Copyright is owned by the Author of the thesis.  Permission is given for 
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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 
CHAPTER ONE 
INTRODUCTION TO THE STUDY 
Engagement in practice-in its unfolding, multidimensional complexity-is both the stage and 
the object-the road and the destination. (Wenger, 1 998, p. 95)  
1 . 1  INTRODUCTION 
Important changes have occurred in how mathematics classrooms are conceptual ised over 
recent years. These changes are in response to the need to consider how mathematics 
education might best be able to meet the needs of students in the 21st century. A central 
hallmark of the changes is a vision of students and teachers actively engaged in shared 
mathematical dialogue in classrooms which resemble learning communities (Manouchehri 
& St John, 2006). The shared dialogue includes use of effective mathematical practices, the 
specific things successful learners and users of mathematic s  do when engaged in reasoned 
discourse during mathematical activity (RAND, 2003).  These changes propose the creation  
of  teaching and learning experiences which are radically different from those which have 
been traditionally offered in New Zealand primary school mathematics classrooms. It is 
against this background that this study, Teachers developing communities of mathematical 
inquiry, was conducted. The main  purpose of this study was to investigate and tell the story 
of the journey two teachers take as they work at developing intellectual learning 
communities i n  primary classroom settings which support student use of effective 
mathematical practices. 
This  chapter identifies the central aim of this study. The background context outli nes how 
the current reforms in  mathematics education in New Zealand are shaped by both political 
goals and the direction taken by the wider international mathematics community. The 
significance of considering the social and cultural nature of learning in mathematics 
c lassrooms is discussed within the context of the current numeracy initiatives within the 
primary sector. 
1 .2 RESEARCH AIM 
This study aims to explore how teachers develop communi ties of mathematical i nquiry that 
supports student use of effective mathematical practices. The focus of the exploration is on 
the different pathways teachers take as they work at developing communities of 
mathematical inquiry, their different pedagogical actions, and the effect these have on 
development of a mathematical discourse community. The study also investigates how 
changes i n  the mathematical discourse reveal themselves as changes in how participants 
participate in  and use mathematical practices. That is ,  the research also seeks to understand 
how the changes in participation and communication patterns in a mathematical classroom 
support how students learn and use effective mathematical practices. 
1 .3 BACKGROUND CONTEXT OF THE STUDY 
During the latter stages of the 20th century New Zealand society experienced rapid and 
widespread technological changes and an economic cl imate of competitive complex 
overseas markets. Knowledge was viewed as i ncreasingly immediate, complicated, and 
constantly changing. This in i ti ated a pol i tical response which involved the establishment of 
ambitious social and cultural goals-goals which aimed to transform New Zealand to a 
knowledge-based economy and society. Education was positioned as a "transformational 
tool which enriches the l ives of individuals, communities and societies" (Clark, 2007, p. 2 ) ;  
an important vehicle to ensure future participation i n  employment and citizenship for al l  
New Zealanders. 
High levels of numerical l i teracy were seen to be in tegral to ful l  partic ipation m a 
knowledge-based, technological ly-oriented society (Maharey, 2006) .  This fuel led increased 
pol i tical attention towards mathematics education, already a source of concern as a resul t  of 
the relatively poor performance of New Zealand students i n  the 1 995 Third International 
Mathematics and Science Study (Young-Loveridge, 2006). A key consequence of those 
results was the development of 'The Numeracy Development Project' (NDP), an on-going 
2 
nation-wide in itiati ve designed to increase student proficiency through strengthening 
teachers' professional capabil i ty in mathematics teaching (Young-Loveridge, 2005 ) .  
The NDP bui lt on more than a decade of  national and international efforts (e.g. ,  Department 
for Education and Employment, 1 999; Ministry of Education, 1 992; National Counci l of 
Teachers of Mathematics, 1 989, 2000; New South Wales Department of Education and 
Training, 1 999) to reform mathematics teaching and learning. Concerns with poor levels of 
achievement of many students in Western society and the disaffection of groups of people 
towards mathematics supported an impetus for reform focused on achieving success for all 
students as successful mathematical thinkers and users . 
The teaching model advocated in the NDP requires that teachers reconceptualise their 
teaching  and learning practices from a previous predominant focus on rote learn ing of 
computational rules and procedures. Inherent in the shift is the repositioning of all members 
of the classroom community as active participants because teachers are required to 
"chal l enge students to think by explaining, l istening, and problem solving; encourage 
purposefu l  discussion, in whole groups, in small groups, and with i ndividual students" 
(Ministry of Education, 2005 , p . l ) . Students are cast as active communicators as they 
col laboratively analyse and validate the mathematical reasoning within solution strategies 
offered by their peers. 
The focus on students ' interactions and communicating of mathematical reasoning within 
the NDP resonates with goals previously established in  Mathematics in the New Zealand 
Curriculum (Ministry of Education, 1 992). This document maintains that "learning to 
communicate about and through mathematics is part of learning to become a mathematical 
problem solver and learning to think mathematically" (p. 1 1  ). This theme is reiterated in the 
current pol icy initiative. The New Zealand Curriculum (draft) (Ministry of Education, 
2006a) states that "students learn as they engage in  shared activities and conversations with 
other people. Teachers can facil itate this process by designing learning environments that 
foster learning conversations and learning partnerships, and where chal lenges, support, and 
feedback are readi ly avai lable" (p. 24). S imilarly, international policy documents (e .g . ,  
3 
Australian Education Council ,  1 99 1 ;  Department for Education and Employment, 1 999; 
National Council of Teachers of Mathematics, 2000) exhort the importance of teachers 
developing mathematics classroom communities which support student communication,  
problem solving and reasoning capacity. Col lectively ,  these documents affirm the 
significance of classroom interactions and the nature of the discourse for the development 
of the kinds of mathematical thinking and learning envisaged within the mathematics 
reform strategies. 
Influenced by contemporary sociocultural learning theories (e.g. , Forman, 2003; Lave & 
Wenger, 1 99 1 ;  Sfard, 1 998;  Wells ,  200 1 a; Wenger, 1 998)  the cunent mathematics reform 
efforts attend to the essentially social and cultural nature of cognition. Within this 
perspective, "mathematics teaching and learning are viewed as social and communicative 
activi ties that require the formation of a classroom community of practice where students 
progressively appropriate and enact the epistemological values and communicative 
conventions of the wider mathematical community" (Goos, 2004, p. 259).  The current  
cuniculum document (Ministry of Education, 2006a) and NDP both advocate pedagogical 
changes that increasingly expect and direct teachers to reconceptualise and restructure thei r 
teaching and learning practices to move towards developing communities of mathematical 
inquiry in  which students are given opportunities to learn, and use, effective mathematical 
practices. 
Under consideration in the context of this study are the mathematical practices which 
emerge and evolve as teachers reconstitute the classroom communication and participation 
patterns .  As discussed in the fol lowing chapter, learning mathematics i s  more than 
acquiring mathematical knowledge. To become competent learners and users of 
mathematics students also need to learn how to "approach, think about, and work with 
mathematical tools and ideas" (RAND, 2003, p. 32). Evidence (e.g. ,  Ball ,  200 1 ;  Goos, 
2004; Lampert, 200 1 ;  Wood, Williams, & McNeal , 2006) shows that students are capable 
of powerfu l  mathematical reasoning when provided with opportunities to engage in  
reasoned collective mathematical discourse . Reasoned discourse in  this form extends past 
explaining mathematical ideas, to include i ntenelated practices of justification, verification 
4 
and validation of mathematical conclusions, coupled with an efficient use of mathematical 
language and inscriptions .  
However, learning and using reasoned discourse reqmres a classroom culture which 
actively supports collective i nquiry. The role the teacher has in developing a culture of 
mathematical inquiry is central to the learning practices which develop in  it  (Goos, 2004). 
Therefore, it  is t imely to explore exactly what pedagogical practices teachers use, the 
communication and participation patterns which result, and how these support students 
learning and use of effective mathematical practices. 
1 .4 RATIONALE FOR THE STUDY 
Wells  (200 1 b) reminds us that in trying to understand as well as to adjust and develop one ' s  
practice, theory emerges from the practice and supports sense-making of  the practice. 
However as We1ls notes, theory is "on ly valuable when i t  shapes and i s  shaped by action" 
(p . 1 7 1 ) . The sociocultural learning perspectives evident in the policy initiatives spanning 
the past decade, provide ample justification for directing specific attention to examin ing 
how teachers establish effective learning partnerships in c lassroom communities i n  which 
the use of reasoned col lective mathematical discourse is the norm. Whi lst we have 
international studies (e.g. ,  Borasi, 1 992; Brown, & Renshaw, 2000; Goos, 2004; Goos et 
al . ,  2004; Lampert, 200 1 )  which i l lustrate how the key sociocultural ideas have been 
understood and applied within mathematics classroom contexts, we have l imited research 
directed towards exploring and examining how New Zealand teachers might develop such 
mathematical c lassroom learning communities, the pedagogical practices they might use to 
develop reasoned discourse, the roles they might take in them, and the reasoning practices 
which emerge and evolve. 
A review of the NDP (Ministry of Education, 2006b) professional development material 
reveals comprehensive guidance for how teachers might teach the mathematical knowledge 
and strategies. But how teachers might organise student participation in communal 
mathematical discourse that extends beyond the teacher leading strategy reporting and 
5 
questioning of strategy solutions sessions to i ndependent cooperative reasonmg Js not 
described (lrwin & Woodward, 2006) .  In reading the many New Zealand NDP evaluation 
reports we learn about enhanced teacher capacity and capability to teach numeracy, and we 
read of student i ncrease in  knowledge and strategy leve ls .  However, what is not revealed is 
how these gains might equate to student growth in the use of the communicative reasoning 
processes, and to mathematical practices from which theorists (Boaler, 2003a; RAND, 
2003;  Van Oers, 200 1 )  suggest deeper mathematical reasoning and conceptual 
understandings develop. A focus on how teachers extend communal exploration of 
mathematical explanations towards reasoning for justification and an investigation into the 
mathematical practices that emerge and are used is appropriate at this time. 
Central to mathematical proficiency is conceptual understanding of mathematical ideas; 
conceptual understanding means not only knowing a mathematical concept but also being 
able  to reason its truth. Although New Zealand policy in i tiatives over the past decade have 
promoted the need for teachers to develop classroom cultures which foster student 
participation in learn ing dialogue-including the presentation of arguments, and chal lenge 
and feedback on mathematical reasoning (Ministry of Education, 1 992, 2006)-these are 
ambitious goals for change (Huferd-Ackles, Fuson, & Sherin,  2004; Si lver & Smith, 1 996). 
Gaps exist in our understanding about the chal lenges New Zealand teachers might 
encounter as they change the classroom interaction norms towards inquiry and 
argumentation. We need to know how teachers themselves might best be supported to learn 
to use reasoned mathematical discourse as they establ ish its use within their classroom 
culture. 
International studies i l lustrate that in primary classrooms where the use of discourse 
extends to justification and argumentation both the cognitive demand and student 
participation in conceptual reasoning i ncreases (Forman, Larreamendy-Joerns, & B rown, 
1 998 ;  Wood et al . ,  2006) .  However, reasoning at complex cognitive levels and working 
through a communal argumentative process is not something many younger students can 
achieve easi ly without explicit adult mediation (Mercer, 2002). In New Zealand there 
appears to be l ittle research which has examined the explicit pedagogical practices used to 
6 
mediate justification and argumentation with young students. A direct focus is needed to 
consider the specific adaptations teachers make in  pedagogical practices which best match 
the social and cultural context of their students. 
Through the col laboration with teachers, this study advances professional development 
beyond the stage of immediate involvement in the NDP. Exploring from the perspectives of 
teachers how they construct communal learning communities within a sociocultural 
learning frame, provides possible models other teachers might use. The research 
partnership used in this study provides space to understand from the teachers' perspective 
what they find important and significant as they explore and examine possible changes in 
c lassroom practices consistent with sociocul tural theory but also appropriate to the complex 
situations of their classroom contexts. Importantly, this focus supports the development  of 
better understanding of how the central theoretical ideas inherent in the sociocultural 
learning perspective can be enacted within the real-l ife messy context of mathematics 
primary classrooms. 
1 .5 OVERVIEW OF THE THESIS 
The thesis is divided into eight chapters . This chapter has set the scene and provided a 
rationale for the study. Chapter Two reviews sociocu ltural learning theories and establ ishes 
its l inks to classroom discourse, communities of mathematical inquiry and mathematical 
practices. Chapter Three discusses the background research on the teaching and learning of 
mathematical practices in c lassroom inquiry communities. Chapter Four provides 
descriptions of different mathematical practices which emerge in the social and 
mathematical norms of classroom communities. Chapter Five outlines the use of a 
quali tative design approach and discusses the methods used to col lect and analyse data. 
Chapters Six and Seven present the results of teachers' developing classroom communities 
of mathematical i nquiry. The teachers' pedagogical practices and the mathematical 
practices which emerge and evolve as a result are discussed in relation to the l iterature. 
Chapter Eight completes this thesis .  The journey of the two teachers is discussed. The 
conclusions and implications and recommendations for future research are presented. 
7 
CHAPTER TWO 
THE BACKGROUND TO THE STUDY 
Learning mathematics involves learning ways of thinking. It involves learning powerful 
mathematical ideas rather than a col lection of disconnected procedures for carrying out 
calculations. But it also entai ls learning how to generate those ideas, how to express them 
using words and symbols, and how to justify to oneself and to others that those ideas are 
true. Elementary school chi ldren are capable of learning how to engage in  this type of 
mathematical thinking, but often they are not given the opportunity to do so. (Carpenter, 
Franke, & Levi, 2003, p. I) 
2.1 INTRODUCTION 
An important observation made in Chapter One was that students learn to use effective 
mathematical practices in ' learning communities' which include articulation of reasoned 
mathematical thinking. For this  reason, close examination wil l  be given in this chapter to 
sociocultural theories of learning that underpin this study. The defining characteristic of 
this theoretical frame shifts the focus from individuals to activity setti ngs-the learning 
environments and communities structured by teachers. 
Section 2.2 provides a description of mathematical practices. It explains why they are 
considered reasoned performative and conversational actions (Van Oers, 200 1 ) used in 
col lective activity. Explanations are offered as to how students gain membership and 
identity through engaging in the discourse of different  activity setti ngs. 
Section 2.3 describes how mathematical practices evolve through socially constructed 
interactive discourse. It reveals that interactive dialogue is central to how mathematical 
practices are developed and used col lectively in c lassroom communities. Theoretical 
justification is provided for the focus on communkation and participation patterns .  
Contemporary views of the zone of proximal development (zpd) are drawn on to  show how 
8 
dialogue supports growth and use of mathematical practices and the development of a 
collective view. 
Section 2 .4 defines communities of mathematical inquiry and the role taken by teachers 
within them. Mathematical inquiry and argumentation are discussed . Reasons are given for 
the importance of this form of discourse. Explanations are offered concerning why 
discursive interaction poses many difficulties for both teachers and students in classroom 
communities. In addition. the concepts of socio-cu ltural' and mathematical norms are 
defi ned . 
2.2 MATHEMATICAL PRACTICES 
This study drew on sociocultural perspectives of learning as a theoretical rationale to 
explain  l inks between mathematical instructional practices and the mathematical practices 
which students construct and use . Theorising from this perspective provides opportunities 
to describe how teachers structure learning environments so that students learn to 
participate in and use mathematical practices as reasoning acts. These acts are embedded 
within the social and communicative context of classroom communities of practice (Lave & 
Wenger, 1 99 1  ). Mathematical practices are specific to and encapsulated within the practice 
of mathematics (Bal l  & Bass, 2003) .  Theorising using a sociocultural perspective offers 
ways to understand how opportunities that students have to construct and use proficient 
mathematical practices are dependent on the mathematical activity structured in the cultural 
and social l ife of c lassroom communities. 
Van Oers (200 1 )  defines mathematical activity as the "abstract way of referring to those 
ways of acting that human beings have developed for deal ing with the quantitative and 
spatial relationships of their cultural and physical environment" (p. 7 1  ). He refers to 
mathematical practices as the mathematical activity 
with the values, rules and tools adopted i n  a specific cultural community we tend to speak of 
a mathematical practice. Any practice contains performative actions and operations that just 
1 Socio-cultural norms refer to the social and cultural norms of the mathematics c lassroom (See p. 25). 
9 
carry out certain tasks which have mathematical meaning within that community ( l i ke 
performing long division). On the other hand, practices also comprise conversational 
actions that intend to communicate about the mathematical operations or even about the 
mathematical utterances themselves (p. 7 1  ). 
Understanding mathematical practices as comprised of not only performative but also 
conversational actions is in accord with the increased curricular attention in the past decade 
to communication in mathematics classrooms (Mini stry of Education, 1992; 2004a; 
National Council of Teachers of Mathematics, 1 989,  2000). Many researchers have 
promoted the need for the development of classroom mathematics communities i n  which 
inquiry and validation of reasoning are considered central (e.g. ,  Brown & Renshaw, 2006; 
Goos, 2004; National Counci l of Teachers of Mathematics, 2000; Wood, et al . ,  2006) .  
The use of  the term 'practice' within the current study i s  taken from the definition provided 
by Scribner ( 1 997). Scribner used the term 'practice' during her research of adults who 
were engaged in everyday work tasks, "guided by a practice approach to cognition" (p. 
299). Scribner clarified that by a practice she was referring to 
a socially-constructed activity organised around some common objects. A practice involves 
bounded knowledge domains and determinate terminologies, including symbol systems. A 
practice is comprised of recurrent and interrelated goal-directed actions. Participants in  a 
practice master i ts knowledge and technologies and acquire the mental and manual skil ls 
needed to apply them to the accomplishment of action goals. (p. 299) 
Mathematical practices in this frame are constructed and used within situated, social and 
cultural activity setti ngs. This theoretical position presents a way to understand how 
teachers, as more knowledgeable members of mathematical communities, social ise students 
to participate in learni ng and to explore how students learn to use the communicative 
reasoning processes-the mathematical practices-from which deeper mathematical 
thinking and understanding develops (Forman & Ansell ,  2002; Van Oers, 200 1 ; Wood et 
al . ,  2006) .  
1 0  
2.2.1 MATHEMATICAL PRACTICES AS REASONED COLLECTIVE 
ACTIVITY 
The notion that mathematics teaching should enable all students to participate i n  the 
construction and communication of powerful ,  connected, and well -reasoned mathematical 
u nderstanding is a common theme in current mathematical research l i terature (e.g. ,  Alr0 & 
Skovsmose, 2002 ; Ball & Bass, 2003 ; Boaler, 2003a; Carpenter et al . ,  2003; Carpenter, 
B lanton, Cobb, Franke, Kaput, & McClain, 2004a; Lampert & Cobb, 2003 ; National 
Council of Teachers of Mathematics, 2000; Romberg, Carpenter, & Kwako, 2005: Sfard, 
2003) .  In keeping with this notion, mathematical learning is mediated through participation 
in reasoned col lective discourse. As Ball and Bass (2003) explain this reasoned col lective 
discourse involves more than individual sense-making. 
Making sense refers to making mathematical ideas sensible, or perceptible, and allows for 
understanding based only on personal conviction. Reasoning as we use it comprises a set of 
practices and norms that are collective not merely i ndividual or idiosyncratic, and rooted i n  
the discipline. Making mathematics reasonable entails making i t  subject to, and the result of, 
such reasoning. (p. 29) 
In Chapter 3, a review of relevant research i l lustrates how teachers' actions m these 
communities can ensure that students are provided with opportuni ties to progressively 
engage in and appropriate a set of reasoning practices .  Through these practices the students 
learn to think, act, and use the mathematical practices of expert mathematical problem 
solvers. 
Mathematical practices are not fixed to specific groups of mathematical users, nor are they 
only invoked in schools ,  or academic mathematical settings (Civil , 2002) .  They are used by 
successful mathematics users of all ages to structure and accomplish mathematical tasks. 
They encompass the mathematical know-how beyond content knowledge which constitutes 
expertise in learning and using mathematics.  Put s imply, the term 'practices' refers to the 
particular things that proficient mathematics learners and users do. Examples of 
mathematical practices i nc lude "justifying claims, using symbolic notation efficiently, 
defin ing terms precisely, and making generalizations [or] the way in which ski l led 
11 
mathematics users are able to model a situation to make it easier to understand and to solve 
problems related to it" (RAND, 2003, p .  xvi i i ) .  The proposal that mathematical practices 
entai l more than what is usually thought of as mathematical knowledge enlarges our view 
of mathematical learni ng. To develop robust mathematical thinking and reasoning 
processes, students need opportunities not only to construct a broad base of conceptual 
knowledge ; they also require ways to build their understanding of mathematical practices; 
these "ways in which people approach, think about, and work with mathematical tools and 
ideas" (RAND, p. xvi i i ) .  
Until the last decade the predominant research focus, derived from psychological models, 
has more generally centred on student construction of cognitive structures. Only recently 
has research explicitly examined how teachers structure mathematical activity to support 
student construction and use of mathematical practices. Viewing mathematical learning as 
embedded within reasoned mathematical discourse offers an alternative way to consider 
student outcomes. From thi s position students' opportunities to construct rich mathematical 
understandings might well be related to the quality or types of classroom di scourse and 
interactions in which they participate. This provides a different explanation from one that 
focuses on individual capabi l ities or the presentation ski l ls  of teachers (Lerman, 200 1 ) . 
Theorising that mathematical learning occurs as a result of sustained participation in 
reasoned mathematical discourse i s  consistent with contemporary sociocultural learning 
perspectives (Forman, 2003; Mercer & Wegerif, 1 999b). 
2.2.2 THE DISCOURSE OF COMMUNITIES OF MATHEMATICAL INQUIRY 
The prominence accorded to discourse in the context of the current study "reflects an 
enterpri se and the perspective of a community of practice" (Wenger, 1 998, p .  289). In this  
form, discourse i s  "a  social phenomenon" (Bakhtin, 1 981, p .  259)  i n  which the past, present 
and the future are encapsulated with in  the contextualised and situated nature of utterances 
of specific genres. Individuals participate with varying levels of expertise across a range of 
speech genres but using an individual voice (Bakhtin ,  1 994). The individual voice, 
however, is a social voice with dialogic overtones of others, given that "thought itself. . .  i s  
born and shaped in  the process of  i nteraction and struggle with others' thoughts, and this 
1 2  
cannot but be reflected in  the forms that verbal ly express our thought as well" (Bakhti n, 
1 994, p .  86) .  
The dialogic nature of discourse provides a usefu l  tool for the current study to explore how 
teachers scaffold students i n  the use of the language of inquiry and argumentation-the 
language which supports mathematical practices. Students learn to question, argue, explain, 
justify and general ise through the models provided by teachers and other participants in the 
dialogue. These models are appropriated and explored as the students learn how to use 
mathematical talk. They try them out, expand and extend them and transform them into 
their own words or thoughts for future use. 
Discourse in this context is not a practice in itself; rather i t  is a resource used in the context 
of a variety of practices which overlap but are often distinctly different (Wenger, 1 998) .  
Membership is provided to the students through their abi lity to participate in, and use ,  the 
discourse of different practices. Identities are constructed in relation to a situation, that is, 
"di fferent identities or social positions we enact and recognise in different setti ngs" (Gee, 
1 999, p. 1 3) .  Thus, when students engage in mathematical practices within a community 
which uses inquiry and argumentation, they not only use a social language, or speech genre, 
they also display the particular identity which is appropriate to the mathematical situation. 
Gee and Clinton (2000) outline how 
each different social language gives us different verbal identities. B ut language never does it 
by itself. It is  always something part and parcel of something bigger, what we will call 
Discourse, with a capital D. Discourses are ways of talking, l istening, reading and writing?
that is using social languages-together with ways of acting, interacting, believing, valuing, 
and using tools and objects, in particular settings at specific times, so as to display and 
recognise particular social ly situated identities. (p. 118) 
For the purposes of the current study the term Discourse with a capital D is not specifically 
used, however the meaning attributed to it by Gee and his col league will be drawn on. 
Framing discourse in this manner provides ways to consider how students may construct 
distinctly different identities as mathematical learners, as a result  of access to participation 
1 3  
in different classroom activity settings. How participants engage in the discourse may be 
affected by the values and beliefs they hold toward their role, and the role of others, in 
using the discipline specific forms of mathematical i nquiry and reasoning. Theorising in 
this way offers explanation for how the mathematical practices of participants in 
mathematical activity may be constrained or enabled depending on how teachers have 
structured students' access to, and use of the discourse. To develop this point further, I wi l l  
draw on sociocultural learning perspectives and activity theory. 
2.3 SOCIOCULTURAL LEARNING PERSPECTIVES AND 
ACTIVITY THEORY 
Sociocultural learning theories offer ways to view mathematical learning as contextuali sed, 
which is to view learning-in-activity. Within this theoretical frame, the social , cultural, and 
institutional contexts are not considered merely as factors which may aid or impede 
learning. Rather, these social organisational processes are integral features of the learning 
itself (Forman, 1 996) .  
The sociocultural perspective on learning is an appropriate means to explain how 
mathematical practices are learnt and used in classroom communities.  This is particularly 
so because "a goal of the sociocultural approach is to explicate the relationships between 
human action, on the one hand, and the cultural, institutional, and historical situations in 
which this action occurs ,  on the other" (Wertsch ,  del Rio, & Alvarez, 1 995 , p. 1 1 ) .  The 
sociocu ltural learning theory described in the next section draws on three themes that 
Wertsch ( 1 985)  developed from Vygotsky' s  work in the early 1 9th century. 
2.3.1 COMMUNICATIVE INTERACTION AND THE MEDIATION OF 
MATHEMATICAL PRACTICES 
To understand the development of a phenomenon Vygotsky ( 1 978)  maintained that it 
should be studied in the process of change, rather than at an end-point  i n  its development. 
Central to this stance is the claim that modes of thinking and reasoning used m 
mathematical practices are not composed of already formed concepts which can be 
1 4  
transmitted; rather they are always in a state of construction and reconstruction (Mercer; 
2000; Sfard, 200 1 ; Wells ,  1 999). Mathematical practices can therefore be thought of as a 
part of a dynamic and evolving transformative process.  That is ,  they are both part of the 
process of developing rich mathematical understanding, and they are also its product. 
The idea that mathematical practices emerge and evolve within a community of users is 
commensurate with Vygotskian thinking. According  to Vygotsky, the social origins of 
higher mental functions-thinking, logical reasoning and voluntary attention are created by 
social processes. They appear first between people on a social plane and then within 
individuals on a psychological plane (Wertsch, 1 985) .  Language and communication that i s  
used initial ly  in the social context is described by Vygotsky ( 1 978) as  fundamental to 
learning. He maintained that language served two functions, "as a communicative or 
cultural tool we use for sharing and jointly developing the knowledge-the 'culture '?
which enables organised human social l ife to exist and continue . . .  and as a psychological 
tool for organising our individual thoughts, for reasoning, planning and reviewing our 
actions" (Mercer, 2000, p. 1 0) .  
The intertwined concepts of mediational means and mediational action are at the core of 
sociocultural thinking (Lerman, 200 1 ; Wertsch, 1 985) .  Al l  activity, i ncluding the mental 
processes of thinking and reasoning, is mediated by tools and signs (Vygotsky, 1 978) .  In  
mathematics classrooms tools i nclude the material and symbolic resources, communicative 
patterns, spoken words, written text, representations, symbols, number systems and 
participation structures .  These mediational means are "embedded in a sociocultural mil ieu 
and are reproduced across generations in the form of collective practices" (Wertsch, 
Tulviste, & Hagstrom, 1 993, p. 344).  Thus, shaped in  sociocultural specific ways, 
mathematical practices evolve within mathematical i nquiry communities in response to the 
communication and partic ipation patterns teachers structure within the activity settings 
(Tharp & Gal l imore, 1 988) .  
In the current study, i n  order to  explain  how the construct of  activity settings relates to  the 
construction of mathematical practices, theories of Vygotsky 's  contemporary, Leontiev 
1 5  
( 1 98 1  ), are drawn on. Activity as it i s  proposed by Leontiev, provides a way to account for 
the transformation of mathematical practices within the complex and goal-directed activity 
enacted by teachers in c lassroom communities. Leontiev, like Vygotsky, defined 
individual s'  thoughts and actions in rel ationship to their goal-directed social and cultural ly 
specific activities. The key elements of activity Leontiev identified as always "material and 
significant" (p. 1 3 ) ,  "primari ly  social" (p. 1 4) and with a "systematic structure" (p. 1 4) .  
Contexts in which collaborative interaction, intersubjectivity and teaching occur are termed 
activity settings (Gal limore & Goldenberg, 1 993; Tharp, 1 993) .  There are many diverse 
forms of activity settings ranging from informal home based groups to the more formal 
school settings . Activity settings "are a construct that unites ( 1 )  objective features of the 
setting and the environment and (2)  objective features of the motoric and verbal actions of 
the participants with (3) subjective features of the participants ' experience, intention, and 
meaning" (Tharp, 1 993, p. 275 ) .  
According to Tharp and Gal l imore ( 1 988),  activity settings, participants in them, and their 
goal-directed activity are not arbitrary; rather they result from "the pressures and resources 
of the larger social system" (p. 73) .  The membership of the activity setting is determined by 
the goals and the setting. In  turn, the activity i s  "performed only when the time i s  congruent 
with the character of the operations and the nature of the personnel" (p. 73) .  The meaning, 
or motivation, for the activity i s  goal dependent although "in the emergent intersubjectivity 
of group performance in its time and place, meaning continues to develop, to emerge, to 
explain ,  and to perpetuate" (p. 73) .  
The construct of 'activity settings' provides a useful explanatory tool to  view how different 
participants transform their beliefs and values of mathematical practices as they strive to 
gain alignment with others in their zpd. Within the everyday contexts of the classroom 
mathematics community there are important variables to consider "( 1 )  the personnel 
present during  an activity; (2) salient cultural values; (3) the operations and task demands 
of the activity itself; (4) the scripts for conduct that govern the participants' actions; (5) the 
purposes or motives of the participants" (Gall imore & Goldenberg, 1 993, p. 3 1 6) .  These 
1 6  
variables present ways to explain connections between participants, the constraints and 
supports provided by others in the activity setti ng, and the cultural patterns, norms and 
values enacted. In turn, changes in the activity settings can be mapped to shifts in the 
communication and participation patterns which teachers enact . Shifts in these structures 
alter the demands on the participants and thus prompt change in the interactional scripts, 
participation roles and their bel iefs and values. 
2.3.3 ZONES OF PROXIMAL DEVELOPMENT 
According to Vygotsky ( 1 986) conceptual reasoning is a result of interaction between 
everyday spontaneous concepts and scientific concepts. Scientific concepts i nvolve higher 
order thinking which is used as students engage in more proficient mathematical practices. 
Vygotsky maintained that "the process of acquiring scientifi c  concepts reaches far beyond 
the immediate experience of the child" (p. 1 6 1  ). Although his research was not centred on  
school ing he  suggested that school was the cultural medium, and dialogue the tool that 
mediated transformation of everyday spontaneous concepts to scientific concepts. 
Articulated reasoning, mqmry and argumentation m the construction of reasoned 
mathematical thought i s  the focus of the current study.  Whi lst the exact nature of how 
external articulation becomes  thought has been extensively debated (Sawyer, 2006a), 
sociocultural theorists are uni ted in their bel ief that col laboration and conversation is the 
key to the transformation of external communication to i nternal thought. This occurs as 
students interact in zpds they construct together. The zpd has been widely i nterpreted as a 
region of achievement between what can be realised by i ndividuals acting alone and what 
can be realised in  partnership with others (Goos, Galbraith, & Renshaw, 1 999; Tharp & 
Gal l imore, 1 988) .  Traditional applications of the zpd were used primari ly to consider and 
explain how novices were scaffolded by experts in problem solving activity (Forman, 
2003) .  Often learning was described as occurring through the expert demonstrating, 
modeling, guiding or explicitly explaining (Forman & McPhai l ,  1 993) .  In this form, 
scaffolding was interpreted as teachers or more expert others "assisting the child i n  
identifying, sequencing, and practising sub-goals for eventual guided assembly . . .  the 
1 7  
asymmetrical structuring of the passive child through a process of breaking down the task" 
(Stone, 1 993, p. 1 80) .  
More contemporary interpretations of the zpd provide ways to consider learning which 
occurs when levels of competence are more evenly distributed across members in the zpd. 
Learning in this form for individuals, groups and whole classroom communities occurs 
during mutual engagement in col lective reasoning activity (e.g. ,  Brown & Renshaw, 2004; 
Enyedy, 2003 ; Goos, 2004; Goos et al . ,  1 999; Lampert, 1 99 1 ;  Mercer, 2000; Wel ls, 1 999; 
Yackel ,  Cobb, & Wood, 1 99 1 ;  Zack & Graves,  200 I ) . Lerman (200 1 )  describes the 
participation in the mathematical discourse and reasoning practices as pu l l ing participants 
forward into their zpds. The zpd is thus defined 
as a symbolic space involving i ndividuals, their practices and the circumstances of their 
activity. This view takes the zpd to be an ever-emergent phenomenon triggered, where it 
happens, by the participants catching each other's activity. It is often fragi le and where it is 
sustained, a process of semiotic mediation and interaction emerges. (Lerman, 200 I ,  p .  I 03) 
Defining the zpd as a symbolic space provides a useful means for the current study to 
explain how participants i n  activity settings mutually  appropriate each others' actions and 
goals (Newman, Griffin, & Cole, 1 989) .  In doing so, teachers and students are required to 
actively engage and work to understand the perspectives taken by other participants. 
Teachers pul led into the zpd work with their students' understandings and attitudes. In turn, 
the students identify with the attitudes and values of the teacher who represents the social 
and cultural practices of the wider mathematical community (Goos, 2004) .  
