginning of the teaching experiment, Nicole was disinterested in being involved

with the problem-posing, perhaps because she felt it was of no interest to her.

However, as she observed her peers, she became drawn into the problem-

posing activities and began to actively participate by posing and sharing her

problems and by providing feedback to her peers about their problems. Her

increased and contextualised socialisation within the research room, created

opportunities for Nicole to engage with the problem-posing activities and resulted

in her writing increasingly sophisticated problems in her workbook. It would

seem that the problem-posing intervention was the contributing factor that

provided this opportunity to Nicole, resulting in socially-oriented learning

development.

Evidence of a number of developmental learning changes was found during the

analysis of the data collected in this study. Some students, like Andrew, became

more self-regulated while others, like Nicole, demonstrated more socially-

oriented developmental learning changes. Paul, through his practice at posing

problems, learnt how to combine existing knowledge with a new knowledge of

problem structures that resulted in substantial increases in his problem-solving

competence. The findings of this study have demonstrated that, in terms of

problem-solving competence, a problem-posing intervention has the potential to

facilitate developmental learning changes amongst middle-year students.

213

6.5 Conclusion

The purpose of this study was to investigate and explain the links between a

problem-posing intervention and the development of problem-solving

competence in middle-year students who had been ascertained as above-

average in standardised intelligence tests, yet underachieving in the problem-

solving criterion of school mathematics tests. In Sections 6.2 and 6.3, evidence

was presented to suggest that a problem-posing intervention could facilitate the

re-engagement of students who were previously disengaged in learning and

could facilitate increased problem-solving competence of some underachieving

middle-year students. The opportunity to be withdrawn from their traditional

classrooms, provided the students in the Intervention Group with a learning

environment that encouraged individuality and one in which they could feel safe

to explore solving problems in a unique way. A focus on posing problems, such

as occurred in this present study, was supported by Brown and Walter (2005, p.

168) who said that “problem generation has the potential to redefine in a radical

way who it is that is in charge of one’s education”.

A number of developmental learning changes were identified in Section 6.4, such

as self-regulation and the creation of new meaning through the combination of

prior and new knowledge that occurred as a result of student involvement in the

problem-posing intervention. From the analysis of the data in Chapter 5 and a

review of the evidence used to respond to the research questions in Chapter 6, a

response can be presented for the overarching question of this study:

How might a problem-posing intervention impact upon the development of

problem-solving competence of underachieving, middle-year students?

214

Involvement in a problem-posing intervention can:

1. facilitate the re-engagement of middle-year mathematics students

2. improve the problem-solving competence of middle-year mathematics

students, and

3. can facilitate developmental learning of middle-year mathematics

students.

215

Chapter 7

Limitations and

Implications for Future

Research

7.1 Chapter Overview

In this final chapter the limitations and implications of this study are discussed.

Consideration is given to limitations regarding the selection of students (see

Section 7.2.1), the timing of the research (see Section 7.2.2), the size of the and

Intervention Groups (see Section 7.2.3), the withdrawal of students from their

usual classroom environment (see Section 7.2.4), the length of the problem-

posing intervention (see Section 7.2.5) and the usefulness of Question Three in

the Student Survey (se Section 7.2.6). In Section 7.3 the implications of this

research are discussed to inform other researchers of potential avenues for

future research. Concluding comments can be found in Section 7.4.

7.2 Limitations of the Study

While every care was taken to ensure the reliability and validity of data obtained

from this study, inevitably as often happens with educational research involving

children, unexpected events occurred that may have had an impact on the

research outcomes of this study. Chapter 4 contains a detailed discussion of the

types of events, such as flooding rains that could not be predicted or planned for

in the design of the teaching experiment. In addition, the number of students

made available for the experiment by the research school, and the dispositional

and situational factors that can influence middle-year students on any given day

for any number of reasons (e.g., Gootman, 2001; Reis & Siegle, 2006), cannot

216

be disregarded as having had an influence on the ‘trustworthiness’ of the data

(Merriam, 1998). While it is less than ideal to work with small sample sizes,

useful analysis of the data was possible through the triangulation of the

qualitative and quantitative data (see Chapters 4, 5 and 6).

7.2.1 Limitations in the Selection of Students

Two specific criteria were used to select participants for this study. Each student

in the Comparison and Intervention Groups was required to have scored above

the 60th percentile in the routinely administered Middle Years Ability Test (MYAT)

(Australian Council for Educational Research, 2005) and they were required to

have scored below the mean score of their cohort in the problem-solving criterion

of their school mathematics tests. While these criteria determined the initial

number of students available for this research, we are reminded that students

who had English as a second language were not considered as potential

participants for this study due to their potential language barrier in undertaking

the tasks contained within the teaching experiment.

When potential participants were identified, they were given a participant

information and permission notice to take home to their parents (see Appendices

A and B). If parents approved of their child’s participation the students were

required to return their signed forms to the school by a nominated date.

Unfortunately, some students did not return their forms by the due date and

hence were unable to become participants in the study. This factor reduced the

sample sizes of both Comparison and Intervention Groups by two students.

Once the groups were confirmed, further reductions in sample sizes occurred

due to some students being at school for the pre-test and not for the post-test.

In addition, throughout the teaching experiment, a number of students in the

Intervention Group were unable to attend all of the teaching sessions due to

illness. Students who had missed more than two of the eight sessions were

deemed to be ineligible for data analysis to maintain a high level of reliability and

217

validity of the data. These factors resulted in the Comparison Group being

reduced to 16 students and the Intervention Group being reduced to 15 students.

7.2.2 Limitations in the Timing of the Research

There were two limitations to the research findings arising as a result of the

timing of the study. In the first instance, senior staff at the School, while being

very supportive of this research project, preferred the students not to miss any

timetabled lessons. In addition, it was deemed inappropriate to ask students to

use their break times or after school to participate in the study. The weekly

assembly, lasting 45 minutes, occurred in the first session of each Monday. This

session was considered to be the most suitable time for the students to be

withdrawn however; the research required a 60 minute block of contact each

week with the students. The compromise chosen was to have participants start

the day 15 minutes earlier than other Year 7 students at the School. All

participating students in the Intervention Group confirmed at the start of the

research they were able and willing to arrive at school every week for the start of

the sessions.

As the teaching experiment progressed, there were a number of unexpected

factors that impacted upon student arrival times at the research room. These

included road works, parents and students who slept in, specialist medical

appointments, school photographs, pupil-free days and students being asked to

accept awards at assembly. Daily notices on the preceding Fridays were used to

remind students of the sessions on each forthcoming Monday and the students

were collected from the playground and classrooms before each teaching

episode began. These initiatives were successful in minimising some of the

disruptions to the start of the sessions however, other disruptions were

unavoidable. On the occasion when students were not at school on the

scheduled Monday due to a pupil-free day, the ‘lost’ session occurred on the

218

following Tuesday. This was a satisfactory exchange of times and resulted in

minimal impact to the flow of the teaching sessions.

The research was conducted in the final term of the year. This time of the year

was chosen so that the students would have completed at least three

mathematics tests, one in each of the preceding three terms. Data from these

mathematics tests were used as a selection criterion for participation in this

study. As detailed in Section 4.3.3, the final session of the teaching experiment

was used for administering the post-test. As this week was the final week of the

year for both the Comparison and Intervention Groups, the atypical excitement

and distraction of the students may have influenced their willingness and ability

to focus on completing the post-test to the best of their abilities. To address this

set of situational distractions the students were encouraged to consider the

significance of their involvement in the study. They were reminded that the

research may be used to help other students to develop their problem-solving

competence. Some students, like Paul, became settled almost immediately on

entering the research room while others, like Andrew and Nicole, found it

extremely difficult to focus on completing the post-test.

7.2.3 Limitations of the Size of the Comparison and Intervention Groups

As a result of the smaller than expected sample sizes, the number of suitable

statistical tests to investigate the data was limited. However, a quantitative

analysis was achieved by comparing the mean scores and standard deviations of

the five aspects of the Profiles of Problem Solving (POPS) test (Stacey et al.,

1993) between the Comparison and Intervention Groups, and by analysing the

number of ‘improved‘ problem-solving aspect scores from the pre-test to the

post-test of individuals in both groups. Additional confidence in findings came

from the evidence contained in the qualitative data collected throughout the

experiment (see Chapter 5). This triangulation of data sought to overcome the

limitations of a smaller than expected sample size in each of the groups.

219

The smaller than anticipated number of students participating in this study was

unexpected and may have impacted upon the trustworthiness of the findings of

this study. Findings from further studies, or findings from a replication of this

study, may benefit from being undertaken in a research site where a larger

number of initial suitable participants were available to allow for attrition of

participants throughout the study.

7.2.4 Limitations of the Withdrawal of Students from their Usual

Classroom Environment

In an ideal research study environment, as many variables as possible would be

kept constant to ensure that changes detected at the conclusion of a study could

be attributed directly to the intervention. However, practically speaking this is not

always possible, particularly when investigations of changes in students are

being conducted. In this present study, there were two factors that necessitated

the withdrawal of students from their usual class groups and rooms.

Firstly, students satisfying the participant requirements were found in all four of

the year 7 classes and all of these students were required to collectively

constitute the Comparison and Intervention Groups. However, students were

familiar with being withdrawn in different groupings, from their usual ‘home’

groups, for language and sport lessons so their withdrawal was deemed to have

minimal impact on the results of the study.

Secondly, withdrawal of the students from their usual ‘home’ rooms to a different

teaching room was required to house the participant group and to facilitate the

teaching experiment. As mentioned in Section 3.5, the classroom in which the

study was conducted, the library classroom, while not being a Year 7 ‘home

room’ was a room with which the students were familiar and to which some of the

students were regularly withdrawn for their language classes. This factor was

also deemed to have minimal impact on the results of the study.

220

The intervention was not directly associated with any part of the students’ usual

mathematics curriculum. That is, there was no intention in this present study for

students to develop specific mathematical skills or knowledge contained in the

Year 7 Mathematics work program. The Semester 1 assessment of Mathematics

at the research school in both 2006 and 2007 provided specific data on the

previous problem-solving competence of the students in the consecutive Year 7

cohorts. These data were used in part as selection criteria to determine students

who met the participant requirements of this study.

It was the intention of this study to determine if, and if so, how can a problem-

posing intervention develop problem-solving competence. The problem-solving

competence was not limited to that used to solve mathematical problems found

within the Year 7 Mathematics Syllabus. However, a number of researchers

have reported that the development of problem-solving competence can be a

powerful way to learn mathematics (e.g., English et al., 2005; Lesh &

Zawojewski, 2007) so it is reasonable to suggest that students may have

developed their understanding of some mathematical concepts as a result of

their participation in the teaching experiment. By demonstrating that a problem-

posing intervention may develop problem-solving competence and by accepting

the proposition that developing problem-solving competence can develop

students’ understandings of some mathematical concepts, it follows that this

present study may also provide useful evidence for researchers investigating

ways to reconsider and redefine appropriate pedagogy for the teaching of

mathematics to middle-year students.

7.2.5 Limitations in the Length of the Problem-posing Intervention

The intervention lasted a total of eight weeks with the post-testing occurring

immediately at the end of the last teaching episode. One could argue that this

immediate post-testing may not have accurately detected significant or

longitudinal changes in engagement, developmental learning or enhanced

problem-solving competence of the participants. In addition, some of the

221

changes recorded between the pre-test and post-test may have been as a result

of familiarity with the testing instrument. This possibility was reported by

Ritchhart (2002) who said we cannot underestimate the impact of test practice on

test scores. However, if we were to accept this position we may not be able to

explain why some student scores fell between the pre-test and the post-test. In

addition, if we were to have used a ‘parallel’ test as the post-test, questions could

be raised about the validity and reliability of the comparative data between the

pre-test results and the post-test scores. Lesh and his colleagues (2000, p. 19)

discussed this type of “trade-off”, to meet conflicting goals, and suggested that it

was a legitimate part of planning the design of an educational research study.

Ideally, this present study would have been conducted over a longer timeframe

than eight weeks. This would have enabled the students to deeper explore

problems, to have additional practice at mathematising different situations and to

become increasingly aware of changes in their own developmental learning, thus

perhaps influencing the engagement of more students. A study conducted over

a longer timeframe may also have produced more consistent and reliable data for

analysis.

Access to student participants in educational research projects is governed by

many factors including the willingness of the research school to participate in the

study and the enthusiasm and willingness of the staff at the school to release the

students for involvement in the intervention. The number of students who meet

the selection criteria and the possible impact of their withdrawal from regular

classes, can also impact upon the time made available for an educational study.

This study was conducted over the longest period made available by the

research school and it could be suggested that had it been a longer timeframe, it

may have added an additional level of ‘trust’ to the research question responses.

222

7.2.6 Limitations of Question Three of the Student Survey

The data collected from this question did not generate detailed information about

why the students thought solving problems was a useful activity. The student

responses clearly showed that students agreed that solving problems was useful

but did not provide detailed information to explain their response. A more helpful

re-wording of the question may have been: If you believe solving problems is a

useful activity write down three reasons why it is useful. The comparison of the

reasons given between the Comparison and Intervention Groups may have been

supportive in interpreting the other data collected in this study.

