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TEACHERS DEVELOPING COMMUNITIES OF 
MATHEMATICAL INQUIRY 
A DISSERTATION PRESENTED IN PARTIAL FULFILMENT OF THE 
REQUIREMENTS FOR THE DEGREE OF 
DOCTOR OF PHILOSOPHY IN EDUCATION 
AT MASSEY UNIVERSITY, AUCKLAND, 
NEW ZEALAND 
ROBERTA KATHLEEN HUNTER 
2007 
ABSTRACT 
This study explores how teachers develop communities of mathematical inquiry which 
faci l i tate student access to, and use of, proficient mathematical practices as reasoned 
col lective activity. Under consideration are the pathways teachers take to change classroom 
communication and participation patterns and the mathematical practices which emerge and 
evolve, as a result. 
Sociocul tural theories of learning underpin the focus of the study. A synthesis  of the 
literature reveals the importance of considering the social and cultural nature of students' 
learning and doing mathematics in  intel lectual learning communities-communities i n  
which shared intel lectual space creates many potential learning situations. 
A col laborative classroom-based quali tative approach-design research-fal ls natural ly  
from the sociocultural frame taken i n  the study. The design approach supported 
construction of a communication and partic ipation framework used to map out pathways to 
constitute i nquiry communities. Study group meetings, participant and video observations, 
interviews, and teacher recorded reflections in three phases over one year supported data 
col lection. Retrospective data analysis used a grounded approach and sociocultural activity 
theory to present the results as two teacher case studies. 
Managing the complexities and chal lenges inherent in constituti ng communication and 
participation patterns each teacher in this study successful ly  developed communities of 
mathematical inquiry within their own classrooms. Important tools that the teachers used to 
mediate gradual transformation of classroom communication  and participation patterns 
from those of conventional learning situations included the communication and 
participation framework and the questions and prompts framework. 
Signifi cant changes were revealed as the teachers enacted progressive shifts in the 
sociocultural and mathematical norms which validated col lective inquiry and 
argumentation as learning tools .  Higher levels of student i nvolvement in mathematical 
dialogue resulted in i ncreased i ntel lectual agency and verbal i sed reasoning. Mathematical 
practices were shown to be in terrelated social practices which evolved within reasoned 
discourse. 
The research findings provide insights i nto ways teachers can be assisted to develop a range 
of pedagogical practices which support the constitution of i nquiry communities. For New 
Zealand teachers, in particular, models for ways teachers can draw on and use their Maori 
and Pasifika students' ethnic social i sation to constitute mathematical i nquiry communities 
are represented in the case study exemplars. 
i i  
ACKNOWLEDGEMENTS 
I would l ike to acknowledge and thank the many people who made this study possible. 
Most importantly I want to thank the teachers who so wil l ingly allowed me to enter their 
world and journey with them as they constructed communi ties of mathematical inquiry. 
The end less time they wil l ingly gave to reflect on their journey, their abi lity to openly 
grapple with change and continue with the journey, and the excitement they expressed as 
their journey progressed was a source of strength which sustained my own journey. I would 
also like to thank the students who allowed me to become a member of their whanau and 
who eagerly developed their own 'voice' and found enjoyment in mathematics in their 
c lassroom mathematics communities. 
I wish to acknowledge and thank Associate Professor Glenda Anthony and Dr. Margaret 
Walshaw my supervisors . They patiently supported me as I took my own circuitous route in 
the development and writing of this study whi le al l  the time providing me with positive and 
invaluable guidance. Their depth of knowledge of ' al l  things mathematics' , their 
questioning and prompts which caused me to reconsider my position, and their guidance 
and wil lingness to step back to provide space for my own development has been a gift. 
Thank you. 
Final ly, I must acknowledge al l the members of my family  who each in their own way have 
supported me and made this  study possible.  
E rima te ' arapaki, te aro ' a, te ko' uko 'u  te utuutu, ' iaku nei .  
Under the protection of caring hands there ' s  feel ing of love and affection . 