As described earlier, the most recognised use of the zpd pertains to the scaffolding 
metaphor used by Bruner ( 1 990, 1 996). In this form, scaffolding supports the learner to 
achieve their goals 'of the moment' (Sawyer, 2006a). However, the learner is not a passive 
recipient, rather, the negotiation of goals is eo-constructed with in the activity setting. For 
example, Mercer (2000) described how students were scaffolded by teachers to participate 
in reasoned mathematical i nquiry eo-constructed in the active contributions of all 
1 8  
participants.  He termed student inquiry of each other' s reasoning in the zpd "interthinki ng" 
(p. 1 4 1 ) . 
Mercer (2000) described interthinking in  shared communicative space as creating an 
intermental development zone founded in the shared knowledge and goals of all community 
members . Connected to the variable contributions of all participants the quality of the 
communicative space, therefore, was "dependent on the existing knowledge, capabil ities 
and motivations of both the learner and the teacher" (p. 1 4 1 ) . The variable contributions 
create the need to continually renegotiate and reconstitute the zpd. Mathematical meaning 
i s  generated through collective inquiry as participants talk, think, and reflectively consider, 
what is being claimed. If the shared dialogue succeeds then the students are extended 
beyond their usual capabil ities but if the dialogue fai ls to sustain al ignment of all members 
then the zpd col lapses and collective mathematical reasoning fails .  
The construct of i nterthinking, pull ing participants i nto a shared communicative space, 
extends a view of the zpd beyond scaffolding. It supports consideration of the learning 
potential for pairs or groups of students working together with others of s imi lar levels of 
experti se in egali tarian relationships (Brown & Renshaw, 2004; Goos, 2004; Goos et al . ,  
1 999; Renshaw & Brown, 1 997) .  The partial mathematical knowledge and ski l ls that 
members contribute support col lective understanding. Through joint  activity, opportunities 
are made avai lable for the group to encounter mathematical si tuations which involve 
erroneous thinking, doubt, confusion and uncertainty (Goos et al . ,  1 999) .  Negotiation 
requires participants to engage in exploration and speculation of mathematical reasoning. 
This is an activity which approximates the actual mathematical practices used by 
mathematicians. 
Within a group the development of a collective view is  dependent on al l members sharing 
goals and values which support mutual engagement i n  mathematical practices. Wenger 
( 1 998) outlines that 
mutual engagement involves not only our competence, but the competence of others. It 
draws on what we do and what we know, as well as on our ability to connect meaningfu l ly 
1 9  
to what we don't do and what we don ' t  know-that is, to the contributions and knowledge 
of others. In this sense, mutual engagement is inherently partial; yet, in the context of a 
shared practice, this partiality is as much a resource as it is a limitation. This is rather 
obvious when participants have di fferent roles . . .  where mutual engagement involves 
complementary contributions. But it is also true . . .  [for those] . . .  who have largely 
overlapping forms of competence. Because they belong to a community of practice where 
people help each other, it is more important to know how to give and receive help than to 
try to know everything yourself. (p. 76) 
Collaboration and construction of a collective view is not always premised on immediate 
consensus. Azmitia and Crowley (200 1 )  explain that "dissension can also serve as a catalyst 
for progress ei ther during or following the collaborative session" (p . 58 ) .  They maintain 
that the essential ingredient is mutual engagement in transactive dialogue. Transactive 
dialogues 
are conversations in which partners critique, refine, extend, and paraphrase each other' s  
actions and i deas or create syntheses that integrate each other' s  perspectives, have been 
l inked to shifts in . . .  scientific reasoning. These transactive dialogues may be the epitome of 
collaborative theory construction because in many cases, individuals walk away with a joint 
product for which they are no longer certain (and may not care) who gets credit for 
particular ideas. (Azmitia & Crowley, 200 1 ,  p .  58)  
Thus, the dialogue functions "as a thinking device . . .  formed as a system of heterogeneous 
semiotic spaces . . .  in which languages interact, interfere, and organise themselves" (Lotman, 
1 988,  p. 36-37) .  In classrooms where teacher structuring of such activity is the norm, 
opportunities are provided for students to be inducted into more discipl ined reasoning 
practices. The "lived culture of the classroom becomes, i n  itself, a chal lenge for students to 
move beyond their established competencies" (Goos et al . ,  1 999, p. 97) to become more 
autonomous participants in classroom communities of mathematical i nquiry. As 
summarised by Burton (2002) 
coming to know mathematics depends upon active participation in  the enterprises so valued 
in that community of mathematics practice that they are accepted as knowing in that 
community . . .  the important words to stress here are 'active participation' .  The identity of the 
20 
learner is as an enquirer and the valued enterprises reflect not only the knowledge of 
mathematical objects and ski lls but also the knowledge of how mathematical enquiry is 
pursued. (p. 1 58) 
Thus, coming to know and do mathematics is described as the outcome of active 
participation in col lective inquiry .  Wi thin an activity setti ng the mathematical practices that 
students come to know and use are directly li nked to the forms of social and cultural 
practices they have access to, within their mathematical c lassroom community .  The use of 
the term ' inquiry' in the current study portrays a view of students col laboratively 
partic ipating tn mathematical practices within explicitly constructed classroom 
communities. In the next section an explanation of communi ties of mathematical i nquiry 
and the discourse of inquiry and argumentation is developed further. The role taken by 
teachers as they implici tly and explicitly facilitate the communication and participation 
patterns and interactional strategies which posi tion students to engage in the di scourse i s  
also described. 
2.4 COMMUNITIES OF MATHEMATICAL INQUIRY 
Within a contemporary sociocultural learning perspective the development and use of 
mathematical practices is matched by the students "increasing participation in communities 
of practice" (Lave & Wenger, 1 99 1 ,  p .  49) .  Describing mathematics classrooms as 
communities of practice points to a group united through common purpose and joint social 
activity .  All participants are considered legitimate; although some members have more 
power and knowledge of valued ski l ls  (for example the teacher as the old-timer), whi le 
other members (often the students as the new-comers) are more peripheral . In this context, 
the constructs of the zpd and intersubjectivity supports the more peripheral member 's  
development and use of  effective and appropriate mathematical practices. Thus "learning as 
legitimate peripheral participation means that learning is not merely a condition for 
membership, but is i tself an evolving form of membersh ip" (Lave & Wenger, p. 53 ) .  
The transformation of mathematical practices i s  part of a dynamic participatory process-a 
process that is aligned with appropriation of the norms and values of the specific 
2 1  
community of practice. Because the students make individual meaning through varied 
forms of participation in particular goals, social systems, and discursive practices 
mathematical practices can be viewed as the outcome of the accepted and codified forms of 
communication and participation practices-the discourse practices (Goos, 2004). 
Van Oers (200 I )  outli ned how a "community commi tted to a particular style of 
conversational actions with regard to a special category of objects can be named a 
community of discourse" (p. 7 1  ). Van Oers differentiated between communities of 
mathematical practice and communities of mathematical di scourse. In  the first form, the 
users of mathematics can be making calculations in supermarkets and shops and often in 
idiosyncratic ways. In contrast, communities of mathematical discourse "mainly includes 
persons interested in  reflectively understanding mathematical actions" (p. 7 1 ) . I t  is 
communities of mathematical discourse which receive particular attention in the current 
study. 
Likewise, based on observations across c lassrooms Cobb, Wood, and Yackel ( 1 993) report 
differences i n  the discourse practices involving "talking about talki ng about mathematics" 
(p. 99) .  For example, the practices and beliefs of students in c lassrooms which use 
memorisation and rote procedures contrast with those which have an inquiry discourse 
approach (Goos, 2004 ). Goos suggested that the practices and beliefs constructed in these 
classrooms 
frame learning as participation m a community of practice characterized by inquiry 
mathematics-where students learn to speak and act mathematically by participating i n  
mathematical discussion and solving new o r  unfami l iar problems. Such classrooms could be 
described as communities of mathematical inquiry. (p. 259) 
Other researchers have described these classroom communities variously as i n  an i nquiry 
mathematics tradition (Cobb, Wood, Yackel ,  & McNeal, 1 992);  an inquiry co-operation 
model (Alr!l} & Skovmose, 2002); and communities of inquiry (Goos et a!, 1 999; Wells ,  
1 999; Zack & Graves, 200 1 ) . In the current study 'communities of mathematical i nquiry' i s  
22 
the term used to consider the communal context i n  which meaning is mutual ly  consti tuted 
as students participate di scursively in reasoned actions and dialogue. 
2.4.1 THE COMMUNICATION AND PARTICIPATION PATTERNS OF 
COMMUNITIES OF MATHEMATICAL INQUIRY 
Within communities of mathematical i nquiry mathematical reasoning is  set within a social 
world, where all participants are bound together in the dialogicality of "mutual regulation" 
and "self regulation" (Sfard, 200 1 ,  p.  27) .  In communities premised on inquiry, value is 
placed on "practices such as discussion and collaboration . . .  valued in building a climate of 
intel lectual chal lenge" (Goos, 2004, p. 259). Engagement i s  founded on 
a communicative relationship between equals that requires participation, commitment, and 
reciprocity. Participation means there are opportunities in the dialogue for the i ndividuals 
to become engaged, question others, try out new ideas, and hear diverse points of view. 
Commitment implies that the participants will be open to hearing the positions of other 
speakers. Reciprocity means a wil lingness to engage in an equi lateral exchange with others. 
In this mode, the structure of discourse is multidirectional and responsive. The content of 
the dialogues is dynamic, connected and unscripted. The purpose of the dialogue is to 
participate and engage others in deep inquiry into the meaning of things. (Manouchehri & 
St John, 2006, p. 545) 
The open and responsive dialogue and the participants' partial ly  overlapping zones of 
mathematical understandings and skil ls are inclusive of multiple levels of knowledge and 
expertise (Goos, 2004) .  This enables the orchestration of many productive interactions. The 
i ncrease and range of activ ity settings offers students additional ways to assume varying 
participatory roles (Forman, 1 996) .  Changes are also evident in  the way members of 
communities relate to each other, to the classroom power and authority base, and to the 
discipline of mathematics i tself (Cornelius & Herrenkohl ,  2004). 
Many  researchers have noted that the roles and scripts of such activity settings are 
chal lenging for both teachers and students (e .g. ,  Bal l  & Lam pert, 1 999; Mercer, 2000; 
Rojas-Drummond & Zapata, 2004; Wel ls, 1 999). The role of the teacher as the 
23 
unquestionable authority is chal lenged. Wider diversity i n  the roles, task demands and 
interactional scripts are also demanded of the students (Forman, 1 996) .  Assuming  
responsibi lity "to propose and defend mathematical ideas and conjectures and to respond 
thoughtfully to the mathematical arguments of their peers" (Goos, 2004, p. 259) is often not 
fami l iar to, nor used automatical ly by students. Moreover, many students on entry into 
inquiry classroom communities may hold contrary beliefs about argumentation, considering  
it e i ther unnecessary or  impolite (Forman e t  al . ,  1 998; Rittenhouse, 1 998; Van Oers, 1 996) .  
We now have considerable evidence of the beneficial effects of students articulating thei r 
mathematical reasoning, and inquiring and challenging the reasoning of others (e.g. , Bal l ,  
1 99 1  ; Cobb et  al . ,  I 993 ; En yed y, 2003; Lampert, 200 1 ; McCrone, 2005 ; Sheri n, Mendez, 
& Louis, 2000; Wood, 2002;  Wood et al . ,  2006) .  It forms a "central role in thinking and 
scientific language" during mathematical activity (Rojas-Drummond & Zapata, 2004, p. 
543 ) .  These researchers claim that argumentation is a powerful reasoning tool in that it 
"al lows individuals to deny, criticise and justify concepts and facts, as well as find 
opposing views and generate a new perspective in social i nteraction or in self-del iberation" 
(p. 543) .  It is through engaging in structured mathematical arguments as a precursor to 
argumentation that "students are able to experience mathematics as a discipl ine that relies 
on reasoning for the validation of ideas" (Wood, 1 999, p .  1 89) .  
Argumentation is defined as the act of presenting grounds for taking a particular posi tion, 
view or conclusion; or to confront the positions, views, or conclusions taken by others 
(Bi llig, 1 996). The diversity of views and disagreement are considered important elements 
in sustaining active engagement of students in inquiry (Wells ,  1 999).  However, being able  
to  engage in  dialogue appropriate to  specific social situations i s  a learnt ski l l  which requires 
explicit scaffolding (Bauersfeld, 1 995 ; O'Connor & M ichaels, 1 996). 
The teacher' s role i s  of prime importance in structuring the activity settings m which 
students are held "accountable to disciplinary standards of inquiry and to fel low students' 
contributions and ideas" (Cornelius & Herrenkoh l ,  2004, p. 477) .  Research studies have 
i l lustrated how enacted participation structures and the use of specific i nteractional 
24 
strategies can enable all classroom members to have a voice in the intellectual activity. 
O 'Connor and Michaels ( 1 993, 1 996) described an interactional strategy used by teachers 
which they termed revoicing. They drew on the seminal work of Goffman ( 1 974; 1 98 1 )  to 
explain how speakers characterised and positioned themselves or animated others through 
words and talk in social situations. In their study of a 3rd and 4th grade teacher and a 6th 
grade teacher revoicing was used in subtle ways to rephrase and expand student talk in 
order to clarify, extend, or move the discussion in di fferent directions. The teachers 
positioned all members in each classroom community to mutual ly engage in intel lectual 
activity. As a result, the students learnt to take various intel lectual roles and to use reasoned 
inquiry and argumentation. Through such means, students become enculturated into 
mathematical dialogue embedded within discursive interaction .  The interactional strategy 
of revoicing offers a way for the current study to explore how teachers can ' social ise' 
students in mathematical situations. 
2.4.2 SOCIO-CULTURAL AND MATHEMATICAL NORMS 
Sullivan, Zevenbergen, and Mousley (2002) describe the socio-cultural norms of a 
mathematics classroom as stable patterns of behaviour or practices, organisational routines 
and forms of communication which "impact on approaches to learning, types of response 
valued, views about legitimacy of knowledge produced, and responsibi lity of individual 
learners" (p. 650). These socio-cultural "norms are inferred by discerning patterns or 
regulari ties in the ongoing interactions of members of a community . . .  patterns in col lective 
activity within a community" (Cobb, 2002, p. 1 89- 1 90) .  All c lassroom mathematical 
communities have their own norms and these differ significantly from one classroom to 
another (Boaler, 2003a). These interaction patterns encapsu late the values and beliefs with 
which the students align in the classroom community (McCiain & Cobb, 200 1 ) . They 
define the expectations and obligations in c lassroom participation structures and are often 
the "hidden regularities" in c lassrooms (Wood, 1 998, p. 1 70) .  
The mathematical norms, i n  contrast, relate specifically to mathematics (McClain & Cobb, 
200 1 ,  Sul livan et al . ,  2002) .  Drawing on Wood' s  (200 1 )  description of the interplay of the 
social nature of student learning, the students' developing cognition and the structure that 
25 
underl ies mathematics, Sulli van and his colleagues define mathematical norms as the 
"principles, generalisations, processes and products that form the basis of the mathematics 
curriculum and serve as the tools for the teaching and learning of mathematics itself' (p. 
650). Alternatively, McClain and Cobb use the term sociomathematical norms-a term 
which has its roots in social psychology-to describe the mathematical norms. Kazemi and 
Stipek (200 1 ) explain that they are the negotiated variables constituted within discursive 
interaction 
For example whereas explain ing one 's  thinking is a social norm, what counts as a 
mathematical explanation is a sociomathematical norm . . .  discussing different strategies is a 
social norm, comparing the mathematical concepts underlying different strategies is a 
sociomathematical norm. Finally, working on tasks in  small groups is a social norm; 
requmng students to achieve consensus using mathematical arguments IS a 
sociomathematical norm. (p. 60) 
These normative understandings evolve within mathematical activity and support higher 
level cognitive activity. They are important elements in the development of a mathematical 
disposition and intellectual agency (McClain & Cobb, 200 1 ). Theorising that mathematical 
practices and mathematical norms are i nterrelated offers us a way to explain how 
mathematical practices are transformed as they are negotiated. At the same time, 
explanations can be made of how subtle differences in communication and participation 
classroom structures affect student opportunities to engage in and appropriate increasingly 
proficient mathematical practices. 
2.5 SUMMARY 
This review has provided a theoretical background to the current study. It began with an 
examination of mathematical practices. I have conceptualised mathematics learning as 
learning ways of thinking and reasoning (Carpenter et al . ,  2003) and I have framed that 
conceptualisation within sociocultural perspectives. Within the school setting, opportunities 
to develop powerful reasoning require the establishment of effective mathematical learning 
communities. It was noted that i nquiry communities provide intel lectual space to support 
26 
the emergence and evolution of increasingly proficient mathematical practices. A gap in the 
li terature was identi fied which related to understanding how teachers structure 
mathematical activity so that their students come to know and use mathematical practices. 
This study aims to examine the action pathways teachers take which facili tate student 
access to, and use of, proficient mathematical practices in inquiry communi ties. 
The construction of inquiry communities presents significant chal lenges for teachers. 
Teachers are required to rethink their roles and responsibil ities and those of their students 
within the discourse patterns they structure in such classroom communities. Both teacher 
and student views are challenged in regard to the place of inquiry and argumentation in  
mathematics c lassrooms. Chapter 3 wil l  outli ne relevant research studies which relate to  the 
pedagogical actions teachers have used to structure communication and participation 
patterns-structures which engaged students in col lective mathematical practices.  The 
various forms of mathematical discourse wi l l  be examined. The importance of col lective 
reasoning wil l  be explained to show its influences on student beliefs and mathematical 
goals .  
27 
CHAPTER THREE 
THE BACKGROUND RESEARCH ON TEACHING AND LEARNING 
OF MATHEMATICAL PRACTICES IN COMMUNITIES OF 
MATHEMATICAL INQUIRY 
For students and teachers, the development of understanding is an ongoing and continuous 
process . . .  the development of understanding takes time and requires effort by both teachers 
and students. Learning with understanding will occur on a widespread basis only when i t  
becomes the ongoing focus of  instruction, when students are given t ime to  develop 
relationships and learn to use their knowledge, when students reflect about their own 
thinking and articulate their own ideas, and when students make mathematical knowledge 
their own. (Carpenter & Lehrer, 1999, p. 32) 
3.1 INTRODUCTION 
Attention was drawn in Chapter Two to the importance of student engagement in reasoned 
mathematical di scourse within learning communities. Particular significance was given to 
the discourse of inquiry and argumentation and its importance in student development and 
use of mathematical practices. The connective thread through this chapter is placed on the 
interactional strategies and the communication and participation structures which teachers 
use to engage students in col lective inquiry and argumentation. 
Section 3.2 outl ines how the structuring of learning environments which involve inquiry 
and argumentation i s  a complex and chal lenging task for teachers. For that reason, this 
review commences with three studies which outli ne the roles taken by exemplary teachers, 
in c lassrooms in which proficient mathematical practices were developed and used. Further 
studies are then drawn on to i l lustrate the differential outcomes for student reasoning 
related to the different classroom interactional patterns.  The section concludes with a 
discussion of socio-cultural norms and mathematical norms and their  rel ationship to student 
agency. Relevant l iterature is drawn on to i l lustrate the importance of teachers developing 
intellectual partnerships within  c lassroom communities. 
28 
Section 3 .3  outl ines the forms of discourse commonly used in mathematics c lassrooms . 
Appropriate li terature is used to outl i ne the difficulties teachers have in restructuring the 
discourse patterns towards inquiry. This li terature also i l lustrates the problems students 
encounter when engaging in communal inquiry and the subsequent effects on student 
values, goals and mathematical identity. To conclude, specific participation structures 
which teachers have used to sociali se students into col lective mathematical discourse in  
zones of  proximal development are described. 
3.2 STRUCTURING COMMUNITIES OF MATHEMATICAL 
INQUIRY 
In Chapter Two, sociocultural and situative learning theories were used to explain and 
justify the focus of the current study. Specifically the focus is on the development of 
classroom communities of mathematical inquiry and participation and communication 
patterns that support construction and use of proficient mathematical practices. Providing 
students with space to engage in  disciplined ways of reasoning and inquiry presents 
considerable challenge (Ball & Lampert, 1 999; Franke & Kazemi, 200 1 ). Teachers are not 
only required to change what they teach, but also how it is taught (Sherin, 2002b) .  Many 
teachers lack experience of learning in such environments (Huferd-Ackles, et al . ,  2004; 
Si lver & Srruth, 1 996) and as a consequence, their fundamental bel iefs about learning and 
teaching are chal lenged (Goos, Galbraith, & Renshaw, 2004; Rousseau, 2004; Weiss, 
Knapp, Hollweg, Burri l l ,  2002).  Moreover, traditional beliefs held by both teachers and 
students about the non-contentious nature of mathematics and about mathematical 
discourse as non-adversarial are put up for chal lenge (Weingrad, 1 998) .  As a result, 
teachers are often required to learn as they teach (Davies & Walker, 2005 ) .  
To effect change, teachers constructing such communities need models to  draw upon.  They 
need to know the pedagogical actions that are appropriate, and have an understanding of the 
mathematical practices which may emerge. Within the l iterature only a l imited number of 
studies have documented the formation of such communities. These have predorrunantly 
documented more senior level classrooms (e.g. ,  Borasi ,  1 992;  Brown, 200 1 ;  Goos, 2004; 
Go os et al . ,  2004; Lam pert, 1 990b, 200 l )  than those classrooms that are under examination 
29 
in the current study. There also appear to be no c lassroom studies with this focus available 
in New Zealand . 
The previous chapter highlighted the importance of the teacher' s role in developing inquiry 
cultures and associated proficient mathematical practices. Constructi ng less hierarchical, 
more interactive learning communities requires explicit teacher actions to ensure that al l 
students can access the di scourse (Goos et al . ,  2004; Lam pert, 200 I ;  Van Oers, 200 I ) . 
Examination of the research li terature reveals a range of studies of teachers who used 
specific pedagogical actions associated with various mathematical practices (e.g. ,  Bal l ,  
1 99 1 ,  1 993;  Bowers e t  al . ,  1 999; Brown & Renshaw, 2004; Cobb et al . ,  200 1 ;  Enyedy, 
2003 ; Kazemi & Stipek, 200 1 ;  Mercer, & Wegerif, 1 999b; Moschkovich, 2002b,  2004; 
O'Connor, 2002; Sherin ,  2002a; Sherin ,  Mendez, & Louis, 2004; Saxe, 2002; Van Oers, 
200 1 ;  Wood et al . ,  2006; Yackel, Rasmussen, & King, 200 1 ;  Zack & Graves, 200 1 ) . But 
the focus of these studies was not placed on the pedagogical actions the teachers took to 
scaffold student participation in construction and use of mathematical practices. Therefore 
this section begins with three studies i n  which the actions and beliefs of the teachers were 
closely documented as they established and worked in communities of mathematical 
inquiry. These studies provide a cohesive view of the teachers' actions, the roles they took, 
their beliefs and the mathematical practices which developed within the communities.  
3.2.1 MODELS OF TEACHERS MEDIATING MATHEMATICAL INQUIRY 
CULTURES 
The seminal work of Lam pert ( 1 990a; 1 990b; 1 99 1 ;  200 I )  has exemplified the key role 
teachers have in i nducting Grade 5 students i nto what Lampert (200 1 )  termed ' studying' . 
By  this is meant learning how to be proficient users of mathematical practices.  Lampert 
described studying as including "activ i ties l ike inquiring, discussing, thinking, reading 
careful ly, and examining closely" (p. 365 ) .  Central to Lampert ' s  notion of studying was 
discursive interaction and the development of productive discourse. She provided 
compel l ing evidence that she, as the teacher, took a significant role in orchestrating a 
respectful exchange of ideas, within extended conversations i nvolving inquiry and 
30 
chal lenge. Lampert, as teacher-researcher, structured her lessons to resemble mathematical 
arguments in which all participants engaged in disciplined public construction and 
evaluation of reasoning. An initial focus was placed on structuring the discourse norms. 
These norms were continual ly revised and reconceptualised so that over time the press for 
explanatory justification and general isation increased. 
Within Lam pert' s (200 I )  c lassroom, developing student autonomy and competence in 
mathematics included ensuring that students gained access to both the mathematical content 
di scussions and the social di scourse process ( i .e . ,  when and how to explain,  question, agree, 
disagree or challenge).  Lampert ' s  role as a skilfu l  l i stener was evident as she used 
instruction that shifted back and forth between mathematical content discussions and 
talking about the social discourse process. Lampert explicitly taught 'politeness' strategies, 
the essential qualities or norms for ways students might disagree. By affirming the 
importance of disagreement and inducting students into ways to question and disagree, 
students were then able to articulate their reasoning, accept opposing views, and negotiate 
at cognitively advanced levels .  In short, through her shaping of discursive interaction the 
students in the community learnt "the practice of mathematics" (p . 5 1 ) . 
Goos (2004) examined another c lassroom mathematical community where through specific 
pedagogical actions the students were inducted into the use of inquiry practices. Goos 
examined a Year 1 1 1 1 2  classroom community of i nquiry over 2 years. She observed that it 
was the teacher who provided predictable patterns for how the students were to engage in 
i nquiry discourse. He "scaffolded the students ' thinking by providing a predictable 
structure for inquiry through which he enacted his expectations regarding sense-making, 
ownership, self-monitoring, and justification" (p. 283). However, as the year progressed 
Goos reported that he "gradually  withdrew his support to pull  students forward into more 
independent engagement with mathematical ideas" (p. 283) .  To facil itate student  
participation in mathematical i nquiry the teacher demonstrated a commitment to  personal 
sense-making and wil l ingness 
to deal with more abstract ideas concerning conjecture, j ustification and proof. Typically he 
modeled this process of i nquiry by presenting students with a significant problem designed 
3 1  
to engage them with a new mathematical concept, eticiting their initial conjectures about the 
concept, withholding his  own judgement to maintain an authentic state of uncertainty 
regarding the validity of these conjectures, and orchestrating discussion or presenting 
further problems that would assist students to test their conjectures and justify their thinking 
to others. (p. 283) 
As students engaged interactively in communicative activi ty in the classroom they learnt 
the mathematical practices associated with inquiry. 
Based on the continuing study of senior classrooms Goos and her col leagues (2004) offered 
"five assumptions about doi ng and learn ing mathematic s  that appear[s ]  to be crucial to 
creating the culture of the community of mathemati cal inquiry" (p. 1 00) .  These 
assumptions (see Table 1 )  u nderpinning teacher and student actions were derived from 
classroom observations and interviews from a 2-year study of four Year 1 1 - 1 2  mathematics 
c lasses. Although construction and use of mathematical practices was not a key focus of the 
research report, evidence is provided that the mathematical practices are subsumed within 
the doing and learning of mathematics. 
32 
Table 1 
Assumptions about doing and learning mathematics implicit in teacher-student 
interactions. 
Assumptions Teacher actions Student actions 
Mathematical thinking The teacher models mathematical Students begin to offer 
is an act of sense- thinking using a dialogic format to invite conjectures and 
making, and rests on students to participate. justifications without the 
the processes of The teacher invites students to take teacher' s prompting. 
specializing, responsibi lity for the lesson content by During whole class 
generalising, providing i ntermediate or final steps in  discussion students ini tiate 
conjecturing and solutions or arguments in itiated by the argumentation between 
convincing teacher. themselves, without 
The teacher withholds judgment on teacher mediation. 
students' suggestions while i nviting 
comment or  critique from other students. 
The process of The teacher asks questions that encourage Students begin to point out 
mathematical inquiry students to question their assumptions and and correct their own and 
are accompanied by locate their errors. The teacher presents each other's errors, and 
habits of individual 'what if scenarios. those made by the teacher. 
reflection and self- Students ask their own 
monitoring 'what if questions. 
Mathematical thinking The teacher calls on students to clarify, Students spontaneously 
develops through elaborate, critique and justify their provide clarification, 
teacher scaffolding of assertions. elaboration, critiques, and 
the processes of The teacher structures students' thinking justifications. 
inquiry by asking questions that lead them Students take increasing 
through strategic steps. responsibility for 
suggesting strategic steps.  
Mathematical thinking The teacher structures social i nteractions Students form informal 
can be generated and between students, by asking them to groups to monitor their 
tested by students explain and justify ideas and strategies to progress, seek feedback on 
through participation each other. ideas, and explain ideas to 
in equal status peer each other. 
partnerships 
Interweaving of The teacher makes explicit reference to Students begin to debate 
fami liar and formal mathematical language, conventions and the appropriateness and 
knowledge helps symbolism, labelling conventions as relative advantages of 
students to adopt the traditions that permit communication. different symbol 
conventions of The teacher Links technical terms to conventions. 
mathematical commonsense meanings, and uses 
communication multiple representations of new terms and 
concepts. 
(From Goos et al . ,  2004, p. 99) 
33 
I n  another paper which described the senior c lassrooms Goos and col leagues (2002) 
provided important evidence of what happened when the teacher structured student 
participation in a joint construction zone. They described how peers of comparable 
experti se worked together in a context in which chal lenge was a common element. They 
outlined how the interplay between transactive dialogue and self-regulated decisions were 
significant in the creation of col laborative zones . Transacti ve reasoning was characterised 
as clari fication, elaboration, justification and cri tique of one ' s  own thinking and the 
thinking of others . Goos and her colleagues noted that the lack of transactive challenge led 
to student fai lure to construct a collaborative zone. 
These important studies i l lustrate that teachers in i nquiry based classrooms fulfi l led 
significant roles . They modeled the practice of inquiry, faci l i tated classroom communities 
of inquirers and were eo-participants in the mathematical practices. Initially they scaffolded 
student thinking toward development and use of proficient engagement in mathematical 
practices and later they faci l i tated student authoring of their own reasoning within the 
established intel lectual partnerships. These research studies have implication for the current 
study. Together, they suggest possibilities of what communities of mathematical i nquiry 
and student participation in mathematical practices might look l ike. They provide a useful 
background with which to view teacher actions as they develop student participation in  
mathematical practices. 
3.2.2 VARIATIONS IN PRACTICES OF CLASSROOM INQUIRY 
COMMUNITIES 
Whilst it  is widely documented that differences in the teaching and learning practices exist 
between more conventional classrooms and those of i nquiry communities, studies also 
report differences within inquiry communities themselves (see Kazemi & Stipek, 200 1 ) . 
The differences, attributed to the varying demands placed on expectations and obligations 
within the structured interaction patterns, have important implications for the mathematical 
practices which result .  
34 
Boaler (2003a) outlined how the varymg interaction patterns students experienced in  
mathematics classrooms resulted in  different students' reasoning patterns .  She examined 
the practices of three inquiry classroom teachers who used simi lar curricular. The 
discernible differences were directly related to how the students were held accountable to 
their own learning and the learning of the col lective within structured communication and 
participation patterns .  For example, one teacher over-structured the problem solving 
activity so that all cogni tive chal lenge was removed. A second teacher under-structured 
activity so that the students were left frustrated and without resources to proceed 
independently. However, the third teacher careful ly structured both the problem solving 
activity and her responses to student questions. She impl icitly directed the students to 
engage in and use mathematical practices to resolve their difficulties. Boaler explained that 
the teacher deflected authority back to the students to validate their own reasoning using 
mathematical practices and the discipli ne of mathematics itself. Through these actions she 
established a c lassroom culture in which all members had agency. Within this structure, 
higher levels of intel lectual reasoning were achieved as the students increasingly assumed 
responsibil ity to validate their own mathematical thinking and that of others. 
Simi larly, Wood and her colleagues (2006) provided persuasive evidence that higher levels 
of complexity in students' articulated mathematical reasoning were "closely related to the 
types of i nteraction patterns that differentiated class discussions among . .  .4 classroom 
cultures" (p. 222) .  Their analysis of 42 lessons in five c lasses of 7 and 8 year olds revealed 
qualitative differences between students in conventional and reform-oriented classrooms. 
These differences were manifest in both the nature of their i nteractions and what they 
articulated. In accord with Boaler' s (2003a) findings, significant differences were directly 
attributed to characteristics of the cultures.  Those which required greater i nvolvement from 
participants resulted in  higher intel lectual levels of verbalised thinking. 
Bui lding on her earlier studies (e.g., Wood, 1 994, 2002; Wood & Turner-Vorbeck, 200 1 ; 
Wood & McNeal , 2003)  Wood' s  (Wood et al . ,  2006) recently reported study clarifies how 
student reasoning is extended to higher intellectual levels in inquiry and argumentative 
cultures.  In a previous study Wood (2002) reported on the research of s ix  Year 2/3 c lasses 
35 
over a period of 2 years. Using analysis of the general ised "patterns of interactive and 
communicative exchanges" (p. 64), Wood constructed a theoretical framework. Key 
dimensions i nvolved the students' 'responsibility for thinking' and ' responsibility for 
participation' .  Four classroom cultures were deli neated: traditional, strategy reporting, 
i nquiry and argument. Wood and McNeal (2003) subsequently reconceptualised these as 
conventional, strategy reporting, and inquiry/argument .  They also added a thinki ng 
dimension . 
In Wood and McNeal ' s  (2003) classroom research they found that teacher use of 
questioning and prompts were significant factors i n  shifti ng student reasoning toward more 
complex intel lectual levels. When comparing the questioning behaviours between 
traditional and reform classrooms they noted that 
questioning in conventional teaching was directed at prompting chi ldren to give teacher 
expected information, while strategy reporting and inquiry/argument emphasised student 
exploration of methods and justification of student ideas. Comparison of strategy reporting 
and inquiry/argument revealed teachers differed in the frequency of prompting for 
mathematical thinking during inquiry interactions and situations involving proof and 
justification. The resolution of differences in students' answers was dealt with differently by 
teachers in reform cultures. Teachers in strategy reporting class cultures emphasised proof 
of a correct answer through the use of concrete objects, while teachers in inquiry/argument 
relied on chi ldren' s  explanations and justification to resolve differences in reasoning.  (p. 
439-440) 
The researchers reported that those classrooms which supported an argument culture had an 
added element of challenge and debate. Dispute or chal lenge from the teacher or students 
i ni tiated further debate and prompted need for justification to support reasoning. 