7.3 Implications of the Research

The findings of this research show that, notwithstanding the limitations mentioned

in Section 7.2, a problem-posing intervention can impact upon the development

of problem-solving competence in some middle-year students who are

ascertained as being of above average intelligence yet underachieving in

problem solving. A group of students with these aforementioned attributes had

not been investigated previously and hence this study serves to add to the body

of research related to problem-posing activities, mathematical pedagogy and

solving novel problems.

While these results would not surprise researchers of the field who confidently

applaud the benefits of problem-posing activities in the pursuit of improved

problem-solving competence (e.g., English, 2003; English et al., 2005), others

such as Silver and Cai (1993b) have been previously unconvinced about a

positive correlation between a student’s problem-solving competence and their

involvement with problem-posing activities. The findings of this study add weight

to the argument that problem-posing activities are worthwhile. However, a

longitudinal study of a larger group of students would be very useful in tracking

the sustainability and consistency of positive impacts that occur as a result of a

problem-posing intervention. Practically speaking, it is difficult to access and

223

track a large group of students for a number of years. If this were undertaken,

the quantitative data may be able to ‘stand alone’ in defending the premise that

incorporating problem-posing activities into a mathematics work program for

middle-year students can result in measurable improvements in their problem-

solving competence. This is a challenge for future research.

During the sixth teaching session of the teaching experiment, students seemed

less engaged by the opportunity to pose worded problems than they had done in

previous weeks (see Section 4.4.5). As a result, a move to a focus on creating

and posing visual problems was planned for the seventh teaching session. This

strategy was employed to increase student motivation by providing diversity to

the teaching methodology while remaining within the parameters of the research

design. The immediate re-engagement of students was clearly evident and was

sustained to the end of the teaching experiment (see Sections 4.4.6 and 4.4.7).

Evidence of the student interest in visual problems was supported by the

substantial increase in the number of students recording visual problems as their

most preferred problems to solve in Question Two of the student surveys (see

Table 5.5 and Section 5.3.2). This was an unexpected finding and one that is

worthy of further research. It may be that it was simply the change in the type of

activity that re-engaged the students or it may be that providing problem-posing

activities, rich in visual stimuli, may be a useful teaching strategy in sustaining

student engagement in problem-solving activities. Further research would clarify

which of these alternatives is more likely.

7.4 Concluding Comments

The education of students in mathematics has been studied over many years by

many researchers (e.g., Anderson, 2007; Brown & Walter, 2005; English &

Larson, 2005). A large body of this research focussed on the teaching practices

used by teachers of mathematics (e.g., English, 2002; Lesh & Zawojewski,

2007), the training and professional development of teachers of mathematics

(e.g., Crespo, 2003; Fosnot & Dolk, 2001) and the importance of problem solving

224

in mathematics work programs (e.g., Cai, 2003; Costa, 2005; NRC, 2004).

Despite a number of studies being undertaken by various researchers in different

countries (e.g., Cai, 2003; Hollingsworth et al., 2003; Shimizu, 2002) and

recommendations for changes to curricula being made, Lester (2003) called for a

fresh view of problem solving as little had changed in teaching practices in

mathematics classrooms. Lesh and Zawojewski (2007) went further and stated

that a serious mismatch still existed between how problem-solving was

experienced by students in schools and the type of skills they required when they

entered the work force.

Over the past decade a number of researchers have begun to address this

anomaly through their research of mathematical modelling as a tool to

contextualise problem-solving experiences (e.g., English et al., 2005; Lesh &

Zawojewski, 2007; Lester & Kehle, 2003). Throughout this research, significant

discussion focussed on the importance of interpreting problems (Lesh &

Zawojewski, 2007) and being able to “see” what a problem is before being able

to solve it (Lesh & Doerr, 2003). English (2003) and Brown and Walter (2005)

were amongst leading researchers who researched the potential of problem-

posing activities to assist students to understand the underlying structure of

problems and to therefore transfer their knowledge between analogous

problems. Their work informed the design and foci of this study.

This study set out to investigate problem-posing activities for a group of students

who often confound their teachers, that is, students who appeared to be

particularly intelligent, but who were not able to perform well at solving problems

in a test situation. The findings of this research have provided some persuasive

evidence to suggest that involvement in problem-posing activities can have a

positive impact for this group of students. It would seem then that this line of

research is fruitful and should be pursued in the endeavour of preparing our

brightest students to be our future problem solvers.

225

References

ACER. (2003). Graduate Skills Assessment (GSA). Camberwell, VIC: Australian

Council for Educational Research.

Adams, K., Brower, S., Hill, D., & Marshall, I. (2000). The components of an

effective mathematics and science middle school: standards, teaching

practices and professional development

Alvesson, M., & Skolberg, K. (2000). Reflexive methodology: New vistas for

qualitative research. London: Sage.

Anderson, C. R. (2007). Still separate, still unequal. In L. D. English (Ed.),

Handbook of International Research in Mathematics Education.

Philadelphia: Taylor and Francis.

Anderson, J. (1997). Teachers' reported use of problem solving strategies in

primary mathematics classrooms. In F. B. K. Carr (Ed.), People in

mathematics education. Proceedings of the 20th Annual Conference of

the Mathematics Education Research group of Australasia. Rotorua, New

Zealand.

Australian Chamber of Commerce and Industry. (2002). Employability skills - An

Employer Perspective: Getting what employers want out of the too hard

basket. Retrieved February 21, 2009, from

http://www.acci.asn.au/text_files/issues_papers/Employ_Educ/ee21.pdf

Australian Council for Educational Research. (2005). Middle years ability test.

Camberwell, Victoria: Australian Council for Educational Research.

226

Baker, J., Bridger, R., & Evans, K. (1998). Models of underachievement among

gifted preadolescents: The role of personal, family, and school factors.

National Association for Gifted Children, 42(1), 5-15.

Ball, D. L., & Cohen, D. K. (1996). Reform by the book: what is - or might be - the

role of curriculum materials in teacher learning and instructional reform?

Educational Researcher, 25, 6-8, 14.

Bandura, A. (1997). Self-efficacy: The exercise of control. New York, NY:

Freeman.

Barrington, F. (2006). Participation in Year 12 Mathematics across Australia

1995-2004. Melbourne: Australian Mathematical Sciences Institute.

Becker, J., Sawada, T., & Shimizu, Y. (1999). Some findings of the US-Japanese

cross-cultural research on students' problem solving behaviours. In E. L.

G. Kaiser, and I. Huntley (Ed.), International comparisons in mathematics

education (pp. 121-139). London: Falmer Press.

Bell, P. (2004). On the theoretical breadth of design-based research in education.

Washington: University of Washington.

Ben-Chaim, D., Fey, J., Fitzgerald, W., Benedetto, C., & Miller, J. (April, 1997).

Development of proportional reasoning in problem-based middle school

curriculum. Paper presented at the Annual Meeting of the American

Educational Research Association, Chicago, IL.

Bentley, B. (1996). An examination of the effectiveness of teaching heuristic

strategies and metacognitive practice to year 6/7 students engaging in

227

problem solving in mathematics. Unpublished Unpublished Masters thesis,

University of South Australia, Adelaide.

Bernardo, A. B. I. (2001). Analogical problem construction and transfer in

mathematical problem solving. Educational Psychology, 21(2), 137-150.

Binet, A., & Simon, T. (1905). Methode nouvelles pour le diagnostic du niveau

intellectual des anormaux. L'Annee Psychologique, 11, 191-336.

Bjorklund, D. F. (2000). Children's thinking: Developmental function and

individual differences (3rd ed.). Belmont, CA: Wadsworth Thomson

Learning.

Bobis, J., Mulligan, J., & Lowrie, T. (2004). Mathematics for children: Challenging

children to think mathematically. Frenchs Forrest, NSW: Pearson

Education.

Boehnke, K. (2007). Peer pressure: a cause of scholastic underachievement? A

cross-cultural study of mathematical achievement among German,

Canadian, and Israeli middle school students. Journal Social Psychology

of Education, 11(2), 149-160.

Boekaerts, M. (1997). Self-regulated learning: A new concept embraced by

researchers, policy makers, educators, teachers, and students. Leiden,

The Netherlands: Elsevier Science Ltd.

Booker, G., & Bond, D. (2001). Booker profiles in mathematics: Thinking

mathematically. Camberwell, Vic.: Australian Council for Educational

Research.

228

Borg, W., Gall, J., & Gall, M. (1999). Applying educational research: A practical

guide (4th ed.). New York: Longman.

Brand, C. (1996). The g factor: General intelligence and its implications.

Chichester, UK: Wiley.

Brody, N. (1997). Intelligence, schooling, and society. American Psychologist, 52,

1046-1050.

Brody, N. (2003). Construct validation of the Sternberg Triarchic Abilities Test:

Comment and reanalysis. Intelligence, 31(319-330).

Broido, E. (2002). Philosophical foundations and current theoretical perspectives

in qualitative research. Journal of College Student Development(Jul/Aug).

Brown, A. (1992). Design experiments: Theoretical and methodological

challenges in creating complex interventions in classroom settings. The

Journal of the Learning Sciences, 2(2), 141-178.

Brown, A., & Campione, J. (1998). Designing a community of young learners:

Theoretical and practical lessons. In N. L. a. B. McCombs (Ed.), How

students learn: Reforming schools through learner-centered education

(pp. 153-186). Washington, DC: American Psychological Association.

Brown, S., & Walter, M. (1983). The art of problem posing. Philadelphia, PA:

Franklin Institute Press.

Brown, S., & Walter, M. (2005). The art of problem posing (3rd ed.). Mahwah, NJ:

Lawrence Erlbaum Associates.

Bruner, J. (1996). The culture of education. In Psychology's next chapter (pp.

160-185). Cambridge, MA: Harvard University Press.

229

Bruner, J. S. (1965). The growth of mind. American Psychologist, 20, 1007-1017.

Burns, R. (1995). Introduction to research methods (2nd ed.). Melbourne:

Longman Australia.

Cai, J. (1997). An Investigation of U.S. and Chinese Students' Mathematical

Problem Posing and Problem Solving.

Cai, J. (2000). Understanding and representing the arithmetic averaging

algorithm: an analysis and comparison of US and Chinese students'

responses. International Journal of Mathematical Education in Science &

Technology, 31(6), 839-855.

Cai, J. (2003). Singaporean students' mathematical thinking in problem solving

and problem posing: an exploratory study. International Journal of

Mathematical Education in Science & Technology, 34(5), 719.

Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and

Chinese students' mathematical problem solving and problem posing.

Journal of Mathematical Behavior, 21(4), 401-421.

Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2003). Patterns of knowledge in

children's addition. Developmental Psychology, 39, 521-534.

Carey, D., Fennema, E., Carpenter, T. P., & Franke, M. L. (1995). Equity in

mathematics education. In E. F. W. G. Secada, & L. B. Adajain (Ed.), New

directions for equity in mathematics education (pp. 93-125). Cambridge,

UK: Cambridge University Press and the National Council of Teachers of

Mathematics.

230

Case, R. (1998). The development of conceptual structures. In D. Kuhn & R.

S.Siegler (Eds.), Cognition, perception and language (Vol. 2). New York,

NY: Wiley.

Cassel, J., & Reid, R. (1996). Use of a self-regulated strategy intervention to

improve word problem-solving skills of students with mild disabilities.

Journal of Behavioral Education, 6(2), 153-172.

Cattell, R. B. (1971). Abilities: Their structure, growth and action. Boston, MA:

Houghton Mifflin.

Ceci, S. (1990). On intelligence...more or less: A bio-ecological treatise on

intellectual development. Englewood Cliffs, NJ: Prentice-Hall.

Ceci, S. J. (1996). On intelligence: A bio-ecological treatise on intellectual

development (2nd ed.). Cambridge, MA: Harvard University Press.

Cobb, P. (1997). Instructional design and reform: A plea for developmental

research in context. In K. G. M. Beishuizen, E. van Leishout (Ed.), The

role of contexts and models in develpment of mathematical strategies and

procedures (pp. 273-289). Utrecht, The Netherlands: Utrecht University.

Collins, A. (1986). Different goals of enquiry teaching (No. 6458). Cambridge,

MA: BBN Laboratories Incorporated.

Confrey, J., & Lachance, A. (2000). Transformative teaching experiments

through conjecture-driven research design. In A. E. Kelly & R. Lesh (Eds.),

Handbook of research design in mathematics and science education.

Marwah, NJ: Lawrence Erlbaum.

231

Contreras, J. N. (2003). A Problem-Posing Approach to Specializing,

Generalizing, and Extending Problems with Interactive Geometry

Software. Mathematics Teacher, 96(4), 270.

Costa, A. L. (2005). Habits of mind. Paper presented at the 12th International

Thinking Conference, Melbourne.

Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes

in preservice teachers' practices. Educational Studies in Mathematics,

52(3), 243-270.

Crotty, M. (1998). Foundations of Social Research: Meaning and Perspective in

the Research Process. Crows Nest, NSW: Allen and Unwin.

Cummins, J., & Sayers, D. (1995). Brave new schools: Challenging cultural

illiteracy through global learning networks. New York: St Martin's Press.

Daniel, G. E. (2003). Effects of cognitive strategy instruction on the mathematical

problem solving of middle school students with learning disabilities. The

Ohio State U., US.

Deary, I. J. (2000). Simple information processing and intelligence. In R. J.