Ill 
TABLE OF CONTENTS 
ABSTRACT 
ACKNOWLEDGEMENTS 
TABLE OF CONTENTS 
LIST OF TABLES 
CHAPTER 1 :  INTRODUCTION 
1 . 1  Introduction 
1 .2 Research aim 
1 .3 Background context of the study 
1 .4 Rationale for the study 
1.5 Overview of the thesis 
CHAPTER 2: THE BACKGROUND TO THE STUDY 
2 . 1 
2 .2  
2 .3  
2 .4  
Introduction 
Mathematical practices 
2 .2 . 1  Mathematical practices as reasoned col lective activity 
2 .2 .2  The discourse of communities of  mathematical i nquiry 
Sociocultural 1earning perspectives and activity theory 
2 . 3 . 1 Communicative i nteraction and the mediation of mathematical 
practices 
2 .3 .2 Zones of proximal development 
Communities of mathematical i nquiry 
2 .4. 1 The communication and participation patterns of communities 
11 
lll 
IV 
X 
2 
2 
5 
7 
8 
8 
9 
1 1  
1 2  
1 4  
1 4  
1 7  
2 1  
of mathematical i nquiry 23 
IV 
2 .4.2 Socio-cultural and mathematical norms 25 
2 .5 Summary 26 
CHAPTER 3: THE BACKGROUND RESEARCH ON THE 28 
TEACHING AND LEARNING OF MATHEMATICAL 
PRACTICES IN COMMUNITIES OF MATHEMATICAL 
INQUIRY 
3 . 1  
3 . 2  
3 . 3  
3.4 
Introduction 28 
Structuri ng communities of mathematical inquiry 29 
3 .2. 1 Models of teachers medi ating mathematical i nquiry cultures 30 
3 .2 .2 Variations in  practices of c lassroom inquiry communities 34 
3 .2 .3 The socio-cultural and mathematical norms of communities of 
mathematical inquiry 37 
3 .2.4 Intellectual partnerships in the mathematical discourse of i nquiry 39 
Forms of discourse used in mathematics classrooms 42 
3 . 3 . 1  Univocal and dialogic di scourse 42 
3 .3 . 2  Inquiry and argumentation 44 
3 .3 . 3  Interactional strategies used by  teachers to engage students in the 
discourse 47 
3 . 3 .4 The mathematical discourse and the development of situated 
identi ties 48 
3 . 3 .5 Examples of frameworks used to structure col lective reasoning 
during inquiry and argumentation 50 
Summary 54 
CHAPTER 4: THE MATHEMATICAL PRACTICES OF 
COMMUNITIES OF MATHEMATICAL INQUIRY 56 
4. 1 Mathematical practices 56 
4.2 Mathematical explanations 57 
4.3 Mathematical justification 59 
4.4 Mathematical generalisations 65 
V 
4.5 Mathematical representations and inscriptions 7 1  
4.6 Using mathematical language and definitions 72 
4.7 Summary 74 
CHAPTERS: METHODOLOGY 76 
5 . 1  
5 . 2 
5 .3  
5 .4 
5 .5 
5 .6 
5 .7 
5 . 8  
Introduction 76 
Research question 77 
The qualitative research paradigm 77 
Design research 78 
5 .4. 1 Testing conjectures :  Communication and participation framework 80 
Ethical considerations 
5 .5 . 1  Informed consent 
5 .5 . 2  Anonymity and confidentiality 
The research setting 
Description of the school 5 .6 . 1 
5 .6 .2 
5 .6 .3  
5 .6 .4 
5 .6 .5 
The participants and the beginning of the research 
Participants in the study groups 
The case study teachers and their students 
Study group meetings 
Data col lection 
5 .7 . 1  Data col lection i n  the classrooms 
5 .7.2 Participant observation 
5 .7 .3  Video-recorded observations 
5. 7 .4 Documents 
5 .7 .5 Interviews with teachers 
5 .7.6 Exit from the fie ld 
Data analysis 
5 .8 . 1 Data analysis i n  the field 
5 .8 .2 Data analysis out of the field 
5 . 8 .3 Sociocultural activity theory data analysis 
vi 
86 
86 
87 
88 
88 
89 
9 1  
92 
92 
94 
95 
95 
96 
97 
98 
99 
99 
1 00 
1 00 
1 03 
5 .9 Data presentation 
5 .9 . 1  Trustworthiness, general i sabi l ity and ecological validity 
1 04 
1 05 
5 . 1 0  Summary 1 06 
CHAPTER 6 :  LEARNING AND USING MATHEMATICAL 
PRACTICES IN A COMMUNITY OF MATHEMATICAL 
INQUIRY: AVA 1 08 
6. 1 Introduction 1 08 
6.2 Teacher case study: Ava 1 09 
6 .3 Establi shing mathematical practices i n  a community of mathematical 
i nquiry 1 1 0 
6.3 . 1 Constituting shared ownership of the mathematical talk 1 1 0 
6 .3 .2  Constituting a safe learning environment 1 1 2 
6 .3 .3  Collaborative construction of mathematical explanations i n  small 
groups 1 1 4 
6.3 .4 Making mathematical explanations to the large group 1 1 6 
6 .3 .5 Learning how to agree and disagree to justify reasoning 1 1 9 
6 .3 .6 Generalising mathematical reasoning 1 2 1  
6 .3 .7 Using and c larifying mathematical language 1 23 
6 .3 .8  Summary of the first phase of the study 1 25 
6.4 Extendi ng the mathematical practices in a mathematical inquiry 
community 1 25 
6.5 
6.4. 1 
6.4.2 
6.4.3 
6 .4.4 
6.4.5 
6.4.6 
6.4.7 
Providing an environment for further intel lectual growth 