Responsibi l i ty for sense-making was most evident in i nquiry and argument cultures. 
Interactions between teachers and students increased as the discussion cultures shifted from 
traditional, to report, to i nquiry and argument. This was matched with an increase i n  
frequency and complexity of  teacher questioning and prompts. 
36 
These significant studies i l lustrate the consequences of teacher structuring of the interaction 
structures-the participation and communication patterns .  Col lectively, the studies draw 
attention to the differential outcomes in both the qual ity of student  mathematical 
understandings and their i nteractions which resulted from the different interactive 
structures. They also provided evidence that only in inquiry/argument cultures were 
students provided with opportunities to construct a shared perspective. In an inquiry culture 
student thinking 
was extended to inc lude whether a method or resu lt is  reasonable . . .  the pulling together of 
ideas for making a judgement . .  . and identifying flaws . . .  and strengthening arguments by 
considering the mathematics from a different perspective . . .  -al l  as a process for 
establishing shared mathematical meaning. (Wood et a l . ,  2006, p. 248) 
Thus, Wood and her col leagues  provided evidence that, through discursive interaction, 
students appropriated the knowledge and ski l ls of when and how to engage in mathematical 
i nquiry and argumentation-key aspects of coming to know and use mathematical 
practices. 
These various research studies have implications for the current study. They i l lustrate the 
importance of teachers developing mathematical learning cultures in which students are 
able to mutual ly engage in  the discourse of inquiry and argumentation .  They also revealed 
the significance of the structured interaction patterns .  Collectively these research studies 
provide convincing evidence that if  students are to engage in inquiry and reasoned 
mathematical talk, then teachers need to explicitly frame classroom interaction patterns.  
3.2.3 THE SOCIO-CULTURAL AND MATHEMATICAL NORMS OF 
COMMUNITIES OF MA THE MA TICAL INQUIRY 
Recent studies have outli ned how differences between c lassroom mathematics 
environments are directly l inked to the enacted social-cultural and mathematical norms of 
classroom communities (Cobb et al . ,  1 993;  Kazemi,  1 998;  Sullivan et al . ,  2002;  Wood, 
200 1 ; Yackel & Cobb, 1 996) .  Both norms are important to the way in which students 
consider and use mathematical practices. Socio-cultural norms affect how students 
37 
construct a mathematical disposition, whereas the mathematical norms, al so referred to as 
sociomathematical norms, affect the development of mathematical agency and autonomy. 
Cobb and his colleagues ( 1 993) report on a year-long teaching experiment in a second?
grade classroom. They describe how the teacher' s explicit scaffolding led to the students 
knowing when and how to talk about 'talking about mathematics' . They outline how other 
i nteraction practices were renegotiated as required. For example, the students learnt that the 
focus of discussion was on solution strategy, not just provision of correct answers . They 
also learnt that all contributions (including errors) had value. The teacher used social 
situations to discuss explicitly her expectations and obligations of the participants in joint 
mathematical activi ty .  Higgins (2005) ,  i n  a New Zealand study, also i l lustrated the 
importance of explicit teacher scaffolding of interaction norms. Higgins outlined the actions 
a Numeracy teacher of predominantly Maori students took to create col lective 
responsibility for mathematical reasoning. The teacher encouraged the students to risk-take 
in reasoning, while at the same time ensuring that their mana, or feeling of self-worth, was 
protected. 
The previous studies, and those described in other studies (e .g. , Blunk, 1 998;  Cobb et al . ,  
200 1 ;  Huferd-Ackles et al . ,  2004; Kazemi,  1 998 ;  Kazemi & Stipek, 200 1 ;  Whi te, 2003 ;  
Wood, 1 999; Yackel & Cobb, 1 996; Zack & Graves, 200 1 ), identify specific conditions 
which supported rich mathematical activi ty in i nquiry environments.  These include risk?
taking, safe learning environments, polite and respectful exchange of ideas, provision of 
thinking time, valuing all contributions, collective analysis of errors and a need for 
agreement and disagreement during discursive exchanges. In all of these studies teacher 
enactment of these socio-cultural norms affected the values and beliefs students held 
towards mathematics-their mathematical disposi tion . 
Mathematical agency or i ntel lectual autonomy however, results from enacted mathematical 
norms. McClai n and Cobb (200 1 ) supplied persuasive evidence of how 1 st Grade students 
gained intel lectual autonomy as they negotiated and renegotiated sociomathematical norms 
(mathematical norms) concerning appropriate mathematical explanation or justi fication. 
38  
Kazemi and Stipek (200 1 )  compared the teaching practices used by four 4th and 5th grade 
teachers. They i l lustrated that although the social norms (socio-cultural norms) in each 
classroom were similar, the sociomathematical norms (mathematical norms) of one 
classroom were markedly different. As a result, mathematical explanations provided by  
students i n  that c lassroom were comprised of  argumentation, justification and verification 
of reasoning. The students in thi s classroom analysed and validated their reasoning, and the 
reasoning of others. Collectively, these studies and many others (e.g., Bowers et al . ,  1 999; 
Cobb et al . ,  200 1 ; McClain, Cobb & Gravemeijer, 2000; Yackel ,  1 995 ; Zack & Graves, 
200 1 )  provide strong evidence that autonomous thinkers emerge when teachers engage 
their students in negotiation and renegotiation of socio-cultural and mathematical norms. 
Autonomy and agency to val idate one ' s  own reasoning and the reasoning of others ' are 
central to the mathematical practices used by successful mathematical learners and users . 
3.2.4 INTELLECTUAL PARTNERSHIPS IN THE MATHEMATICAL 
DISCOURSE OF INQUIRY 
The development of intellectual partnerships i s  fundamental to student engagement i n  
mathematical practices at proficient levels of  expertise. Intel lectual partnerships evolve 
when students are required to use their own autonomy or authority i n  critical analysis and 
validation of their own and others ' mathematical reasoning. Within these partnerships all 
participants have a voice and shared authority in mathematical i nquiry and the validation of 
reasoning (Amit & Fried, 2005 ; Povey, Burton, Angier, & Boylan, 2004). 
Stigler and Hiebert' s ( 1 999) large scale comparative study of Japanese and American 
teachers demonstrated that many of the American teachers in their study perceived 
themselves, or were perceived by their students as the main source of authority. As external 
authority figures they had the responsibility to ensure that their students learnt certain 
mathematical rules and procedures. They also assumed that it  was their responsibi l i ty to ask 
all the questions and validate answers. Hamm and Perry (2002) investigated the practices of 
six American 1 st Grade teachers and reported the same findings. The teachers, despite their 
reported intent to require students to validate their reasoning, seldom stepped away from 
their authoritarian role .  These studies and others (e.g. ,  Skott, 2004; Stipek, Givven, Salmon, 
39 
MacGyvers, & Valanne, 200 1 ;  Weiss et al . ,  2002) i l lustrate the way in  which the beliefs 
held by the teachers and students affect how authori ty was viewed in the c lassrooms. 
Intel lectual partnerships are grounded in col laborative dialogue (Amit & Fried, 2005) .  Amit 
and Fried explained that when specific students or the teacher are considered the authority 
in discussion, collaborative dialogue is difficult . They used two case studies of 8th Grade 
classrooms to i l lustrate the effects of externally attributed authority. When teachers and 
more expert peers were given immense and unquestioned authority thi s resulted in a lack of 
reflective analysis on their part or chal lenge to students' sense-making. These researchers 
argue that authori ty should be dynamic and fluid and that teachers must work "to make 
their  students i nto colleagues who final ly  wi l l  completely share authority with them" (p. 
1 65 ) .  Researchers have noted however, that construction of intel lectual partnerships is a 
lengthy process which requires explicit teacher actions (Boaler, 2003a; Goos et al . ,  2004; 
McCrone, 2005 ; Povey et al . ,  2004) 
In examining inquiry environments that provide space for intellectual partnerships to 
develop, research studies highl ight the importance of rich chal lenging mathematical 
activity. Researchers argue that teachers need to make mathematical activity problematic in 
a form which represents real disciplinary specific inquiry (Cornelius & Herrenkohl,  2004; 
Engle & Conant, 2002; Forman, et al . ,  1 998; Lampert, 1 990b; 200 1 ;  O'Connor, 2002; 
Peressini  & Knuth, 2000; Smith & Stein, 1 998;  Stein & Smith, 1 998) .  Stein (200 1 )  
maintained that rich problematic mathematical tasks are ones which promote 
argumentation. These problems, which are often open-ended and el icit  multiple strategies, 
force students to examine the general ity of their reasoning, and often modify their 
reasoning.  In a study of middle school students' engagement in rich problematic tasks Stein 
i l lustrated students' use of justifi cation i ncluding ini tiating and backing claims 
appropriately, critically evaluating their own and others' arguments and validating their 
reasoning during extended dialogue .  
Lampert ( 1 990b, 1 99 1 )  and O'Connor (2002) both used position driven discussions to elicit 
multiple strategies and solution pathways. Lampert ( 1 99 1 )  i l lustrated how through 
40 
discursive dialogue the students learnt disciplinary inquiry and explanatory justification .  
Position driven discussions involve a teacher facil itating discussion which centres on a 
framing question .  For example, in O'Connor ' s  study in  a 5 th Grade classroom the teacher 
posed the question 'can al l fractions be turned into a decimal ' ?  The teacher ski lfu l ly  
managed the tension between pursuit of collective reasoning and the diverse individual 
contributions using the "divergences to create a discussion with some of the properties of 
real mathematical discovery" (p. 1 80) .  The discussion concluded with a shared 
understanding which had been val idated by the col lective community. 
Shared ownership and active engagement i n  the discourse i s  central to students' developing 
reflective val idation of their own reasoning and the reasoning of others. The work of Boaler 
(2003) and Wood and her colleagues (2006) (see section 3 .2 .2)  i l lustrated that a press 
towards inquiry and argumentation created a community of validators . Likewise, Huferd?
Ackles et al. (2004) outlined how an elementary teacher shifted classroom discourse 
patterns and created a community of validators. The teacher explicitly scaffolded the 
discourse toward reasoned coll aborative talk. Shifts in the discourse were matched with 
increased levels of student investment i n  their own and col lective reasoning. As the teacher 
progressively shifted toward a more faci l i tative position, student thinking became more 
central and the students learnt to talk with authoritative mathematical understanding. 
Teacher researcher Ball ( 1 99 1 ,  1 993)  also i l lustrated in an elementary classroom how the 
teacher' s  structuring of social i nteractions provided the students with increased 
opportunities to col lectively engage in and learn the reasoning discourse of mathematical 
practices. Bal l  noted that the students' growth in use of the discourse was matched with 
their sense of agency in the classroom. 
Cobb et al .  (200 1 )  also l inked teacher scaffolding of shared ownership of i nquiry discourse 
with student perceptions of authority and agency. Students in their study progressively 
shifted from peripheral to more substantial participatory positions as they participated in  the 
reasoning discourse related to measure activities. The students assumed responsibil ity to 
validate their own mathematical thinking, rather than using the teacher, or text, as sources 
of external authority. Hunter (2006) and Young-Loveridge (2005a) in two recent New 
4 1  
Zealand studies also provide convincing evidence that students in inquiry focused cultures 
viewed both mathematics and their role as mathematical learners differently. In both studies 
involving students in middle school,  the students recognised that it  was their responsibili ty 
to validate their own reasoning and the reasoning of others. They indicated that they took 
communal responsibil ity for sense-making and described a proficient use of mathematical 
practices to do so. 
Together, these studies of inquiry communities i l lustrate that both intel lectual autonomy 
and shared understanding are supported when there is a press toward inquiry. Mathematical 
communities develop when c lassroom cultures are premised on the reciprocal responsibi l i ty 
of all participants to engage actively in col lective mathematical reasoning. Implications for 
the current study include the need to consider how teachers and students negotiate the 
socio-cultural and mathematical norms and how these affect the mathematical practices. 
Likewise, consideration i s  needed of the actions teachers take to develop intel lectual 
partnerships in which students become successively more proficient in the use of 
mathematical practices. There appears to be l imited research studies available in New 
Zealand related to these factors . 
3.3 FORMS OF DISCOURSE USED IN MATHEMATICS 
CLASSROOMS 
The forms of talk that students participate in shape the mathematical practices they come to 
know and use. They al so affect the students' bel iefs and influence their goals during 
mathematical activity. 
3.3.1 UNIVOCAL AND DIALOGIC DISCOURSE 
How teachers socialise students' use of the di scipline-specific mathematical discourse is of 
prime importance. The function of discourse in more traditional forms of mathematical talk 
focuses on transmitting meaning (univocal discourse) framed around what the teacher 
wants to hear. Mehan ' s  ( 1 979) examination of traditional pedagogies noted the typicality of 
the univocal form in which the teacher i nitiated the discourse, selected students to respond, 
42 
evaluated responses and provided feedback or questions unti l  the answers sought were 
provided. In contrast, inquiry discourse is focussed on generating meaning (dialogic 
discourse) .  It requires a more balanced partnership in the dialogue (Knuth & Peressini ,  
200 1 ;  Lotman, 1 988 ;  Manouchehri & St. John,  2006; Wertsch, 1 99 1 ) . 
Knuth and Peressi ni (200 1 )  provide v ignettes from a 4 year professional development 
project aimed to support teachers to foster meaningful mathematical discourse. The first 
vignette i l lustrates how a teacher structured univocal discourse through unintentional ly  
shaping the di scourse so that it aligned with her own perspective. The same teacher in the 
second vignette structured dialogic discourse so that multiple layers of meaning were 
generated. The vignettes i l lustrated the complexities and chal lenges for teachers to 
orchestrate i nquiry community interactional patterns .  This is of particular interest because 
both forms of discourse are always on a continuum-univocal discourse ceases at the point  
where listeners have received an intended message and dialogic discourse commences 
within the structure of "give-and-take communication" (Knuth & Peressini ,  p.  3 2 1  ) .  
Restructuring di scourse patterns is an activity fraught with many pitfal ls .  This i s  
particularly so  because the more conventional forms of classroom discourse in which 
teacher talk has dominated is likely to be the most common form of talk both students and 
teachers have experienced in their former mathematics c lassrooms (Lampert & Cobb, 
2003) .  Nathan and Knuth (2003) analysed data from a 2 year study of a 6th Grade c lassroom 
teacher as she reconstructed classroom interactional strategies to support dialogic discourse. 
In the first year, although she appeared to scaffold more open communication norms, her 
actions i ndicated that she retained a central position and dominated the flow of discourse. 
In the second year, she faci l i tated increased mathematical communication but removed 
herself from scaffolding the participants' mathematical reasoning. This resulted in 
discourse which lacked rigorous mathematical analysis and argumentation. 
These studies and others (e.g., Kazemi & Franke, 2004; Sherin, 2002a, 2002b; Sherin et al . ,  
2004; Steinberg, Empson ,  & Carpenter, 2004) i l lustrate the multiple issues teachers face in  
managing the discourse of  inquiry and argumentation .  One challenge to teachers i s  the 
43 
messy dialogue which evolves when student thinking is central . Teachers also face conflict 
when structuring learning opportunities which may i nitial ly give a sense of being 
unsuccessful (e.g., Schwan Smith, 2000). Other studies identified problematic situations are 
related to teachers' need to finely balance their role in the dialogue (Chazan & Bal l ,  1 999; 
Lobato, Clarke, & Ellis ,  2005 ) .  For example, when and how should teachers i nsert 
questions and chal lenge? When should they i nsert their mathematical ideas and 
explanations to ensure that student mathematical understandings are being analysed and 
advanced? All these issues have implications for teachers in the current study as they 
orchestrate the discourse of inquiry and argumentation .  
3.3.2 INQUIRY AND ARGUMENTATION 
Framed in a sociocultural learning perspective, proficient mathematical practices evolve 
through collective participation in discursive i nteraction which has as its focus  the 
emergence of shared meaning (Forman, 2003; Mercer, 2000; Wells, 1 999) .  In  this frame, 
student access to and valuing of the discourse of i nquiry and argumentation is recogni sed as 
key to their participation i n  mathematical practices. 
In  separate studies involving New Zealand students, Meaney (2002) and B icknel l  ( 1 999) 
both reported the ambivalent beliefs some students held toward the value of communicating 
mathematical explanations, or the importance these had for learning mathematics.  
Simi larly, Young-Loveridge (2005b) and her colleagues (Young-Loveridge, Taylor, & 
Hawera, 2004) reported considerable variation i n  student v iews towards the value of 
explaining their strategies and listening to others . Interview data of the views of 1 80 nine to 
eleven year-olds included students who had been in discussion-intensive New Zealand 
Numeracy Development Project classrooms. Although the students stated that they liked to 
explain their reasoning, many did not recognise the value of explaining their reasoning and 
discounted the value of attending to the thinking  of others. Nor did they realise the 
importance of discursive exchange as a means to advance their own thinking or in the 
construction of a col lective view. Young-Loveridge suggested that the responses of the 
majority of i nterviewees indicated that "they continued to regard mathematics from an 
individualistic perspective as being a private activ i ty of l ittle or no concern to others in their 
44 
class" (p. 28) .  Their  identity as mathematical learners remained focused on a model of 
mathematics learning as acquiring knowledge rather than participating in  the construction 
of understandings through interactions in communities of learners (Cobb & Lampert, 2003 ;  
Sfard, 1 998, 200 1 ,  2003) .  
Numerous other studies have i l lustrated the difficulties students encounter when engaging  
in communal discussions (e.g. ,  Gee & Clinton, 2000; Lampert, Rittenhouse, & Crumbaugh, 
1 996; McCrone, 2005 ; Mercer, Dawes, Wegerif, & Sams, 2004; Rojas-Drummond & 
Mercer, 2003 ; Rojas-Drummond, Perez, Velez, Gomez, & Mendoza, 2003; Rowe & 
Bicknell ,  2004; Wegerif & Mercer, 2000; Wegerif, Mercer, & Rojas-Drummond, 1 999).  
Irwin and Woodward (2006) ,  i n  a New Zealand study, examined the talk that Year 5-6 
students used when working in  small cooperative problem solving groups independent of 
the teacher. Whilst the teacher consistently modeled inquiry strategies during large group 
discussions Irwin and her col league observed that when the students worked together 
beyond the teacher' s c lose scrutiny they exhibited difficulties using i nquiry talk. The 
teacher verbally i nstructed them to work cooperatively but did not discuss why nor provide 
explicit scaffolding. At times i ndividual students appeared to adopt aspects of i nquiry talk 
but predominantly competitive talk and the use of procedural rules prevailed. As a result ,  
the use of unproductive forms of talk prevented the construction of communal views. 
A lack of expl icit teacher guidance on student use of shared language may explain why 
students do not real i se the benefits which accrue through l i stening closely to  the reasoning 
of others, or developing a col lective view. For example, Sfard and Kieran (200 1 )  i llustrated 
the difficulties two 1 3  year old students encountered during a 2 month teaching sequence. 
Sfard and Kieran examined their mathematical discussion, within the context of the 
immediate mathematical content, to explore the effectiveness of their communication. 
Although, both students' ach ievement results significantly improved, c lose data analysis 
i l lustrated that this enhanced performance could not be attributed to their conversations. 
Analysis showed that the students had placed l ittle value on the other' s contributions during 
discussion. 
45 
Mercer ( 1 995, 2000) and his colleagues (e.g. ,  Wegerif, Mercer, & Dawes, 1 999; Wegerif et 
al . ,  1 999) examined the forms of mathematical talk students used when working with their 
peers in small groups .  In their many studies in English and Mexican schools they have 
consi stently noted that students use three different forms of talk, and these in turn use 
different degrees of engaging constructi vely with each others ' reasoning. The most 
productive form of talk the students engage in ,  they termed exploratory talk. Exploratory 
talk they characterised as language which included such phrases as 'because ' ,  'if' , 'why ' ,  ' I  
think' and 'agree' .  However, without explicit teacher intervention and scaffolding they 
noted that the students consistently used the two other forms of unproductive talk the 
researchers termed disputational and cumulative talk. 
Disputational talk, Mercer (2000) has described as discussion characteri sed by cyclic forms 
of assertions and counter-assertions which remain unexamined by the participants. Short 
utterances and lack of expl icit reasoning are used as the participants primari ly  focus on self?
defence. Rather than trying to reach joint agreement individuals struggle to hold control .  
Although actions are not overtly uncooperative Mercer noted participants ' perspectives 
"compete with rather than complement each other" (p. 97) .  
Cumulative talk, Mercer (2000) described as lacking the element of confrontation, that i s  
questions and argument are avoided. During this form of talk the participants build on each 
others' thinking adding their own ideas and "in a mutual ly  supportive, uncritical way 
construct together a body of shared knowledge and understanding" (p. 97). The 
construction of a col lective view is accomplished but not evaluated by the students. 
Exploratory talk i nvolves students' mutual investment in developing col lective reasoning 
and occurs as a result of reasoned debate. Exploratory talk supports students engaging in 
each other' s reasoning in  mutually constructed zones of proximal development. 
Exploratory talk is  that in which partners engage critically but constructively with each 
other' s ideas. Statements and suggestions are sought and offered for joint consideration. 
These may be challenged and counter-challenged, but chal lenges are justified and 
46 
alternative hypotheses are offered. In exploratory talk, knowledge is made public ly  
accountable and reason ing is visible in  the talk. (Mercer & Wegerif, 1 999b, p. 96-7) 
Teacher actions to ensure that students engage at high levels of inquiry make certain that 
mathematical reasoning is visible and therefore avai lable for challenge and debate. At the 
same time, the accessibi lity of the reasoning means that a col lective view can be 
constructed (Mercer, 2002; Wel ls, 1 999). 
As previously discussed, the forms of talk that students use in  mathematical activity 
influence their goals and attitudes and the mathematical practices they use. Whilst recent 
New Zealand policy documents (e.g. ,  Ministry of Education, 1 992, 2006) have promoted 
notions of students working interactively, making explanations, justification and 
general isations of their mathematical reasoning, there is a gap in the research l i terature in  
New Zealand on the forms of  talk and actions teachers should take which best support these 
mathematical practices. 
3.3.3 INTERACTIONAL STRATEGIES USED BY TEACHERS TO ENGAGE 
STUDENTS IN THE DISCOURSE 
The previous section outl ined the complexities i nvolved in teacher orchestration of learning 
partnerships with students i n  the discourse. Beyond ensuring that the students understand 
the rights and responsibi l ities enacted in c lassrooms, teachers are also required to "align 
students with each other and with her, as proponents of a particular hypothesis or position" 
to ensure collective intellectual reasoning (O'Connor & Michaels, 1 996, p. 66) .  
O'Connor and Michaels ( 1 996) i l lustrated that higher levels of reasoning were fostered 
when teachers scaffolded students to participate in collective dialogue. Forman and her 
col leagues ( 1 998)  i l lustrated ' teacher revoicing' in a middle school classroom investigation 
of area measure. The teacher revoiced and thus aligned and realigned the students to engage 
in collective mathematical argumentation .  The study provided clear evidence that the 
teacher use of revoic ing positioned the students to use mathematical practices to examine 
47 
and explore the generality of mathematical concepts-rather than rely on mathematical 
rules and algorithms. 
Teachers have a central role in inducting their students i nto the construction of persuasive 
mathematical arguments. Drawing on episodes from a middle school mathematics lesson 
Strom, Kemeny, Lehrer, and Forman (200 1 )  i l lustrated how the teacher enculturated the 
students i nto the mathematical practices. The model s  of teacher revoicing showed the 
complexities involved in the posi tioning of students as intel lectual owners of their 
reasoning. In the extended dialogue, all forms of reasoning were seriously considered, 
mathematical terms were clarified, and the contexts of arguments were integrated with 
previously introduced ideas and terms. Through the teacher' s choice of specific words and 
indirect speech, students were animated and positioned to intel lectually engage in col lective 
argumentation . 
Similarly, Empson (2003) i l lustrated how the si tuated identities of two low performing 
Grade One students were enhanced through explicit teacher revoicing, and animating 
activity. Revoicing was a powerful tool the teacher used to ensure students were considered 
mathematically competent by their peers. This resu lted, in them contributing productive 
mathematical ideas within a variety of roles. These included the roles of problem solvers 
and claim makers. Empson ' s  study highlights a key equity issue of relevance to the current 
study-that teachers ensure that all students are able to engage authoritatively in 
mathematical activity and gain intel lectual agency. 
Collectively these studies and many more l ike them (e.g. ,  Cobb et al . ,  200 1 ;  Forman & 
Ansel l ,  2002; Huferd-Ackles et al . ,  Kazemi & Stipek, 200 1 ;  Yackel ,  2002; Zack & Graves, 
200 1 )  report posi tive outcomes for students when their teachers al igned them to take 
specific roles in the use of inquiry and argumentation in the development of col lective 
thinking. The studies demonstrated that through explicit teacher positioni ng the students 
gained a sense of authorship and authority toward their mathematical activity. Positive 
mathematical dispositions were developed in  which they saw themselves as users and doers 
of mathematics. 
48 
3.3.4 THE MATHEMATICAL DISCOURSE AND THE DEVELOPMENT OF 
SITUATED IDENTITIES 
In recent years there has been considerable discussion and debate as to how diverse student 
groups are positioned i nto taking-or not taking-appropriate intellectual roles in 
mathematics classrooms. Explanations have varied. One theme in  the li terature of relevance 
to the current study, relates to the patterns of interaction and how these mediate identities 
for diverse students (e.g. ,  B lack, 2004; Baxter, Woodward, Voorhies, & Olson, 2002 ; 
Boater, 2006a, 2006b; Civil  & Planas, 2004; Cobb & Hodge, 2002; Empson, 2003; Khisty 
& Chval, 2002; Lubienski, 2000a, 2000b; Martin, Pourdavood, & Carignan, 2005 ; 
Moschkovich, 2002b; Planas & Civil ,  2002 ; Planas & Gorgori6, 2004; Pourdavood, Svec, 
& Cowen, 2005 ; Varenne & McDermott, 1 998; White ,  2003) .  
Planas and Gorgori6 (2004) i l lustrated how a teacher' s actions i nfluenced how students 
perceive themselves and others as valid contributors to mathematical dialogue. Space to 
engage in mathematical activity provided differential ly  by the teacher resulted in different 
groups of students constructing different mathematical meaning and identity. These 
researchers revealed how a teacher unintentional ly i nfluenced how and when the local and 
immigrant students participated in mathematical activity. The teacher posi tioned the local 
students to discuss and argue their reasoni ng through to sense-making. The immigrant 
students were required to enumerate numbers rather than engage in explanation or 
argumentation .  The differential treatment meant that immigrant students ' were unable to 
contribute to the discourse; their contributions were perceived as inval id by local students. 
Unlike the local students the immigrant students were not socialised i nto how to participate 
i n  col lective argumentation and construct communal reasoning. This  resulted in their 
construction of different identities as mathematicians. 
Pourdavood et al . (2005) maintain that teachers must ensure that al l students are explicitly 
scaffolded to have a 'voice' and agency in the discourse . For example, White (2003)  
showed how 3rd Grade students were positioned as agents responsible for validation of their 
own thi nking and the thinking of others . Pivotal to their agency were norms which placed 
49 
value on sharing and respecting each others' contribution within mathematical dialogue. 
The actions of two teachers in White ' s  study emphasised to the classroom community that 
all ideas would be seriously considered and constructively examined and explored. 
Martin et al. (2005) also i l lustrated how a pedagogical focus on engaging students in 
col lective reasoned inquiry discourse resulted in the development of a positive 
mathematical disposition and sense of agency. These researchers examined the 
mathematical experiences of two low socioeconomic status African American children.  
They drew on classroom observations, interviews with the students and their parents and 
teachers. They also observed the two students during mathematical activity paired with 
students from more conventional schools. Significantly, their observations revealed that the 
two students maintai ned their sense of 'voice ' i ndependently of their teacher as they drew 
the 'outsider students ' into working collaboratively. 
Collectively, these studies i l lustrated the different identities students construct as an 
outcome of their participation in collective mathematical reasoning. Persuasive evidence 
was provided that the salient features of teachers ' actions affected the forms of discourse 
the students participated in .  Likewise, it affected how they considered the outcome of 
discursive i nteraction on their knowledge construction . 
Ensuring that al l students are socialised into the discourse of mathematical i nquiry i s  a key 
equity issue (Boaler, 2006a, 2006b; Martin, et al . ,  2005 ; Gutierrez, 2002 ; MacFarlane, 
2004; Moschkovich, 2002b). In addition, specific to the New Zealand context of this study 
MacFarlane (2004) promotes the need for teachers to be "culturally responsive" (p. 27)  to 
Maori and other diverse students. Cultural ly responsive teaching provides space "which i s  
cultural ly, as  wel l  as  academical ly  and socially, responsive" (p .  6 1  ) .  This i s  of particular 
relevance to the current study because the school has a wide range of diverse learners i n  the 
classrooms. 
3.3.5 EXAMPLES OF FRAMEWORKS USED TO STRUCTURE COLLECTIVE 
REASONING DURING INQUIRY AND ARGUMENTATION 
50 
This chapter has focused on the close connections between student participation in i nquiry 
and argumentation and the development of mathematical practices .  How teachers scaffold 
student participation in mathematical practices and the communication and participation 
structures they use are significant. The complexities involved in achieving this and the 
many issues involved have been identified. This fi nal section reviews a number o f  
international studies that i l lustrate the posi tive outcomes for i ndividual and communal 
learning when specific frameworks are used which scaffold student inquiry and 
argumentation in zones of proximal development. These studies were influential i n  
informing the implementation o f  the current study. 
The first framework is provided by Mercer (2000) and his col leagues. They constructed a 
programme they termed 'Talk Lessons ' . The ground ru les teachers used specified how the 
students interacted . They included the following criteria 
( I )  all relevant information is shared; (2) the group seeks to reach agreement; (3) the group 
takes responsibi lity for decisions; (4) reasons are expected; (5) challenges are accepted; (6)  
alternatives are discussed before a decision is taken; and (7) al l in the group are encouraged 
to speak by other group members. (Mercer & Wegerif, 1 999a, p.99) 
In the planned intervention of nine lessons students were explicitly scaffolded to engage 
productively in col lective argumentation. These lessons focused on developing a communal 
view. In order to achieve interthinking lessons began with "integrated teacher-led whole 
class dialogue and group activity, so that chi ldren could be expected to begin their activ i ty 
and discussion with a shared conception of relevant knowledge and of how they shou ld talk 
and think together effectively" (Rojas-Drummond & Mercer, 2003, p .  1 03) .  
Rojas-Drummond and Mercer (2003) reported that the students in both the target and 
control classes were assessed using the Ravens Progressive Matrices psychological test 
prior to being trained in the use of ' talk lessons ' .  Post assessment used the problem solv ing 
section of the Ravens Progressive Matrices. Completed individual ly  and in  small problem 
solv ing groups,  the results i l lustrated significant differences. The target c lass students used 
more exploratory talk, and through using exploratory talk were more successful at solving 
5 1  
the Raven' s puzzles. What was more significant was that they were also more successful at 
solving problems alone. Their improved relative performance indicated that they had 
appropriated the grounds rules of the exploratory talk, and so were "able to carry on a kind 
of si lent rational dialogue with themselves" (p .  1 05) .  
Simi lar results were attained in a larger study, which used the alternative British official 
Standard Attainment Task Mathematics and Science assessments (Rojas-Drummond & 
Mercer, 2003). Studies using the 'Talk-Lesson ' format have also been replicated in Mexico 
with comparable results. In the large Mexican study Rojas-Drummond and Zapata (2004) 
outl ined how expl icit teacher scaffolding of students in the use of exploratory talk resulted 
in progressive shifts toward increased levels of exploratory talk and argumentation .  
Through the use of talk-lessons the students negotiated more frequently, constructed more 
arguments and provided a variety of perspectives for consideration. Elaborated reasoning 
was articulated in that they 
tended to present their arguments in a more explicit way, and to provide more supports to 
sustain thei r opinions, making their reasoning more expl icit i n  their language. They also 
used a greater number and variety of l inks to mark the logical organisation between 
different components of arguments, such as the assertions, the supports and the conclusions, 
thus making their interventions more understandable to others. (p. 554) 
Another international study which reported on the use of a structure which scaffolded 
students to participate in mutual inquiry of open-ended problems is described by Alr0 and 
Skovmose (2002) .  Teachers applied their Inquiry-Cooperation Model across age groups . 
The students were specifical ly taught to actively l isten,  and identify each others ' 
perspectives. Claims were proposed as the tentative positions and thinking aloud made 
reasoning visible and avai lable for scrutiny. Ideas were clarified through teacher or student  
reformulation and then subjected to  challenge. These researchers maintained that the 
participation structure scaffolded the students' development of "mathemacy" (p. 1 36) .  
Their understanding shifted beyond a focus on numbers and rules to also exhibit a reflective 
autonomy toward considering the validity of their reasoning. 
52 
Brown (200 1 ,  2005) and his colleague (Brown & Renshaw, 1 996, 1 999, 2000, 2004; 
Renshaw & Brown, 1 997) reported on collective argumentation, a framework that they 
used to structure col lective reasoning. Thi s  framework was designed for teachers to 
scaffold student participation in i nquiry and argumentation in zones of proximal 
development. The collective argumentation  was organised around the following key 
strategies that require students to 
represent the task or problem alone, compare representations within a small group of peers, 
explain and justify the various representations to each other in the small group, and final ly 
present the group' s  ideas and representations to the class to test their acceptance by the 
wider community of peers and teacher. (Brown & Renshaw, 2004, p. 1 36) 
Brown (2005) reported on the outcome of a teacher ' s  use of col lective argumentation in a 
Year 7 longitudinal classroom study. He outl ined how the students engaged in col laborative 
interactions using the structured format and as a result constructed and reconstructed 
dynamic and generative zpds . The zpd as an intel lectual or social space made the reasoning 
visible and supported critical examination and evaluation of key mathematical concepts . 