Sternberg (Ed.), Handbook of intelligence (pp. 267-284). New York, NY:

Cambridge University Press.

Delisle, J. R. (1992). Guiding the social and emotional development of gifted

youth: A practical guide for educators and counselors. New York:

Cambridge University Press.

232

DeLoache, J. S., Miller, K. F., & Pierroutsakos, S. L. (1998). Reasoning and

problem solving. In D. Kuhn & R. S. Siegler (Eds.), Cognitive, language,

and perceptual development (Vol. 2). New York, NY: Wiley.

Department of Education Training and the Arts. (2007). New basics project,

2007, from http://education.qld.gov.au/corporate/newbasics/

DEST, & ANTA. (2004). Employability skills final report: Development of a

strategy to support universal recognition and recording of employability

skills - A skills portfolio approach. Canberra, ACT: Department of

Education, Science and Technology and Australian National Training

Authority.

Einstein, A., & Infeld, L. (1938). The evolution of physics. New York: Simon and

Schuster.

English, F. W., & Larson, R. L. (2005). Chapter 5: Curriculum validation and

abandonment. Retrieved 16 August, 2005, from

http://www.angelfire.com/theforce/shu_cohort_viii/images/currch5.pdf

English, L. D. (1996). Children's problem posing and problem solving

preferences. In M. M. J. Mulligan (Ed.), Children's number learning: A

research monograph of MERGA/AAMT (pp. 227-242). Adelaide:

Australian Association of Mathematics Teachers.

English, L. D. (1997a). The development of fifth-grade children's problem-posing

abilities. Educational Studies in Mathematics, 34(3), 183-217.

English, L. D. (1997b). Promoting a problem-posing classroom. Teaching

Children Mathematics, 4(3), 172.

233

English, L. D. (2002). Priority themes and issues in international research in

mathematics education. In L. D. English (Ed.), Handbook of international

research in mathematics education (pp. 3-15). Mahwah, NJ: Laurence

Erlbaum.

English, L. D. (2003). Problem posing in the elementary curriculum. In F. Lester

& R. Charles (Eds.), Teaching mathematics through problem solving.

Reston, Virginia: National Council of Teachers of Mathematics.

English, L. D., Cudmore, D., & Tilley, D. (1998). Problem posing and critiquing:

How it can happen in your classroom. Mathematics Teaching in the Middle

School, 4(2), 124-129.

English, L. D., Fox, J. L., & Watters, J. J. (2005). Problem posing and solving

with mathematical modelling. Brisbane: Queensland University of

Technology.

English, L. D., & Halford, G. S. (1995). Mathematics education: Models and

processes. Hillsdale, NJ: Lawrence Erlbaum Associates.

Feuerstein, R. (1979). The dynamic assessment of retarded performers: The

Learning Potential Assessment Device, theory, instruments, and

techniques. Baltimore, MD: University Park Press.

Fosnot, K. T., & Dolk, M. (2001). Young mathematicians at work: Consructing

number sense, addition, and subtraction. Portsmouth, NH: Heinemann.

Freire, P. (1970). Pedagogy of the oppressed. New York, NY: Herder & Herder.

Galton, F. (1883). Inquiries into human faculty and its development. London:

Macmillan.

234

Gardner, H. (1999a). Are there additional intelligences? The case for naturalist,

spiritual, and existential intelligences. In J. Kane (Ed.), Education,

information and transformation. Englewood Cliffs, NJ: Prentice-Hall.

Gardner, H. (1999b). Intelligence reframed: Multiple intelligences for the 21st

century. New York, NY: Basic Books.

Gentry, M., Gable, R. K., & Springer, P. (2000). Gifted and nongifted middle

school students: Are their attitudes toward school different as measured

by the new affective instrument, My Class Activities...? Journal for the

Education of the Gifted, 24(1), 74-96.

Gergen, K. J. (1996). Social psychology as social construction: The emerging

vision. In C. M. a. A. Haslam (Ed.), The message of social psychology:

Perspectives on mind in society. Oxford: Blackwell.

Glesne, C., & Peshkin, A. (1992). Becoming qualitative researchers: An

introduction. White Plains, NJ: Longman.

Goldman-Segall, R. (1998). Points of viewing children's thinking. Hillsdale, NJ:

Lawrence Erlbaum and Associates.

Gonzales, N. A. (1994). Problem posing: A neglected component in mathematics

courses for prospective elementary and. School Science & Mathematics,

94(2), 78-86.

Gonzales, N. A. (1998). A Blueprint for Problem Posing. School Science &

Mathematics, 98(8), 448-456.

235

Gootman, M. (2001). The caring teacher's guide to discipline: Helping young

students learn self-control and responsibility (2nd ed.). Thousand Oaks,

CA: Corwin Press.

Goswami, U. (1996). Analogical reasoning and cognitive development. In H. W.

Reese (Ed.), Advances in child development and behavior (Vol. 26). San

Diego: Academic Press.

Goswami, U. (Ed.). (2002). Blackwell handbook of childhood cognitive

development. Malden, MA: Blackwell.

Gottfredson, L. (2003). On Sternberg's "Reply to Gottfredson". Intelligence, 31,

415-424.

Gozales, N. (1994). Problem posing: A neglected component in mathematics

courses for prospective elementary and middle school teachers. School

Science & Mathematics, 94(2).

Graham, S., Harris, K. R., & Larsen, L. (2001). Prevention and intervention of

writing difficulties for students with learning disabilities. Learning

Disabilities Research & Practice, 16(2), 74-84.

Gravemeijer, K. P. E. (1994). Developing realistic mathematics education.

Utrecht: CD-B Press, Freudenthal Institute.

Greeno, J. G. (2003). Situative research relevant to standards for school

mathematics. In J. Kilpatrick, W. Martin & D. Schiffer (Eds.), A research

companion to principles and standards for school mathematics (pp. 304-

332). Reston; VA: The National Council of Teachers of Mathematics.

236

Grigorenko, E. L. (2000). Heritability and intelligence. In R. J. Sternberg (Ed.),

Handbook of intelligence (pp. 53-92). New York, NY: Cambridge

University Press.

Groves, S., Mousley, J., & Forgasz, H. (2006). Primary numeracy: A mapping,

review and analysis of Australian research in numeracy learning at the

primary school level. Canberra: Centre for Studies in Mathematics,

Science and Environmental Education: Deakin University.

Guilford, J. P. (1988). Some changes in the structure-of-the-intellect model.

Educational and Psychological Measurement, 48, 1-4.

Halford, G. S. (2002). Information processing models of cognitive development.

In U. Goswami (Ed.), Blackwell handbook of cognitive development (pp.

555-574). Malden, MA: Blackwell handbook of cognitive development.

Halpern, D. F. (1997). Sex differences in intelligence: Implications for education.

American Psychologist, 52, 1091-1102.

Hamilton, E., Lesh, R., Lester, R., & Yoon, C. (2006). The use of reflective tools

in building personal models of problem solving. In R. Lesh, E. Hamilton &

J. Kaput (Eds.), Models and modeling as foundations for the future in

mathematics education. Marwah; NJ: Lawrence Erlbaum.

Hart, K. (Ed.). (1981). Children's understanding of mathematics. London: John

Murray.

Hernstein, R., & Murray, C. (1994). The bell curve: Intelligence and class

structure in American life. New York: Simon and Schuster.

237

Hollingsworth, H., Lokan, J., & McCrae, B. (2003). Teaching mathematics in

Australia: Results from the TIMSS video study. Camberwell.

Hughes, J. (2002). Harold Wyndham and educational reform in Australia, 1925–

1968. Educational Research and Perspectives, 29(1), 1-65.

Hutton, U., Wilding, J., & Hudson, R. (1997). The role of attention in the

relationship between inspection time and IQ in children. Intelligence, 24,

445-460.

Impara, J. C., & Plake, B. S. (Eds.). (1998). The thirteenth mental measurements

yearbook. Lincoln, NE: Buros Institute of mental Measurements.

Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to

adolescence. New York, NY: Basic Books.

Jensen, A. R. (1998). The g factor: The science of mental ability. Westport, CT:

Praeger/Greenwood.

Jensen, A. R., & Miele, F. (2004). Intelligence, race, and genetics: Conversations

with Arthur R. Jensen. Boulder, Colorado: Westview Press.

Johnson, D. W. (2000). Teaching mathematics to gifted students in a mixed-

ability classroom. Retrieved 25th September, 2006, from

http://ericec.org/digests/e594.html

Jones, S., & Myhill, D. (2004). Seeing things differently: Teachers' constructions

of underachievement. Gender and Education, 16(4), 531-546.

Kanevsky, L., & Keighley, T. (2003). To produce or not to produce?

Understanding boredom and the honor in underachievement. Roeper

Review, 26.

238

Kelly, A. E., & Lesh, R. A. (Eds.). (2000). Handbook of research design in

mathematics and science education. Marwah, NJ: Lawrence Erlbaum.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping

children learn mathematics. Washington, DC: National Academy Press.

Kincheloe, J., Steinberg, S., & Gresson, A. (1996). Measured lies: The bell curve

examined. New York: St Martin's Press.

Klahr, D. (1992). Information processing approaches to cognitive development. In

M. H. Bornstein & M. E. Lamb (Eds.), Developmental psychology: An

advanced textbook (3rd ed.). Hillsdale, NJ: Laurence Erlbaum Associates.

Kline, M. J. (Ed.). (1996). Einstein's 1912 manuscript on the special theory. New

York: George Braziller, Inc. in association with the Jacob E. Safra

Philanthropic Foundation and the Israel Museum.

Knuth, E. J., & Peterson, B. E. (2002). Fostering Mathematical Curiosity.

Mathematics Teacher, 95(2), 126.

Korniszewski, P., & Mallet, J. (1948). A survey of educational psychology

services in 41 countries. Paris: United Nations Educational, Scientific and

Cultural Organization.

Kramarski, B., Mevarech, Z. R., & Arami, M. (2002). The effects of metacognitive

instruction on solving mathematical authentic tasks. Educational Studies in

Mathematics, 49, 225-250.

Lavy, l., & Bershadsky, I. (2003). Problem posing via "what if not?" strategy in

solid geometry: A case study. Journal of Mathematical Behavior, 22(4),

369-387.

239

Leder, G., & Forgasz, H. (2003). Measuring mathematical beliefs and their

impact on the learning of mathematics: A new approach. In E. P. G. Leder,

G. Torner (Ed.), Beliefs: A hidden variable in mathematics education.

Netherlands: Springer.

Lemke, M., Sen, A., Pahlke, E., Partelow, L., Miller, D., Williams, T., et al. (2004).

International outcomes of learning in mathematics, literacy and problem

solving: PISA 2003 results from the U.S. perspective - highlights.

Washington, DC: National Center for Educational Statistics.

Lesh, R., & Clarke, D. (2000). Formulating operational definitions of desired

outcomes of instruction in mathematics and science education. In R. Lesh

& A. Kelly (Eds.), Handbook of research design in mathematics and

science education. Marwah, NJ: Lawrence Erlbaum.

Lesh, R., & Doerr, H. (2003). Foundations of a models and modeling perspective

on mathematics teaching, learning, and problem solving. In R. D. Lesh, H.

(Ed.), Beyond constructivism: Models and modeling perspectives on

mathematics problem solving, learning and teaching. marwah: NJ:

Lawrence Erlbaum.

Lesh, R., Lovitts, B., & Kelly, A. (2000). Purposes and assumptions of this book.

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

science education. Marwah, NJ: Lawrence Erlbaum.

Lesh, R., & Zawojewski, J. (2007). Problem solving and modelling. In F. Lester

(Ed.), Handbook of international research in mathematics education (2nd

ed.). Mahwah; NJ: Lawrence Erlbaum.

240

Lester, F. (1994). Musings about mathematical problem-solving research: 1970-

1994. Journal for Research in Mathematics Education, 25(x), 660-675.

Lester, F. (Ed.). (2003). Teaching mathematics through problem solving. Reston,

Virginia: National Council of Teachers of Mathematics.

Lester, F., & Kehle, P. (2003). From problem solving to modeling: The evolution

of thinking about research on complex mathematical activity. In R. L. H.

Doerr (Ed.), Beyond constructivism: Models and modelling perspectives

on mathematical problem solving, learning and teaching. Mahwah:

Erlbaum.

Lincoln, Y., & Denzin, N. (2000). The seventh movement: Out of the past. In N.

D. a. Y. Lincoln (Ed.), Handbook of qualitative research (2nd ed., pp.

1047-1065). Thousand Oaks, CA: Sage.

Lipman, P. (2004). High stakes education: Inequality, globalization, and urban

school reform. New York: Routledge/Falmer.

Lohman, D. F. (2000). Complex information processing and intelligence. In R. J.

Sternberg (Ed.), Handbook of intelligence (pp. 285-340). New York, NY:

Cambridge University Press.

Love, T. (2002). Multiple theoretical perspectives in the long PhD: A foundation

problem in PhD education. Perth: Edith Cowan University.

Lowrie, T. (2002). Young children posing problems: The influence of teacher

intervention on the type of problems children pose. Mathematics

Education Research Journal, 14(2), 87-98.

241

Luke, A., Elkins, J., Weir, K., Land, R., Carrington, V., Dole, S., et al. (2003).

Beyond the middle: A report about literacy and numeracy development of

target group students in the middle years of schooling. Canberra:

Department of Education, Science and Training.