Col lectively constructing and making mathematical explanations 
Engaging in explanatory justification and mathematical 
argumentation 
Problem solving and inscribing mathematical reasoning 
Justifying and generalising mathematical reasoning 
Using mathematical language 
Summary of the second phase of the study 
Owning the mathematical practices i n  a community of mathematical 
inquiry 
6 .5 . 1  Maintaining i ntel lectual partnerships in col lective i nquiry 
1 26 
1 30 
1 32 
1 36 
1 38 
1 4 1  
1 42 
1 42 
and argumentation 1 43 
VII 
6.6 
6 .5 .2 Transforming informal i nscriptions to formal notation schemes 1 48 
6.5 .3  Increasing the press for generalising reasoning 149 
6 .5 .4 Summary of the third phase of the study 1 52 
Summary !52 
CHAPTER 7: LEARNING AND USING MATHEMATICAL 
PRACTICES IN A COMMUNITY OF MATHEMATICAL 
INQUIRY: MOANA 1 54 
7 .I Introduction !54 
7 .2  Teacher case study two: Moana 1 55 
7 .3  Changing the in teraction norms towards a community of mathematical 
7.4 
i nquiry 1 57 
7 .3 . 1 The initial start to change the communication and participation 
patterns 
7 .3 .2 Constituting shared mathematical talk 
7 .3 .3  Constructing more inclusive sharing of  the talk i n  the community 
7 .3 .4 Learning to make mathematical explanations 
7 .3 .5 Learning how to question to make sense of mathematical 
explanations 
7.3.6 Summary of the first phase of the study 
Further developing the communication and participation patterns of a 
community of mathematical inquiry 
7.4. 1 Col lectively constructing and making mathematical explanations 
7 .4.2 Providing a safe risk-taking environment to support intel lectual 
growth 
7 .4.3 Positioning students to participate i n  the classroom community 
7 .4.4 Providing explanatory justification for mathematical reasoning 
7 .4.5 Exploring relationships and pattern seeking 
7 .4.6 Using mathematical language 
7 .4.7 Summary of the second phase of the study 
7 .4 Taking ownership of mathematical practices on a community of 
1 57 
1 59 
1 62 
1 65 
1 68 
1 70 
1 7 1  
1 72 
1 75 
1 78 
1 8 1  
1 85 
1 87 
1 88 
mathematical inquiry 1 89 
7 .5 . 1  Using model s  of cultural contexts to scaffold student engagement 
in i nquiry and argumentation 1 89 
7 .5  .2  Further developing student  agency of  the mathematical discourse 1 92 
7 .5 .3  Problem solving and a shift towards generalis ing 1 94 
viii 
7.6 
7 .5 .4 
7 .5 .5  
Justifying explanatory reasoning through inscriptions 
Summary of the third phase of the study 
Summary 
CHAPTER 8: CONCLUSIONS AND IMPLICATIONS 
8 . 1 
8 .2 
8 .3 
8.4 
8 .5 
8 .6 
8 .7 
Introduction 
The pathways to developing communities of mathematical inquiry 
8 .2 . 1 Scaffolding student communication and participation in 
mathematical practices 
Supporting students to become members of communities of mathematical 
i nquiry 
Supporting teachers to construct communities of mathematical inquiry 
Limitations 
Implications and further research 
Concluding words 
1 96 
1 98 
1 98 
200 
200 
20 1 
203 
207 
209 
2 1 2  
2 1 3  
2 1 5  
REFERENCES 2 1 7  
APPENDICES 244 
Appendix A: Teacher information sheet and consent form. 244 
Appendix B :  Student information sheet and consent form. 247 
Appendix C :  Board of Trustees I nformation sheet and consent form. 249 
Appendix D: Parent and care-giver information sheet and consent form. 252 
Appendix E:  The framework of questions and prompts. 254 
Appendix F: Sample transcript with teacher annotations. 256 
Appendix G:  Problem example developed in the study group to support early 257 
algebraic reason ing.  
ix  
Appendix H :  Problem examples developed in the study group which required 258 
multiple ways to validate the reasoning. 
Appendix I :  Problem examples developed in the study group which supported 259 
exploration of partial understandings. 
Appendix J :  Moana' s chart for the ground rules for talk. 260 
Appendix K: Examples of expansions of the communication and participation 26 1 
framework. 
Appendix L: A section of the table of data for the activity setting in the classroom.262 
List of Tables 
Table 1 :  Assumptions about doing and learning mathematics implicit in 
teacher-student i nteractions. 
Table 2: The communication and participation framework 
Table 3 :  A time-l ine of data col lection 
Table 4: Example of the codes 
Table 5 :  Examples of the themes for teacher and student actions 
Table 6: Examples of the evidence for one theme 
X 
33 
84 
94 
1 02 
1 02 
1 03