The learning situations and the joint partnerships created were not dependent on 'expert?
novice ' partnerships but rather were often compri sed of individuals with i ncomplete but 
relatively equal knowledge. In that situation the interactions created multi-directional zpds. 
For example, Brown and Renshaw (2004) outlined how the teacher used col lective 
argumentation to scaffold students to "adopt different speaking posi tions or voices" (p. 1 2) .  
They described how students began from 'my voice ' ,  shifted to  ' your voice' and final ly  
developed 'our voice ' .  Thus, ownership of explanations were shared and defended within 
intel lectual partnerships. The reciprocal interaction required that participants explained and 
justified their reasoning. In turn, they expected clarification and justification of reasoning 
from others. Chal lenge through disagreement and conflict was as important i n  this process 
as agreement and consensus (Brown & Renshaw, 2004). 
The previous studies reported on structures used extensively with l arge numbers of students 
and for lengthy periods of time. Rowe (2003) however, i l lustrated positive learning 
outcomes in a short 4-week New Zealand study. The intervention was designed to develop 
53  
verbal interactions at higher cogni tive levels .  It was framed around students working with 
these key points 
Wait and give individuals time to think for themselves; Be specific with feedback and 
encouragement; Give help when asked in the form of a specific strategy, idea or question 
rather than an answer; and support agreement or disagreement with evidence. (Rowe & 
Bicknel l ,  2004, p. 496) 
Collectively, these studies provide compell ing evidence that students can and do learn to 
participate in inquiry and argumentation through explicit scaffolding. Structuring discursive 
interaction so that al l  participants are able to access a shared intellectual space creates many 
potential learning situations for the participants.  The partnerships and learning situations 
are conducive to the students learning and using mathematical practices. Although none of 
these studies focused directly on mathematical practices and how these were transformed, 
the intel lectual c l imate created a context for their u se. 
3.4 SUMMARY 
This review recognises the complexities and chal lenges teachers encounter in constructing 
inquiry learning communities. The review began with specific studies which depicted 
particular teachers and their actions and beliefs as they developed and worked within 
inquiry communities . Important l i terature i l lustrated how variations i n  the interactional 
patterns of i nquiry communities affected how the students in them participated in  i nquiry 
and argumentation. In  the second section, l iterature related to the forms of talk used m 
classrooms i ncluding the discourse of inquiry and argumentation was rev iewed. 
Woven through the review of l iterature were descriptions of specific interactional strategies 
and communication and participation structures used by teachers to i nduct their students 
into intellectual c l imates. Studies drawn on in this review provided convincing evidence 
that teachers can establish such intel lectual c l imates through a range of pedagogical 
practices. There i s  however a gap in  the l iterature which describes more specifical ly how 
mathematical practices as i nterrelated reasoned verbal and performative actions are taught 
54 
and learnt with in intel lectual communities. This finding is consistent with that of The 
RAND Mathematics Study Panel (2003) who noted how little attention had been given to 
the understanding of mathematical practices "such as problem solving, reasoning, proof, 
representation . . .  " (p. 33) as a coordinated group which are used ski lful ly and flexibly by 
expert users of mathematics. Some of these various studies wil l  be drawn on in the 
fol lowing chapter to outline background research for the mathematical practices. How these 
evolve and are transformed through teacher guidance wil l  be described. These are presented 
as s ingle practices in the following review because there appear to be few studies avai lable 
which consider them as i ntegrated practices. 
55 
CHAPTER FOUR 
THE MATHEMATICAL PRACTICES OF COMMUNITIES OF 
MATHEMATICAL INQUIRY 
School mathematics should be viewed as human activity that reflects the work of 
mathematicians-finding out why given techniques work, inventing new techniques, 
justifying assertions, and so forth. It should also reflect how users of mathematics 
investigate a problem situation, decide on variables, decide on ways to quantify and relate 
the variables, carry out calculations, make predictions, and verify the utility of the 
predictions. (Romberg & Kaput, 1 999, p. 5 )  
4.1 MATHEMATICAL PRACTICES 
The observations made in the previous chapters draw attention to the fact that mathematical 
practices are grounded within collective practices.  These practices i nvolve reasoned 
performative and conversational actions and occur in social and cultural activity systems 
and amongst multiple participants (Saxe, 2002; Van Oers, 200 1 ) . Mathematical practices 
evolve and are transformed within the community they are developed. For that reason the 
literature reviewed in this chapter draws on studies which correlate wi th the age of the 
primary school students (aged 7 - 1 2) in the current study, and the mathematical practices 
students within this age group learn and use during mathematical activity.  
Previously I noted that student learning and use of proficient mathematical practices were 
both dependent on how teachers structured classroom participation and communication 
patterns. Likewi se, in this chapter recognition is given to how i nquiry and chal lenge 
supports both emergence and change in  the mathematical practices students use. Relevant 
l iterature is drawn on to discuss and describe what mathematical practices are and how 
these are used by successful mathematical learners to develop collective reasoning. Links 
are made to the actions teachers take to engage student participation in those practices and 
how participation mediates development and use of mathematical practices. 
56 
4.2 MATHEMATICAL EXPLANATIONS 
Mathematical explanations are statements which commence from well reasoned conjectures 
(Whitenack & Yackel ,  2002).  These conjectures, although provisional, are statements 
which are used to present a mathematical position the explainer is taking. They make 
visible and avai lable for c larification, or challenge, aspects in the reasoning which may not 
be obvious to l isteners (Carpenter et al . ,  2003) .  
An extensive body of research undertaken with primary aged students has outl ined criteria 
for what is accepted as well structured mathematical explanations (e.g . ,  Bowers et al . ,  
1 999; Carpenter et al . ,  2003 ; Cobb, Boufi ,  McClain & Whitenack, 1 997; Cobb et al . ,  200 l ;  
Cobb et al . ,  1 992; Forman & Larreamendy-Joerns ,  1 998; Kazemi & Stipek, 200 1 ;  Perkins, 
Crismond, Simmons and Unger, 1 995 ; Reid, 200 1 ;  Whitenack, Knipping, & Novinger., 
200 1 ; Whitenack & Yackel, 2002; Yackel ,  1 995 ; Yackel & Cobb, 1 996). The criteria 
i nclude the need for explainers to make explanations as explicit as required by the 
audience, relevant to the situation, and experientially real for the audience. Explainers also 
have to supply sufficient evidence to support the claims. This may require that the 
explainer provides further elaboration or re-presentation of the explanation in multiple and 
rich relational ways. Concrete material or graphical, numerical, or verbal contextual support 
may also be needed. 
Teachers have a key role to play in the construction and use of mathematical explanations 
within c lassroom communities. The important studies of Wood and her col leagues (2006) 
discussed in the previous chapter, i l lustrated that is the teacher who establ ishes how 
students participate in developing, using and analysing mathematical explanations. 
Moreover, these researchers showed how the questions and prompts teachers use shape the 
explanations students make. Other studies undertaken within Grade 1 to Grade 5 c lasses 
(e.g. ,  Bowers et a l . ,  1 999; Kazemi & Stipek, 200 1 ; McClain & Cobb, 200 1 ;  Whitenack et 
al . ,  200 1 )  have i l lustrated important pedagogical actions. Effective teachers pressed for 
acceptable explanations. These were differentiated between the different, more 
sophisticated, more efficient and easi ly understood explanations. The teachers also ensured 
57 
that explanations were accessible and understood by the community. They revoiced 
explanations to ensure that conceptual explanations were maintained. Other actions 
included ensuring  that their questions and those of their students' were framed so that 
solution strategies were directed toward specifi c  clarification of mathematical explanations. 
For example,  the fol lowing vignette from McCiain and Cobb (200 I )  i l lustrates how the 
teacher faci l i tated an efficient explanation. Clearly evident is the students' implicit 
recognition that acceptable explanations should be easi ly understood by all participants and 
that they should be of a conceptual not calculational nature. The teacher' s mediational 
actions supported the students to autonomously validate their own reasoni ng. 
Pressing the explanatory reasoning towards easy and efficient solution strategies 
Towards the end of a 6 month teaching experiment in which norms for what made an 
explanation different and notions of easy and hard had been established in the community 
the fol lowing task was presented: There is fourteen cents in the purse. You spend seven 
cents. How much is left? Initially a student  Kitty used an arithmetic rack and made two 
rows of seven beads. Then she moved four beads on one rod and three on the other leavi ng 
groups of three beads and four beads respectively. When she finished another student 
interjected: 
Teri 
Teacher 
Teri 
Teacher 
I think I know a way that might be a little easier for Kitty. 
You think so? 
[comes to the board] We know that seven plus seven equals fourteen 
because we have seven on the top and seven on the bottom . . .  It might just be 
easier if we just moved one of the sevens on the top or the bottom (points to 
each group separately) .  
You mean move a whole group of seven altogether? 
Kitty nods in agreement as she looks at Teri . The term 'easy '  used in this context referred 
to a thinking strategy or grouping solution (structuring the 1 4  as a composite of two sevens) 
that Teri considered easily accessible for the classroom community. 
From McClain and Cobb (200 1 )  
Learning to construct knowledge of what constitutes an explanation which is conceptually 
not procedurally based and matching it appropriately to an audience may initially pose 
considerable chal lenges for some students . The various studies of Cobb and his colleagues 
and other researchers (e .g . ,  Cobb, 1 995 ;  Cobb et al . ,  1 992; Cobb et al . ,  1 993;  Kazemi,  
1 998;  Kazemi & Stipek, 200 1 ;  Yackel ,  1 995 ; Yackel et al . ,  1 99 1 )  documented the 
58 
difficulties students had in elementary grade classrooms as they developed the abi l i ty to 
construct appropriate explanations. For example, Yackel and Cobb ( 1 996) outli ned how a 
group of 2nd Grade students in their study were unable to assess what needed to be said and 
what could be assumed as ' taken-as-shared' in their community. The researcher also noted 
the difficulties some students had in maintaining focus on mathematical reasoning. They 
reported that these students changed their explanations in response to peer or teacher 
reaction because they interpreted the social situation as more important. 
A group of New Zealand researchers (e.g. ,  Anthony & Walshaw 2002; Meaney, 2002, 
2005) also drew attention to the difficulties younger students have in applying appropriate 
evaluative criteria regarding sound explanations .  At the secondary level Bicknell ( 1 999) 
described how many students lacked an understanding of what was required in writing 
explanations. Despite their use of "varying modes of representations for an explanation 
from purely symbolic to a combination of symbols, d iagrams, and words" (p. 8 1 )  they 
lacked confidence to assess the adequacy of their explanations. These studies have 
important implications for the current study. Given the recent focus on students explaining 
their reasoning in the New Zealand Numeracy Development Project (Ministry of 
Education, 2006b) ensuring that students can make and evaluate mathematical explanations 
is a key task for teachers. Providing explanatory reasoning i s  an important precursor for 
explanatory justification and argumentation (Cobb et al . ,  1 997). 
4.3 MATHEMATICAL JUSTIFICATION 
Explanations become explanatory justification when explainers are required to provide 
further evidence in order to address disagreement or challenge to their reasoning 
(Whitenack & Yackel, 2002) . The fol lowing vignette i l lustrates the subtle differences 
between mathematical explanation and justification .  The vignette i l lustrates how the 
teacher scaffolded the shift between two mathematical practices. Initially she prompted the 
students to examine the explanation .  S he then facili tated the use of it as a scaffold for other 
participants to develop arguments-justification that supported, refined or refuted the ideas 
in the explanation. As part of the socially i nteractive process of argumentation the questions 
59 
used by both the teacher and the students shifted from asking 'what did you do' to those 
which encompassed challenge l ike 'but why would you ' .  
Providing explanatory reasoning and justification of the reasoning 
Ms Jones has reconvened the Grade-2 class to l i sten to an explanation for the fol lowing 
problem : Aunt Mary has 31 pieces of candy on the counter and Uncle Johnny eats 15 
pieces of candy. Show how much candy Aunt Mary has on the counter now. Casey has 
recorded on the board 30 - 1 5  = 1 5 ; 1 5  + 1 = 1 6 . He then explains: 
Casey Urn I took that away from the 3 1 .  And plus 1 equals 1 6 . If you take that 1 
and add it onto the 30 to make 3 1 , i t ' s  just when you minus you just have 1 
higher number, and if you take the 1 off the 30, add onto the 1 5 , you get 1 6. 
And that ' s  where I got the 1 6 . 
The teacher then opened the discussion by asking  for comments or questions. 
Shari That was really cool. Because sometimes people know that 1 5  plus 1 5  
equal s 30. So if you know that i t  might be easier. 
Ms Jones confirmed with Casey that he had used the doubles fact then continued: 
Teacher 
Teri 
Casey 
Teri 
Ms Jones 
Teri 
Ms Jones 
So that was something Casey knew and he worked with that first. Does 
anybody e lse have a comment or question? 
Why would you add it [the 1 ]  onto 30 and put it  onto the 1 6? The 1 to the 
1 6? 
I added i t  onto the 1 5  to get the 1 6 . 
Huh? 
Does that make sense Teri? 
But why would you add it to the 30 first and then the 1 5 ?  
Can you show u s  your original thinking? What did you have up  first? I think 
that might help her to see what you did. 
From Whitenack and Yackel (2002) 
In this v ignette, the teacher supported continued questioning until the explainer had 
persuaded the listeners of the legitimacy of his  mathematical actions. At the same time, she 
directed him to re-evaluate and re-present his original reasoning. Through these actions the 
teacher ensured that justification was also used to ascertain or convince the explainer 
himself of the validity of what he was arguing. 
Krummheuer ( 1 995 ) described justification as reasoned and logical argumentation which 
consists of a combination of conjectures or c laims, and one or more pieces of supporting 
60 
evidence. Included in the claims are supporting evidence which might be comprised of 
analogies, models, examples and possibly beliefs and a conclusion. Krummheuer drew on 
the theories of Toulmin ( 1 958)  to examine the reasoning process of Grade Two students ' 
analysis and verification of their own and their peers' reasoning. He presented 
schematical l y  how two students together constructed reasoned argumentation . The in itial 
statement (data) provided the foundation for an assertion (conclusion). A further statement 
in the form of an inference (warrant) was made which verified reliabi l i ty of the assertion. 
Further warrants (backing) were then provided. These strengthened the conclusion and 
provided specific evidence which i l lustrated how the data led to the specific claim.  Cobb 
(2002) also analysed a middle school statistics class drawing on Toulmin ' s  work. He 
i llustrated that it was the teacher' s expectation for additional warrants and backing that led 
to the use of conceptual rather than calculational discourse. 
Many studies have shown that explanatory justification is constructed and reconstructed 
when teachers have pressed their students to take specific positions to make reasoned 
claims (e.g. ,  Enyedy, 2003;  Manouchehri & St. John, 2006 ; Martin et al . ,  2005; Saxe, 2002; 
Sherin, 2002a; Wood et al . ,  1 993) .  The previous chapter discussed the importance of 
teacher use of position statements and problems which engage students i n  disciplinary 
specific i nquiry. O'Connor (2002) outlined how a teacher used a posi tion driven statement 
related to rational numbers with her 5th Grade class. The i nteractional strategies the teacher 
used supported individual students to make reasoned claims . O'Connor described the 
discursive conversational turns the dialogue took. Positions would be maintained or 
changed from turn to turn as the students rethought their c laims in l ight of 
counterexamples. Thus, the teacher created an intel lectual c l imate in which the students had 
space to evaluate all perspectives used in the arguments. 
Constructing classroom cultures which provide participants in the dialogue with space for 
extended thinking is important for the development of j ustified claims. Explanations are 
extended to justification when the students are required to suspend judgment to examine the 
perspectives taken by others and construct supporting evidence for either agreement or 
disagreement. For example, Lam pert ( 1 990b) in her c lassroom studies outlined how she 
6 1  
used strategy solutions as "the site of mathematical argument" (p. 40) .  The strategies were 
considered hypotheses and used to faci l i tate student questioning which took the "form of a 
logical refutation rather than judgment" (p . 40). Kazemi and Stipek (200 I )  also i l lustrated 
how a 5th Grade teacher required that students suspend judgment (agreement or 
disagreement with solutions) until they had constructed justification for their stance. 
The following vignette taken from Sherin et al . (2004) i l lustrates the teacher' s orchestration 
of open dialogue which required the middle school students to use extended thinking time 
to evaluate the positions taken in a claim.  The students were asked to discuss various 
mathematical explanations for six graphs. Each graph represented a different way a flag 
might be hoisted to the top of the flagpole. The teacher engaged the students in each others' 
perspectives and pressed them to validate their positions. He withheld his own evaluative 
judgement but required that they validate either agreement or disagreement. In the extended 
discourse, he allowed the students to control the flow but also slowed it to provide close 
examination of claims. The provision of time enabled the students to not only resolve their 
differing views but also to work from confusion to sense-making. 
Joey 
Mr Louis  
Joey 
Mr Louis 
Ben 
Sam 
Lis a 
Mr Louis  
Sam 
I agree. 
Why? 
Justifying agreement or disagreement 
Because it shows like that he, he waited a while, then it went straight up and 
it didn ' t  take time. 
Ben? 
Wel l  I agree because it would be real ly difficult to do that because he' s  
real ly small l ike me and so probably the only way he could do i t  would have 
to be l ike stay on the top of the flagpole and then jump down l ike holding 
the rope and . . .  
It ' s  possible if  you have a real long flag. 
It sti ll takes time ! 
That ' s  an interesting point .  Are you guys with this conversation? Did you 
hear what Sam said? Sam, do you want to make that point  again?  
If you have a real long flag and it' s the length of  the pole, it' s in both places 
at one time. 
Mr Louis  What do people think about that idea? Can someone rephrase what Sam said 
in a different way that might help clarify it for people? 
From Sheri n  et al .  (2004) 
62 
In  the above vignette, the teacher' s  request for Sam to revoice the c laim in an alternative 
form i ndicated that what was considered acceptable justification for a claim had sti 11 not 
been establ ished. This i l lustrated the role teachers have in scaffolding what is accepted in 
the community as valid mathematical justification. 
Specific c lassroom social norms may also demand that mathematical explanations be 
extended to justification. These include norms which place value on the individual ' s  right 
to disagree until convinced. For example, White (2003) described how a teacher in her 
study of 3rd Grade learners pressed the community towards col lective thinking. Through 
extended discussion she positioned and repositioned the students to accept or reject answers 
freely and work towards mathematical c larity and consensus. White, like a number of other 
researchers (e.g. ,  Blunk, 1 998; Rittenhouse, 1 998; Weingrad, 1 998) i l lustrated the 
importance of teachers positioning students to engage 'pol itely '  with each others' thinking 
whether agreeing or disagreeing. 
Other studies have i l lustrated the importance of classroom cultures where the students are 
confident that they can express their  lack of understanding or i nquire about their own 
erroneous thinking (e.g. ,  Kazemi.  1 998; Simon & B lume, 1 996; Whitenack & Knipping, 
2002;  Yackel ,  200 1 ) . Situations which chal lenge reasoning are often caused by the 
introduction of higher levels, or new ways of thinking. To gain agreement requires 
extended and open discussion and debate. For example, Whitenack and Knipping i l lustrated 
how Grade 2 students accommodated the reasoning provided by various participants. 
Introduction of notation by one student shifted the reasoning to a more advanced level 
which chal lenged many participants' thinking. Consequently, in the extended dialogue 
which fol lowed, justification was required in many forms and models (concrete material, 
pictures and number sentences) before the claim gained explanatory relevance for al l 
partic ipants. 
Enyedy (2003) showed how ih Grade students' claims in the context of computer 
simulated probability problems were structured and explained logically with each assertion 
bui lding from those previously understood. However, chal lenge to the validity of one pair 
63 
of students ' public prediction led to social conflict which resulted in  the need for 
')ustification and evidence" (p . 385) .  Enyedy maintained that the communal nature of the 
argument "encouraged the students to make their reasoning expl ic i t  and public so that i t  
could be challenged, tested and modified" (p .  391  ) .  Likewise, Kazemi ( 1 998) showed how 
the teacher used open dialogue, discussion and debate to reconstruct enoneous reasoning in 
a Grade 4-5 c lassroom, when a student explained and notated that 1 12 + L /8 + L /8 + 1 18 + 
1 18 + 1 18 equated both I and 1 18 and the teacher engaged the entire class in  analysi s of the 
reasoning and scaffolded the development of justification using fraction equivalence. 
Teacher construction of norms which promote an expectation of mathematical sense?
making and development of a collective view have been described as important in pressing 
students towards constructing justifications (Mercer, 2000; Rojas-Drummond & Zapata, 
2004; Simon & Blume, 1 996; Yackel ,  200 1 ). For example, I nagaki , Hatano and Morita 
( 1 998) provided persuasive evidence of how the press toward a shared perspective supports 
development of justification. In their study of 1 1  Grade 4-5 classes the students were asked 
to develop collective consensus during a session of whole class argumentation in each 
c lassroom. I ndividual students were asked to select and provide persuasive mathematical 
justification for the most appropriate solution strategy for addition of fractions with 
different denominators . In their findings the researchers outli ned how different students 
considered and reconsidered their choices. They also i ncorporated other students ' 
utterances i nto their subsequent justification for their own change in choice of solution 
strategy. Fol lowing each classroom session specific students were interviewed including 
some who had remained s i lent throughout the extended discussion. All students were able 
to nominate individual students who were proponents of particularly plausible arguments. 
They also provided clear reasons as to why these students' justification had either 
convinced them or caused them to reconsider their own selected solution strategy. Inagaki 
and his col leagues suggested that the s i lent partic ipants had selected 'agents ' who 'spoke' 
for them and even when they were unable to align their view with an ' agent' they 
responded to "proponents '  and opponents' arguments in  their mind" (p. 523) .  
64 
The study of Inagaki et al .  ( 1 998) i l lustrated that i n  the act of developing a collective view 
all participants were required to take into consideration each other' s perspective. In 
identifying with al l participants' perspectives, the l isteners and interlocutors were bound in 
communality in the views they held. Not only were they required to accommodate the 
opinions of others they were also required to accommodate their own views. Their personal 
perspective already coloured by the dialogic overtones of others, acted as a thinking tool 
which they used to accommodate, or negotiate alternative views, on their path to 
construction of col lective reasoning. 
4.4 MATHEMATICAL GENERALISATIONS 
The previous sections have il lustrated that teacher actions which scaffolded student 
participation in reasoned dialogue were of significance in promoting a shift from 
mathematical explanation to justification. The need to validate claims, in turn, scaffolds the 
development of generalised models of mathematical reasoning. 
Inherent in general i sed models are the processes, procedures, patterns, structures and 
relationships . New cases are not created ;  rather existing concepts are further developed and 
extended through reflective and evaluative reasoning (Lobato, El lis, & Mufioz, 2003; 
Mitchelmore, 1 999; Mitchelmore & White, 2000; Romberg & Kaput, 1 999; Skemp, 1 986). 
For example, Enyedy (2003) i l lustrated how a teacher used computer simulations with 7th 
Grade students to construct probabilistic reasoning. In the search for reasoned justification 
the students shifted from "local models of a situation" (p. 375) to "models for" (p. 375) the 
generalised situation. S imi larly, Gravemeijer ' s  ( 1 999) Grade One classroom study showed 
how a ruler was the initial model .  In the discursive dialogue the teacher scaffolded the 
ruler' s gradual transformation "from measuring to reasoning about the results of the 
measuring" (p. 1 68) .  Gravemeijer explained that in using the ruler as a model for 
"reasoning about flexible mental-arithmetic strategies for numbers up to 1 00" (p. 1 68) the 
teacher enabled the students to generalise from a model of the numbers as referents, to a 
model for numbers as mathematical enti ties i n  their own right. 
65 
Skemp ( 1 986) explained that engaging in generalising is a 
sophisticated and powerful acti vity. Sophisticated, because it involves reflecting on the form 
of the method while temporari ly ignoring its content. Powerful, because it makes possible 
conscious, controlled and accurate reconstruction of one ' s  existing schema-not only in 
response to the demands for assimi lation of new situations as they are encountered but 
ahead of these demands, seeking or creating new examples to fit the enlarged concept. (p. 
58 )  
Teachers are required to support their students into assumi ng a 'mi ndful '  approach to 
recognising patterns, combining processes, and making connections at an elevated level of 
awareness (Fuchs, Fuchs, Hamlett, & Appleton, 2002) .  The RAND Mathematics Study 
Panel (2003) describe thi s as using "intel lectual tools and mental habits" (p. 38) to pattern 
seek and confirm. Generalisations evolve through cycles of reflective pattern finding as 
theories are publicly proposed, tested, evaluated, justified, and revised (Carpenter et al . ,  
2004b). Although the forerunner to  generalisations may have begun as  an intuitive leap (as 
many generalisations do) they have then undergone critical reflective analysis to verify 
their validity (Skemp, 1 986). 
The fol lowing v ignette from Lanin (2005) in a 6th Grade classroom offers an example of 
how a teacher used patterning tasks to scaffold the students to construct a numeric 
generalised scheme. In the social mil ieu of the classroom the teacher provided time and 
space for the students to construct and reconstruct their  thinking as they searched for and 
reflectively tested patterns.  In the class discussion the teacher faci l itated wide ranging 
discussion of possibilities. This made each contribution a p latform for other participants to 
build from. As a result, the community resolved the problem situation using a well 
reasoned mathematical general isation. 
Constructing and justifying a generalisation 
The students had been given the fol lowing problem to solve using a computer spreadsheet: 
The Jog Phone Company is currently offering a calling plan that charges 10 cents per 
minute for the first 5 minutes for any phone call. Any additional minutes cost only 6 cents 
66 
per minute. The students were required to determine a general rule for phone cal ls  that were 
5-minutes or longer. Dirk made links with previous problems and focused on number 
sequences generated in the problem situ ation and using a rate-adjust strategy derived the 
rule C = 6n + 20 (C is the cost of the phone call in cents and n is the number of minutes). 
When questioned he explained: 
Dirk 
Questioner 
Dirk 
I did 6 because i t  [costs 6 cents] for 6 minutes after and then, like the 20 was 
the number that I needed to get to 56. 
Oh, I see so you adjusted. So when you took 6 times 6 you got 36 and then 
you said, "it needs to be 56 cents", so you add 20 back on. Did you assume 
it was right or did you try some other numbers? 
No, I tried other numbers like 7 and if it went up 6 then I was right, and i f  it  
didn't  then it was wrong. 
In  the fol lowing whole class discussion a range of generalisations were presented which 
l i nked back to the context of the situ ation through the use of generic examples. Dirk 
presented the final general isation further describing his rate-adjust strategy: 
Dirk Wel l  I just did times 6 cents because it' s 6 cents every minute and when I 
put in 6 it said 6 times 6 equals 36 and then I just added 20 to get 56 because 
I knew 50 plus 6 was 56. And then I wanted to make sure that it didn' t  work 
for only 6. So I did 7 and it went up 6, and i t ' s  supposed to do that and then I 
did 8 and it went up 6 and it' s supposed to do that. 
Questioned further Dirk could not relate how his rule related to the context of the phone 
cost problem. However another student provided further justification for the generalisation .  
Teacher 
Student 
Where is that plus 20 coming from? 
Because on the first 5 minutes he added 6 instead of l O  and there are 4 cents 
left over [for each minute] and then you need to add 4 for every minute and 
4 times 5 for 5 minutes, 4 extra cents is 20. And that ' s  where the 20 extra 
cents comes from. 
This  student had i l lustrated conceptual understanding of the relationship between adding 6 
repeatedly and multiplying by 6 .  
From Lanin (2005) 
A number of international studies have reported the difficulties many students encounter 
when asked to construct and justify generalisations (e.g. ,  Falkner, Levi & Carpenter, 1 999; 
Smith & Phillips, 2000; Warren & Cooper, 2003) .  For example, Saenz-Ludlow and 
Walgamuth ( 1 998) and Carpenter, Levi ,  Berman and Pligge (2005) report difficulties which 
include a lack of knowledge of the meaning of operations, relationships between 
operations, the equivalence relationship and equals sign . Anthony and Walshaw (2002) in a 
New Zealand study described simi lar findings .  These researchers outlined the difficulties 
67 
the groups of Year 4 and 8 students i n  their study had when asked to produce generalised 
statements related to number properties and relationships. Although many of the students 
drew on the basic properties of addition most did not exhibit explici t awareness of 
commutativity; neither its mathematical structure, nor its mathematical properties.  These 
researchers argued the need for teachers to provide students with opportunities to make 
"explicit their understanding of why number properties such as commutativity and identity 
'hold good"' (p. 46). 
Numerous studies have shown that explicit focusing of student discussion on the 
relationships between numbers properties and operations resulted in their powerful 
construction of generalisations (e.g. B lanton & Kaput, 2002, 2003, 2005 ; Carpenter & Levi ,  
2000; Carpenter et al . ,  2005 ; Kaput & B l anton, 2005 ; Schifter, 1 999; Vance, 1 998) .  For 
example, Carpenter and Levi (2000) i l lustrated how a Grade 2 teacher directed student 
attention toward number properties and operations i ncluding the properties of zero. As a 
result they constructed a number of robust general isations .  B lanton and Kaput (2005) also 
outl ined how a 3rd Grade teacher explicitly integrated algebraic reasoning into her 
classroom practice. As a result the students 
generalized about sums and products of even and odd numbers; generalized about 
properties such as the result of subtracting a number from i tself, expressed as the 
forrnulization a - a = 0; decomposed whole numbers i nto possible sums and 
examined the structure of those sums; and generalised about place value properties 
(p. 420). 
Other studies in early and middle primary classrooms (e.g. ,  Carpenter & Levi ,  2000; 
Carpenter et a l . ,  2003 ; Carpenter et al . ,  2005 ; Falkner et al . ,  1 999; Kaput & Blanton, 2005 ; 
Lampert, 1 990; 200 1 ; Saenz-Ludlow & Walgamuth, 1 998) have i l lustrated teacher use of 
mathematical tasks which focus on relational reasoning (e.g. ,  relational statements which 
use the equals sign). This provoked rich c lassroom dialogue and provided teachers with 
i ns ight into students' reasoning. 
68 
In order for students to develop generalised understandi ngs Watson and Mason (2005 ) 
promote teacher use of 'example' spaces. In these intellectual spaces, the students are able 
to suspend judgement as they search for and identify plausible patterns or counterexamples 
to challenge and di sprove their hypothesis. These researchers maintain that the abi l ity to 
generalise quickly and broadly through a cyclic search for mathematical patterns using 
conceptual reasoning is a hallmark of what proficient mathematics problem solvers do. But 
to make this happen students need rich conceptual understanding of the underlying 
structures and properties of mathematical procedures (Kaput & Blanton ,  2005 ; Skemp, 
1 986).  Some researchers (e .g. ,  B lanton & Kaput, 2003; Watson & Mason, 2005 ) argue that 
teachers should transform existing instructional materi al from problems with single 
numerical responses, to ones which lead to students providing a range of conjectures. They 
explain that variations on problem parameters (often by size) decrease chances of the 
students using simple model ing or computing of answers. Rather, they are nudged in the 
direction of thinking about the problem in general . 
Teacher use of open-ended mathematical problems within a c lass of problems is promoted 
in many studies (e .g . ,  Kaput, 1 999; Kaput & Blanton, 2005 ; Krebs, 2003;  Lanin,  2005 ; 
Lesh & Yoon, 2004; Mitchelmore, 1 999; Sadovsky & Sessa, 2005 ; Smith, 2003 ; Sriraman, 
2004) .  These studies outlined how use of open-ended problems, not easi ly  solved 
empirically, drew sustained effort and supported rich dialogue. Krebs (2003) i l lustrated 
how gth Grade students were structured to work within 'example' spaces. She outl ined how 
they persevered with challenging problems, deliberated at length, explored and tested a 
number of hypotheses before constructing their generali sations. In doing so, they also made 
connections both between the representations (tables, symbols, graphs) and to previous 
problems and tasks. 
Teacher use of such problems increases opportunities for students to engage in inquiry and 
argumentation and to develop powerful models of their reasoning (Carpenter & Levi ,  2000; 
Carpenter et a l . ,  2005 ; Kaput, 1 999; Warren & Cooper, 2003) .  For example, in the 
fol lowing vignette from B lanton and Kaput the teacher increased the numbers beyond the 
students' immediate arithmetic capacity. This action el iminated their capacity to compute 
69 
and so they were pressed to examine the properties of the numbers. As a result, the student  
showed that she knew that she needed to  on ly  consider the last digits of  the two numbers to 
validate the general isation. 
Generalising the properties of odd and even numbers 
The teacher asked the students whether the sum of 5 + 7 was even or odd :  
How did you get that? Teacher 
Tory I added 5 and 7, and then I looked over there [indicating a list of even and 
odd numbers on the wal l ]  and I saw it was even. 
Teacher What about 45 ,678 plus 856,3 1 7, odd or even? 
Mary Odd. 
Teacher Why? 
Mary Because 8 and 7 is even and odd, and even and odd is odd. 
From B lanton and Kaput (2003) 
Also i l lustrated in the vignette was how the teacher 's  use of the question 'why' supported 
further explanation of the general isation. 
B lanton and Kaput (2003) maintain that teachers should use specific questions like "Tel l  
me what you were thinking? Did you solve this a different  way? How do you know this i s  
true? Does this always work?" to focus student attention and draw justification about the 
properties and relationships of numbers (p. 72). For example, Strom et al . (200 1 )  revealed 
the importance of teacher questioning. Within a Grade Two c lassroom episode of dialogic 
talk the students' everyday measurement and area experiences were progressively 
mathematised as the teacher pressed the students toward generali sation and certainty. The 
teacher' s use of questions l ike 'why' ; 'does it work for al l  cases ' ; 'can you know for sure' 
nudged the students to search for patterns and to consider underlying generalised 
mathematical structures. Similarly, Zack ( 1 997, 1 999) in an elementary class showed that 
challenge through teacher and student questioning acted as a catalyst which promoted a 
search for patterns and need for generalised reasoning. In the student i nitiated dialogue, 
patterns of conjecture and refutation were constructed as the participants worked towards 
col lective agreement. The logical connectives (because, but, if . . .  then) used in response to 
the specific questions supported the construction of general ised explanations. 