Malara, N. (2002). The problematic relationship between theory and practice. In

L. English (Ed.), International Research in Mathematics Education (pp.

574). LEA.

Marat, D. (2005). Assessing mathematics self-efficacy of diverse students from

secondary schools in Auckland: Implications for academic achievement.

Issues in Educational Research, 15.

McLellan, M. J. (1998). Review of the profiles of problem solving. In 13 Mental

Measurements Yearbook. Lincoln, NE: Buros Institute of Mental

Measurements.

McNeil, L., & Valenzuela, A. (2000). The harmful impact of the TAAS system of

testing in Texas: beneath the accountability rhetoric. Cambridge, MA:

Harvard Civil Rights Project.

Medina-Diaz, M. (1998). Review of the profiles of problem solving. In The

Thirteenth Mental Measurements Yearbook. Lincoln, NE: Buros Institute of

Mental Measurements.

Merriam, S. (1998). Qualitative Research and Case Study Applications in

Education. Revised and Expanded from "Case Study Research in

Education." San Francisco, CA: Jossey-Bass Publishers.

242

Mestre, J. (2000). Progress in research: The interplay among theory, research

questins and measurement techniques. In R. Lesh & A. Kelly (Eds.),

Handbook of research design in mathematics and science education.

Marwah, NJ: Lawrence Erlbaum.

Ministerial Council on Education, Employment, Training, & Affairs, Y. (2008a).

National assessment program: Literacy and numeracy. Retrieved March 1,

2009, from www.naplan.edu.au

Ministerial Council on Education, Employment, Training, & Affairs, Y. (2008b).

National declaration on educational goals for youung Ausralians - Draft.

Retrieved March 1, 2009, from

http://www.curriculum.edu.au/verve/_resources/Draft_National_Declaratio

n_on_Educational_Goals_for_Young_Australians.pdf

Naglieri, J. A., & Kaufman, J. C. (2001). Understanding intelligence, giftedness,

and creativity using the PASS theory. Roeper Review, 23(3), 151-156.

National Curriculum Board. (2008). The national mathematics curriculum:

Framing paper. Retrieved February 22, 2009, from

http://www.ncb.org.au/verve/_resources/National_Mathematics_Curriculu

m_-_Framing_Paper.pdf

National Mathematics Advisory Panel. (2008). Final Report: US Department of

Education.

National Numeracy Review. (2008). National numeracy review report. Canberra:

Human Capital Working Group.

243

NCTM. (2000). Curriculum and evaluation standards for school mathematics.

Reston, VG: National Council of Teachers of Mathematics.

NRC. (2001). Adding it up: helping children learn mathematics. Washington, DC:

National Academy Press.

NRC. (2004). On evaluating curricula effectiveness: Judging the quality of K-12

mathematics evaluations. Washington, DC: National Academy Press.

NRC. (2005). How students learn: History, mathematics, and science in the

classroom. Washington, DC: Division of Behavioural and Social Sciences

and Education, National Research Council of the National Academies.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New

York: Basic Books.

Papert, S. (1991). Situating constructionism. In I. H. a. S. Papert (Ed.),

Constructionism. Norwood, NJ: Ablex Publishing.

Paris, S., & Newman, R. (1990). Developmental aspects of self-regulated

learning. Educational Psychologist, 25(1), 87-102.

Patton, J. D. (2002). The role of problem pioneers in creative innovation.

Creativity Research Journal, 14(1), 111-126.

Pejouhy, N. H. (1990). Teaching math for the 21st century. Phi Delta Kappan,

72(1), 76-78.

Piaget, J., & Beth, E. (1966). Mathematical epistemology and psychology.

Dordrecht, Netherlands: D. Reidel.

Piaget, J., & Inhelder, B. (1969). The psychology of the child. New York, NY:

Basic Books.

244

Piirto, J. (2007). Talented children and adults: Their development and education

(3rd ed.). New York: Prufrock Press Incorporated.

Polya, G. (1957). How to solve it (2nd ed.). New York, NY: Doubleday.

Price, J. (1996). Building bridges of mathematical understanding for all children.

Teaching Children Mathematics, 27(5), 603-608.

Queensland Studies Authority. (2005). 2005 Queensland year 7 tests in aspects

of literacy and numeracy. Brisbane, QLD: Queensland Studies Authority.

Queensland Studies Authority. (2009). Queensland Core Skills Test.

Reis, S., & Siegle, D. (2006). Degrees of underachievement (Report for

conference in Basel, Switzerland). Conneticut: University of Conneticut.

Richardson, J. (1999). The concepts and methods of phenomenographic

research. Review of Educational Research, 69(1), 53-82.

Ridgeway, J., Zawojewski, J., & Hoover, M. N. (2000). Problematising evidence

based policy and practice. Evaluation and research in Education, 14(3,4),

181-192.

Ritchhart, R. (2002). Intellectual character: What it is, why it matters, and how to

get it. San Francisco, CA: Jossey-Bass.

Roberts, P. (1996). Structure, direction and rigour in liberating education. Oxford

Review of Education, 22(3), 295-316.

Rogoff, B. (1998). Cognition as a collaborative process. In D. Kuhn & R. S.

Siegler (Eds.), Cognition language, and perceptual development (Vol. 2).

New York, NY: Wiley.

245

Rose, G. (2007). Visual methodologies: An introduction to the interpretation of

visual methods. London: Sage.

Rumelhart, D. E. (1980). Schemata: The building blocks of cognition. In R. J.

Spiro, B. C. Bruce & W. F. Webster (Eds.), Theoretical issues in reading

comprehension: Perspectives from cognitive psychology, linguistics,

artificial intelligence, and education (pp. 33-58). Hillsdale, NJ: Lawrence

Erlbaum Associates.

Ryan, A., & Patrick, H. (2001). The classroom social environment and changes in

adolescents' motivation and engagement during middle school. American

Educational Research Journal, 38(2), 437-460.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,

metacognition and sense making in mathematics. In D. A. Grouws (Ed.),

Handbook on research on mathematics teaching and learning (pp. 334-

370). New York, NY: Macmillan.

Schultz, R. A. (2000). Flirting with underachievement. The Hollingworth Center

for Highly Gifted Children, 13(2), 42-48.

Schunk, D. H. (2001). Social cognitive theory and self-regulated learning. In B. J.

Zimmerman & D. H. Schunk (Eds.), Self-regulated learning and academic

achievement: Theoretical perspectives (2nd ed., pp. 125-152). Marwah,

NJ: Lawrence Erlbaum Associates.

Schwandt, T. (2001). Dictionary of qualitative inquiry (2nd ed.). Thousand Oaks,

CA: Sage Publications.

246

Secada, W., & Berman, P. (1999). Equity as a value-added dimension in

teaching for understanding in school mathematics. In E. Fennema & T. A.

Romberg (Eds.), Mathematics classrooms that promote understanding

(pp. 33-42). Marawah, NJ: Lawrence Erlbaum.

Shimizu, Y. (2002). Capturing the structure of Japanese mathematics lessons:

Some findings of the international comparative studies. Paper presented

at the Ninth Southeast Asian Conference on Mathematics Education,

Singapore.

Siegler, R. S. (1991). Children's Thinking (2nd ed.). Englewood Cliffs, NJ:

Prentice-Hall.

Siegler, R. S. (1996). Emerging minds: The process of change in children's

thinking. New York, NY: Oxford University Press.

Silver, E. (1994). On mathematical problem posing. For the Learning of

Mathematics, 14(1), 19-28.

Silver, E. (1997). Fostering creativity through instruction rich in mathematical

problem solving and problem posing. Analyses, 3, 75-80.

Silver, E., & Cai, J. (1993a). Mathematical problem posing and problem solving

by middle school children. Paper presented at the Annual meeting of the

American Educational Research Association, Atlanta, GA.

Silver, E., & Cai, J. (1996). An analysis of arithmetic problem posing by middle

school children. Journal for Research in Mathematics Education, 27(5),

521- 534.

247

Silver, E., & Jinfa, C. (1996). An analysis of arithmetic problem posing by middle

school students. Journal for Research in Mathematics Education, 27, 521-

539.

Silver, E., Mamona-Downs, J., Leung, S., & Kenney, P. (1996). Posing

mathematical problems: An exploratory study. Journal for Research in

Mathematics Education, 27(May), 293-309.

Silver, E. A., & Cai, J. (1993b). An analysis of arithmetic problem posing by

middle school children. Journal for Research in Mathematics Education,

27(5), 521- 534.

Simons-Morton, B., Crump, A., Haynie, D., & Saylor, K. (1999). Student-school

bonding and adolescent problem behavior. Health Education Research,

14(1), 99-107.

Skovsmose, O., & Valero, P. (2002). Democratic access to powerful mathematics

ideas. In L. English (Ed.), Handbook of international research in

mathematics education (pp. 383-408). Mahwah, NJ: Lawrence Erlbaum.

Spearman, C. (1904). "General intelligence", objectively determined and

measured. American Journal of Psychology, 15(2), 201-293.

SPSS Inc. (2007). Statistical package for social sciences (student version)

(Version 15). Melbourne: Prentice Hall.

Stacey, K., Groves, S., Bourke, S., & Doig, B. (1993). Profiles of problem solving.

Hawthorne, Vic: Australian Council of Educational Research.

Stecher, B. M. (2002). Consequences of large-scale, high stakes testing on

school and classroom practice. In L.S. Hamilton, B.M. Stecher & S. P.

248

Klein (Eds.), Making sense of test-based accountability in education.

Santa Monica CA: RAND.

Steffe, L. P., Thompson, P. W., & von Glasersfeld, E. (2000). Teaching

experiment methodology: Underlying principles. In A. Kelly & R. Lesh

(Eds.), Handbook of research design in mathematics and science

education. Marawh, NJ: Lawrence Erlbaum.

Stein, M., Boaler, J., & Silver, E. (2003). Teaching mathematics through problem

solving: Research perspectives. In H. S. R. Charles (Ed.), teaching

mathematics through problem solving: Grades 6-12 (pp. 245-256).

Reston, VA: National Council of Teachers of Mathematics.

Stern, W. (1912). The Psychological Methods of Intelligence Testing (G. Whipple,

Trans.). Baltimore: Warwick and York.

Sternberg, R. J. (1985). Beyond IQ: A triarchic theory of human intelligence. New

York, NY: Cambridge University Press.

Sternberg, R. J. (1988). The triarchic mind: A theory of human intelligence. New

York, NY: Viking.

Sternberg, R. J. (1997a). Successful intelligence. New York, NY: Plume.

Sternberg, R. J. (1999a). Successful intelligence: Finding a balance. Trends in

Cognitive Sciences, 3, 436-442.

Sternberg, R. J. (2002). Individual differences in cognitive development. In U.

Goswami (Ed.), Blackwell handbook of cognitive development (pp. 600-

619). Malden, MA: Blackwell.

249

Sternberg, R. J. (Ed.). (2000). Handbook of intelligence. New York, NY:

Cambridge University Press.

Stigler, J. W., Gonzales, P., Kawanaka, T., Knoll, S., & Serrano, A. (1999). The

TIMSS videotape classroom study: Methods and findings from an

exploratory research

project on eighth-grade mathematics instruction in Germany, Japan, and the

United States. Washington, D.C.: U.S. Government Printing Office.

Stipeck, D., & Gralinski, J. H. (1996). Children's beliefs about intelligence and

school performance. Journal of Educational Psychology, 88(3), 397-407.

Stoyanova, E. (2003). Extending students' understanding of mathematics via

problem-posing. Australian Mathematics Teacher, 59(2), 32.

Tate, W., & Rousseau, C. (2002). Access and opportunity: The political and

social context of mathematics education. In L. D. English (Ed.), Handbook

of International Research in Mathematics Education (pp. 271-300).

Marwah, NJ: Lawrence Erlbaum.

Taylor, S., & Bogdan, R. (1998). Introduction to qualitative research methods: a

guidebook and resource. Canada: John Wiley & Sons.

Teasley, S. D. (1995). The role of talk in children's peer collaborations.

Developmental Psychology, 31, 207-220.

Terman, L. M. (1916). The measurement of intelligence: An explanation of a

complete guide for the use of the Stanford revision and extension of the

Binet-Simon Intelligence Scale. Boston: Houghton Mifflin.

250

Thelan, E., & Smith, L. B. (1998). Dynamic system's theories. In R. M. Lerner

(Ed.), Handbook of child psychology: Vol. 1. Theoretical models of human

development (5th ed.). New York, NY: Wiley.

Thomas, M. S. C., & Karmiloff-Smith, A. (2002). Modelling typical and atypical

cognitive development: Computational constraints on mechanism of

change. In U. Goswami (Ed.), Blackwell handbook of cognitive

development (pp. 575-599). Malden, MA: Blackwell.

Thurstone, L. L. (1938). Primary mental abilities. Chicago: University of Chicago

Press.

Tierney, W. (1997). Academic outlaws: Queer theory and cultural studies in the

academy. Thousdand Oaks, CA: Sage.

Van de Walle, J. (2004). Elementary and middle school mathematics (5th ed.).

New York: Pearson.

Van Den Brink, J. F. (1987). Children as arithmetic book authors. For the

Learning of Mathematics, 7(2), 44-48.

van Essen, G., & Hamaker, C. (1990). Using self-generated drawings to solve

arithmetic word problems. Journal of Educational Research, 83(6), 301-

312.