70 
4.5 MATHEMATICAL REPRESENTATIONS AND 
INSCRIPTIONS 
Mathematical representations and inscriptions are important social tools which are used to 
mediate individual and collective reasoning within classroom communi ties (Carpenter & 
Lehrer, 1 999; Cobb, 2002; Forman & Ansell ,  2002; Greeno, 2006; Saxe, 2002).  In the 
current study the term inscriptions is used to include a range of representational forms 
which proficient users of mathematics employ and understand. These include symbolic and 
invented notation and other "signs that are materially embodied in some medium . . .  such as 
graphs, tables, l ists, photographs, diagrams, spreadsheets and equations" (Roth & McGinn, 
1 998,  p .  37). 
Concrete material and problem situations grounded in i nformal and real world contexts 
potential ly provide a starting point to develop multiple forms of representation (Smith, 
2003) .  These serve as reference points which are then mathematised-progressed toward 
abstraction and generalisation. For example, McCiain and Cobb ( 1 998) in a Grade One 
c lass described an instructional sequence which began as "experientially real" (p. 6 1  ). This 
i nitial context served as a means to both "fold back" (p. 59) and "drop back" (p. 67) when 
needed, to maintain conceptual understanding. They described how the students abstracted 
and generali sed a form of quanti tative reasoning which they represented symbolical ly on a 
number li ne. 
Heuser (2005) depicted a study in which 1 st and 2nd Grade students developed fluid use of 
symbolic representation for computation procedures. Heuser described how hands-on?
activity embedded in problem situations laid the foundations. Reflective discussion 
advanced the students from modeling problems through drawings to development of 
numerical procedures to explain their invented strategies. Heuser outlined how many 
students generalised their i nvented numerical computational strategies to solve other more 
complex and non-routine problems. Many of the invented notation schemes c losely 
resembled more standard procedures but the students retained a rich sense of place value. 
7 1  
When students are engaging in inquiry and argumentation effective teachers draw on the 
different representational forms individuals use to make the reasoning publ ic and accessible 
for community exploration and use (Sawyer, 2006a). Likewise, the obligation to make 
avai lable multiple explanations of reasoning influences how students learn and use ways of 
representing or inscribing their current reasoning (Cobb, 2002; Lehrer & Schauble, 2005 ; 
McClain,  2002; Roth & McGinn, 1 998;  Saxe, 2002). For example, Whitenack and 
Knipping (2002) i l lustrated the interdependent relationship of inquiry and challenge that led 
to more advanced models for conceptual reasoning being developed. The 2nd Grade 
students devised invented notational inscriptions for two digit addition and subtraction. 
However, the need for further clarification and justification led to the c laims bei ng recast 
flexibly as pictures and invented symbolic schemes. These resulted in generalised symbolic 
schemes which served as explanatory tool s  which clarified, explained and interpreted and 
substantiated the claims. 
Strom et al . (200 1 )  outlined how discursive dialogue was used to transform students' 
i nformal knowledge depicted as i nvented representations i nto notation schemes which 
included formal algorithms. Teacher actions were integral to ways in  which the students ' 
representational models supported explanation, general isation, and development of 
collective certainty. The teacher press led to inscriptions being used to flexibly "re-present" 
(Smith, 2003, p. 263), re-explain and re-construct the concepts embodied in the models .  In  
the extended discussion the students learnt ways to  examine, explore, and debate the 
notation schemes and assess their adequacy. Thus learning to make use of, and 
comprehend, representational forms was achieved through social interactive activity 
(Greeno, 2006) .  The representational models had also been used as reflective tools which 
facili tated student thinking about their learning and the learning process (Sawyer, 2006a). 
4.6 USING MATHEMATICAL LANGUAGE AND DEFINITIONS 
Negotiating mathematical meaning is dependent on students ' access to a mathematical 
discourse and register appropriate to the c lassroom community. Students who display 
mathematical l i teracy are able to use the language of mathematics to maintain meaning 
72 
within the context of its construction, in its form or mode of argumentation and matched to 
audience needs (Moschkovich, 2002b; Thompson & Rubenstein, 2000) .  Forman ( 1 996) 
proposes that these include the differing use of mathematical terms and ways of defining 
these terms .  Included also are use of a particular syntax, use of brevity and precision to 
present oral and representational forms of the discourse or to respond to chal lenge (Mercer, 
1 995) .  
Gaining fluency and accuracy in mathematical talk requires a shift from an informal use of 
terms and concepts to a more narrow and precise register (Meaney & Irwi n, 2003) .  Meaney 
and Irwin maintain that "if students are not encouraged to use mathematical language (both 
terms and grammatical constructions) then eventually their mathematical learning wi l l  be 
restricted" (p. 1 ) . Drawing on a study of Year 4 and 8 students in New Zealand these 
researchers showed the students more readi ly suppl ied answers which conformed to a social 
not mathematical register. Likewise, i n  Australia Warren (2003) documented 87  eight year 
olds' responses to explaining and connecting meanings for words commonly used i n  
addition and subtraction . Warren noted that students neither comprehended the many 
nuances of mathematical language nor made connections with how words overlapped in 
meaning. In a 2005 study Warren (2005) reported results of a longitudinal study of 76 Year 
3-5 students. The study investigated the development of student understanding of words 
common ly l inked to equivalent and non-equivalent situations. She indicated that student 
construction of meaning for terms l ike 'more' or ' less' remained tied to their meanings in 
arithmetic operations and 'equal' denoted the answer. Warren i l lustrated that the narrow 
use of the concepts was establ ished by Year 3 and remained firmly embedded in the Year 5 
students' thinking. 
Different social groups and communities make use of varying discourse and 
communication patterns specific to their si tuated context (Gee, 1 999; Moschkovich, 2003 ;  
Nasir, Rosebery, Warren ,  & Lee, 2006). For example, Irwin and Woodward (2005),  i n  a 
New Zealand study, noted the prevalence of "col loquial terms and conversational 
conventions" (p. 73) .  The teacher promoted mathematical explanations of reasoning in the 
predominantly Pasifika Year 5/6 discussion i ntensive Numeracy classroom. However, the 
73 
researchers concluded that the lack of emphasis on "the use of the mathematics register, 
both the terms and the discourse of premise and consequence" (p. 73)  affected how the 
students talked together mathematically. Latu (2005 ), in another New Zealand study, 
showed the difficulties encountered by a group of secondary Pasifika students. The group 
used mathematical terms but these were restricted to exact contexts in which they were 
learnt .  Relational statements in word problems were least understood. However, Latu 
demonstrated that those Pasifika students able to code switch between a fi rst language and 
the language of mathematics (in English), performed better than those who had only 
restricted forms of Engl ish as their first language. 
Other studies have i l lustrated that diverse students are able to gain fluency in mathematical 
discourse when teachers focus specifical ly  on the rich use of mathematical language and 
terms. For example, Khisty and Chval (2002) outlined how a teacher engineered her 51h 
Grade Latino student ' s  learning environment. Her engagement of students in lengthy 
periods of collaborative problem solving included an expectation that they explain and 
justify their mathematical reasoning. She explicitly modeled the use of mathematical ly rich 
language and complete mathematical statements. As a result, the students were inducted 
into fluent use of the specialised discourse of mathematics. 
Moschkovich ( 1 999) used a c lassroom episode in a 3rd grade classroom of English second 
language learners. The teacher "did not focus primari ly on vocabulary development but 
instead on mathematical content and arguments as he interpreted, clarified and rephrased 
what students were saying" (p. 1 8) .  The teacher careful ly l istened to the students, probing 
and revoicing what they said to maintain focus on the mathematical content of their 
contributions .  As a result, they gradual ly  appropriated both the use of the mathematical 
register and knowledge of how to participate in mathematical dialogue. 
4.7 SUMMARY 
This l iterature review of mathematical practices used by primary aged students i l lustrated a 
range of mathematical practices and described how each evolved and was transformed 
74 
within the social and cultural context of classrooms. Di scursive interaction and its role i n  
the development of  communal sense-making wove across the mathematical practices. In the 
discursive dialogue, explaining mathematical reasoning was extended to justifying it. In  
turn, the need for multiple forms to  validate the reasoning led to  generalisi ng. Representing 
or inscribing reasoning and using mathematical language appropriately were also embedded 
in the di scursive activity of col lective reasoning. Learning and doing mathematics was 
outl ined as an integrative social process. Each mathematical practice engaged in, within 
classroom communities is i ntrinsical ly  interconnected and related reciprocally to the other 
mathematical practices. 
Gaps in the l iterature correlated with those identified by the RAND Mathematics Study 
Panel (2003) .  This group of researchers noted the paucity of l iterature which outli ned how 
mathematical practices, as a col lection of connected and wel l-coordinated reasoning acts, 
were used ski lful ly and flexibly by expert users of mathematics.  There also appeared to be a 
scarcity of New Zealand l iterature which explicitly related to mathematical practices or 
understanding how these developed within c lassroom communities. The current study 
moves away from a focus on si ngle mathematical practices to consider how mathematical 
practices are used as integrated practices. How the teacher structures the environment 
which provides students with cognitive space to participate in the mathematical practices 
wil l  be examined. 
The following chapter describes the methodology of design research used in the current 
study. Design research fal ls  natural ly from the sociocultural frame taken in the current 
study. 
75 
CHAPTER S 
METHODOLOGY 
What teachers do i s  strongly influenced by what they see in  given teaching and learning 
situations. For example, as teachers develop, they tend to notice new things . . .  these new 
observations often create new needs and opportunities that, i n  turn, require teachers to 
develop further. Thus the teaching and learning situations that teachers encounter are not 
given in nature: they are, in large part, created by teachers themselves based on their current 
conceptions of mathematics, teaching, learning, and problem solving. (Lesh, 2002, p. 36) 
5.1 INTRODUCTION 
This study aims to investigate how teachers develop mathematical inquiry communities in 
which the students participate in proficient mathematical practices. A classroom-based 
research approach-design research-was selected. This approach fol lowed natural ly from 
the theoretical stance taken by the researcher and used in this study. 
Section 5 .2 states the research question. In section 5 .3  an interpretive qualitative research 
paradigm is examined. I l lustrations are provided to show how a qual i tative paradigm 
supports the aims of this research .  Section 5 .4 provides an explanation of design research as 
it is used in this study. Explanation is given of how design research is congruent with 
col laborative partnerships between teachers and researchers within the situated context of 
schools and classrooms. How the framework of communicative and participatory actions 
was designed i s  outlined. In Section 5.5 ethical considerations are discussed in relation to 
school based col laborative research. 
Section 5 .6 outlines how participation by the school and teachers in the design research was 
establ ished. Section 5.7 describes data col lection in  the c lassrooms and the process of data 
analysis. The different research methods used in the study are outli ned and an explanation 
is given of how these were used to maintain the voice of the teachers and capture their 
perspectives of the developing discourse and mathematical practices. Section 5 .8  describes 
76 
the approach taken to analyse the data. Finally, Section 5 .9 outl i nes how the findings are 
presented. 
5.2 RESEARCH QUESTION 
The study addresses one key question : 
How do teachers develop a mathematical community of inquiry that supports student 
use of effective mathematical practices ? 
The focus of exploration is on the different pathways teachers take as they develop 
mathematical communities of inquiry; their different pedagogical actions, and the effect 
these have on development of a mathematical di scourse community .  
Implicit in the question is how changes in  the mathematical discourse reveal themselves as 
changes in how teachers and students participate in and use mathematical practices. That is ,  
the research also seeks to understand how the changes in participation and communication 
patterns in a mathematical classroom support how students learn and use proficient 
mathematical practices. 
5.3 THE QUALITATIVE RESEARCH PARADIGM 
This research is guided by a qualitative interpretive research paradigm (Bassey, 1 995) and 
draws on a sociocultural perspective. The interpretive and sociocultural approaches adopted 
in  this study are grounded in "si tuated activity that locates the observer in the world" 
(Denzin & Lincoln, 2003, p. 3) .  They share the notion that real i ty is a social construct and 
that a 'more-or-less' agreed interpretation of l ived experiences needs to be understood from 
the view of the observed (Merriam, 1 998;  Scott & Usher, 1 999) .  Recognition is given to 
descriptions based within social meanings which are always subject to change during social 
interactions (Bassey, 1 995 ) .  Such a v iew enables one to report on the ' l ived' experiences of 
the teachers and their students, and use their 'voice' to interpret the multiple realities of the 
construction of mathematical i nquiry communities and their complementary mathematical 
practices. 
77 
Qualitative researchers acknowledge that the 'distance' between the researcher and 
researched is minimised as each individual interacts and shapes the other (Creswel l ,  1 998;  
Denzin & Lincoln, 2003) .  A report of mathematical practices in inquiry communities needs 
to be understood as an interpretation of the researcher and the teachers involved in the 
study. A collaborative and participatory approach was appropriate given that the purpose 
was "to describe and interpret the phenomena of the world in attempts to get shared 
meanings . . .  deep perspectives on particular events and for theoretical insights" (Bassey, 
1 995, p. 1 4) .  Moreover, the quali tative design research methods provided "a formal , c lear 
and structured place for the expertise of teachers to be incorporated withi n the production 
of artefacts and interventions designed for use in the classroom" (Gorard, Roberts, & 
Taylor, 2004, p .  580).  
5.4 DESIGN RESEARCH 
Design research has been gaining momentum as a classroom-based research approach 
which supports the creation and extension of understandings about developing, enacting, 
and maintaining innovative learning environments (Design Based Research Col lective 
(DBRC), 2003) .  Design research is commonly  associated with the important work of 
Brown ( 1 992) and Coll ins ( 1 992) who situated their research and design within the real?
life, messy, complex situations of classrooms. However, the foundations for it  span the past 
century. It has its roots within social constructivist and sociocultural dimensions and the 
theories of Piaget, Vygotsky, and Dewey who shared a common view of classroom l ife "not 
as deterministic, but as complex and conditional" (Confrey, 2006, p. 1 39) .  
Design research provides a useful and flexible approach to examine and explore 
innovations in teaching and learning in the naturali stic context of real world settings 
(Barab, 2006; Col l ins, Joseph, & Bielaczyc, 2004) .  Its methods emphasise the design and 
investigation of an entire range of "innovations: artefacts as wel l as less concrete aspects 
such as activity structures, institutions, scaffolds and curricula" (DBRC ,  2003, p. 6) .  
Moreover, these methods offer ways to create learning conditions which current theory 
78  
promotes as productive but which may not be commonly practised nor completely  
understood. This  has relevance for th i s  study which sought to  explore how teachers develop 
mathematical i nquiry communities which support student use of proficient mathematical 
practices. 
Design research is an approach which links theoretical research and educational practice. 
There are many varied forms and terminology for design research accepted within 
education, each with its own different goals ,  methods, and measures (Kelly, 2006) .  Variants 
include classroom teaching experiment-multi-tiered and transformative; one-to-one 
teaching experiment; teacher and pre-service teacher development experiment; and school 
and di strict restructuring experiments . Irrespective of the form design research is described 
as 
experimental, but not an experiment. It is  hypothesis generating and cultivating, rather than 
testing; it is motivated by emerging conjectures. It i nvolves blueprinting, creation, 
intervention, trouble-shooting, patching, repair; reflection, retrospection, reformulation, and 
reintervention. (Kelly, p. 1 1 4) 
Within design research the "central goals of designing learning environments and 
developing theories or 'prototheories' of learning are i ntertwined" (DBRC, 2003, p. 5 ) .  
This sits easi ly with the current study given that a key a im is  to explore and broaden 
knowledge of how teachers enact participation and communication patterns i n  
mathematical communities that support student use of efficient mathematical practices. 
Although this research study was conducted in a single setting the intent is to add to 
broader theory through offering insights about the process. This is attempted through the 
provision of rich accounts of the pedagogical actions teachers take to develop i nquiry 
communities and the corresponding mathematical practices the participants engage in .  
Barab (2006) maintains that through providing "methodological precision and rich 
accounts . . .  others can judge the value of the contribution, as well as make connections to 
their own contexts of innovation" (p. 1 54 ) . 
A central feature of design research i s  that it leads "to shareable theories that help 
communicate relevant implications to practi tioners and other educational developers" 
79 
(DBRC, 2003, p .  5 ) .  Of importance to the current study is how design research is able to 
"radical ly increase the relevance of research to practice often by involving practitioners in 
the identification and formulation of the problems to be addressed, and in  the interpretation 
of results" (Lesh, 2002, p. 30). The col laborative and ongoing partnership between the 
teachers and myself as a researcher was a significant feature of this study. Engaging in  
partnership with the teachers as a study group and with each individual teacher in h i s  or her 
c lassroom setti ng, supported the wider evaluation, reflection and re-defining of specific 
aspects of the intervention. It also required the use of methods commonly used in design 
research to develop accounts that documented and connected the processes of enactment to 
the outcomes under investigation. 
An important aspect of the design research approach is the account provided of how the 
designed intervention interacts with the complexities within authentic educational settings 
(Cobb et al ., 2003 ; Confrey, 2006).  Due to the many dependent variables within the 
educational setting many regard design research as messier than other forms of research. 
Gorard and his col leagues (2004) justify the messiness of design research explaining that it  
evolves through its characterisation of contexts. It does this by revising the procedures at 
wi l l ,  which "al low[s] participants to i nteract, develop profi les rather than hypotheses, 
i nvolve users and practi tioners in the design, and generate copious amounts of data of 
various sorts" (p. 5 80). 
5.4.1 TESTING CONJECTURES: COMMUNICATION AND PARTICIPATION 
FRAMEWORK 
Design research is an interventionist and i terative approach which uses on-going 
monitoring both in the field (and out of the field) of the success or fai lure of a designed 
artefact to gain immediate (and accumulating) feedback of the viabi l ity of the theory it is 
grounded in (Col lins, 1 992;  Kel ly, 2006) .  A collaborative relationship between the teachers 
and researcher and the trial l ing of artefacts in the fie ld within iterative cycles has 
similarities to the approach used in action research. An important difference is that it  is the 
researcher who introduces the problem and takes a central position in the collaborative 
80 
design, trial l ing, systematic documentation, and the reflection and evaluation of an 
i ntervention artefact as it  i s  implemented within an authentic context. In the current study, 
the designed artefact took the form of a communication and participation framework (see 
Table 2 ,  Section 5 .4 .2) .  The framework outlines a conjectured set of communicative actions 
(verbal and performative) and a conjectured set of participatory actions that progressively 
constitute effective mathematical practices i n  an i nquiry classroom. The intent was that the 
framework of communicative and participatory actions would guide teachers as they 
scaffolded students to engage in col lective reasoning activity within communities of 
mathematical i nquiry. 
According to Confrey and Lachance (2000) conjectures are inferences "based on 
inconclusive or incomplete evidence" (p . 234-5) which provide ways to reconceptuali se 
either mathematical content or pedagogy. In  the current study, these conjectures related to 
possible communication and participation patterns ,  which i f  teachers pressed their students 
to use, could result in them learning and using proficient mathematical practices. Steffe and 
Thompson (2000) maintain that generating and testing conjectures "on the fly" (p. 277) is 
an important element of design methodology. The process of designing tool s  such as 
frameworks or trajectories and then subjecting them to on-going cri tical appraisal is a 
crosscutting feature of design research. Cobb and his  col leagues (2003) explain  that design 
research "creates the conditions for developing theories yet must place these theories in 
harm' s  way" (p. 1 0) in order to extend them and benefit from other potential pathways 
which may emerge as the research unfolds. 
The beginning point for the design of the communication and participation framework used 
in the current study drew on a theoretical framework proposed by Wood and McNeal 
(2003) .  Their two-dimensional hierarchical structure (discussed previously in Chapter 
Three, Section 3 .2 .2 )  i l lustrated that student responsibility for thinking and participating in  
col lective activity deepened in a shift from conventional to  strategy reporting discussion 
contexts, deepening again in the further shift to inquiry and argument contexts. Their 
descriptions of the communicative and participatory actions teachers prompt students to use 
in each discussion context, which l ink to different levels of student engagement i n  
8 1  
col lective reasoning practices, provided an in itial basis for the communication and 
participation framework. It was then extended through the use of many studies previously 
discussed in  Chapter Three and Four. In  these studies, the specific  communication and 
participation patterns teachers scaffolded their students to use, which gradually inducted 
them from peripheral to more central positions in the collective discourse, were identified. 
These were added to the communication and participation framework as conjectures of 
possible actions teachers could scaffold students to use, to provide them with opportunities 
to learn and use mathematical practices within i nquiry cultures. 
The structure of the communication and participation framework was comprised of two 
separate components-communication patterns and participation patterns. Vertically, the 
framework outl ined a set of collective reasoning practices matched with conjectures 
relati ng to the communicative and performative actions teachers might require of their 
students, to scaffold their participation in learning and using mathematical practices. 
Likewise, conjectures of a set of participatory actions teachers may expect of their students 
to promote their i ndividual and col laborative responsibility in the col lective activity were 
added.  The horizontal flow over three phases sketched out a possible sequence of 
communicative (and performative) actions, and participatory actions, teachers could 
scaffold their students to use, to gradually deepen their learning and use of proficient 
mathematical practices within reasoned inquiry and argumentation . The design of the 
communication and participation framework recognised the possibility that the 
participating teachers might be novices with in i nquiry environments and not have had many 
opportunities to engage in col lective reasoned discourse. Therefore, c lear and descriptive 
detai ls of communicative and participatory actions teachers could press students to use 
were provided. 
The communication and participation framework was designed and used by the teachers as 
a flexible and adaptive tool to map out and reflectively evaluate their trajectory or pathway. 
They drew on the communicative and participatory actions outli ned in the framework, to 
p lan individual pathways of pedagogical actions to use, to guide the development of 
col lective reasoned discourse in i nquiry classroom communities. Although the teachers 
82 
reached simi lar endpoints their individual pathway they mapped out using the framework 
was unique. The flexible design of the framework responsively accommodated each 
teacher' s need for less or more time to unpack and manage the changing participation and 
communication patterns requirements at different points. Whi le the framework was 
sufficient in itse lf  as a guide some teachers required c larification and further e laboration 
and more detai l ed explanation of the communication and participation actions they might 
press the students to use (see Section 7 .3 .2 for how the framework was used as a tool to 
support specific teacher needs and Appendix K for a more detai led explanation and 
expansion of one section of it) .  
An additional tool which was developed in conjunction with the communication and 
participation framework during the current study took the form of a framework of questions 
and prompts (see Appendix E). The framework of questions and prompts was designed in 
co l laboration with the teachers in the study group setti ng and complemented the 
communication and participation framework. The need for and use of this tool emerged and 
evolved through col l aborative reflective examination of c lassroom communication and 
participation patterns and the communicative and participatory actions outli ned in the 
communication and participation framework. The bas is  for the framework of questions and 
prompts also drew on the important work of Wood and McNeal (2003) .  It evolved and was 
extended during the current study as col l aborative observations were made of how the use 
of specific questions and prompts influenced how the students participated in 
communicating their mathematical reasoning. Conjectures were made that teacher 
modeling of the questions and prompts as wel l  as a press on students to use them would 
deepen student agency in the mathematical discourse. 
Further discussion of how the framework of questions and prompts deve loped 
col laborativel y  in the study group setting, how it was used in c lassrooms, and how it was 
u sed as an additional evaluative tool by the teachers is provided in this chapter (see Section 
5 .6 .5) .  The communication and participation framework used by the teachers is presented 
on the fol lowing page. 
83 
Table 2 
The communication and participation framework: An  outline of the communicative 
and participatory actions teachers facilitate students to engage in to scaffold the use of 
reasoned collective discourse. 
Phase One 
Use problem context to 
make explanation 
experiential ly real 
Indicate agreement or 
disagreement with an 
explanation. 
Look for patterns and 
connections. 
Compare and contrast 
own reasoning with that 
used by others. 
, , Discuss and use a range c c 
c .? of representations or 
? B .e inscriptions to support 
?-
c tl an explanation. "' Q.l "' ? "'  c t ?-C. "'  
Q.l c 
r.. ell 
, Use mathematical words c 
-;j .? to describe actions. 
(.J .... 
? ?= eo:? ?;:  e Q.l QJ "'  
..C -o  
.... 
c ell ell e Q.l 
bll bll c ell 'iii ? ? c 
ell 
Phase Two 
Provide alternative ways 
to explain solution 
strategies. 
Provide mathematical 
reasons for agreeing or 
disagreeing with solution 
strategy. 
Justify using other 
explanations. 
Make comparisons and 
explain the differences 
and simi larities between 
solution strategies. 
Explain number 
properties, relationships. 
Describe i nscriptions 
used, to explain and 
justify conceptually  as 
actions on quantities, not 
manipulation of symbols. 
Use correct mathematical 
terms. 
Ask questions to c larify 
terms and actions. 
84 
Phase Three 
Revise, extend, or 
elaborate on sections of 
explanations. 
Validate reasoning usi ng 
own means. 
Resolve disagreement by 
discussing viabil ity of 
various solution 
strategies. 
Analyse and make 
comparisons between 
explanations that are 
different, efficient, 
sophisticated. 
Provide further examples 
for number patterns, 
number relations and 
number properties. 
Interpret inscriptions used 
by others and contrast 
with own. 
Translate across 
representations to clarify 
and justify reasoning. 
Use mathematical words 
to describe actions 
( s trate gi es). 
Reword or re-explain 
mathematical terms and 
solution strategies. 
Use other examples to 
i l lustrate. 
Active listening Prepare a group explanation and Indicate need 
and questioning justification collaboratively.  to question 
for more during and 
information. Prepare ways to re-explain or justify the after 
selected group explanation. explanations. 
Collaborative 
support and Provide support for group members when Ask a range of 
responsibi l ity for explaining and justifying to the large group questions 
reasoning of all or when responding to questions and including 
group members. chal lenges. those which 
draw 
Discuss, interpret Use wait-time as a think-time before justification 
and reinterpret answering or asking questions. and 
problems. Indicate need to question and challenge. generalised 
models of 
Agree on the Use questions which challenge an problem 
"' construction of explanation mathematical ly and which situations, c one solution draw justification. number .? - strategy that al l  patterns and C.J ? members can Ask clarifying questions if representation properties. ...... J. explain .  and inscriptions or mathematical terms are 0 -? Indicate need to not clear. Work together c. 
:g question during collaborati vely -J. large group in small ? 
Q., sharing. groups 
examining and 
Use questions exploring al l  
which clarify group 
specific sections members 
of explanations or reasoni ng. 
gain more 
information about Compare and 
an explanation. contrast and 
select most 
proficient (that 
all members 
can 
understand, 
explain and 
justify). 
85 
5.5 ETHICAL CONSIDERATIONS 
The ethics of social research focuses on the need to protect al l participants from possibility 
of harm (Babbie, 2007 ; Berg, 2007) .  Of central importance in entering school and 
classroom communities for research purposes is the need to consider potential harm to 
teachers and their students. Berg outlines the many difficulties social researchers encounter 
when engaging with issues of "harm, consent, privacy and confidentiality" (p. 53) .  Mindful 
of these difficulties, this section examines and discusses the principles and practices which 
underwrite this c lassroom study. 
5.5. 1 INFORMED CONSENT 
Integral to ethical research is the fundamental principle of informed consent (Bogdan & 
Biklen, 1 992) .  Participants need to be "fu l ly  informed about what the research is about and 
what participation wil l  involve, and that they make the decision to participate without any 
formal or informal coercion" (Habibis, 2006, p. 62). In this study, it was particularly 
important that the teachers had complete understanding of the research process, for many 
reasons.  These included the collaborative nature of the research,  the extended length of the 
research project and the extensive time commitment required for discussion and review of 
pedagogical practices, study group activity and viewing video-captured observations. 
The teachers' informed decision to participate was integral to the study.  Robinson and Lai 
(2006) maintain that in school situations the "power differentials for gaining free and 
informed consent needs to be careful ly considered" (p. 68) .  In this study, i ndividual consent 
was gained through comprehensive discussions . Anticipation of issues which posed the 
possibility of harm were careful ly considered and sensitively discussed.  These included the 
possibility of embarrassment on being observed and videoed in c lassrooms and talking 
about and viewing video-captured observations with me and other study group members. 
The benefits of participating in the study were explored so that the teachers could better 
balance risks with possible outcomes. The teachers were also clear about their right to 
withdraw at anytime. This was particularly important given the length of the design-based 
research project. 
86 
Informed consent i s  also a key issue when working with chi ldren.  Important factors include 
ful l  cognisance of what participation in the research impl ies for them (Habibis, 2006) .  In  
thi s  study, the students first discussed the project in their classrooms and were provided 
with student and parent information sheets which provided additional information about the 
research. 
5.5.2 ANONYMITY AND CONFIDENTIALITY 
Anonymity and confidentiality are key ethical issues (Habibis, 2006). Although al l 
participants in this study were allocated pseudonyms, assuring anonymity remained 
problematic given that the teachers in the school community were aware who the 
participants (teachers and students) in the study were. Also the involvement of the school i n  
a larger projed placed the research within scrutiny of  a wider audience. 
Habibi s (2006) outl ines how confidentiality overlaps with anonymity-both are concerned 
with maintaining the privacy of respondents. 
However, whi le anonymity is concerned with the identification of individual respondents, 
confidentiality is concerned with ensuring that the information they provide cannot be 
linked to them. Even if the respondents in a study are identified, the pri nciple of 
confidentiality means that their specific contribution cannot be identified. (p. 67) 
In  this research, although the participants in the study discussed and viewed selected video 
excerpts, beyond this immediate group the specific contributions of individuals could not 
be l inked back to i ndividual participants. In  addition, i nterviews and further discussions of 
lessons occurred on an i ndividual basis so that each person ' s  confidential ity in relation to 
others in the research was protected. 
1 This project was one of four nested projects of a larger Teaching Learn i ng Research Initiative study. 
87 
5.6 THE RESEARCH SETTING 
This research began with my approach to Tumeke School in the final term of the year 
before the research was planned. I was invited to meet with the senior management team to 
discuss further the possibili ties of the staff participating in the col laborative research 
project. At a staff meeting to outl ine the proposed research involvement for the teachers 
and students I responded to the teachers' many questions and confirmed that it was each 
teacher' s right to choose to participate in the study. Fol lowing thi s exploratory staff 
meeting the Principal confirmed that in subsequent smaller team meetings the staff had 
independently confirmed their wish to participate in the research .  The Principal explained 
that the teachers at Tumeke School regarded their involvement as professional 
development. They considered that their involvement would provide them with 
opportunities to extend their understanding of numeracy teaching beyond the NDP in which 
they had all recently participated. The Principal also notified the Board of Trustees and 
verified their support. 
5.6. 1 DESCRIPTION OF THE SCHOOL 
Tumeke School2 is a smal l New Zealand suburban ful l-primar/ school situated on the city 
boundary. The school ' s  deci le ranking4 of 3 reflects the low socioeconomic status of 
individual fami l ies within the community. The school roll fluctuates between 1 70 and 250 
students. Students of New Zealand Maori and Pacific Nations backgrounds comprise 70% 
of the school rol l .  The remainder include 2 1 %  New Zealand European students and 9% 
Thai, Indian, and Chinese students. The majority of students from the Pacific Nations and 
other backgrounds are New Zealand born, bi l ingual and with English spoken as a second 
language in their home. The school has a high pattern of transience; approximately 30% of 
students in each c lass enter and leave during the school year. 
2 Tumeke is a code name for the school 
3 A school which includes both primary and in termediate aged students 
4 Each state and i ntegrated school i s  ranked into deciles, low to high on the basis of an indicator. The indicator 
used measures the socio-economic level of the school comm u nity. The lowest decile ranking i s  a decile l .  
88 
For organisation and team planning purposes, Tumeke School is organised in  three clusters : 
Junior, Middle, and Senior teams. Altogether, there are nine composite5 classes, each with 
students within age bands of two or three years . 
5.6.2 THE PARTICIPANTS AND THE BEGINNING OF THE RESEARCH 
At the beginning of the school year seven teachers expressed interest in participating in the 
study (three teachers from the senior school level and four teachers from the middle school 
level) .  The teachers were provided with information sheets (see Appendix A) and we met to 
discuss in more depth the research aims, the nature of the study, and their role as eo?
researchers. In discussing ethical issues I explained that anonymity was not possible 
because other teachers at the school knew their identity. Issues of confidentiality were 
discussed and I outli ned the need for "an environment of trust" (Robinson & Lai, 2006, p .  
204 ), especially in relation to expected col laborative critiquing of classroom excerpts. I also 
reinforced that participation in the research was voluntary and that the teachers could 
withdraw from the study at anytime. 
In this first meeting other issues were discussed and explored. The teachers expressed 
concern about the time they might be expected to commit to the research.  They also 
discussed how to ensure that the focus of the research on mathematics did not detract from 
their professional growth in other curriculum areas . They were also concerned about how 
we could manage the participation of all their students, i ncluding those who did not have 
parental consent. Having another adult in the classroom and taking part in video recorded 
observations posed concerns for some members of the group. These teachers expressed 
personal fear about what the researcher and video recorded observations might reveal about 
both their mathematical understandings and their pedagogical expertise. The outcome of 
this wide ranging discussion was a shared understanding that the ultimate purpose of the 
research was a col laborative examination of ways to enhance student engagement m 
mathematical discourse which supported student achievement of higher levels of 
mathematical understandings. The openness of the discussion provided the beginnings of a 
5 Classes comprised of more than one age group of students 
89 
joint partnership in  the research and ensured that the teachers saw that they were i n  a 
posi tion in which they had "power over knowledge" (Babbie, 2006, p .  30 1 ) .  
In the first term of  the school year I was invited to present at two staff meetings .  In  the first 
meeting I provided an overview of the Numeracy Development Project (Ministry of 
Education, 2004a) including content focused on the strategy and knowledge stages .  In the 
second staff meeting I faci l i tated discussion and exploration of the participation and 
communication patterns of mathematical inquiry classrooms. I also provided the staff with 
a number of artic les which described elements of i nquiry classrooms. These research 
articles6 were selected with the intention of seeding ideas of what inquiry classrooms and 
the use of different col lective reasoning practices in them might look l ike. Subsequently, 
the articles supported on-going di scussion among the team and were useful in  the eventual 
implementation of the previously described communication and participation framework 
(see Table 2). 
During Term One I was in Tumeke School two mornings a week. This provided 
opportunities for the teachers to further discuss the study with me on an individual and 
informal basis. In response to their requests I worked collaboratively alongside the teachers 
6 The research articles included: 
Fra i v i l lig, J . ,  Murphy, L., & Fuson, K. ( 1 999) .  Advancing c h i ldren's mathematical thinking in everyday 
classrooms. Journal for Research in Mathematics Education, 30(2), 1 48 - 1 70. 
Huferd-Ackles, K . ,  Fuson, K .  C., & Sherin, M .  G .  (2004) .  Describing levels and components of a math-talk 
learning community. Journal for Research in Mathematics Education, 35(2), 8 1 - 1 1 6. 
Kazemi ,  E., & Stipek, D. (200 1 ) . Promoting conceptual thinki ng in four upper-elementary mathematics 
c lassrooms. The Elementary School Journal, 102( 1 ) , 59-79 
R ittenhouse, P. S .  ( 1 998).  The teacher's role in mathematical conversation: Stepping in and stepping out. In 
M .  Lampert, & M .  L .  Blunk (Eds.), Talking Mathematics in School (pp. 1 63- 1 89). Cambridge: Uni versity 
Press. 
Ste in, M .  (200 1 ) . Mathematical argumentation: Putting umph i nto c lassroom discussions. Mathematics 
Teaching in the Middle School, 7(2), 1 1 0- 1 1 3 . 
Wood, T. ( 1 999). Creating a context for argument in mathematics class. Journal for Research in Mathematics 
Education, 30(2), 1 7 1 - 1 9 1 .  
Wood, T.,  & McNea l ,  B .  (2003 ) .  Complexity in teaching and c h i ldren's mathematical thinking. Proceedings 
of the 2003 Joint Meeting of PM? and PMENA Hono l u l u, H I :  CRDG, Col lege of education, University of 
Hawaii .  
Mercer, N . ,  & Wegerif, R .  ( 1 999b). Children's talk and the development o f  reasoning in the c l assroom. British 
Educational Research Journal, 25( I ), 95- 1 1 2. 
W h i te, D. Y. (2003). Promoting productive mathematical c lassroom discourse with diverse students. The 
Journal of Mathematical Behaviour, 22( 1 ) , 37-5 3 .  
90 
during their numeracy lessons. At times the teachers asked me to observe them teaching 
numeracy lessons and give them feedback. My on-going involvement i n  the school 
supported my "access to the site under investigation" (Scott & Usher, 1 999, p .  1 00) .  My 
acceptance by the teachers and students as an 'adopted' member of the staff establi shed the 
foundations for our subsequent collaborative role .  As Bogdan and Biklen (2007) explain, 
"qualitative researchers try to interact with their subjects i n  a natural, u nobtrusive and 
nonthreatening manner" (p. 39). My presence i n  the school throughout the first term had 
established me as an accepted person around the school and in al l the classrooms, with staff 
members and the students. 
Towards the end of the first term, I reconfirmed with the seven teachers that they wanted to 
participate in the research. The teachers distributed information sheets and consent forms 
(see Appendix B and Appendix D) to their students and the students' parents or caregivers. 
These were returned to a col lection box in the school office. 
5.6.3 PARTICIPANTS IN THE STUDY GROUPS 
The seven teachers who participated in the study were a diverse group. Thi s  group included 
two Maori teachers, one teacher who was part Cook Islands and part European, two New 
Zealand European teachers and two Indian teachers and their teaching experience ranged 
from 3 years to 20 years. Early in the study, three of the teachers withdrew due to medical 
and family reasons leaving four participants in the study, two of whom are included as 
detailed case studies in this research report. These two teachers were selected to reveal the 
distinctly different  ways in which they transformed their classrooms into communities of 
mathematical i nquiry. Whi lst these two teachers had engaged in simi lar activities in the 
study group setti ng and while they appeared to fol low a simi lar path guided by the 
communication and participation framework, c lose examination of their goals and motives, 
and those of their students, led to important differences in the pedagogical practices the 
teachers took to construct discourse communities. These two case study teachers and their 
class composition are described in the fol lowing section. 
9 1  
5.6.4 THE CASE STUDY TEACHERS AND THEIR STUDENTS 
A va described herself as part New Zealand Maori-part New Zealand European . She was in 
her ninth year of teaching. The students in her Year 4-5 class were ethnical ly diverse : 47% 
were New Zealand Maori, 25% were from the Pacific Nations, 24% were New Zealand 
Europeans and 4% were other groupings. Based on the results from the New Zealand 
Numeracy Development Project assessment tool (Ministry of Education, 2004c) ,  24% of 
the students were achieving at significantly lower numeracy levels than that of comparable 
students of a simi lar age grouping in New Zealand school s ;  55% were one level below and 
2 1% were working at the level appropriate for their age group. No students were working at 
a level above the level attained by their age group national ly. 
Moana described herself as a New Zealand Maori married to a Cook Is lander and socially 
i nvolved in both cultural groups. She was in her fifth year of teaching. The students in her 
Year 4-5 -6 class were predominantly New Zealand Maori or from the Pacific Nations. 40% 
of the students were New Zealand Maori , 58% were Pacific Nations  and 2 %  were of New 
Zealand European ethnicity. In Moana' s c lass 3 1 %  of the students were achieving at 
significantly low numeracy levels ;  59% were one level below and I 0% were at the level 
average for their age group. Like Ava' s students, none of the students were achieving at a 
level above the average for their age group.  
5.6.5 STUDY GROUP MEETINGS 
Study group meetings with participating teachers continued at regular intervals throughout 
the duration of the research .  In this setting, I had many roles i ncluding being a participant, 
and acting as a resource for the teachers within a collaborative sharing of knowledge 
environment. 
Study group activities i ncluded examining research articles (see Appendix E) for 
descriptions of the communication and participation patterns of i nquiry c lassrooms, looking 
at  how teachers changed the forms of mathematical talk towards that of inquiry and 
argumentation, and reading about mathematical practices students used in i nquiry 
92 
environments. A DVD 'Powerful practices i n  mathematics and science' (Carpenter et al . ,  
2004b) from an international research study stimulated an examination of  inquiry practices. 
The teachers also re-examined the materials and learning activities in the NDP (Mini stry of 
Education, 2004b) and discussed which learning tasks best supported their goals to change 
the classroom discourse patterns. A joint decision was made that the group would adapt this 
material for use within the study. 
The video c lassroom records (see section 5 .7 .4) were used extensively in the study group 
context. Sections were selected to view and examine and through repeated viewing critical 
i ncidents were able to be identified; so too were the antecedents and consequences of these 
events. During viewing the discussion centred on the communication and participation 
patterns students used and how these supported student engagement in mathematical 
practices, the use of the different mathematical practices, and the pedagogical actions 
associated wi th the students' engagement in  increas ingly proficient mathematical practices .  
Other activities included col laborative review (with a peer or the group) of selections of 
their classroom video data and transcripts i n  order to ' tel l i ts story ' .  As part of thi s activity 
the teachers recorded explanations on their matching transcripts (see Appendix F). These 
i ncluded their intentions at the outset of the lesson and what they noticed as the students 
engaged in mathematical activity. It also i ncluded how as teachers they adjusted their 
actions or facil itated discussion to match what the students were saying and doing as they 
engaged in mathematical practices. 
In the third study group meeting (Phase 1 ,  Term 2, Week 7)  the teachers used the video 
observations to support the design of a framework (see Appendix E) of the questions and 
prompts to support the development of specific mathematical practices. As described 
previously the questions and prompts suggested by Wood and McNeal (2003) provided the 
foundations for the framework. Examination of the teachers' classroom video records 
supported identification of further questions and prompts. This  framework became a useful 
tool for the teachers to help model the types of questions and prompts they wanted their 
students to use. 
93 
5.7 DATA COLLECTION 
This section describes how the design experiment methodology and data col lection was 
used in this research.  Table 3 traces the study research programme over the research year 
from entry to the school to the point of withdrawal from the school and outli nes the 
schedule of observation phases. 
Table 3 
A time-li ne of data collection 
2003 Term 4 
Oct-Nov. Initial contact with Tumeke School and meeting with Senior Management. 
Staff meeting to outl ine and discuss the proposed research. 
2004 Term 1 
Week 2- 1 0  Researcher present In the school two mornings a week working 
collaboratively with 7 teachers to provide in-class and out-of-class support. 
Week 2 Meeting with 7 teachers to discuss the aims of the research. Information 
sheets and consent forms given to teachers. 
Week 4 Staff meeting: A summary of the NZNDP strategy and knowledge stages. 
Week 7 Staff meeting: Discussion and exploration of the discourse patterns of 
inquiry communities. Research artic les provided for background reading. 
Teacher participation reconfirmed. Information sheets and consent forms 
provided for students and caregivers . 
Week 9 Initial study group meeting to negotiate a tentative communication and 
participation framework. 
Week 1 0  Unstructured interviews with the teachers . 
Term 2 
Week 1 -4 Data col lection in  the senior school in Years 6-8. 
Week 3 Senior school study group meeting. 
Week S Middle and senior school study group meeting. 
Week S- 1 0  Data col lection i n  the middle school i n  Years 3-6 .  
Week 7 Study group meeting with the case study teachers . 
Week 1 0  Study group meeting with case study teachers. Unstructured interviews. 
Term 3 
Week 2-S Data col lection with the case study teachers in Years 3-8. 
Week 4 Study group meeting with case study teachers. 
Week 7- 1 0  Data col lection with the case study teachers in Years 3-8. 
Week 7 Study group meeting with case study teachers. 
Week 1 0  Study group meeting_ with case study_ teachers. 
Term 4 
Week 3-6 Data col lection w ith the case study teachers i n  Years 3-8. 
Week S Study group meeting with case study teachers. 
Week 7 Unstructured interviews with case study teachers. 
94 
5.7 .1  DATA COLLECTION IN THE CLASSROOMS 
Data col lection began at the start of the second school term in the senior level of the school .  
In the first 4 weeks two to three video observations of hour- long mathematics lessons were 
made in each senior classroom. In week 5 of the school term data col lection began in three 
middle school classrooms and continued for the fi nal five weeks of the school term. In the 
third term of the school year data collection occurred in the classrooms of the four case 
studies during weeks 2-5 and weeks 7 - 10. In the final term data collection began in week 3 
and was completed in week 6.  
5.7.2 PARTICIPANT OBSERVATION 
In any classroom study observations occur on a continuum. They may consist of less 
structured descriptive data which attends to minute detail or they may be tightly structured 
to focus on specific and selected events or persons. The degree to which the researcher 
participates in observations is also on a continuum with participation at its highest level 
when observations are least structured and the researcher i s  a participant observer 
(Creswell ,  1 998). 
A sociocultural perspective in which the "world i s  understood as consisting of individuals 
and col lections of individuals in teracting with each other and negotiati ng meanings in the 
course of their dai ly activities" (Scott & Usher, 1 999, p .  99) fi ts wel l  with participant 
observation techniques. Participant observation i s  grounded in establishing "considerable 
rapport between the researcher and the host community and requiring the long term 
immersion of the researcher in the everyday life of that community" (Angrosino, 2006, p .  
732) .  Developing a balanced relationship between al l participants was an  essential feature 
of the current study and so I took the role of participant observer, shifting back and forth 
from total immersion i n  the learning  context to complete detachment. This role is consistent 
with that taken by many researchers who have col laborated with teachers i n  design-based 
research (e.g. , McClain & Cobb, 200 1 ;  Whi tenack & Knipping, 2002) .  
95 
5.7.3 VIDEO-RECORDED OBSERVATIONS 
The use of video-recording as a tool  which flexibly captures and then supports study of the 
moment-to-moment unfolding of complex classroom interactions has made it a "powerful 
and widespread tool in the mathematics education research community" (Powel l ,  Francisco, 
& Maher, 2003, p .  406). Within design based research it has become widely used to collect 
and archive large amounts of both visual and aural data within the naturalistic contexts of 
c lassrooms (Roschel le, 2000; Sawyer, 2006b) .  
Video-data capture was used extensively in this  research primari ly due to two features 
which Bottorff ( 1 994) identi fies as its strengths: density and permanence. The notion of 
density refers to the way in which it is able to capture in depth observations of simultaneous 
events and provide audio and visual data in real time. This  included capturing antecedents 
and consequences of events as wel l  as nuances of tone and body language of participants. 
Each mathematics session was video recorded using one camera. Within each lesson video 
recorded observations targeted specific lesson events. The first section of each lesson was 
recorded as the teacher discussed and outli ned the mathematical activity with the students . 
Then the camera was positioned to focus on one small group (selected by the teacher) to 
capture the students' interactions as they problem solved. Final ly, the concluding l arge 
group sharing session was video recorded. 
The video-recorded data became a permanent record readi ly avai lable for subsequent 
review and iterative analysis .  In this research,  it  was used extensively by all members of the 
team to examine multiple factors on both an immediate and longitudinal basis .  Together we 
discussed and analysed many aspects in the emergence of each classroom mathematical 
i nquiry community. 
Although video is a valuable methodological tool for gathering data, its use i s  not problem 
free (Bottorf, 1 994; Hal l ,  2000; Roschelle, 2000) .  Roschelle explains that the introduction 
of the video camera in a naturali stic context causes change in the ways participants interact 
and behave. To minimise unwanted effects caused through use of the video, prior practice 
96 
runs, explai ni ng and discussing the purpose of taping, and the teacher modeling of 
everyday behaviour took place in each classroom. 
Hall (2000) and Roschelle (2000) also caution that video data is neither a complete account, 
nor theory free. "These realities both constrain and shape later analyses and presentation of 
resu lts" (Powell et al . ,  2003, p .  408) .  This research drew on multiple sources of data to 
supplement and triangulate the video observations in recognition that video-recording can 
be mechanical and theory loaded. These i ncluded field-notes of aspects of the classroom 
context i ncluding the physical setting, the organi sation of groups, the activities and 
discussion, the interactions and dialogue of the teacher with the students, with me, and of 
student to student and documents. 
5.7.4 DOCUMENTS 
Documents are an important source of evidence to augment the audit trail (Bogdan & 
Biklen, 1 992). I n  this study the documents included the mathematics problems the teachers 
used with the students, student written work samples, the charts recorded within large 
group explanations and those the teachers used to support the communication and 
participation patterns. Also included were the informant diaries completed by the teachers. 
Some teachers regularly completed diaries; others were constrained by lack of time and 
pressure in their dai ly life as teachers. Although the teacher recorded reflections reduced as 
the study progressed, discussion with me became more frequent. S imi larly, their 
conversations with their peers became more frequent. They discussed interesting or 
puzzling aspects of student reasoning, student participation in mathematical practices, or 
the outcome of their use of specific problems or activities. 
Teacher reflections about their videoed lessons were another important source of data. The 
video observations were transcribed as soon as possible after each observation and made 
avai lable for the teachers to read. The teachers added reflective comments to the transcripts 
as they read them (see Appendix F). Additionally, they would watch sections of the video 
observations which had attracted their attention in the transcriptions and add comments. 
The video excerpts and notes would be discussed in the study group context. Such activity 
97 
supported richer development of taken-as-shared interpretations of what was evolving in 
the classroom communities. It enabled me to gain "better access to the meanings of the 
participants in the research" (Scott & Usher, 1 999, p .  1 00). Maintaining the voice of all 
participants and capturing the teachers' perspectives about emerging inquiry discourse 
patterns and their corresponding mathematical practices was a central aspect in this study. 
5.7.5 INTERVIEWS WITH TEACHERS 
Where possible, fol lowing each lesson observation, the teacher and I briefly di scussed the 
learning activities, and explored our shared view of the communicative interactions and 
emerging mathematical practices. On a micro-level these short discussions resulted in  
small modifications of the learning activities and enactments in  the communication and 
participation framework. 
Three scheduled interviews were held with the case study teachers. In the first interview the 
teachers described their students and the achievement levels of their c lasses. They also 
outl i ned their attitudes towards mathematics and the approaches they used when teaching 
mathematics. The second interview took place with the four case study teachers as a group 
at the end of the second term. This interview explored how the shifts in the interaction 
patterns were challenging the teacher and students beliefs about their roles in the classroom. 
The teachers also made direct comparisons with the previous mathematical classroom 
cultures they had enacted, the classroom communication and participation patterns and how 
they had organised mathematical activity. At the conclusion of the study the four case study 
teachers participated in a semi-structured in terview to capture their reflections on their 
involvement in the research study. 
Interviewing offers flexibil ity for the researcher to probe deeper for alternative or richer 
meanings. Kvale ( 1 996) suggests that the interviewer s hould be a "miner" (p. 3 )  or 
"wanderer" (p. 5) and engage in i nterviews which resemble conversations which lead 
participants to recount their "stories of their l ived world" (p. 5 ) .  However, whilst more 
informal interviews can act as 'conversations' Babbie (2007) notes the need for 
interviewers to be aware that "they are not having a normal conversation" (p. 307). Rather 
98 
they need to listen closely and promote participant talk. This was an important 
consideration in this research. The interviews supported my intention of understanding, 
from the perspective of the teachers, the actions they took to construct an i nquiry 
community and how they perceived the effect of these actions on the mathematical 
practices the students used. On one hand, I was required to be a design research partner and 
eo-share knowledge of the developing inquiry cu ltures and mathematical practices. On the 
other hand, I needed to balance this with acting as a "social ly  acceptable incompetent" 
(Lofland & Lofland, 1 995, p. 56-57) .  This requi red setti ng aside a personal knowledge of 
teaching mathematics in an inquiry classroom and offering myself in the role as "watcher 
and asker of questions" (p. 56). 
5.7.6 EXIT FROM THE FIELD 
Withdrawing from the close relationship of the year-long collaborative design based 
research was an important process. In addition to the exit i nterviews, the teachers met with 
my research supervisors to debrief. The fol lowing year I partic ipated in a team planning 
meeting and provided support for the teachers as they mapped out ways to develop 
mathematical inquiry communities with their new classes. Then at regular intervals I 
continued to informal ly  visit and provide on-going support. The teachers were also 
provided with conference papers related to the research project to read and confirm. They 
have become members of a local mathematical professional group and have eo-presented 
workshops at local professional development days and one eo-presented a workshop with 
me at a national conference. 
5.8 DATA ANALYSIS 
Consistent with the use of a design approach, data col lection and analysis held 
complementary roles in that one informed the other in an iterative and cycl ic manner. 
99 
5.8.1 DATA ANALYSIS IN THE FIELD 
Data analysis began concurrently with data col lection in an intensive on-going process to 
identify tentative patterns and relationships in the data. Memos supported the 
contextualisation of the data, the development of theoretical insights, and noted connections 
between the immediate situation and broader theory. Initial coding themes were used to 
feed-back to shape subsequent decisions in the design research.  Of particular importance in 
the iterative and on-going analysis in the field was the review of the individual teachers' 
progress in enacting aspects of the communication and participation framework?
confirmation of aspects of its utility or need for extension, expansion, or revision of 
particular teacher actions and supports were noted. This iterative and recursive process is 
consistent with design research in which the researcher conducts "on-going data analyses in 
relation to theory" (Barab, 2006, p .  1 67) .  
5.8.2 DATA ANALYSIS OUT OF THE FIELD 
At completion of data collecting I began the daunting task of analysis of the data set in its 
entirety-a key element of design research methodology used to maintain trustworthiness 
of the research findings (Cobb, 2000; McClain,  2002b). Qualitative data analysis i s  
described as  an art form which includes an important "creative e lement" (Wi llis, 2006, p .  
260) .  It is a complex, time-consuming and intu iti ve process which requires a constant 
spiral l ing forward and backward from concrete chunks of data toward larger and more 
abstract levels (Creswell ,  1 998) .  In this study, a search for explanatory patterns drew on 
constant comparative methods. Using the general approach of Glaser and Strauss ( 1 967) to 
manage a volume of data and bui ld theory inductively from it is consistent with other 
quali tative and design-based research studies (e.g. ,  Bowers et al . ,  1 999; Cobb, 2000; Cobb 
& Whitenack, 1 996; McClain, 2002b; Moschkovich & Brenner, 2000) .  
Glaser and Strauss ( 1 967) contend that the  development of theoretical ideas should occur 
alongside the collection and analysis of data. These scholars used the term 'grounded' to 
indicate that the theoretical constructs are embedded in  the data col lection and analysis 
process. Important to this quali tative approach i s  the i terative use of coding and memo-
1 00 
writing for the generation of larger and more abstract concepts (Will is ,  2006) . Wil l i s  
out li nes coding as a process in which the qualitative researcher attributes meaning toward 
chunks of text, identifies the key concepts in the text and labels those which are most 
salient. In this process, i ncidents in the data are coded and constantly compared in order to 
develop and define their properties. Then, so that the codes and their properties can be 
integrated, the relationships among the concepts are closely examjned. Eventual ly as 
patterns of relationships in the concepts are clarified, some which were initially noted are 
discarded as irrelevant to the emerging theoretical constructs (Glaser & Strauss). 
In this research, coding began with the data col lected in one classroom. To confirm or 
refute in itial codes and memos recorded in the field, transcripts and field notes were read 
and reread and video observations repeatedly viewed. Hunches grounded in the data were 
formulated using a process Bogdan and Biklen ( 1 992) term "speculating" (p. 1 57) .  Patterns 
in communication and participation and incidents and events which occurred and 
reoccurred during classroom interactions were noted. The use of codes and memos to 
maintain a way of developing the emerging 'hunches' into a more coherent form continued. 
These were added to field notes,  teacher reflections, the transcribed video observations and 
the c lassroom artefacts. Follow up questions and words or incidents circ led to be 
considered further were added. 
In order to bring meaning and structure to the volume of data from the four case study 
c lassrooms I needed to use a data reduction process (Scott & Usher, 1 999). Data reduction 
involved sampl ing from the data set by including what was relevant and excluding what 
was irrelevant to the research question. The analysis of one classroom was used to create an 
initi al set of codes, simple descriptions which sought to give meaning to the mathematical 
discourse; the participatory and communicative acts takjng place in the mathematics 
lessons. The codes covered both single l ines of words and action or chunks of text (see 
Table 4) .  
l 0 l 
Table 4 
Examples of the codes 
Teacher te lls students to ask for an explanation which is di fferent. 
Teacher asks group_ members to support each other in exglanation. 
Teacher asks for col lective support when student states lack of understanding. 
Teacher asks students to predict guestions related to strategy_ solution. 
Student expects support when stating lack of understanding or aski ng for help. 
Student adds ideas or questions that advance col lective thinking. 
Student tel ls group members that group work involves team work. 
Student makes sure that everyone else understands strategy solution. 
After coding al l the data I then began the task of manually sorting the codes into coherent 
patterns and themes to use to guide analysis of the other three case study classrooms. At the 
conclusion of thi s first iteration twelve themes for the teachers' actions and ten themes for 
students' actions had been constructed (see Table 5) .  
Table 5 
Examples of the themes for teacher and student actions 
Teacher actions Student actions 
The teacher asks for col lective The students use strategies which support 
responsibi l ity to the mathematics collective responsibi lity to the mathematics 
community. community. 
The teacher provides models for how The students use questioning, clarification 
students are to question, c larify, and argue and argumentation to support sense making 
to draw explanatory justification of and the development of explanatory 
mathematical thinking. justification. 
The teacher positions specific students so The students position themselves and 
they are able to engage equitably in others to engage in mathematical activity 
mathematical activity. and validate their reasoning. 
The teacher asks the students to make clear The students make clear conceptual 
conceptual explanations of their thinking. explanations of their mathematical 
thinking. 
The teacher emphasises the need for sense- The students use specific questions which 
making and scaffolds students to ask support sense-making of mathematical 
questions to make of a mathematical explanations. 
explanation. 
Fol lowing the first level of coding I engaged in refinement of the codes, testi ng and 
retesting these, discarding some, and verifying or adding others as they emerged in the data. 
1 02 
The evidence of teacher actions and student actions which supported each code was closely 
examined, added to, redefined or deleted. Evidence which supported each theme was 
developed and used to examine the data of each setti ng. (See Table 6) 
Table 6 
Examples of the evidence for one theme 
The theme: The teacher emphasi ses the need for sense-making and scaffolds students to 
ask questions to make sense of a mathematical explanation. 
1 .  The teacher asks students to track each step of an explanation. 
2 .  The teacher explicitly affi rms students for fol lowing step by step. 
3 .  The teacher asks for explanation to be re-explained i n  a different way. 
4. The teacher model s  questions which lead to clear conceptual explanation (what, 
where, is that, can you show us, explain what you did) .  
5 .  The teacher asks for use o f  materials to back up explanation. 
6. The teacher asks listeners to clarify thinking behind steps in an explanation . 
In the final iteration the underlying patterns and themes were used to develop conceptual 
understanding of how the teacher' s  pedagogical practices guided by the communication and 
participation framework, initiated students i nto the use of effective mathematical practices 
within inquiry communities. Data analysis tables of each classroom activity setting 
supported the exploration and sense-making of the themes in early drafts of the findings. 
Gabriel (2006) maintains that embedding the writing of sections of the findings as part of 
the data analysis process is an important tool which supports theorising and sense-making 
of the data. In  this study, writing based on the prel iminary analysis developed in the field 
began immediately .  Through writing conference presentations (Hunter, 2005 , 2006, 2007a, 
2007b) and chapter revisions, a richer sense of the relationships i nvolved in changing 
communication and participation patterns i n  classroom communities emerged. 
5.8.3 SOCIOCULTURAL ACTIVITY THEORY DATA ANALYSIS 
As i l lustrated in the previous section a grounded approach provided a useful tool to code 
the data in an ordered and systematic way. In turn, sociocultural activity theory provided an 
additional conceptual means to examine and analyse the interaction between various 
elements of the activity system and reveal significant differences in  each of the teacher' s 
1 03 
development of a discourse community. Of particular relevance were the subjects-the 
teachers and students-and their "expanding involvement-social as well as i ntel lectual" 
(Russell, 2002, p. 69) in communicating and partic ipating in proficient mathematical 
practices in inquiry mathematics classroom communities. Also under consideration was the 
object or focus of activ ity-the overall direction of activity and its shared purpose or 
motive. Russell notes that the "direction or motive of an activity system and its object may 
be understood differently or even contested, as participants bring many motives to a 
collective interaction and as conditions change" (p.  69) .  As the communicative and 
participatory patterns changed in classroom communities, of interest was evidence of the 
resistance, discontinuities and deep contradictions produced in the activity system.  These 
related particularly to the actions the teachers took in enacti ng the communication and 
participation patterns and how they interpreted these actions in their interviews when they 
reviewed and discussed the lessons and video observations. 
5.9 DATA PRESENTATION 
The findings are reported in the form of case studies of two classroom teachers and the 
pedagogical actions they took to construct communities of mathematical inquiry and their 
corresponding mathematical practices. Direct quotations are used in the findings, drawn 
from the interview data, discussions with teaching colleagues and me, and the teachers' 
own verbal or written statements as they examined v ideo observations and transcripts or 
reviewed and reflected on lessons .  The teachers ' voice provides a way to gain deeper 
understanding of the teachers' goals and motives as they reconstructed the communication 
and participation patterns towards inquiry in their classrooms. 
The findings are reported in  three distinct phases. These phases relate to the three school 
terms i n  which data col lection took place, although there is some overlap which 
corresponds with changes in communication and participation patterns i n  the classrooms 
and their related mathematical practices. Vignettes of mathematical activity are provided to 
i l lustrate the actions of the teachers and students. They provide detail of the mathematical 
practices as these emerged and were refined. 
1 04 
5.9 .1  TRUSTWORTHINESS, GENERALISABILITY AND ECOLOGICAL 
VALIDITY 
Although objectivity, re liabil ity and validity are factors which make design research a 
"scientifically sound enterprise" (DBRC, 2003, p. 7) these attributes are managed 
di fferently within this form of research. Gravemeijer and Cobb (2006) argue that a central 
aim of the design research approach is to achieve ecological validi ty-that is that 
descriptions of the results should offer the starti ng point for adaptation to other situations. 
These researchers explain that an aim of the approach i s  to construct local i nstructional 
theory which can act as a frame of reference (not direct replication) for teachers who want 
to adapt it  to their own classroom situations and personal objectives. To achieve this 
requires thick description of al l detai ls including the participants, the learning and teaching 
context, and what happened in the design research supported by analysis of how the 
different elements may have influenced the process. In the current study thick description i s  
provided of al l e lements in the research. The study's ecological validity i s  further 
strengthened through use of teacher input i nto how best to adjust their i ndividual pathways 
towards enacting the communication and participation patterns of inquiry while also 
accommodating their particular circumstances . 
In the design research approach the conventionally accepted factors of val idity and 
reliabil ity are replaced by need to establ ish credibi l ity and trustworthiness (Lincoln & 
Guba, 1 985) .  Trustworthiness of the research centres on the need for the credibil ity of the 
analysis (Cobb, 2000; Cobb & Whitenack, 1 996; McClain ,  2002b). McClain outli nes the 
need for a "systematic analytical approach,  in which provisional claims and conjectures are 
continual ly open to modification" (p. 1 08) .  Documentation in all the phases should include 
"the refin ing and refuting of initial conjectures.  Final c laims and assertions can then be 
justified by backtracking through the various levels of the analysis" (p. 1 08 ) .  Other 
criterion which i ncreases credibi l ity and trustworthiness of analysis include prolonged 
engagement with participants in the field by the researcher (Lincoln & Guba, 1 985)  and 
peer cri tiquing of analysis .  This research was driven by the need for a systematic, thorough, 
and auditable approach to the analysis of the l arge sets of video records, observations and 
1 05 
transcriptions generated during its l ife. Prolonged engagement with the teachers as eo?
researchers, and their students in c lassroom communities was a strong feature of the study. 
The on-going collaborative teacher-researcher partnership provided peer cri tique and 
important insights into understanding and interpreting acti vity from the teachers ' personal 
perspectives. 
In design research the issue of generali sabi l i ty is addressed by v1ewmg events as 
"paradigmatic of broader phenomena" (Cobb, 2000, p. 327). Cobb explains that 
considering "activi ties and events as exemplars or prototypes . . .  gives rise to 
generalisabi li ty" (p. 327) .  However, Cobb clarifies his position outlining that this is not 
generalising in a traditional view where particulars of individual situations "are e ither 
ignored or treated as interchangeable" (p. 327) with s imi lar situations. Instead, he argues 
that the post hoc theoretical analysis which occurs after data collection is completed is 
relevant for i nterpretation of cases across wider situations.  Generalisabi lity was supported 
in this research through the use of retrospective analysis, the careful detai l ing of the setting 
and participants, and the use of multiple classrooms and teachers as cases. 
5.10 SUMMARY 
This chapter began by outl in ing the broad research question used in this study. The 
selection of the research paradigm and use of design research fol lowed natural ly from the 
sociocultural perspective of this study. The key characteristics of design research 
methodology were explored in relation to the collaborative relationship between the 
teachers and myself. 
Descriptions were provided of the data col lection methods and how these captured the 
teachers' perspective on the emergence of a discourse community and i ts corresponding 
mathematical practices. Consistent with the use of design methodology, data collection and 
analysis held complementary roles in that one informed the other in an iterative and cyclic 
manner. Data analysis occurred concurrently with data collection, then after data collection 
1 06 
had concluded, of the complete data set. The use of sociocultural activity theory provided a 
way to establish important differences between the case study teachers. 
In the fol lowing chapters I present the findings of two case study teachers. The literature 
signalled the many complex situations which teachers may encounter as they develop 
mathematical discourse communities. These chapters i l lustrate the different pathways 
teachers take as they engage with the complexities of developing classroom communities of 
mathematical inquiry. 
1 07 
CHAPTER SIX 
LEARNING AND USING MATHEMATICAL PRACTICES IN A 
COMMUNITY OF MATHEMATICAL INQUIRY: AVA 
One needs an identity of participation in order to learn, yet needs to learn in order to 
acquire an identity of parti cipation. (Wenger, 1 998,  p. 277) 
6.1 INTRODUCTION 
The l iterature chapters drew attention to the need for student engagement i n  reasoned 
mathematical discourse in l earning communities if they are to learn and use proficient 
mathematical practices. Persuasive evidence was provided that teachers can establish such 
intel lectual cl imates through the application of a range of pedagogical actions. In  this 
chapter and the next, each case study is organised in  three distinct phases which directly 
relate to the three school terms in which data col lection occurred. Each section reports on 
the pedagogical actions of the two teachers (A va and Moana), and then the transformative 
changes these caused in the mathematical practices used in the classroom community. A 
commentary that links the changes in mathematical practices to the l iterature accompanies 
each section. 
For Ava' s case study Section 6.2 describes Ava' s beliefs about doing and using 
mathematics and how these beliefs shaped the initial c lassroom learning context. Section 
6.3 outlines the different tool s  Ava used to mediate the foundations of an inquiry classroom 
culture. Descriptions are provided of how the safe collaborative classroom culture 
supported student participation in questioning and explaining mathematical reasoning.  
Section 6.4 describes the actions A va took to engage the students i n  i nquiry and 
argumentation .  A close relationship is i l lustrated between changes in the participation and 
communication patterns and i ncreased student engagement in collaborative argumentation 
1 08 
(Andriessen, 2006) .  Evidence of increased use of exploratory talk (Mercer, 2000) and more 
efficient use of mathematical practices is provided. 
Section 6.5 outlines how A va' s actions to further transform the participation and 
communication patterns resulted in an intel lectual cl imate where students mutually  engaged 
in proficient mathematical practices. How learning partnerships withi n multi -dimensional 
zones of proximal development emerged is described. 