Vernon, P. A., Wickett, J. C., Bazana, P. G., & Stelmack, R. M. (2000). The

neuropsychology and psychophysiology of human intelligence. In R. J.

Sternberg (Ed.), Handbook of intelligence (pp. 245-264). New York, NY:

Cambridge University Press.

251

Vygotsky, L. S. (1981). The genesis of higher mental functions. In J. V. Wertsch

(Ed.), The concept of activity in Soviet psychology. Armonk,NY: Sharpe.

Vygotsky, L. S. (Ed.). (1978). Mind in society: The development of higher

psychological processes. Cambridge,MA: Harvard University Press.

Wechsler, D. (1991). Wechsler Intelligence Scale for Children (3rd ed.). San

Antonio, TX: Psychological Corporation.

Younger, M., & Warrington, M. (1996). Differential achievement of girls and boys

at GSCE: Some observations from the perspective of one school. British

Journal of Sociology of Education, 17(3), 299-313.

Zawojewski, J., Lesh, R., & English, L. (2003). A models and modeling

perspective on small group learning activity. In R. Lesh & H. Doerr (Eds.),

Beyond constructivism: Models and modeling perspectives on

mathematics problem solving, learning and teaching (pp. 337-358).

Mahwah; NJ: Lawrence Erlbaum.

252

Appendices

253

APPENDIX A

PROJECT INFORMATION SHEET AND PARENT CONSENT FORM FOR

COMPARISON GROUP

254

PARTICIPANT INFORMATION for QUT RESEARCH PROJECT

“A problem-posing intervention in the development of problem-solving

competence of underachieving, middle-year students.”

Researcher Contacts

Assistant Professor Deborah Priest Chief Investigator

Mobile Number xxxxxxxx Email xxxxxxx

Description This project is being undertaken as part of a Doctor of Philosophy (PhD) research study by Deborah Jean Priest. The purpose of this project is to investigate the issues surrounding the acquisition of problem-solving skills of middle-year students. In particular, it will look at the links between problem-posing skills development and improvements in problem-solving competence.

The data collection for this research project will occur in Term 4 of 2006, with selected students from the current Year 7 cohort, and again in Term 4 of 2007, with selected students from next years Year 7 cohort.

The students from the current Year 7 cohort will simply be asked to complete a 40 minute problem solving test, containing 6 questions, in week 1 of Term 4 and then the same test again at the end of week 8 of Term 4. The students from the current Year 7 cohort will represent the “comparison” group.

Next year, the selected students (current Year 6 students) will also undertake the same problem solving test in weeks 1 and 8 of term 4, but those students will receive a weekly problem-posing lesson, from the researcher, throughout Term 4 of 2007. The selected students from 2007 will represent the “intervention” group. The changes, if any, in problem-solving competence between the two groups participating in the two eight week periods, will then be compared to determine if the a problem-posing intervention has resulted in improved problem-solving competence.

255

Participation Your child has been selected to participate in the study as part of the “comparison” group.

The school has given the researcher permission to conduct this research and to approach you to request permission for your child to take part. Your child’s participation in this project is voluntary. If you do agree for your child to participate, your child can withdraw from participation at any time during the project without comment or penalty. Your decision for your child to participate will in no way impact upon their current or future relationship with QUT or with your child’s grades for school assessment.

Each student participating in this project will complete the test in class time, in a classroom at the school, and will be supervised by a teacher. The test has proven to be a valid and reliable measure of students’ problem-solving ability and looks at five aspects of problem solving: correctness of answer, method used, accuracy, extraction of information and quality of explanation. Please contact the researcher directly if you wish to ask any questions about the test. Expected benefits Results from this study will provide the researcher with crucial insights into how teachers and parents can assist their children to become more confident and competent problem solvers. Staff at the school have taken a keen interest in the project to date and will be reviewing the findings of the researcher.

Problem-solving competence is at the centre of contemporary curricula, including the International Baccalaureate Middle Years Program, which is currently being implemented at the school. Through this research project, the researcher hopes to demonstrate that particular forms of teaching intervention can be used by teachers to assist students to become more confident and competent at solving non-routine problems.

Risks There are no risks beyond normal day-to-day living associated with your child’s participation in this project. Confidentiality All comments and responses will be treated in strictest confidence. The names of individual students and their personal results in the testing will only be available to the researcher. The information obtained from the data will be discussed in the researcher’s thesis, but pseudonyms (replacement names) will be used to reference students and their results. This will ensure the anonymity of the participating students. All tests will be stored in a locked cabinet at the office of the researcher at the University and no other person, other than the researcher, will have access to the individual tests. Consent to Participate I would like to ask you and your child to sign the written consent form (enclosed) to confirm your agreement for your child to participate. Please ask your child to return the complete consent form to their class teacher as soon as possible before the end of Term 3.

256

Questions / further information about the project Please contact the researcher, Deborah Priest, to have any questions answered or if you require further information about the project. Please retain this sheet for future reference. Concerns / complaints regarding the conduct of the project QUT is committed to researcher integrity and the ethical conduct of research projects. However, if you do have any concerns or complaints about the ethical conduct of the project you may contact the QUT Research Ethics Officer on 3864 2340 or ethicscontact@qut.edu.au. The Researcher Ethics Officer is not connected with the research project and can facilitate a resolution to your concern in an impartial manner.

257

CONSENT FORM for QUT RESEARCH PROJECT

“A problem-posing intervention in the development of problem-solving

competence of underachieving, middle-year students.”

Statement of consent By signing below, you are indicating that you:

• have read and understood the information document regarding this project;

• have had any questions answered to your satisfaction;

• understand that if you have any additional questions you can contact the researcher;

• understand that your child is free to withdraw at any time, without comment or penalty;

• understand that you can contact the Research Ethics Officer on 3864 2340 or ethicscontact@qut.edu.au if you have concerns about the ethical conduct of the project;

• have discussed the project with your child;

• agree for your child to participate in the project. Parent/Guardian Name

Signature

Date / /

Statement of Child consent Your parent or guardian has given their permission for you to be involved in this research project. This form is to seek your agreement to be involved.

By signing below, you are indicating that the project has been discussed with you and you agree to participate in the project. Student’s Name

Signature

Date / /

258

APPENDIX B

PROJECT INFORMATION SHEET AND PARENT CONSENT FORM FOR 2007

INTERVENTION GROUP

259

2007 PARTICIPANT INFORMATION for QUT RESEARCH PROJECT

“A problem-posing intervention in the development of problem-solving

competence of underachieving, middle-year students.”

Researcher Contacts

Assistant Professor Deborah Priest Chief Investigator

Mobile Number xxxxxxxx Email xxxxxxx

Description This project is being undertaken as part of a Doctor of Philosophy (PhD) research study by Deborah Jean Priest. The purpose of this project is to investigate the issues surrounding the acquisition of problem solving skills of middle-year students. In particular, it will look at the links between problem-posing skills development and improvements in problem-solving competence.

The data collection for this research project began in Term 4 of 2006, with selected students from last years Year 7 cohort, and it will occur again in Term 4 of 2007, with selected students from this years Year 7 cohort.

The students from last year were simply asked to complete a 40 minute problem-solving test, containing 6 questions, in week 1 of Term 4 and then the same test again at the end of week 8 of Term 4. The students from the 2006 Year 7 cohort represent the “comparison” group.

This year, the selected Year 7 students will undertake the same problem-solving test in weeks 1 and 8 of term 4, but, in addition, will receive a weekly problem-posing lesson, from the researcher, throughout Term 4 of 2007. The selected students from 2007 will represent the “intervention” group. The changes, if any, in problem-solving competence between the two groups participating in the two eight week periods, will then be compared to determine if the a problem-posing intervention has resulted in improved problem-solving competence.

260

Participation Your child has been selected to participate in the study as part of the “intervention” group.

The school has given the researcher permission to conduct this research and to approach you to request permission for your child to take part. Your child’s participation in this project is voluntary. If you do agree for your child to participate, your child can withdraw from participation at any time during the project without comment or penalty. Your decision for your child to participate will in no way impact upon their current or future relationship with QUT or with your child’s grades for school assessment.

Each student participating in this project will complete the test in class time, in a classroom at the school, and will be supervised by a teacher. The test has proven to be a valid and reliable measure of students’ problem-solving ability and looks at five aspects of problem solving: correctness of answer, method used, accuracy, extraction of information and quality of explanation. Please contact the researcher directly if you wish to ask any questions about the test. Expected benefits Results from this study will provide the researcher with crucial insights into how teachers and parents can assist their children to become more confident and competent problem solvers. Staff at the school have taken a keen interest in the project to date and will be reviewing the findings of the researcher.

Problem-solving competence is at the centre of contemporary curricula, including the International Baccalaureate Middle Years Program, which is currently being implemented at the school. Through this research project, the researcher hopes to demonstrate that particular forms of teaching intervention can be used by teachers to assist students to become more confident and competent at solving non-routine problems.

Risks There are no risks beyond normal day-to-day living associated with your child’s participation in this project. Confidentiality All comments and responses will be treated in strictest confidence. The names of individual students and their personal results in the testing will only be available to the researcher. The information obtained from the data will be discussed in the researcher’s thesis, but pseudonyms (replacement names) will be used to reference students and their results. This will ensure the anonymity of the participating students. All tests will be stored in a locked cabinet at the office of the researcher at the University and no other person, other than the researcher, will have access to the individual tests. Consent to Participate I would like to ask you and your child to sign the written consent form (enclosed) to confirm your agreement for your child to participate. Please ask your child to return the complete consent form to their class teacher as soon as possible before the end of Term 3.

261

Questions / further information about the project Please contact the researcher, Deborah Priest, to have any questions answered or if you require further information about the project. Please retain this sheet for future reference. Concerns / complaints regarding the conduct of the project QUT is committed to researcher integrity and the ethical conduct of research projects. However, if you do have any concerns or complaints about the ethical conduct of the project you may contact the QUT Research Ethics Officer on 3864 2340 or ethicscontact@qut.edu.au. The Researcher Ethics Officer is not connected with the research project and can facilitate a resolution to your concern in an impartial manner.

262

2007 CONSENT FORM for QUT RESEARCH PROJECT

“A problem-posing intervention in the development of problem-solving

competence of underachieving, middle-year students.”

Statement of consent By signing below, you are indicating that you:

• have read and understood the information document regarding this project;

• have had any questions answered to your satisfaction;

• understand that if you have any additional questions you can contact the researcher;

• understand that your child is free to withdraw at any time, without comment or penalty;

• understand that you can contact the Research Ethics Officer on 3864 2340 or ethicscontact@qut.edu.au if you have concerns about the ethical conduct of the project;

• have discussed the project with your child;

• agree for your child to participate in the project. Parent/Guardian Name

Signature

Date / /

Statement of Child consent Your parent or guardian has given their permission for you to be involved in this research project. This form is to seek your agreement to be involved.

By signing below, you are indicating that the project has been discussed with you and you agree to participate in the project. Student’s Name

Signature

Date / /

263

APPENDIX C

STUDENT SURVEY SHEET

264

Student Survey Sheet:

Name: ____________________________ Date: ____________________

Question One:

Do you enjoy solving problems? Please explain your answer in a few sentences.

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

Question Two:

What type of problems do you prefer to solve? Please explain your answer in a

few sentences.

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

265

Question Three:

Do you think learning to solve problems is a useful thing to do? Please explain

your answer in a few sentences.

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

Question Four:

What things could teachers do to assist you to become better at solving

problems?

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

266

APPENDIX D

TEACHING EXPERIMENT LESSON ONE

267

TEACHING EXPERIMENT LESSON 1 – YEAR 7

Date: October 8, 2007 Topic: Student Survey and Pre-testing

Year Level: 7 Duration: 60 minutes

Rationale:

The purpose of this lesson is to allow the students to complete a survey containing 4 attitudinal questions and then a problem-solving pre-test, before they are involved in a teaching experiment during Term 4. Both the survey and the test are identical to the survey and test that the students will complete in their final session at the end of the teaching experiment. Students WILL NOT be made aware that they will be completing the same survey sheets or tests at the start and end of the teaching experiment. This is to ensure that as many variables as possible are controlled throughout this experiment.

The results from both the surveys and the tests will allow the researcher to determine if there are noteworthy changes and/or emergent themes in student attitudes to problem solving or their problem-solving competence, as a result of the students’ involvement in the teaching experiment.

Key Investigations:

Establish student attitudes and/or problem-solving competence before the students are involved in the teaching experiment.

268

THE MULTIPLE INTELLIGENCES CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

� Charts � graphs � diary √ visual metaphors

√ 3D

√ sketching

√ mind maps

� stories � retelling � book making � research � speeches √ reading

� drama

� activities � hands on � body language � crafts � drama � mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

√ coding

√ geometry

√ measuring

√ classifying

√ money

√ time

√ individual study

� journals √ reflection

� self-esteem activities √ individual reading

√ individual projects

� cooperative learning � sharing � group work � discussion � peer editing � brainstorming

269

Time Lesson Plan - Learning Activities Resources

5

mins

Introduce myself, the research I am undertaking, the students’ participation and the 8 week teaching experiment.

13

mins

Ensure students are well spread out around the room and that they each have a pen with which to write.

Hand out a survey sheet to each student and ask them to write their names on the top of the sheet and then complete the attitudinal surveys individually and in silence.

Collect the survey sheets.

One survey sheet

for each

participant. Spare

pens.

40

mins

Pre-tests to be handed out face down while students remain in silence. Inform students about the administration process of the test. Ask the students to write their names on the top of the front page of the

test and turn the first page over. Administer the test according to the requirements specified in the POPS

Administrator’s Handbook. Collect the test booklets from the students.

Pre-tests for each

student.

2

mins

Ask students to assist in returning the furniture in the room to the way it usually is.

Thank the students, remind the students to arrive on time next lesson, and allow students to return to their classes.

270

APPENDIX E

TEACHING EXPERIMENT LESSON TW O

271

TEACHING EXPERIMENT LESSON 2 – YEAR 7

Date: October 15, 2007 Topic: Introduction to Problem Posing

Year Level: 7 Duration: 60 minutes

Rationale:

The students begin the teaching experiment by considering a problem that is novel in nature but one to which they can relate. This problem does not have one “correct” answer, although it is likely that a number of students will choose to follow a similar path. The student’s justifications of their own answers will be a focus of the lesson. This problem has been chosen to begin the process of thinking “outside” the box and empowering and validating students to do so. It is also designed to dispel the notion that every problem has only one answer.

The lesson will continue with the students being asked to modify the problem to generate a different problem for their peers to solve. They will be reminded that it must be a problem that they can solve. This will begin the process of giving the students some autonomy in their learning as supported by the Critical Theory (Tierney, 1997).

Key Investigations:

• Is there always one correct answer to a problem? • Is one answer “better” than another? • How many problems can be generated from one base problem?

272

THE MULTIPLE INTELLIGENCES: CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

� charts � graphs � diary � visual metaphors � 3D √ sketching

√ mind maps

√ stories

√ retelling

� book making � research � speeches √ reading

√ drama

� activities √ hands on

√ body language

� crafts √ drama

� mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

� coding � geometry � measuring √ classifying

√ money

√ time

√ individual study

� journals √ reflection

√ self-esteem activities

√ individual reading

√ individual projects

√ cooperative learning

√ sharing

√ group work

√ discussion

√ peer editing

√ brainstorming

273

Time Lesson Plan - Learning Activities Resources

7 mins

Introduction.

Welcome students back and once again show appreciation for their involvement.

Hand out a new exercise book for each student and ask them to write their names on the front cover.

Remember to emphasise no summative assessment contribution for their school grades.

Remember to emphasise we are going to have loads of fun! Ask students if there are any questions.

One new exercise book for each student.

3 mins

Tell students they can work individually, in pairs or a small group for the first activity.

Ask students to move places if they need to, but remind them that they must all write their own solutions in their own exercise book.

Show students a variety of “concrete” resources at the front table if they want to use them at any time.

Bags of “matchsticks”, plastic discs, connectable blocks, small timber blocks, and fake coins

7 mins

Hand out the problem about Jane and Tom (see Problem for Lesson One on next page ).

Ask students to open their exercise books and write today’s date on the first page.

Ask students to read the problem, determine a solution, and write down their solution, including some notes to justify it, in their exercise books. I will go to each student and stick the problem into their exercise books (to save time).

as above +

glue stick, photocopies of Problem for Lesson One for each student

8 mins

Ask students to stop work. Ask students to write their final answer on the yellow post-it-notes, that I

handed out while they were working, and ask a member of each group, or individual, to stick the note on the whiteboard.

I will quickly sort the solutions to put those with the same answer together in a visual way on the board.

Ask students why there are different answers. (expect students to say incorrect process used, but hope for students to say different interpretations were made)

Ask students if there is a problem with having different answers. Ask for volunteers to justify their answers for the class (validate different

solutions that have appropriate, but alternate interpretations).

Pad of yellow post-it-notes

10 mins

Ask students to look again at the problem and ask them if they could modify it in some way to create a new problem that they could solve.

Ask students to write their new problem down, as well as the solution to it, stating any assumptions they made along the way.

All resources that were available at the start of the lesson

20 mins

Ask students to stop work. Ask students for volunteers to share their problems. (either they read, or a

friend or I read – their choice) Ask each volunteer to share their thinking that resulted in the new problem

being posed.

274

Repeat previous two steps for as long as time allows.

5

mins

Thank all students for their contributions of posed problems and collect exercise books.

Ask students what they think they have learned during the lesson and record them on the board (visual reinforcement).

Add any outcomes that they have missed. Tell students to look forward to some more creative fun next lesson and

then allow students to leave.

Problem for Lesson Two

Tom and Jane are brother and sister. They live next door to Mr and Mrs Lee who

are about to leave on a two week trip to China. Mr Lee has asked Tom and Jane

to look after Fifi (their cat) and Pluto (their dog) while they are away. Mrs Lee

has just given Tom and Jane’s parents $200 to share between the children

based on how much work they each contribute to the care of the pets.

Fifi needs combing every second day for 15 minutes and needs fresh food and

water daily.

Pluto needs walking every day for 30 minutes and also needs fresh food and

water daily.

If Jane looks after Pluto and Tom looks after Fifi, and they both do an equally

good job, how should Tom and Jane’s parents share the $200 between the

children? Remember to fully justify your solution stating clearly any assumptions

you have made.

275

APPENDIX F

TEACHING EXPERIMENT LESSON THREE

276

TEACHING EXPERIMENT LESSON 3 – YEAR 7

Date: October 29, 2007 Topic: Problem Posing

Year Level: 7 Duration: 60 minutes

Rationale:

The lesson will begin with a review of the previous lesson. In particular, we will discuss the following attributes of a “good problem”: interest factor, challenge level and “do-ability”. We will look at some problems and discuss whether they meet the criteria we have discussed. In particular, I want to draw the students’ attention to the different types of problems; such as long, wordy problems compared to pictorial or design problems and what makes each type challenging or interesting. This process will allow the students to decide on what types of problems they would like to create, and should liberate them from institutionalised thinking patterns.

I will then draw the students’ attention to the problems that they wrote last lesson and I will ask them to self-rate their problem. This is designed to develop some reflective and metacognitive skills which will inform their future problem posing. I will then tell the students that, in addition to the students’ self-ratings, I will also rate their problems from this week. I have chosen to do this because I want to maximise the motivation of the students during their problem posing. In the final week I will announce some awards; such as Most Entertaining Problem Poser, Most Improved Problem Poser; Best Overall Problem Poser.

Key Investigations:

• What constitutes a “good problem”? • Identify different types of problems.

277

THE MULTIPLE INTELLIGENCES: CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

� Charts � graphs � diary √ visual metaphors

√ 3D

√ sketching

√ mind maps

√ stories

√ retelling

� book making � research � speeches √ reading

√ drama

√ activities

√ hands on

√ body language

√ crafts

√ drama

� mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

√ coding

� geometry √ measuring

√ classifying

� money √ time

√ individual study

� journals √ reflection

√ self-esteem activities

√ individual reading

√ individual projects

√ cooperative learning

√ sharing

√ group work

√ discussion

√ peer editing

√ brainstorming

278

Time Lesson Plan - Learning Activities Resources

5 mins

Introduction.

Welcome students to class. Ask students what they liked and disliked about the last lesson. Introduce the 1,2,3 card system of peer feedback.

Video cameras in place and all resources laid out.

5 mins

Hand out exercise books to students. Discuss what attributes were displayed by the students’ problems from

last week. Reminding students about interest factor, challenge level and “do-ability”. Write these up on the board.

Ask students to open their books and look at the criteria rating sheet that I have stuck into each book. Tell students about the self-rating and teacher-rating that will occur for the remainder of the term.

Ask students to self-rate their problem from last week. Discuss briefly with the students and be positive about improved ratings

as improved problem-posing competence is achieved.

Student exercise books with self-rating criteria sheets already attached to the first weeks problem.

15 mins

Ask students if they would like to work individually, in pairs or a small group for the activity.

Ask students to move places if they need to, but remind them that they must all write their own solutions in their own exercise book.

Ask students to open their exercise books and write today’s date on the next new page.

Ask students to write a problem based on the stimuli provided, determine a solution, and write down their solution, and some notes to justify their solution, in their exercise books. Emphasise the attributes of a good problem that were discussed earlier.

Sets of stimuli on each table:

Mini packets of Skittles; set of three different sized bulldog clips; one new Post-it-note pad (enough for each student)

25 mins

Ask students to stop work. Ask students for volunteers to share their problems. (either they read, or a

friend or I read – their choice) Ask other students to “rate” the problem by holding either a 1, 2 or 3 card

in the air. Repeat previous two steps for as long as time allows. Ask students to self-rate their problems on the criteria sheets that have

been attached to their books before the lesson began.

Cards with 1, 2 and 3 written on them.

8 mins

Summarise the main points that arose from the lesson. Points to summarise may include imagination of the students, length of problems, similarity or difference of problems, or student engagement. Use language that will ensure the students leave the room feeling valued, empowered and heard.

Ask students what they think they have learned during the lesson.

2 mins

Thank all students for their contributions of posed problems and collect exercise books.

Remind students that I will rate their problems before next lesson. Tell students to look forward to some more creative fun next lesson and

then allow students to leave.

279

PHOTOGRAPH OF LESSON STIMULI PROVIDED TO EACH STUDENT

280

APPENDIX G

TEACHING EXPERIMENT LESSON FOUR

281

TEACHING EXPERIMENT LESSON 4 – YEAR 7

Date: October 30, 2007 Topic: Problem Posing

Year Level: 7 Duration: 60 minutes

Rationale:

The lesson will begin with a review of the previous lesson and a reminder of the self-rating and teacher-rating for the students’ problems. In particular, I will summarise my thoughts on their problems in general. This may draw the student’s attention to the attributes of a “good” problem that have been discussed over the last two lessons.

We will re-visit the notion of different types of problems as most students constructed similar problems during the last lesson. For this lesson I will provide an unexpected object for stimuli. I hope that students will think beyond a simple addition or multiplication type of problem to write. The students still seem to be thinking superficially and without imagination.

The lesson will continue with the students being asked to write a problem for their peers to solve that is based around the stimuli provided. I expect some students to be confounded by the stimuli, but I will assist them during the lessons with prompts and starting point ideas for them to develop.

Key Investigations:

• How can I write a problem about a seemingly unquantifiable object?

282

THE MULTIPLE INTELLIGENCES: CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

� Charts � graphs � diary √ visual metaphors

� 3D √ sketching

√ mind maps

√ stories

√ retelling

� book making √ research

� speeches √ reading

� drama

� activities � hands on √ body language

� crafts √ drama

� mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

√ coding

√ geometry

√ measuring

√ classifying

√ money

√ time

√ individual study

� journals √ reflection

√ self-esteem activities

√ individual reading

√ individual projects

√ cooperative learning

√ sharing

√ group work

√ discussion

√ peer editing

√ brainstorming

283

Time Lesson Plan - Learning Activities Resources

3 mins

Introduction.

Welcome students to class. (Exercise books, scoring cards and photographs to be on desks before students enter the room)

Briefly discuss the problems written from the last lesson. Give students a few minutes to read the comments I have written about

their problems and ask any questions they may have.

Room to be prepared with video cameras in place and all resources laid out.

7 mins

Ask students to look at the sheet of three questions. Discuss with students about the attributes of each problem and what

makes them interesting, challenging and do-able, or not. Draw students’ attentions to the fact that the final problem is do-able if the

original painting were available, however, discuss ways of getting an answer that may be close to the real answer.

A sheet with 3 questions on and an answer sheet for each student.

10 mins

Discuss today’s stimuli asking students what their initial thoughts are. Ask students to open their exercise books and write today’s date on the

next new page. Ask students to write a problem, determine a solution, and write down

their solution, and some notes to justify their solution, in their exercise books.

Emphasise the attributes of a good problem that were discussed earlier.

Student exercise books with rating criteria sheets already attached to the first weeks problem. Photograph of some grass.

30 mins

Ask students to stop work. Ask for volunteers to share their problems. (either they read, or a friend, or

I read – their choice) Ask other students to rate the problems with the 1, 2 or 3 card system. Repeat previous two steps for as long as time allows. Ask students to self-rate their problems on the criteria sheets that have

been attached to their books before the lesson began.

Cards with 1, 2 and 3 written on them.

8 mins

Summarise the main points that arose from the lesson. Points to summarise may include imagination of the students, length of problems, similarity or difference of problems, or student engagement. Use language that will ensure the students leave the room feeling valued, empowered and heard.

Ask students what they think they have learned during the lesson.

2 mins

Thank all students for their contributions of posed problems and collect exercise books.

Remind students that I will rate their problems before next lesson. Tell students to look forward to some more creative fun next lesson and

then allow students to leave.

284

PROBLEM 7.1

CONSIDER THE INFORMATION GIVEN BELOW TO WORK OUT THE NUMBER OF JELLY BEANS

IN A FULL JAR.

PROBLEM 7.2

THE FIGURE BELOW IS MADE UP OF 5 CONGRUENT SQUARES. THE PERIMETER OF THE

FIGURE IS 72 CM. FIND THE NUMBER OF SQUARE CM IN THE AREA OF THE FIGURE.

285

PROBLEM 7.3

TO THE BEST OF YOUR ABILITY DETERMINE HOW MANY DOTS MAKE UP THIS PICTURE.