6.2 TEACHER CASE STUDY: A V A 
In an interview at the beginning of the study A va outl ined how she had always liked 
teaching mathematics.  However, she reported that her earlier participation in  the New 
Zealand Numeracy Development Project (NDP) (Ministry of Education, 2004a) had caused 
a loss of confidence in her abi l ity to meet her students' mathematical needs. She outli ned 
how the new instructional materials and strategies in the NDP, and its focus on students 
explaining their mathematical thinki ng, conflicted with how she had previously taught 
mathematics. Formerly she believed that it was her responsibi l ity to explain the rules and 
procedures and that she had felt confident about her abi l ity to do so. The concerns A va 
voiced are consistent with those other researchers (e.g. , Rousseau, 2004; Sherin,  2002b; 
S ilver & Smith, 1 996; Weiss et al . ,  2002) have identified as teachers change their 
pedagogical practices to imp lement i nquiry based learning. As an experienced teacher Ava 
had been secure in  her routines and in her understanding of the nature of mathematics. But 
her involvement i n  the professional development had led to dissonance in her previously 
held beliefs about both teaching mathematics and what she was being asked to implement. 
Despite the reservations Ava had toward the NDP she had implemented aspects of what she 
had learnt in the professional development. She reported that she used or adapted the NDP 
lesson outlines to teach the students solution strategies.  She had also encouraged the 
students to generate and explain their strategy solutions to her and a group of l isteners. 
However, ini tial observations at the start of the study showed that these sharing sessions 
took a format in which a student would explain a strategy and the other students would sit 
1 09 
l i stening in  s ilence. If questions were asked, it was A va who more often asked them. A va 
had done what Sherin (2002b) reports many teachers do when given new curricula designed 
to change the content of instruction-she had transformed the material to fi t wi th her 
familiar routines . As a result, the classroom context was consistent with a 'conventional 
culture'  (Wood, 2002) .  Although Ava had constructed a classroom cl imate in  which the 
students talked more in the mathematics lessons,  the teacher- led questioning elicited 
teacher-expected answers . A va had retained a central position in the classroom as the main 
source of mathematical authority. 
6.3 ESTABLISHING MATHEMATICAL PRACTICES IN A 
COMMUNITY OF MATHEMATICAL INQUIRY 
Through discussion with me and the study group Ava established the immediate actions she 
wanted to take. To do this she drew on the communication and participation framework 
(see Figure 1 ,  section 5 .4. 1 )  and the research articles described previously i n  Chapter 5. The 
art icles offered her ways to consider how she might establish an i nquiry environment. They 
also provided models for her to consider of students engaging in mathematical practices. 
The communication and part ic ipation framework provided Ava with a tentative pathway to 
structure her students' partic ipation in reasoned mathematical activity .  In the first instance, 
the mathematical practice she aimed to extend focused on students' participation in 
mathematical explanations . Thi s  participation i ncluded how they constructed and explained 
their arguments. It also included how the other students as l isteners actively engaged with 
the reasoning being used. A va indicated that to achieve this she would need to build a more 
collaborative classroom environment-a classroom culture where student reason ing was a 
key focus. 
6.3.1 CONSTITUTING SHARED OWNERSHIP OF THE MATHEMATICAL 
TALK 
At the beginning of the study Ava and her students held distinctly different roles with in  the 
classroom discourse community. The communicative rights and responsibilities were what 
Knuth and Peressini (200 1 )  described as asymmetrical .  Ava was the dominant voice and 
1 1 0 
she used univocal discourse as the most prevalent form of communication. To enact 
changes to the discourse patterns Ava directly addressed the new ' rules' for talk. She made 
explicit the changes being enacted by commencing each mathematics session with a 
discussion of expectations. She outlined and examined with the students how they were 
requi red to work together to bui ld a mathematical community. She emphasi sed that 
working together involved an increase in col laborative participation in mathematical 
dialogue involving them both as li steners and as talkers. She repositioned herself from the 
central position of 'mathematical authority' to that of 'participant in the dialogue' .  She 
modeled the shift explicitly by placing emphasi s on the use of the words 'we' and 'us'  as 
she participated in discussions .  For example, during a mathematical discussion at the end of 
the first week of the study she asked: Can you show us with your red pen what would 
happen ? We want to know. Through consistent use of simi lar statements she indicated to 
her students that she, too, was a member of the classroom community. 
Changing the rules and roles in the community 
The classroom interactions had previously been shaped by asymmetrical patterns of 
discourse and by the mathematical authority which Ava implicitly held . To re-mediate how 
they all participated in communicating mathematical reasoning required the reconstitution 
of the interactional norms. When we reviewed the video observations A va explained her 
reasons for the explicit discussions. In  the first instance, I interpreted that her overarching 
purpose was directed toward developing active student engagement in i ncreased 
collaborative interaction and dialogue within classroom community . She wanted the 
students to be a key part of the changes and so her discussions were designed to motivate 
the students and help them to understand and adopt the changes. In an informal discussion 
during the first month of the study she commented that many of the students were finding 
the changes challenging. She related these specifical ly to the expectations they had of her 
role as 'mathematical authority' .  The gradual repositioning of herself from the role of 
authority to that of participant i ndicated her shift toward a more "dynamic and fluid" (Amit 
& Fried, 2005 , p .  1 64) view of authority in the classroom. Ava had taken the first step 
toward the development of an i ntel lectual partnership in the c lassroom community based on  
shared ownership of  the classroom talk. 
1 1 1  
6.3.2 CONSTITUTING A SAFE LEARNING ENVIRONMENT 
Ava, in her discussions with me, noted a need to establ ish a safe learning environment. She 
considered that in a safe environment her students would be wi l l ing to engage in 
mathematical dialogue more readi ly. She anticipated that the increased focus on their need 
to explain and justify their reasoning might cause a temporary reduction in her students' 
confidence . Therefore in these beginning stages of the study, in order to provide support for 
the students, she made many statements to them which affirmed her belief in their abi l i ty to 
cope with the changes. She confirmed to them that she understood their need for time to 
learn how to engage in mathematical dialogue. She regularly expressed belief in their 
mathematical abi lity and used thi s to explain why she believed in their abi l i ty to participate 
in mathematical conversations. For example, she told them:  Sometimes you share that 
mathematics magic really quietly with someone next to you or near you . . .  I know and you 
know that it is simmering, that things are happening and that 's all right. 
At the same time, she explicitly outlined how explaining or questioning reasoning required 
the students to take both social risks and intellectual risks. She also indicated that the 
students needed to take risks in their own reasoning when making sense of other' s 
explanations and justification. 
The following vignette i l lustrates the explicit attention A va gave to the development of a 
c lassroom climate as she engaged the students in discussion of their responsibi l ities within 
their changing roles. 
Intellectual risk taking 
Before the students began their mathematical activi ty Ava discussed a need to risk-take. 
Ava Remember how yesterday we talked about i n  maths learning how you go 
almost to the edge? So therefore I am going to move you out of your 
comfort zone. It' s lovely being in a comfortable cosy place. Even as an adult 
we love to be there too. But if you are already there then i t ' s  time to move 
1 1 2 
on, out a l i ttle bit .  . .  so you go out there . . .  maybe a bit more . . .  a bit further 
next time and come back in again . . .  
Sandra And when you are out there you wil l  make that your comfort zone. Then 
move on and make that your comfort zone. 
A va So your comfort zone as Sandra said will move. You may have been here. It 
wil l  go out a li ttle bit. You will get used to this over here and you wi l l  think 
oh that ' s  cool ,  I am quite comfortable here. I don ' t  feel threatened. I am not 
stressed. I can do this. Hey so if I can do this and I was here before 
as Sandra said I will go over here. 
(Term 2 Week 5) 
Changing beliefs, affective relationships 
Early in the study, examjnation of the data reveals that Ava was very aware of tensions and 
contradictions between the interactional norms she wanted to establish and the previous 
norms which sti l l  shaped the community ' s  current learning behaviour and beliefs .  
However, if  student beliefs about 'doing and usi ng mathematics' were to be re-mediated the 
tensions and contradictions were expected and indeed necessary components to activate 
changes in the current learning environment. Russell (2002) explains that activity systems 
constructed by humans are continual ly subject to change; change which is driven by 
contradictions within their various elements. A va recognised that affective needs were an 
important consideration to thi s change. The actions Ava took in attending to the students' 
affective needs when establishing collaborative learning are explained by a sociocultural 
learning perspective. During col laborative interaction and eo-construction of meaning, 
intel lect and affect are interdependent elements "fused in a unified whole" (Vygotsky, 
1 986, p .  373) .  
Ava 's  actions i n  placing direct focus on how the students would be assisted to make 
changes within a safe supportive environment aimed to provide them with affective support 
to take the required social and intel lectual risks . The establi shment of a safe, supportive, 
positive learn ing environment is reported as an essential pedagogical component for 
engendering learning competence (Alton-Lee, 2003 ; Cobb, Perlwitz, Underwood-Gregg, 
1 998 ;  Povey et al . ,  2004; Wells ,  1 999). Her carefully crafted care and support i ncorporated 
the key elements Mahn and John-Steiner (2002) identify as components of emotional 
1 1 3 
scaffolding including the "gift of confidence, the sharing of risks i n  the presentation of new 
ideas, constructive criticism and the creation of a safety zone" (p. 52) .  
6.3.3 COLLABORATIVE CONSTRUCTION OF MATHEMATICAL 
EXPLANATIONS IN SMALL GROUPS 
Ava stated that in line with the NDP she regu larly used small activity groups m her 
mathematics lessons. These groups of 3-4 students were required to construct solution 
strategies for explaining at a larger sharing session .  However, in reviewi ng of videotapes 
with me A va recognised that the students working in their smal l groups predominantly 
engaged in use of either cumulative or disputational talk (Mercer, 2000) . Cumulative and 
disputational talk, as we saw in Chapter 3, has been characteri sed as an unproductive form 
of talk which l imits how group members explore each other 's  mathematical reasoning. Ava 
acknowledged that her students needed more specific guidance on how to work 
col laboratively. 
In the first i nstance, A va focused on how the students participated together in smal l group 
activity. She outlined to the students her requirement that they actively engage in l istening, 
discussing, and making sense of the reasoning used by others. To develop their ski l l s  to 
work col lectively she stressed that all group members needed to engage in construction of 
mathematical explanations and be able to explain them to a wider audience . 
In  accord with the pathway she had mapped out, Ava init iated an immediate shift towards 
establishing that the mathematical explanations the students constructed should be wel l  
reasoned, conceptually clear, and logical . She explicitly scaffolded how they were to 
provide an explanation :  Talk about what you are doing . . .  so whatever number you have 
chosen don 't just write them. You say I am going to work with . . .  or I have chosen this and 
this because . . .  and this is what I am going to do. She outlined not only how these 
explanations needed to make sense for a l istening audience but also how listeners needed to 
make sense of the explanations offered by others. To develop their ski l l  in the examination 
and analysis of explanations she provided opportunities for the small group members to 
construct, explain, and, in turn, question and clarify explanations step-by-step. The 
1 1 4 
fol lowing vignettes from three separate observations early in  the study i l lustrate how Ava 
inducted the students into public construction and evaluation of conceptual explanations 
within collaborative zones of proximal development (Vygotsky, 1 978) .  
Collaborative interaction and sense-making 
The students had individual time to think about a solution strategy then Ava said: 
Ava You are going to explain how you are going to work i t  out to your group. 
They are going to l isten.  I want you to think about and explain what steps 
you are doing, each step you are doing, what maths thinking you are using. 
The others in  the group need to listen careful ly  and stop you and question 
any time or at any point where they can ' t  track what you are saying. 
(Term 2 Week5) 
The students were asked to construct i ndividual conjectures then Ava directed them to 
examine and explore each solution strategy: 
Ava They might say I think it is  59. That ' s  cool but they have to back i t  up, 
explain how they came up with it . They have to say why. So I want you 
before you even begin to go around in your group and actual ly talk about i t .  
Someone in  your group may ask you a question . For example, that' s an 
interesting solution . Why do you think that? Could you show us how you 
got it? 
(Term 2 Week 6) 
Ava explained and explored the group roles then directed them:  
Ava Argue your maths. Explore what other people say. Listen careful ly bit by bi t  
and make sense of each bit. Don ' t  just  agree. Check i t  al l out  first. Ask a lot 
of questions. Make sure you can make sense that you understand. What ' s 
another important thing i n  working in  a group? 
Alan Share your ideas . Don ' t  just say I can do i t  myself that adds on to teamwork. 
A va That ' s  right. We do need to  use each other' s thinking . . .  because we are very 
supportive and that 's  the only way everyone wi l l  learn. So we have to be 
discussing, talking, questioning, and asking for clarifi cation. Whatever i t  
takes to c larify what you understand i n  your mind. 
(Term 2 Week 5) 
Interaction scripts, peer collaboration, making and clarifying mathematical explanations 
Ava had assumed that the students would construct appropriate knowledge because they 
were asked to interact cooperatively in a small group. Thi s  is  a common mjsconception 
(Mercer, 2000). The i nit ial examination of lessons in Ava' s c lassroom confirmed what 
1 1 5 
Mercer and other researchers report (e.g. ,  Irwin & Woodward, 2006; Rojas-Drummond et 
al . ,  2003 ; Wegerif et al . ,  1 999):  the students had many difficul ties engaging in and using 
mathematical talk in the smal l group si tuation 
To transform how the students interacted in mathematical activity in their small groups 
required expl icit renegotiation of both what the task required and the scripts for conduct?
the ground rules which shape the interaction (Gall imore & Goldenberg, I 993) .  A va's  use of 
clear directives to outl ine her expectations gave the students a working knowledge of what 
their obl igations were. It also provided them with understanding of the learning potential of 
small group interactions, offering them a motive to participate appropriately and develop 
ownership of the ground rules. She laid the foundations for peer col laboration (Forman & 
McPhai l, 1 993)  as she structured the group activity .  The students were introduced to and 
practised ski l l s  for explaining their reasoning and examining the reasoning of others . Many 
of these actions paral lel the pedagogical actions taken by Lampert (200 1 )  when she 
instituted small group col laboration. These included : establishing with the students 
recognition of themselves as valuable sources of knowledge;  emphasising mutual 
responsibility for sense-making; and the requirement of individual responsibility for 
understanding, thus removing any possibil ity that a lack of understanding cou ld be 
attributed to others. 
6.3.4 MAKING MATHEMATICAL EXPLANATIONS TO THE LARGE GROUP 
Each mathematics lesson inc luded small group activity and a larger sharing session. In the 
sharing session groups of students provided the mathematical explanations they had 
constructed together. In these sessions A va took a key role. She questioned and provided 
prompts to ensure that the explanations were tied to their problem form. When required, she 
specifical ly asked questions to make the explanations experiential ly real for the l isteners. 
To promote c larification of explanatory reasoning A va discussed and modeled the use of 
specific question starters (e .g. ,  what, where, i s  that, can you show us  and explain what you 
did) and monitored how they questioned each other. She would ask them to rephrase a 
question if she judged that i t  had not elicited the information required to make sense of an 
explanation .  She directly focused their attention on paradigmatic model s  of students using 
1 1 6 
questioning effectively to make sense of an explanation .  For example, Ava asked: Did you 
see that? For example, as we are saying add two, that 's what Runt was saying, okay so 
we 've got these two here. Then Alan asked a very good question, why aren 't we adding on 
three each time ? 
Through analysing and discussing video observation excerpts with me and the study group, 
Ava became increasingly aware of the potential learning opportunities student contributions 
to discussions offered. In  c lass she began to li sten more closely to the reasoning used in 
small groups in order to structure who presented explanations to the larger group sessions. 
For example, the fol lowing vignette shows how Ava specifically selected a group to share 
'fau lty' thinking. During the presentation she directly i nterceded at regular intervals to 
facil itate space for li steners to think about, question, and clarify each section .  But, she 
herself did not evaluate it .  
Providing space to question and clarify mathematical explanations 
Ava selected a group to explain their solution strategy to the large group. She stated:  
Ava 
Sandra 
Ava 
Pania 
Ava 
Pania 
Sandra 
Ava 
Pania 
Ava 
I know for some of you, it pushed you out to the edge. I can see some semi 
and that means partly confused looks and that' s okay . . .  you wi ll need to 
question anything you don ' t  understand . We are going to l isten step by step 
and when you finish one step put the l id on the pen and see if anyone has got 
any questions . . .  Remember you need to talk about what your group did. 
[records 52 on  a sheet] The rule was timesing it by two and i t  goes up. 
Right. Can you stop there and wait for questions. 
How did you get that? 
[prompts] How did you get what? 
How did you work out it was 52 sticks? 
We went 25 and 25 equals 50. No. 1 2  and 1 2  equals 24 plus 2 would equal 
25.  
You people need to be real ly l istening to what i s  being  said here .  You have 
got to ask yourself does it make sense to you. 
I don ' t  think i t  will make sense because 24 plus 2 would equal 26. 
Wel l  tel l  you what. You have done reall y  well .  There are some adults who 
would not even know how to work through that sort of problem so well 
done. 
Ava then selected another group to provide the solution strategy for the problem used in the 
vignette in section 6 .3 .6 .  
(Term 2 Week 7) 
1 1 7 
After the teaching session she explained her selection to me. She said that she knew that 
this group ' s  explanation would chal lenge the reasoning of others and require that the 
listeners make sense of i t, section by section.  She saw it as a tool which provided practice 
for the other students to question and clarify sections as needed. 
Changing roles and rules, mediating sense-making and the norms for mathematical 
explanations 
The observational data shows that A va' s expectations of both how mathematical 
explanations were given and the student ' s  role in l i stening to them, ran counter to the 
students' previous experiences. As discussed previously, Ava had voiced the potential 
conflicts she knew the students would have between their perception of her ' teacher' role 
and the more faci l i tative role she planned to develop. Hence, she structured the large group 
interactions directly.  Her requirement that explanations be explained in sections, with each 
section supported by space to 'think' , made the reasoning open and visible and available for 
clarification and challenge. The models of how to make an explanation and the questions 
she explicitly modeled and used during the sessions provided the students with tools to 
sense-make. Through these actions Ava began to pul l  forward all participants i n  the 
dialogue into what Lerman (200 1 ) termed a symbolic space-a zone of proximal 
development in which she was scaffolding sense-making. 
Kazemi and Franke (2004) noted the difficulties many teachers experience when attending 
to the different reasoning their students use. The discussions Ava had with me and her 
col leagues, of the ways her students reasoned, were the tools which mediated her growth in 
understanding and her value of them. A va' s use of student contributions and erroneous 
reasoning shifted thinking past a focus on correct answers to look at and explore the 
solution strategies. Thus the students were 'pressed' (Kazemi & Stipek, 200 1 ) to assume 
their new role, to analyse the reasoning and be accountable for their own sense-making. In 
addition, Ava' s requirement that explanations be conceptuall y  based and her emphasis on 
construction, explanation and clarification of them led to the constitution of mathematical 
norms for what constituted c lear and logical explanations in the classroom community. 
1 1 8 
6.3.5 LEARNING HOW TO AGREE AND DISAGREE TO JUSTIFY 
REASONING 
In  the early stages of the study the participation structure that Ava made avai lable to 
students operated as a scaffold for the development of argumentation .  Although 
mathemati cal argumentation was not a strong feature of how the students interacted A va 
i nitiated and maintained discussion about the need for both agreement and disagreement i n  
the construction of  reasoned explanations. For example, when a student stated that working 
as a group required agreement she responded by asking: Yes you could be agreeing with 
what the person says . . .  but are you always agreeing, do you think? She carefu l ly  structured 
ways in which the students could approach disagreement and challenge. As i l lustrated in 
the fol lowing vignette, Ava scaffolded ways in which they could disagree with an 
explanation so that justification became necessary. 
Scaffolding ways to disagree 
As the students worked together she reminded them:  
Ava Arguing is not a bad word . . .  sometimes I know that you people thi nk to 
argue i s  . . .  I am talking about arguing in a good way. Please feel free to say if 
you do not agree with what someone else has said. You can say that as long 
as you say it in an okay sort of way.  If you don ' t  agree then a suggestion 
could be that you might say I don ' t  actual ly  agree with you . Could you show 
that to me? Could you perhaps write it in numbers? 
Could you draw something to show that idea to me? That' s fine because 
sometimes when you go over and you do that again you think . . .  oh maybe 
that wasn' t  quite right and that ' s  fine. That 's  okay. 
(Term 2 Week 6) 
Beliefs about disagreement and argumentation 
Inducting studen ts into the use of j ustification and col laborative argumentation 
(Andriessen, 2006) was begun through discussions of the need for not only agreement but 
also disagreeing and arguing. As Mercer (2000) and his col leagues (Rojas-Drummond, & 
Zapata, 2004; Wegerif, & Mercer, 2000) have shown, when students maintain constant 
agreement often cumulative talk results-a less productive form of mathematical reasoning. 
1 1 9 
Therefore, Ava had to ensure that the students understood that even when working 
collaborati vely agreeing with the reasoning of  others is  not  always productive 
mathematical ly .  Her regular references of the need for both agreement and disagreement, 
and her specifi c  scaffolding of ways to chal lenge, provided assistance to the students to 
shift the discourse patterns from cumulative or disputational forms of talk, toward use of 
exploratory patterns (Mercer, 2000). 
Although Ava had in itiated discussion of the need for di sagreement she noted the tensions 
these expectations placed on the students. During analysis of a video excerpt Ava stated:  
Disagreeing is so hard for these students so I am supporting them and ensuring that they 're 
okay with the concept of agreement and disagreement, also how to approach each other 
when voicing their opinions. Her statement reveals her recognition of the contradictions 
between the new rules for tal k, those used previously, and those which governed the forms 
of talk used in their wider community beyond school. As Gallimore and Goldenberg ( l 993 ) 
explain ,  chi ldren "are shaped and sustained by ecological and cultural features of the fami ly 
n iche" (p. 3 1 5 ) .  Ava' s statements and carefu l structuring of ways to argue i l lustrated that 
she was aware that for many of her students from non-dominant groups ' (Nasir, et al . ,  
2006) mathematical argumentation as a specific speech genre (Gee, l 999) was not 
necessari ly  with in  their current repertoire of cultural practices (Gee, 1 992) .  In addition, she 
recogni sed that the form many of her students might have participated in  previously was an 
appositional or aggressive form of argument (Andriessen, 2006) which she thought would 
have shaped negative bel iefs about ' arguing' . As a result, she knew that transforming their 
beliefs and constituting mathematical argumentation as a si tuated discourse practice in the 
classroom might i nitially be problematic.  She knew that to shift to an i nquiry environment 
required that the students learn the discourse of inquiry and challenge and therefore she 
purposely continued to lay the foundations for its development. 
1 Non-dominant groups include students who speak national or language variations other than standard 
English and students from low income communities. 
1 20 
6.3.6 GENERALISING MATHEMATICAL REASONING 
In conjunction with a focus to shift the students towards generalising Ava chose to use a 
series of problems which required the students to engage in active and extended discussion 
of their reasoning. These problems2 were intentional ly  devised in the teacher' s study group 
to support the use of early algebraic reasoning. They required exploration of patterns and 
relationships and the construction of a rule  as an unexecuted number sentence to describe 
the relationship. The problems began as single tasks but col lectively they provided a means 
to scaffold the students to construct and voice generalisations. During task enactment A va 
emphasised the physical representation of the problems to i l lustrate the recursive geometric 
patterns .  However, discussion with me led to Ava varying the task parameters so that the 
students were also encouraged to use analytical approaches. As the problem contexts were 
extended A va pressed the students to look beyond the concrete constraints of the immediate 
situation.  She directed them to make and test their conjectures on a range of numbers and 
explore whether they worked on all numbers. She began teaching sessions with discussion 
and direct modeling of the use of diagrams,  ordered pairs and tables of data. These operated 
as a structure in which the students were able to explore and develop systematic strategies 
to test out the patterns that produced the data. 
As I described in Section 6 .3 .4 A va carefu l ly  selected groups to provide explanations to the 
larger group. In the fol lowing vignette she had observed functional reasoning in one group. 
She used this group' s  explanation as a way to shift the wider groups' thinking towards 
generalis ing. Continuing from the vignette in section 6 .3 .4 Ava takes a different role,  
recognising that this group are presenting an argument which many of the l isteners may 
have difficulty accessing. The explainers had established and recorded a functional 
relationship between the number of sticks needed to develop a triangular pattern and the 
number of triangles. She faci l itates a prolonged discussion of their model and how they 
came to develop it. She participates in questioning and directs attention to their notations. 
Her questioning prompted the l isteners to reflectively analyse their reasoning. Her 
questions drew further explanation of how the group had reflectively constructed the 
2 For examples of the problems see Appendix G 
1 2 1  
generali sation together through persistent pattern seeking and examining and re-examining 
the reasoning they were working with. 
Rachel 
Ava 
Sandra 
Rachel 
Generalising a functional relationship 
[recording] We started off with a table and we stuck a picture of a li ttle 
triangle and we wrote sticks. We know that one triangle has three sticks but 
from then on it is two. So two triangles have five sticks . . .  We put a sign that 
said plus one here then we went from one to ten plus one. 
Put the lid on the pen for now. Anybody got any questions? 
Why did you do that? 
Well it  helps us with our rule. We put a sign up here that says times two. 
Rachel explains and records x 2 and + I ,  records from 1 to 1 0  and records the value of each 
number x 2 + 1 underneath 
Ava 
Wang 
Ava 
Sandra 
Ava 
Sandra 
Ruru can you fol low what Rachel ' s  group have done so far? Can you 
explain what you see happening here? Wang what about you? 
The first one is the one triangle makes up to three because they are adding 
two on. 
[points at the ordered pairs] Why though have Rachel and their group . . .  
done this? Why have they put these numbers down here as well as these two 
numbers? You are right in that they are adding two on. But why have they 
done it l ike this? Why? Maybe somebody else can help? 
To help them know what their strategy is .  
[asking the explaining group] You will accept that? 
[interjecting] Most of us haven ' t  done the two times and the one plus. 
Ava faci l itated further discussion and questioning related to the use of the systematic 
recording of the table and ordered pairs. Then she asked the explainers to continue. 
Aorangi Wel l  six times two equals twelve . . .  plus one equals thirteen. Then Rachel 
when she was thinking of this, that' s when she saw Pania writing it. She 
thought of this [points at the recorded rule ] .  She thought six times two plus 
one equals thirteen . . .  Then what we did, we did brackets around the six and 
the two. We started thinking it was times two inside and plus one outside. 
A va Can you stop there just for a moment. Think about this. All of you think 
about your own explanations. Wang said they are adding two on . . .  i s  th is 
group adding two . . .  where are the two? We need to think about what they 
are doing? 
Alan Instead of plus two, plus two . . .  they are timesing by two. So they didn't have 
to go long . . .  i t  was six times two plus one. Or i t  could be one hundred times 
two plus one. 
(Term 2 Week 8) 
1 22 
Open-ended problems, example spaces, reflective pattern finding 
Through construction and exploration of problems in the study group, Ava was introduced 
to possible ways she could mediate her students' early development of algebraic reasoning. 
However, it was the informal discussions after teaching sessions which shifted Ava ' s  
understanding that the students' attention on  use (and enjoyment) o f  manipulatives can i n  
some instances potential ly hinder their development o f  underlying mathematical 
understandings. Moyer (200 1 )  identified the tension many teachers have when using 
manipulatives as one of balancing student confidence and enjoyment with maintai n ing a 
press towards more generalised mathematical understandings. In  the lesson the shift in 
emphasis from the use of manipulatives and the concrete si tuation extended the problems 
so that they became more open-ended. Ava provided the students with example spaces 
(Watson & Mason, 2005 ) ;  these became i ntellectual spaces in which they then had 
opportunities to search for and test examples and counterexamples of numerical patterns 
and relationships. 
Using student explanations in which there was evidence of a 'mindfu l '  approach (Fuchs et 
al . ,  2002) to reflective pattern seeking provided a foundation for i nducting them into the use 
of more "intellectual tools and mental habits" (RAND Mathematics Study Panel, p .  3 8) .  
The reasoning contributed by the explainers provided a p latform for other participants to  
make connections and to  build from.  The use A va made of student explanation, her 
modeling of careful l istening, and the positioning of other students to access the reasoning, 
are important actions which influenced student beliefs about themselves as mathematical 
doers and users . 
6.3.7 USING AND CLARIFYING MATHEMATICAL LANGUAGE 
Ava placed direct attention on scaffolding how the students used talk to describe and 
question explanations and provide appropriate responses. The fol lowing vignette i l lustrates 
how Ava accepted the students' use of short utterances and informal terms in the 
mathematical discussions but then facilitated further discussion which focused direct ly on  
1 23 
how the language of mathematics was being used . Ava revoiced, rephrased, or elaborated 
on sections of explanations to clarify the mathematical meanings of words 
Revoicing to clarify and define mathematical terms 
After a group had modeled an explanation using equipment and described the action  of 
repeatedly adding three sticks as equivalent to squares times three they are challenged. 
Jo 
Pania 
AI an 
Ava 
Jo 
Sandra 
Ava 
AI an 
Ava 
Isn ' t  that just plusing three sticks not timesing it? You are not timesi ng 
you' re adding. 
Wel l  what she sort of means it is l ike it is going up. 
Is that t imesing going up? 
When we talk about timesing what do we actual ly mean? 
We mean multiplying not adding. Adding is plus [indicates a +  with fingers] 
that sign. 
You mean when you add two more squares on, that is multiplying? 
Rachel was saying she is adding three, adding another three, so that ' s  three 
plus three plus three. So if you keep adding three al l  the time what is another 
way of doing it? 
You can just times instead of adding. It won ' t  take as long and it is more 
efficient. 
Yes you are right. Did you all hear that? Alan said that you can just times it, 
multiply by three because that is the same as adding on three each time. 
What word do we use instead of timesing? 
Alan Multiplication, multiplying. 
(Term 2 Week 7) 
Multi-levels of language 
Examination of the data reveals that Ava regularly attended not only to how the students 
used mathematical talk to explain their reasoning, but also to their use of the mathematical 
language. Ava accepted the students' use of colloquial terms but revoiced (O'Connor & 
Michaels, 1 993)  and rephrased what they said, using rich multi-levels of word meanings. 
Her actions in providing multiple levels of words and revoic ing and extending the students' 
mathematical statements paralleled the actions of a teacher Khisty and Chval (2002) 
described. Ava, l ike that teacher, ensured through her revoicing and provision of multiple 
layers of meaning that her students could access the specialised discourse of mathematics.  
1 24 
6.3.8 SUMMARY OF THE FIRST PHASE OF THE STUDY 
In this section I have outli ned how Ava maintained a focus on constructing a safe 
collaborative classroom environment where the students gradual ly gained confidence to 
participate i n  explaining, questioning and clarifying mathematical reasoning. The trajectory 
Ava mapped out using the communication and participation framework guided shifts in the 
classroom interaction patterns and establishment of the foundations for an inquiry 
classroom culture (Wood & McNeal, 2003).  Ava ' s  immediate focus was placed on 
scaffolding the mathematical discourse. She mediated the establi shment of classroom 
norms concerning how students participated in the mathematical talk and what 
mathematical thinking was discussed. 
This initial focus is consistent with what a number of other researchers have noted (e.g. ,  
Rittenhouse, 1 998; Si lver & Smith, 1 996; Wood, 1 999); teachers when constituting inquiry 
communities first establish the norms for discourse, and then they turn their attention to the 
content. In this environment the students began to learn to use more proficient forms of 
mathematical practices as they learnt to talk about ' talking about mathematics' (Cobb et al . ,  
1 993). 
6.4 Extending the mathematical practices in a mathematical inquiry 
community 
Our reflections on the success of the initial establishment of the interaction patterns 
prompted us to move forward drawing on further steps on the communication and 
participation framework. Ava stated that the students were ready: to move from the 'what ' 
to the 'why. She wanted the students to engage in meaty mathematical arguments using 
various methods to convince others or to justify why they agree or disagree with a solution. 
The communication and participation framework was used to plan shifts toward increased 
student use of i nquiry and chal lenge. 
1 25 
6.4. 1 PROVIDING AN ENVIRONMENT FOR FURTHER INTELLECTUAL 
GROWTH 
As Ava and the students repositioned themselves in the mathematical discussions the 
students gradual ly assumed more agency over when and how they engaged in the discourse. 
In accord with the communication and participati on framework, Ava introduced the use of 
a 'thinking time ' ,  a pause in mathematical dialogue designed to support all participants' 
active cogni tive engagement. Its purpose was to offer opportunities for participants to 
analyse arguments, frame questions, or structure their own reasoning and explanations. A va 
explicitly outli ned to her students how 'think time' was a sense-making tool which 
provided them with an opportunity to reconsider and restructure their arguments . She 
regularly reinforced its use, halting discussions to outl i ne how she observed i ts use: I am 
pleased to hear the way you are all exploring each others ' thinking. Stopping and 
thinking . . .  having a think time. It 's good to hear things like "I thought it was that one but 
now I think it is this one and this is the reason why ". Or "I thought that but now I think this 
because we " . . .  and explaining it and using drawings to back up what you are saying. 
Learning to participate in mathematical dialogue continued to be a risk-taking act for some. 
'Think time' was used to scaffold those students who were i nitially diffident or unconfident 
about publicly explaining or responding to questions and challenge. Rather than accepting  
si lence from a student as  reason to  bring questioning or explaining to an  end, Ava would 
suggest a 'think time ' .  But she would clearly indicate that they were sti l l  responsible to 
explain :  You have a think about it while I pop over to this group so we can hear their 
thinking then we will come back to you. Examination of the data reveals that through 
balancing 'think time' with an expectation of accountabi lity the students became more 
wi l ling to express partial understandings or outli ne difficulties they had understanding 
sections of an argument. 
Earlier in the study, the vignette in section 6 .3 .4 i l lustrated how Ava used an explanation 
based on faulty thinking to provide the students with practice in questioning and sense?
making. However, often the reasons for the misconceptions remained unexamined. 