286

PROBLEM 7.1 SOLUTION

LET THE NUMBER OF JELLY BEANS IN A FULL JAR BE N. SINCE THERE ARE 7 JARS ON

EACH SIDE OF THE SCALES WE CAN CONSIDER ONLY THE CONTENTS OF THE JARS OR

LEFTOVER JELLY BEANS.

6N + ¼ N = 375

6.25N = 375

N = 375/6.25

N= 60

THERE ARE 60 JELLY BEANS IN A FULL JAR.

PROBLEM 7.2 SOLUTION

SINCE THIS SHAPE IS MADE UP OF 5 CONGRUENT (SAME) SQUARES, THEN WE CAN

CONSIDER EACH EDGE TO BE OF EQUAL LENGTH. LET THE LENGTH OF ONE EDGE BE L.

TOTAL NUMBER OF OUTSIDE EDGES IS 12.

TOTAL PERIMETER = 72CM.

12L = 72

=> L= 6CM

FIVE SQUARES MAKE UP THE SHAPE. EACH SQUARE HAS AN AREA OF LXL= 6X6 CM2

= 36 CM2

THE AREA OF THE ENTIRE SHAPE = 5X36 CM2

= 180 CM2

PROBLEM 7.3 SOLUTION

? - Discuss

287

LESSON 3 STIMULI

288

APPENDIX H

TEACHING EXPERIMENT LESSON F IVE

289

TEACHING EXPERIMENT LESSON 5 – YEAR 7

Date: November 5, 2007 Topic: Problem Posing

Year Level: 7 Duration: 60 minutes

Rationale:

The lesson will begin by providing students with the opportunity to read the comments that I wrote in their books from last lessons problem. I will summarise my thoughts on their problems in general and make mention of some students who have improved or whose problem had addressed the attributes of a “good” problem. This is important to make a link between last weeks lesson and where we are headed today.

I have written a problem based on the photo of grass from last lesson. I want the students to read it and deconstruct it in order to get some additional ideas about how to create an interesting story line and embed the information for solving the problem. I also want them to experience what looks like a formidable problem but in actual fact is very straight forward and “do-able”.

The lesson will continue with the students being asked to write a problem, for their peers to solve, that is based around the three shapes I have glued in the students’ books. Once again, some students may be unsure of how to start, but I will assist them during the lessons with prompts and starting point ideas for them to develop.

This week I will ask students to use the 1, 2, 3 card system three times to rate their peer’s (volunteers) problems in each of the three separate attributes of do-ability, challenge and interest factor. I hope this will provide students with some immediate, yet more specific, feedback from their peers. Last week I felt the students where getting a little restless, so each week I will ask for volunteers and try to share the opportunities across all students in the remainder of the teaching experiment and encourage those who may be less willing to volunteer.

Key Investigations:

• What makes a problem look difficult? • What makes a problem interesting to read and do? • How can students make their problems “do-able” yet challenging?

290

THE MULTIPLE INTELLIGENCES: CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

� charts � graphs � diary √ visual metaphors

� 3D √ sketching

√ mind maps

√ stories

√ retelling

� book making � research � speeches √ reading

� drama

� activities √ hands on

√ body language

� crafts √ drama

� mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

√ coding

√ geometry

√ measuring

√ classifying

� money √ time

√ individual study

� journals √ reflection

√ self-esteem activities

√ individual reading

√ individual projects

√ cooperative learning

√ sharing

√ group work

√ discussion

√ peer editing

√ brainstorming

291

Time Lesson Plan - Learning Activities Resources

3 mins

Introduction.

Welcome students to class. (Exercise books, scoring cards and my problem to be on desks before students enter the room)

Briefly discuss the problems written from the last lesson and mention notable problems and posers.

Give students a few minutes to read the comments I have written about their problems.

Room to be prepared with video cameras in place and all resources laid out.

10 mins

Give students a problem that I wrote about last weeks stimuli and have students deconstruct it and then try to solve it.

Ask for two volunteers to use the whiteboard to share their solutions. Write up other students’ final answers and then discuss.

Consider Mrs Priest’s solution and discuss the setting out example and the importance of good quality written communication.

Ask students to place both their completed worksheet and my printed solution in the back of their books so that I may glue them in before next lesson.

Researcher problem on grass and the associated worksheet. (see below)

20 mins

Direct students’ attention to the three shapes I have glued in their books (triangle, square and circle).

Ask students to write a problem, determine a solution, and write down their solution, in their exercise books. Remind students to think of the three attributes of a good problem and the need to communicate their answers more effectively.

Student exercise books with rating criteria sheets and printout of 3 shapes triangle, square and circle already attached.

20 mins

Ask students to stop work. Ask students for volunteers to share their problems. This week we will focus on each attribute of a quality problem individually.

I will ask other students to rate the problems with the 1, 2 or 3 card system for interest factor, then do-ability, and then challenge.

Repeat previous two steps for as long as time allows.

Cards with 1, 2 and 3 written on them.

5 mins

Summarise the main points that arose from the lesson. Ask students to self-rate their problems on the criteria sheets that have

been attached to their books before the lesson began.

2 mins

Thank all students for their contributions of posed problems and collect exercise books.

Remind students that I will rate their problems before next lesson.

292

Lesson 4 Stimuli

293

Researcher Problem (stimuli – photograph of grass)

Mrs Priest noticed brown patches appearing in her lawn. According to Mr White, her neighbour, the rain has brought lawn grubs to a lot of lawns in the area and the lawn will need to be sprayed. Mrs Priest rang a man at a lawn maintenance company and asked him how much it would cost to remove the grubs. The man offered to give Mrs Priest a quote over the phone and asked her what size lawn she had. She replied, “it is twice as long as it is wide and it takes 24 of my paces to walk the full length”. The man on the phone then asked Mrs Priest how long each of her paces was. She excitedly replied “I have just got a new pedometer and had to calculate that! My pace is about ¾ of a metre”. If the cost of lawn grub removal is $5.20/m2 what will the total cost of removing the lawn grubs be if the man does his maths correctly?

What makes this problem challenging? (Challenge)

Write down the facts that will help you solve this problem. (Do-ability)

What makes the problem interesting? (Interest Factor)

Write your solution below:

294

POSSIBLE SOLUTION:

Length in paces: 24 paces

Length in metres: 24 x ¾ = 18m

Width: ½ of 18m = 9m

Area of lawns = LxW

= 18 x 9 m2

= 162 m2

Cost to remove grubs = Area x cost/ m2

= 162 x 5.20

= $842.40

The total cost to remove the lawn grubs is $842.40.

295

APPENDIX I

TEACHING EXPERIMENT LESSON S IX

296

TEACHING EXPERIMENT LESSON 6 – YEAR 7

Date: November 12, 2007 Topic: Problem Posing

Year Level: 7 Duration: 60 minutes

Rationale:

To start this lesson I have written a very short problem based on the three shapes from last lesson. I want the students to read it and consider it in terms of the three attributes of a “good” problem. Last week I asked the students to look at a long, wordy problem and this problem is designed to provide a contrast so that students may become open to cleverly designed, short problems. It will be worth discussing that the interest factor may be minimised by posing a short problem.

The lesson will continue with the students being asked to write a problem for their peers to solve that is based around the pictures that I have put on the desks. The pictures are of a skateboard cartoon and an iPod. I chose these objects as they are meaningful artefacts for the students and most of them will have had first-hand experience with either or both objects.

This week I will again ask students to use the 1, 2, 3 card system to rate their peer’s problems in each of the three separate attributes of do-ability, challenge and interest factor. The immediate, yet more specific, feedback from their peers appeared to be appreciated last week. I will again ask for volunteers to read their problems (different to those who read their problem out last week).

Key Investigations:

• Can a short problem, even a one-line problem, be challenging, interesting and do-able all at the same time?

297

THE MULTIPLE INTELLIGENCES: CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

� charts � graphs � diary √ visual metaphors

� 3D √ sketching

√ mind maps

√ stories

√ retelling

� book making � research � speeches √ reading

� drama

� activities √ hands on

√ body language

� crafts √ drama

� mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

� coding � geometry � measuring √ classifying

√ money

√ time

√ individual study

� journals √ reflection

√ self-esteem activities

√ individual reading

√ individual projects

√ cooperative learning

√ sharing

√ group work

√ discussion

√ peer editing

√ brainstorming

298

Time Lesson Plan - Learning Activities Resources

3 mins

Introduction.

Welcome students to class. (Exercise books, scoring cards and my problem to be on desks before students enter the room)

Briefly discuss the problems written from the last lesson and mention notable problems and posers.

Give students a few minutes to read the comments researcher has written about their problems.

Room to be prepared with video cameras in place and all resources laid out.

10 mins

Give students a problem that I wrote about last weeks stimuli and have students deconstruct it and then try to solve it.

Ask for two volunteers to use the whiteboard to share their solutions. Write up other students’ final answers and then discuss.

Consider researcher’s solution and discuss the setting out example and the importance of communication.

Ask students to place both their completed worksheet and my printed solution in the back of their books so that I may glue them in before next lesson.

Researcher’s problem on the triangle, square and circle and the associated worksheet. (see below)

20 mins

Direct students’ attention to the two pictures I have placed on their desks of a skateboard cartoon and an iPod.

Ask students to write a problem, determine a solution, and write down their solution, in their exercise books.

Remind students to think of the three attributes of a good problem. Emphasise the need to communicate their answers more effectively in a

written format when they have completed posing their problem.

Student exercise books with rating criteria sheets already attached. Pictures of a skateboard cartoon and an iPod. (5 sets)

20 mins

Ask students to stop work. Ask students for volunteers to share their problems (different students

from last week). Again this week we will focus on each attribute of a quality problem

individually. I will ask other students to rate the problems with the 1, 2 or 3 card system for interest factor, do-ability and challenge.

Repeat previous two steps for as long as time allows.

Cards with 1, 2 and 3 written on them.

5 mins

Summarise the main points that arose from the lesson. Ask students to self-rate their problems on the criteria sheets that have

been attached to their books before the lesson began.

2 mins

Thank all students for their contributions of posed problems and collect exercise books.

Remind students that I will rate their problems before next lesson and that I look forward to providing them with some feedback.

REMIND STUDENTS ABOUT THE ROOM CHANGE NEXT LESON DUE TO THE STOCKTAKE IN THE LIBRARY 19/11.

Room booking for Ray 2.2

299

Researcher’s Problem (stimuli – THREE SHAPES)

If I cut the largest circle out of a square with area 36 m2, what percentage of the square remains?

What makes this problem challenging? (Challenge)

Write down the facts that will help you solve this problem. (Do-ability)

What makes the problem interesting? (Interest Factor)

Write your solution below:

300

SOLUTION (with example of setting out)

Let side length of square = S

Area of square = SxS

SxS = 36 m2

S=6m

Radius of square = 3m � 6m �

Area of circle = ∏ r2

= 28.27 m2

Area remaining = 36 – 28.27

= 7.72 m2

% area remaining = 7.72 x 100

36

=21.46%

301

Lesson 6 Stimuli

OR

302

APPENDIX J

TEACHING EXPERIMENT LESSON SEVEN

303

TEACHING EXPERIMENT LESSON 7 – YEAR 7

Date: November 19, 2007 Topic: Problem Posing

Year Level: 7 Duration: 60 minutes

Rationale:

I noticed last week that some students were starting to become a little restless after either finishing their problem posing early or simply with the sameness of each lessons format. This week I am changing the lesson and will need the students to move around the room. As with all teaching, lesson delivery needs to respond to students and their needs. I sense a need for change and will provide it with an opportunity to work quite differently. This week we have also been moved to a different classroom, so it is an ideal time for this weeks activity.

The lesson will begin with the students being asked to consider some quizzles on the board (see examples at the end of the lesson plan). In my experience, students are highly motivated to solve this type of visual problem. To further encourage students to make their own visual puzzles and to allow the students to move around the room, I will show them 16 visual picture puzzles that have been created by some Year 7 students I taught a number of years ago. The puzzles will be placed around the room so students can move freely and come back to a problem at any time. We will review the intended answers and consider the validity of any other answers that students may have come up with.

In the second half of the lesson I will ask the students to begin thinking of a visual, 3D puzzle of their own. They can brainstorm with me, their neighbour or in small groups. Each student or group of students will need to create a problem of their own for the class to do next week. This public, visual sharing will allow students to use their own initiative, creativity and interest area which is consistent with Critical Theory. I hope the students will find this activity to be an exciting way to round-off the teaching experiment.

Key Investigations:

• Can visual problems have the three attributes of a good problem? If so, what do they “look” like?

304

THE MULTIPLE INTELLIGENCES: CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

√ charts

� graphs � diary √ visual metaphors

� 3D √ sketching

√ mind maps

√ stories

√ retelling

� book making √ research

� speeches √ reading

� drama

√ activities

√ hands on

√ body language

√ crafts

√ drama

� mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

√ coding

� geometry � measuring √ classifying

� money √ time

√ individual study

� journals √ reflection

√ self-esteem activities

√ individual reading

√ individual projects

√ cooperative learning

√ sharing

√ group work

√ discussion

√ peer editing

√ brainstorming

305

Time Lesson Plan - Learning Activities Resources

3 mins

Introduction.

Welcome students to class. Briefly discuss the problems written from the last lesson and mention

notable problems and problem posers. Give students a few minutes to read the comments I have written about

their problems.

Video cameras in place, all resources laid out. 3D puzzles laid out in order on tables behind the students’ desks.

10 mins

Talk to the students about visual problems such as quizzles. Ask students how these problems might rate with the three attributes of a “good” problem. Discuss which attribute they feel is particularly served well by the visual presentation.

Give students an opportunity to do 8 quizzles from the board. Discuss the answers and the strategies the students used to solve them.

Selection of quizzles.

20 mins

Direct students’ attention to the 3D, visual puzzles that are laid out on the desks behind them and the answer sheet stuck in their exercise books.

Ask students to move along the line of puzzles and record answers to them in their books. They may go back to any, do them in any order, or change their answer if they need to.

Student exercise books, answer sheets attached. Set of 16 x 3D visual puzzles.

20 mins

Ask students to stop work and return to their seats. Ask students to share their answers to each puzzle as I hold them up one

by one. Discuss what made the problem obvious or difficult, or what caught students’ attention and focus.

Ask students to write a final comment about their enjoyment, or otherwise, of this activity on the bottom of their answer sheets.

5 mins

Ask students to begin thinking about their own visual problem. Tell students they will each need to bring a constructed problem to next weeks lesson for everyone to share and attempt to solve.

Encourage students to talk to other students, their parents or friends in order to get some really interesting ideas that will satisfy the three attributes of a “good” problem.

2 mins

Thank all students for their contributions of posed problems and collect exercise books.

Remind students that they should all try to be on time for the session.

306

Some examples of quizzles:

Solution: ‘Once upon a time’

Solution: ‘A balanced meal’

Solution: ‘Makeup’

Once

____

Time

P

U

E

K

A

M

Meal

307

3-D V ISUAL PROBLEMS STUDENT ANSW ERS

1. ______________________________________________________________

2. ______________________________________________________________

3. ______________________________________________________________

4. ______________________________________________________________

5. ______________________________________________________________

6. ______________________________________________________________

7. ______________________________________________________________

8. ______________________________________________________________

9. ______________________________________________________________

10. _____________________________________________________

11. _____________________________________________________

12. _____________________________________________________

13. _____________________________________________________

14. _____________________________________________________

15. _____________________________________________________

16. _____________________________________________________

308

3-D Visual Problems Correct Answers

1. Tulip

2. Chain Letter

3. Apricot

4. Middle Man

5. Phone Number

6. Zebra Crossing

7. Address

8. Monkey

9. Cross-country

10. Six Feet Under

11. Centigrade

12. Sticks and stones can break my bones

13. Accountant

14. Time flies when you are having fun

15. Bibliography

16. Abandon Ship

309

APPENDIX K

TEACHING EXPERIMENT LESSON E IGHT

310

TEACHING EXPERIMENT LESSON 8 – YEAR 7

Date: November 26, 2007 Topic: Pictorial Problem Posing

Year Level: 7 Duration: 60 minutes

Rationale:

This lesson is the final lesson of the teaching experiment before the students undertake their post-test and final attitudinal survey. It will close with the presentation of awards for the Most Improved, Most Entertaining and Best Overall problem posers, as well as awards for the best 3D puzzles as judged by the students themselves. The students should be highly motivated during this lesson as a result of these culminating activities.

In previous weeks, students have only been able to share their problems by reading them out. Written problems are often best read and re-read rather than just heard. Some students have told me that they felt other students would have liked their problems more if they had more time to read them. With time being limited for each teaching episode, I needed to find an alternative format for problem posing that would allow all of the students to share each others problems within a teaching-episode (60 minutes) timeframe. Hence, last week I introduced visual problems to the students and gave them an opportunity to investigate what makes them interesting and challenging, but still do-able. The students enjoyed the session enormously (as can be seen from the comments in their work books). For example,

Oliver wrote, “It was fun and very challenging. It was a lot better than normal work.”

Hayley wrote, “I thought it was a great idea and it was so much fun.”

Felicia wrote, “I thought this lesson was so much fun! I think it was the best lesson so far! It was so cool!”

The students will have a chance to pose and present their own problems for each other during this lesson.

Key Investigations:

• Students will investigate if they can create problems for their peers that are interesting, challenging, and do-able.

• They will investigate other student’s 3D problems.

311

THE MULTIPLE INTELLIGENCES: CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

� charts � graphs � diary √ visual metaphors

� 3D √ sketching

√ mind maps

√ stories

√ retelling

� book making � research � speeches √ reading

√ drama

√ activities

√ hands on

√ body language

√ crafts

√ drama

� mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

√ coding

� geometry √ measuring

√ classifying

� money √ time

√ individual study

� journals √ reflection

√ self-esteem activities

√ individual reading

√ individual projects

√ cooperative learning

√ sharing

√ group work

√ discussion

√ peer editing

√ brainstorming

312

Time Lesson Plan - Learning Activities Resources

5 mins

Introduction.

Welcome students to class. (Exercise books to be laid out on front desk before students enter the room)

Ask students to place their 3D problem on top of one of the sequentially numbered pieces of paper that have been set out on the row of desks.

Settle students onto the chairs at the front of the room. Briefly remind the students of what is going to happen in this lesson

including the viewing of their 3D problems.

Video cameras in place and all resources laid out. Sheets with 1, 2, 3,…15 to be laid across 15 desks in the room ready for student puzzles.

15 mins

Remind the students about the three attributes of a “good” problem. Ask the students to consider this for each problem as they attempt to work out the answers of each others problems.

Hand out student’s exercise books and ask students to open to the answer sheet that has already been glued in and then to move to the puzzles and attempt to “solve” them. They may go back to any, do them in any order, or change their answer if they need to.

Student puzzles and student exercise books with answer sheets already attached.

15 mins

Ask students to stop work and return to their seats at the front of the room. Ask the students to turn their chairs around so they are facing the desks on which the puzzles are placed.

Ask students to share their answers to each puzzle as I hold the puzzles up one by one. Discuss what made the problem obvious or difficult or what caught students’ attention and focus. If the puzzle was not able to be solved, ask the author of the problem to share the answer with the group.

Ask students to comment on their thoughts about the problems; in particular, what made them interesting, challenging and do-able.

18 mins

Tell students that their puzzles are now going to be peer assessed. All students will leave the room and collect a set of 1, 2, and 3 cards from the desk outside of the classroom. Students will be asked to enter the classroom one by one to place their voting cards on the problems they thought were the best. 3 for the best problem, 2 for the next best problem and 1 for the third best problem.

Students remaining outside of the classroom will not be able to see on which puzzles the students place their cards as the blinds will be drawn in the classroom. A research assistant will remain in the room to ensure that no students remove/change other students’ cards. I will remain outside of the classroom with the students to ensure they remain orderly and quiet while the voting takes place inside the classroom.

When the voting is completed, all students will re-enter the room to witness the tallying. A student volunteer to write scores on the board.

Sets of three cards with 1, 2 and 3 written on them. One set for each student.

5 mins

Announce the “best”, second “best” and third “best” problems as reviewed by the student peers.

Announce the three major awards; Most Improved Problem Poser, Most Entertaining Problem Poser and Best Overall Problem Poser as reviewed by me and hand out certificates.

Certificates for student winners.

2 mins

Collect exercise books, thank students for their puzzles and input today. Have students help to quickly rearrange the room.

313

3-D Visual Problems Student Answers

1. ______________________________________________________

2. ______________________________________________________

3. ______________________________________________________

4. ______________________________________________________

5. ______________________________________________________

6. ______________________________________________________

7. ______________________________________________________

8. ______________________________________________________

9. ______________________________________________________

10. -______________________________________________________

11. _____________________________________________________

12. _____________________________________________________

13. _____________________________________________________

14. _____________________________________________________

15. _____________________________________________________

314

APPENDIX L

TEACHING EXPERIMENT LESSON N INE

315

Date: November 26, 2007 Topic: Student Survey and Post-testing

Year Level: 7 Duration: 60 minutes

Rationale:

The purpose of this lesson is to allow the students to complete a survey containing 4 attitudinal questions and then a problem-solving post-test, following the completed teaching experiment. Both the survey and the test are identical to the survey and test that the students completed in their first session at the start of the teaching experiment. The results from both of these items will allow the researcher to determine if there are noteworthy changes and/or emergent themes in student attitudes to problem solving or their problem-solving competence, as a result of the PPI.

Key Investigations:

• Have student attitudes and/or problem-solving competence changed as a result of the teaching experiment in which the students participated?

THE MULTIPLE INTELLIGENCES: CHECKLIST

Visual/Spatial Verbal/Linguistic Bodily/Kinaesthetic

� Charts � graphs � diary √ visual metaphors

√ 3D

√ sketching

√ mind maps

� stories � retelling � book making � research � speeches √ reading

� drama

� activities � hands on � body language � crafts � drama � mime � PE

Logical/Math Intrapersonal Interpersonal

√ timeline

√ coding

√ geometry

√ measuring

√ individual study

� journals √ reflection

� self-esteem activities √ individual reading

� cooperative learning � sharing � group work � discussion � peer editing � brainstorming

316

√ classifying

√ money

√ time

√ individual projects

Time Lesson Plan - Learning Activities Resources

15 mins

Welcome students back and tell them about the significance of this lesson.

Ensure students are well spread out around the room and that they each have a pen with which to write.

Hand out a survey sheet to each student and ask them to write their names on the top of the sheet and then complete the attitudinal surveys individually and in silence.

Collect the survey sheets.

One survey sheet for each participant. Spare pens.

40

Mins

Post-tests to be handed out face down while students remain in silence. Remind students about the administration process of the test. Ask the students to write their names on the top of the front page of the

test and turn the first page over. Administer the test according to the requirements specified in the POPS

Administrator’s Handbook. Collect the test booklets from the students.

A set of post-tests for the group and a POPS Administrator’s Handbook.

5 mins

Ask students to assist in returning the furniture in the room to the way it usually is.

Thank the students for their valued input, participation and enthusiasm throughout the teaching experiment and allow students to return to their classes.

317

APPENDIX M

PROFILES OF PROBLEM SOLVING ASSESSMENT INSTRUMENT

318

319

320

321

322

323

324

325

326

327

APPENDIX N

PROBLEM CRITERIA SHEET

328

PROBLEM CRITERIA SHEET

DATE :

Interest Factor Challenge Level “Do-ability” Comments

Self -

rating

□ Very entertaining,

original and everyone wanted to solve it (3 points)

□ At least half the

class seemed interested and wanted to solve it (2 points)

□ Not many people

seemed interested in solving my problem or it was not original (1 point)

□ Not too hard and not

too easy. More than 3 steps had to be taken in order to solve the problem. (3 points)

□ Some felt it was too

hard or too easy. At least 2 steps had to be taken in order to solve the problem. (2 points)

□ Very little challenge

for most Year 7 students. (1 point)

□ Problem was very

well presented and clear. All information required to solve the problem was provided and there was no ambiguity. (3 points)

□ Most students felt the

problem was clear and felt they could solve it without extra information or clarification (2 points)

□ The problem was

confusing and not many students felt confident they could do it without more help or information. (1 point)

Self-rating: /9 points

Teacher

-rating

□ Very entertaining,

original and everyone wanted to solve it (3 points)

□ At least half the

class seemed interested and wanted to solve it (2 points)

□ Not many people

seemed interested in solving my problem or it was not original (1 point)

□ Not too hard and not

too easy. More than 3 steps had to be taken in order to solve the problem. (3 points)

□ Some felt it was too

hard or too easy. At least 2 steps had to be taken in order to solve the problem. (2 points)

□ Very little challenge

for most Year 7 students. (1 point)

□ Problem was very

well presented and clear. All information required to solve the problem was provided and there was no ambiguity. (3 points)

□ Most students felt the

problem was clear and felt they could solve it without extra information or clarification (2 points)

□ The problem was

confusing and not many students felt confident they could do it without more help or information. (1 point)

Teacher-rating: /9 points

329

APPENDIX O

PARTICIPANT PSEUDONYM CODE TO PSEUDONYM NAME CONVERSION FOR

COMPARISON GROUP

330

COMPARISON GROUP

Pseudonym Code Pseudonym Name

ao6 Adam

b06 Ben

c06 Clare

d06 Diane

e06 Ellen

f06 Fay

g06 Gayle

h06 Helen

i06 Imogen

j06 Jack

k06 Kyle

l06 Laura

m06 Matt

n06 Nola

o06 Olive

p06 Penny

331

APPENDIX P

PARTICIPANT PSEUDONYM CODE TO PSEUDONYM NAME CONVERSION FOR

2007 INTERVENTION GROUP

332

2007 INTERVENTION GROUP

* Please note that data from students I07, M07, and Q07 were excluded from this study as these students

were absent for more than two of the seven teaching episodes.

Pseudonym Code Pseudonym Name

A07 Andrew

B07 Blair

C07 Courtney

D07 Danielle

E07 Ethan

F07 Felicia

G07 Georgia

H07 Hayley

J07 Joanne

K07 Kelly

L07 Leah

N07 Nicole

O07 Oliver

P07 Paul

R07 Rodney

333

APPENDIX Q

MARKING SCHEME FOR THE PROFILES OF PROBLEM SOLVING TEST

334

335

336

337

338

339