Through reflective discussion with me, Ava' s awareness of the potential learning 
1 26 
opportunities in examining and analysing erroneous reasoning increased. In  c lass she began 
to use problems which required students to examine and justify their selection of a solution 
strategy. These included solution strategies which had common misconceptions students of 
their age consistently make and which appear reasonable when seen from the perspective of 
the learner (see Appendix I ) .  Then in sharing sessions Ava would purposefully select a 
group who had an argument which had the potential to progress al l their thinking. Thi s is 
i l lustrated in the fol lowing vignette when Ava selects a group who are arguing that an 
erroneous solution strategy is correct. Despi te two previous groups' c lear and wel l argued 
explanations Ava hears the on-going questions and chal lenges. Her response is to faci l i tate 
c lose examination of the reasoning using student challenge to explore the many partial 
understandings they all hold. 
Making the reasoning clear, visible, and open to challenge 
For the problem3 under discussion A va selected a group to explain that she knew had 
developed an incorrect explanation. 
Sandra 
Ava 
Pania 
Sandra 
Rachel 
Ava 
[records 1 12 ,  1 15, 3/ 1 0] My argument  is Col in  is correct on his one and I am 
going to show you how it goes . .  . I  think that Col in  took al l the first numbers 
from the top numbers and he has added them together which is five. Then 
he' s got all the bottom numbers and he ' s  picked one which is the five to get 
to his whole. 
Stop Sandra. You have a mountain of questions. 
Why did you choose the bottom one the five? 
Because that' s how he probably got a whole. Because if  he went five and 
[poi nts at the 3/ 1 0] ten he would get a half and if he went five and [points at 
the 1 /2 ]  two he would get two wholes and a half. But he said he had a whole. 
In other words you are sayi ng that you are adding to five and using the other 
five [points at the denominator of 5] to get a whole? 
So can you just use one from there? Can you show us using pictures to 
convince us? 
3 Col i n  and J u nior are having an argument about their homework. In the problem the clown eats 1 I 5 of one 
frui t  square, Y2 of another frui t  square and 3/ 10 of the last fru it square. The clown then puts the remainder of the 
frui t  squares o n  one plate i n  the fridge to eat later. Their teacher has asked them to say how much fruit square 
as a fraction, the clown ate altogether. Coli n  says that the clown altogether ate I frui t  square but Junior says 
that the clown ate 5/1 7. Who is correct and why? Can you work out what each of them has done and what they 
were thinking? Work together and discuss how you would explain the correct answer to them. You w i l l  need 
to work out a number of different ways to explain so that they are convinced. Think about using pictures as 
well as words and n umbers to convince them 
1 27 
Sandra 
Rangi 
Sandra 
[draws a rectangle and colours in each sl ice as she explains] They basical ly 
ate one fifth, another fifth and then three fifths. So that made them into fifths 
i nstead of them [poi nts at 31 1 0] .  So that would make five and five [at the 
bottom] would equal one whole. 
How did you make them into fi fths? 
[points at the denominators in 1 12 and 31 1 0] Just pretended those were not 
there .  
Further discussion takes place and A va directs the students to  reread the problem. 
Rangi [points at the numerators then denominators] Can I show you something? 
Junior was wrong in  the problem because first he added those together 
which was five. Then he added that together and it was seventeen. 
Rachel What Sandra is doing is adding the top three which equals five . . .  and taken a 
five from the bottom to make a whole. 
A va [records a list of different fractions] Can you do that? Does that make sense? 
Would it sti ll work the same if for example you have different numerators 
which will not add up to any of your denorn.jnators? I am going to leave that 
thought with you. Wou ld it sti ll work? If you think it won ' t  explain it to us. 
If  you think yes you prove it ei ther way. 
(Term 3 Week 4)  
Thinking space, increased independence, errors as a resource, mathematical analysis, 
learning in the act of teaching 
Evident i n  the data is a shift in positioning of Ava and her students as they grew in what 
McCrone (2005)  terms communicative competence-knowing when and how to initiate 
and participate in interactions using appropriate mathematical talk. Introducing a 'think 
time' as a tool  to use during social i nteraction provided the students with a thinking 
device-a space to reconceptual ise the reasoning under consideration . In thi s space, their 
explanations and the explanations of others became reflective tools (Cobb et al . ,  1 997) 
which contributed to developing richer levels of mathematical discourse. 
Providing a gap in the interactions also acted as a form of social nurturing (Anthony & 
Walshaw, 2007) for those less confident members of the community. In  addition to the 
previously described affective support that Ava provided in the first phase, provision of a 
space to think and the added press to participate indicated that she "cared for" (Hackenberg, 
2005 , p. 45) her students and their engagement in the mathematical discourse. But the 
'caring' meant that a hesitant response, or no response at al l ,  did not signal a dependent 
1 28 
relationship, nor remove responsibi lity for participation. Rather, it led to increased 
independence indicated in the expectation that they take ownership of their explanation or 
identify aspects of reasoning they were unsure of. 
The classroom observations revealed that problems which required extended dialogue and 
negotiation of sense-making, including those that examined misconceptions, were 
important tool s  which mediated how the students participated in the mathematical 
discourse . The enors became "resources for learning" (Airl?l & Skovsmose, 2002, p. 22) .  
As Brodie (2005) explains, many misconceptions are reasonable errors which "make sense 
when understood in relation to the current conceptual system of the learner, which i s  
usual ly a more limited version of  a mature conceptual system" (p . 1 79) .  The enors in this 
si tuation served as a point of continuity for the community to discuss and grapple with 
complex ideas . Building on errors as an entry point for further discussion is a feature of 
many studies located in inquiry communities (e .g. ,  Fraivil l ig, 200 1 ;  Fraivil l ig, Murphy, & 
Fuson, 1 999; Kazemi, & Stipek, 200 1 ;  Whi te, 2003).  
Ava analysed and noted her own increased pedagogical ski l l  to ' notice' (Sherin ,  2002b) 
student thinking and learning when she reviewed the video excerpt: I saw here that they 
wanted to continue asking questions and Sandra was really starting to rethink and I think 
that there would have been more than Sandra in the rethinking. I am giving them food for 
thought trying to challenge their thinking to see if they can understand and discuss why and 
why not adding different denominators. It doesn 't hurt to leave them hanging. What a shift 
for us all . . .  before it was about the right answer and now . . .  In  collaborative discussion with 
me she voiced how she was learn ing to faci l itate productive discourse which maintained a 
fine balance between providing knowledge and allowing the group to struggle through to 
eo-construction of arguments . A va was "learning in  the act of teaching" (Davies & Walker, 
2005 , p. 273),  drawing together her knowledge of the mathematical content, the learning 
task, and knowledge of how student reasoning could be used to progress or deepen 
understanding. 
1 29 
6.4.2 COLLECTIVELY CONSTRUCTING AND MAKING MATHEMATICAL 
EXPLANATIONS 
A va continued to monitor how the students constructed and explained their reasoning. To 
faci l i tate the group processes Ava introduced the use of one l arge sheet of paper and one 
pen to assist the recording of a strategy solution to explain to the larger group. Individual 
group members were each required to explain a solution strategy then after close 
examination select the most efficient one. The students also knew that during large group 
sessions A va would halt an explainer and require that other members of the smal l group 
continue or clarify the explanation.  This encouraged members to closely examine and 
explore their reasoning, ask each other questions, and search for alternative ways to explain 
and engage in reasoned debate about their thinking. 
The fol lowing vignette i l lustrates how A va constructed collaborative group processes to 
i nc lude students' responsibi l i ties to each other and their l is tening audience. In  turn, the 
students extended their responsibi l ity for each other' s sense-making but also anticipated 
how the explanation would be u nderstood by the larger group. The attention participants i n  
small groups placed o n  their analysis of their explanations scaffolded support for each other 
when explaining and justifyi ng their explanations in the larger sessions. 
Constructing a collaborative explanation and justification 
Before the small groups began to work Ava described their group responsibil ities . 
Ava You need to make sure that everyone in your group all work together to 
make sense of it. Together you need to know what you're . . .  saying and what 
you are doing. You may need to use your fractions pieces and lots of 
different ways to make it make sense to al l of you in the group . . .  When it 
comes to the sharing time you need to be able to explain and justify what 
you are saying i n  lots of different ways. We are all going to need to be able 
to see what you are saying, see your reasons  behind your explanations. I am 
going to ask anybody in  the group to explain .  So you have to make sure that 
everybody in  the group can explain anything you are asked. 
After this group of students have explored three different solution strategies for the 
problem4 they discuss which one to provide to the larger group. 
4 See problem on p. 20 
1 30 
Rachel 
Tipani 
Rachel 
About this one, it ' s a bit hard to understand because it was so fast. 
Okay. The truth i s  th is is the most efficient way. [Points] That ' s  a good way. 
That' s a good way.  But that 's  the most efficient. 
[points] Yeah but that one is the most efficient because it's easier to 
understand. Th is i s  more confusing even if it  is the fastest. So let' s go with 
the one we know everyone wil l  understand. 
When Rangi provides their group' s  solution strategy in the sharing session he is chal lenged: 
Tipani Why did you shade in two tenths and call i t  one fifth? You didn' t  explain 
why. You just said there was only one fifth left and you just shaded it in?  
Rangi The clown ate one whole and it says he ate half. Then he ate three tenths and 
then there was only one fifth . So I had to shade in two because that was the 
only squares left. 
Josefina [steps in to support her group member] I can explain why. An easier 
way . . .  because when you divide tenths i nto fifths there is two of the tenths 
resembles one of the fifths. So that 's  why she shaded in two because two 
tenths equals one fi fth. 
(Term 3 Week 4) 
Communal discourse, mathematical explanations, objects of reflection 
Both the communication and participation framework and our discussions  were tools which 
mediated the continued press toward constituting an i nquiry classroom patticipation 
structure where col laborative interaction in  the discourse was central . As a communal 
activity, classroom discourse involves both individual and shared accountabil ity (McClain 
& Cobb, 1 998) .  Col l ins (2006) explains that the key idea "is  to advance the collective 
knowledge of the community . . .  to help individual students learn" (p . 55) .  Multiple zones of 
proximal development were constructed as participation  patterns shifted from single 
explainers representing groups, to all members assuming responsibility to track what their 
explainer outlined and when required, intervene to respond to inquiry and challenge. 
I previously described in Chapter 4 how many researchers reported that teacher 
expectations were vital to the quality and types of explanations constructed. In this study i t  
was Ava' s expectations which scaffolded the development of the classroom community' s 
criteria for an acceptable mathematical explanation. Evident in the communal discourse 
was the requirement that explanations be experiential ly  real and conceptually c lear and 
1 3 1  
provide as much information as the l istening audience required. In  the act of comparing and 
analysing explanations to find those which were different, more sophisticated, or efficient, 
the explanations became explicit objects of reflection (Cobb & Yackel ,  1 996). 
6.4.3 ENGAGING IN EXPLANATORY JUSTICA TION AND MATHEMATICAL 
ARGUMENTATION 
The requi rement for conceptual explanations provided the foundations with which to build 
mathematical argumentation. We noted during evaluative discussion of the classroom video 
excerpts that when the students provided explanations they often drew on more than one 
form to validate their explanatory reasoning. To extend this practice A va required that all 
the groups construct multiple ways to validate their thinking. Problems (see Appendix H) 
which required multiple ways to convince others were developed in the study group and 
used by A va. In class, Ava explicitly discussed 'maths arguing' and how mathematicians 
use it as a tool to make mathematical reasoning clear: Mathematicians engage in arguing. 
They do it all the time. It is good to have a healthy mathematical argument. You actually 
learn a lot more from arguing about your maths than not arguing about your maths. It just 
opens it all up . . .  all the thinking. 
In the study group, the teachers and I had examined the video records of each classroom 
and recorded the questions and prompts the teachers and students used during mathematical 
activity. Building on the questions and prompts Wood and McNeal (2003) used to delineate 
classroom cultures, we developed a framework (see Appendix E) based on questions and 
prompts which had emerged in the classroom talk. As I explained in Chapter Five (see 
section 5 .4. 1 )  these were the questions and prompts used by participants in the discussion. 
These questions and prompts were identified as tool s  which mediated further examination 
and extension of the mathematical reasoning and the use of different mathematical 
practices .  In class A va used the models to guide how she scaffolded the students to use 
specific questions and prompts so that they engaged in the mathematical practices. Separate 
wall charts were constructed relating to the different types of questions to use with 
mathematical explanations and mathematical justification. In  subsequent lessons A va added 
questions which arose during the dialogue-questions which asked why and prompted for 
1 32 
justification and validation of reasoni ng. Ava also assisted the students by asking them to 
prepare group responses to questions they might be asked in the larger situation :  Think 
about the questions that you might be asked. Practise using some of those questions like 
why does that work or how can you know that is true. Try to see what happens when you 
say if I do that . . .  then that will happen. 
Ava's emphasis on the need for justification and validation of reasoning was accepted and 
modeled by the students. They recognised that it supported possibilities for confirmation or 
reconstruction of their reasoning. They regularly appropriated Ava' s words to prompt 
argumentation from their fel low students. For example, one student observed another 
student frown and directed: Argue against them if you need to, if you disagree with their 
maths thinking. So they can think about it again and maybe change. Or you might agree in 
the end with their stuff 
The fol lowing vignette i l lustrates how the press toward justification and val idation of 
reasoning led to student engagement in exploratory talk (Mercer, 2000). Each step in  the 
argument was closely examined and rehearsed. Then, for clarity, or ease of sense-making, 
the sections were reworked, reformulated and re-presented. 
Using multiple means to justify and validate an argument 
After sustained discussion of drawings and possible solution strategies for a problem5 one 
member of a small group provides a verbal explanation : 
Tipani 
Ava 
Pania 
Ava 
I know what to do. It might be a l i ttle complicated and we might have to do 
something else to show that, to prove it. Half  of the fruit square pretend that 
the half is i n  tenths . So one half i s  equal to five tenths plus three tenths 
[points at the 3/ 1 0  on the problem sheet] which equals eight tenths plus one 
fifth which i s  two tenths, so that means Colin was correct, one fifth which is 
two tenths which equals one whole which is one fruit square. 
Are you convinced Pania? 
It was a l i ttle fast. 
Perhaps recording i t  Pania  wi l l  mean that you get to see the bits that were 
too fast .  But  remember to keep asking those why questions. You need to be 
convinced one hundred percent. 
5 See problem p. 20 
1 33 
Tipani records the fraction symbols and draws a rectangle which she divides i nto 1 0  
segments . She colours each segment and points at the symbol as she explains :  
Rachel Pretend I don ' t  understand so explain it to me. 
Tipani [colours in the segments and points at symbols] So this is  one half, this one. 
Then I got three tenths and I am pretending that instead of i t  being just l ike 
that, the one half I am doing it in  tenths .  So because of that I am going three 
tenths and five tenths which is . . .  
Rachel/Pania Eight tenths .  
Tipani Then I know that one fifth is  equivalent to two tenths . So I shaded that in ,  in 
my head and that 's  how I got one whole. 
Rachel So Tipani what about the fifth? Shouldn ' t  we use that too? 
Tipani [poi nti ng at the segments] No because eight, that ' s  the first mark, that 's  the 
second mark because that ' s  eight and then one fifth is equivalent to two 
tenths so that 's  it . 
Rachel 
Tipani 
Rachel 
Pania 
Rachel 
So what you 've done is you made one fi fth . . .  
Into two tenths. 
So you are changing the one fifth into two tenths because that is  the 
equivalent fraction and then using the equivalent fraction? [She records 1 15 
= 2/ 10  on a sheet] . 
Yeah. One fifth equals two tenths .  
Why . . .  yeah she has taken one half . . .  and three tenths and she is using one 
fifth as an equivalent fraction. So it is one whole. 
Pania Ten tenths .  
Rachel [records I 0/ 10  = I ]  One whole. 
Tipani Because that 's the equivalent fraction .  
(Term 3 Week 4) 
Accessing a social language, interthinking, maths arguing 
An analysis of c lassroom observations suggests the framework of questions and prompts 
were tools which mediated further change i n  the classroom discourse patterns .  Directly 
focusing the students on their need to use specific forms of questions to gain particular 
mathematical information had the effect of shaping the form and content of the discourse. 
These actions assisted the students to access what Wood ( 1 998) described as the form, 
"knowing how to talk" and content, "knowing what to say" (p. 1 70) i n  the mathematical 
discourse. This provided the students with access to what Bakhtin ( 1 994) terms a soci al 
language or speech genre and a specific identity. 
1 34 
Reviewi ng the video taped lessons provided insight i nto how the students appropriated and 
used the various models of ways to question and chal lenge. To begin, we observed that the 
students appeared to do what Ava described as "parroting questions" as they tried out ways 
to inquire and challenge. But later students were observed to rephrase and reformulate the 
questions, expand and extend them and make them ' their own ' .  
Ava' s actions, in providing students with expl icit ways to frame close examination and 
exploration of the reasoning of others, resonates with many other studies which have used 
specific structures to frame the classroom interactions (e.g. ,  Brown, 2005 ; Mercer, 2000; 
Rojas-Drummond & Zapata, 2004; Rowe & Bicknel l ,  2004). Like the students in these 
studies, the students in Ava' s room used the social language to engage in joi nt activ ity, to 
do what Mercer (2000) described as interthinking. The press toward a shared perspective 
shifted explanations to justification and validation of reasoning, particularly when other 
participants in the di scussion either did not understand what was being explained, or 
disagreed with the reasoning. In these situations, the students were required to negotiate a 
shared perspective using what Ava and her students called 'maths arguing' . They li stened 
closely to the reasoning of others, questioned then re-explained, re-modeled and rehearsed 
arguments unti l  they became integrated as 'our' voice. 
Although the students became more ski l led at 'maths arguing' Ava continued to carefu l ly  
structure the affective components of  the classroom culture .  She outlined to  me toward the 
end of the second phase how she consistently aimed: to establish a supportive environment 
in which it is okay and mathematically sound practice to disagree with someone else 's 
thinking. This puts the onus on the explainer to clearly demonstrate how they worked 
through a solution. It also puts the responsibility of seeking clarification and asking 
analytical questions onto the other group members. However, she recognised the inherent 
on-going difficu lties this held for some of the students, commenting: I feel a bit like a 
cracked record going over all these norms all the time but we always have new students 
joining. Also I am aware that these students are growing into this behaviour now but 
disagreeing can be so hard for these students so I find that I have to keep almost giving 
1 35 
them permission to disagree or argue. Her reasons for the tensions were discussed 
previously in section 6 .3 .5 .  
6.4.4 PROBLEM SOLVING AND INSCRIBING MATHEMATICAL REASONING 
When the study began, the study group discussed the need for the students to engage in, 
interpret, and develop different ways to solve problems independently. On the 
communication and participation framework it was also planned that the students would 
develop their own ways to record their solution strategies .  But two terms into the study Ava 
continued to closely structure what they did; reading the problems to them then leading 
discussion on possible strategies and outl ining these step by step. Her detailed directives 
l imited the students ' opportunities to engage in pattern seeking and testing of conjectures. 
She placed little focus on the use of notational schemes other than to require that students 
match their verbal explanations with some form of representation which varied between 
informal notations, concrete material, or prepared representational recording forms. 
At the end of the first phase our analysis of video excerpts led A va to reconsider how the 
students could be scaffolded to engage in problem solving more autonomously.  In accord 
with the communication and participation framework, she began to give the problems to the 
groups to read and work with i ndependently. She closely monitored how they worked with 
the problem contexts and how they planned to model their solution strategies. She would 
halt group work briefly to examine possible conjectures and different pathways. At the 
same time, she began to structure how she wanted the students to notate. In the larger 
session she initial ly recorded: I am just going to record . . .  someone in your group can 
explain it and I will record it here and I will check with you that that 's what you mean, so I 
understand where you are coming from because I don 't want to be putting down an idea 
which isn 't yours. Okay ? Thus she modeled how she wanted the students to record their 
reasoning col lectively. A pattern of recording names of groups or individuals on the drafts 
to denote ownership of the reasoning was also establ ished. Draft work was placed on 
display where they were accessible for later use. 
1 36 
Initially, the inscriptions were used as exploratory tools to explore and explain reasoning 
but in conjunction with shifts i n  the pathway A va had mapped out they became tools used 
to justify and further validate mathematical thinking. For example, the fol lowing v ignette 
shows how a solution strategy is outli ned using an informal notation scheme. In the 
explanation the terms are revoiced and expanded on. In response to questioning, 
explanation is provided of how the group explored the appl ication of their solution beyond 
the immediate problem si tuation to problem situations drawing on a general isation to verify 
their own thinking. 
An inscription used as a public reasoning tool to generalise 
Jo presents an explanation in the large sharing session as Josefina notates: 
Jo 
Tipani 
Rachel 
Josefina 
Jo 
What Josefina did was instead of using times five she timesed it by ten .  
Remember when I was talking about levels and how I timesed the place 
value? Well instead of 756 it went up to 7560. What she has done now 
because it is ten times ten now she i s  halving it because you have to times 
that by five and half of ten is five. So she halved i t  and 7000 in half equals 
3500 and 500 in half equals 250 and 60 in half equals 30. Then what she did 
was added 3500 and 250 so she had 3750 then 30 so she got 3780. The 
strategy she is using is division. 
It was multiplication and division. 
Halving. She halved it, divided i t  by two. The thing is would that work al l 
the time that strategy? 
We did talk about that. We knew it works real ly well with five because ten i s  
easier to  multiply because as  Jo says it goes up in levels l ike in place value. 
But I thought we could use other numbers l ike six and do five and then add 
on one more lot of marbles. 
(Term 3 Week 9) 
Inscriptions used as tools to validate reasoning publicly, over-structuring, authorlity, 
Ava' s initial notating provided a model that inscriptions were tools which publ icly 
complemented and validated reasoning. Her explicit attention to verifying that she recorded 
what was explained emphasised the need for accuracy in recording communal reasoning. 
The classroom observations clearly i l lustrated that the manipulatives and problem situations 
were a beginning point from which many different inscriptions were developed and re?
developed in the discursive interaction. Research l iterature (e.g. ,  Lehrer & Schauble, 2005 ; 
1 37 
Roth & McGinn, 1 998; Sawyer, 2006) suggests that the requirement to make avai lab le 
multiple ways to validate reasoning influences how inscriptions are developed and used in  
classroom communities. This was evident i n  this study. 
Reflecting on her role as teacher A va noted the tensions and contradictions she encountered 
when we analysed a video excerpt: Did I need to read the problem ? Here I was focusing on 
what and how I wanted the students to problem solve . . . / think that giving them the sheet 
with the four bags meant that they were already being shaped. Over-structuring of activ ity 
is consistent with what many teachers do when they restructure the discourse towards 
inquiry. Bal l  (200 1 ) and Schifter (200 1 )  outli ne that often teachers focus on their own 
perspective of how a problem should be solved or they focus on pedagogical issues rather 
than the students' mathematical reasoning. In this situation, Ava expressed doubts about the 
students' abi l ity to read and make sense of the problems together. 
Close listening to student reasonmg and 'noticing' (Sherin,  2002b) the students' 
col laborative problem solving ski l ls when analysing the video excerpts with me, mediated 
the shift in her beliefs.  When she gave a problem to a group without discussion she stated: 
This is the struggle I have been working through . . .  I thought it was my role to do the 
'explaining the problem bit '. In actual fact I may have been doing the students a disservice 
unintentionally by removing the ownership of the problem from them. Previously Ava had 
continued to be what Povey et al .  (2004) describe as an "external authority" (p. 44 ) .  Her 
trusting that the students could autonomously read and problem solve together provided 
them with space in a rel ationship of shared "author/ity" (Povey, 1 995 , cited i n  Povey et al .  
p .  45 ) .  
6.4.5 JUSTIFYING AND GENERALISING MATHEMATICAL REASONING 
Generalisations students often use implicitly had been discussed in the study group. A va  
knew that these required explicit exploration with the students. However, her growth in  
recognising and using opportunities which emerged in  the  c lassroom dialogue developed 
s lowly. The requirement that the students develop multiple ways to j ustify their reasoning 
resulted in  them often spontaneously drawing on prior understandings which related to the 
1 38 
underlying structures and properties of mathematical numerical relationships. Through 
examination of student generated numerical generalisations within video excerpts A va  
became more attuned to hearing them i n  class. She began to  use them to explicitly develop 
and refine the thinking. For example, when a student talked about odd and even numbers 
she in itiated further exploration of numbers beyond the students' immediate number 
capacity. Thus she eliminated the possibi l ity that they could compute to verify their 
reasoning. In the fol lowing vignette Ava has intervened to use the explanation to examine 
the commutative property of multiplication applied in a fractional number context. 
Generalising using the commutative property of multiplication 
After extended discussion a group explain a solution strategy for the problem6. 
Tipani 
Hinemoa 
Ava 
Tipani  
Ava 
A dam 
Hone 
Ava 
Hone 
Ava 
[draws a rectangle she divides into 5 sections] I have divided the cake into 
fifths and she took two fifths of the cake so I am going to shade in two 
fifths. But she could only eat half of what she took . . .  given that there are two 
pieces, two divided by two equals one so she could only eat one. 
One piece of what she took. 
One half of what she took. So how much did she eat? 
I think that they both ate the same amount  because Alex ate one fifth which 
i s  equivalent to two tenths and Danielle ate . . .  yeah they both ate one fifth or 
two tenths and so they are both equivalent fractions. So they both ate the 
same amount .  
Using multiplication how could you word that. . .  if  you just looked at the 
numbers? 
I have got an explanation. 
It is because each one is using the same fraction and they have just turned it 
around. 
Just turned it around? 
Just l ike yesterday. It was the same for yesterday. So they are just turned 
around. 
Think about the numbers and think about multiplication. There are two 
numbers there. What are you actually  doing  with the half and the two fifths? 
6 Faith took 7/8 of a cake b u t  could only e a t  1 /3 o f  what s h e  took. Danielle took 1 /3 o f  a cake t h e  same s i z e  as 
Fai th' s  cake but could only eat 7/8 of what she took. Which of them ate more? Explain why the answer works 
out as it does. Can you work out a number of ways to conv i nce everybody else? Be ready to answer any 
questions other people w i l l  ask you. 
1 39 
Hinemoa You are multiplying them because it is just what Hone explained, they are 
just turned around. The numbers are flipped around but it' s the same like we 
sti l l  end up with the same answer. 
(Term 3 Week 7) 
Using student voiced generalisations, learning in the act of teaching 
A va' s earlier lack of attention to numerical general isations confi rmed Carpenter et al . ' s  
(2003) contention that many teachers despite extensive experience learning and teaching 
number have not abstracted the fundamental structures of ari thmetic. The analysis reveals 
that it was my nudging and the explicit exploration of video excerpts of student voiced 
general isations that mediated Ava ' s  "algebra eyes and ears as a new way of both looking at 
the mathematics they are teaching and l istening to students ' th inki ng about it" (Blanton & 
Kaput, 2005 , p. 440).  As in B lanton and Kaput' s  study, this study showed that Ava ' s  abi lity 
to spontaneously transform the mathematical talk into one which used algebraic reasoning 
grew and this was matched with an increased search by the students for wider ways to 
justify and validate their arguments. 
As I described in the previous phase Ava was learning in  the act of teaching and teaching in 
the act of learning. In the midst of lessons she was required to negotiate among the aspects 
of her own current knowledge and make on the spot decisions about how to adapt and use 
this to extend her students'  thinking. These requirements created pedagogical tensions for 
her. After she watched the video excerpt of the preceding vignette she said: I wondered 
after why I didn 't get them to work with whole numbers to explore it. That 's a problem I 
often meet up with. I don 't grab the opportunity to do things like this because I am juggling 
too many balls in the air at once. These tensions and the dilemma they caused her have 
simi larly been identified by many researchers (e.g. , Bal l ,  1 993 ;  Heaton, 2000; Sherin et al . ,  
2004) .  Ava i s  responding to the fact that her transformed role within the inquiry 
environment has different intellectual demands (Hammer & Schifter, 200 1 ) . 
1 40 
6.4.6 USING MATHEMATICAL LANGUAGE 
The shift evident in the classroom community toward increased use of justifications and 
generalisations was matched with a more proficient use of the mathematical language. A va 
required that the students be specific i n  describing their actions and solution strategies .  To 
extend their repertoire of mathematical talk she used a range of cues-hesitation, body 
language and facial expressions. Also, she would revoice, rephrase and extend the 
description, naming the strategy specifical ly :  you are plussing as your strategy, adding two 
each time so using addition as your strategy. In turn, as i llustrated in the following 
vignette, the students would appropriate Ava' s actions. They would scaffold their fel low 
students when required or re-explain, revoice, and extend the argument. 
Using the mathematical talk to scaffold others 
In a sharing session Hemi explains a solution strategy for a problem7 : 
Hemi 
Hone 
So Peter was wrong because he had five e ighths which equals two quarters 
[shades in 1 12 and 1 18 of a shape] so Jack was right because he has three 
quarters [Shades in 3/4 of a second shape] which is six eighths. 
How can you prove it to us? 
Hemi' s  body l anguage indicates discomfit so Pania intercedes : 
Pania I can show them. This group, that 's  Hemi and his group are saying that Jack 
is arguing that five eighths is less than three quarters [Pania draws a 
rectangle and sections it in eighths .  Three times she records 2/8 as she 
shades two sections then she records 6/8 .  She draws another rectangle and 
this time shades in  5/8 ] .  They are saying the three quarters is an equivalent 
fraction to six eighths. Like them I can prove it i s  bigger because look thi s  i s  
only five eighths, which is smaller than s ix  eighths. Is  that right you guys? 
(Term 3 WeekS) 
7 Problem: Peter and Jack are having an argument over their homework. They have been asked i f % is b i gger 
than ?.!. Peter is arguing that because the numbers are bigger in o/s then % must be bigger than %. Is he right or 
wrong? Can you work out some different ways to explain how you would solve their argument? You need to 
work together to convince either Peter or Jack using a range of ways including drawings, diagrams and 
symbo l s .  Discuss the mathematics argument you would use and the different challenges Peter or Jack might 
make to your argument. 
1 4 1  
Appropriating and using the specialised discourse of mathematics 
According to Forman and McPhai l ( 1 993) if students are to participate in peer collaboration 
they need a shared discourse . Ava' s actions made avai lable a shared understanding of how 
to participate in the more specialised di scourse (Gee, 1 992;  Gee & Clinton, 2000) .  This 
involved the students not only becoming more precise in  their use of mathematical words, 
but also more proficient in collaborating in mathematical argumentation. Within the 
constructed zones of proximal development, developed in the i nquiry environment, not 
only did Ava pull the students forward (Lerman, 200 1 )  but the students pul led each other 
forward . Through their appropriation of Ava' s actions and words they extended and 
rephrased each other' s short utterances and used their mathematical explanations, terms, 
and defini tions as their own. 
6.4.7 SUMMARY OF THE SECOND PHASE OF THE STUDY 
A close relationship is evident in the data between the communication and partic ipation 
framework, Ava and her student' s changed roles, and transformation of the participation 
and communication patterns. The emphasis placed on the use of i nquiry and argumentation 
and the i ntel lectual space A va provided had the effect of scaffolding consistent use of 
exploratory talk (Mercer, 2000) during col laborative argumentation (Andriessen, 2006) .  
Increasingly the students engaged publ icly in  collaborative construction and critical 
analysis  of their mathematical reasoning. A community of inquirers had been established in  
which the students were using many i nterrelated mathematical practices with increasing 
proficiency. Closely aligned to the i ncrease in the efficient use of mathematical practices 
was A va' s focus on scaffolding discourse in  which student thinking was central . 
6.5 OWNING THE MATHEMATICAL PRACTICES IN A 
COMMUNITY OF MATHEMATICAL INQUIRY 
Our collaborative discussion and verbal and written analysis of classroom observations 
prompted Ava to plan further shifts in  the classroom i nteraction patterns. A va voiced her 
observations that as the s tudents gained confidence to chal lenge and debate their reasoning 
1 42 
all their roles had changed. She noted that she was increasingly faci l i tati ng the dialogue and 
the students were taking ownership of the discourse. Maintaining a press toward 
mathematical i nquiry and argumentation continued to be an emphasis but A va wanted to 
further increase the flow of mathematical discourse driven by student thinking. She wanted 
to develop ways to : support students when they want to inject, butt in with another idea 
which helps all our thinking. She explained that this remained : a learning curve for all of us 
because we were so used to participating by taking turns. 
6.5.1  MAINTAINING INTELLECTUAL PARTNERSHIP IN COLLECTIVE 
INQUIRY AND ARGUMENTATION 
Whilst maintaining the social norms previously constituted Ava now expected the students 
to actively assume responsibi lity for their sense-making: It is not up to me to actually go 
around and focus on certain people and say ah that one looks confused . . .  It is not my job to 
focus in on somebody and say you look like you don 't understand . . .  You look like you need 
to be asking a question. It is your job to be listening, to be watching and trying to make 
sense of what someone is explaining. If it is not clear to you, you need to jump in straight 
away. Ava wanted the students to act autonomously in the discourse. She voiced an 
observation that when she participated in the small and the larger discussions the students 
looked to her for permission to question or justify the reasoning. She considered that this 
often i nterrupted the interactive flow. So, she i ntroduced the concept of ' no hands up ' 
particularly when there were many questions and chal lenges for an explanation. Although 
the students were initially hesi tant she prompted them with directives : Hone you wanted to 
ask a question ? Come on just jump in and do it. The students also appropriated the words 
"just butt i n" and used them to prompt their less confident peers. I l lustrated in the following 
vignette i s  how this resulted in an interactive flow of conversation in  which the students 
used proficient forms of mathematical practices to c losely examine, analyse, and validate 
their mathematical reasoning as they worked to solve the fol lowing problem. 
1 43 
Facilitating the flow of interactive student led discourse 
Beau i s  doing a T.V. contest and this is a q uestion w h i c h  is asked . 
Which of the fol l owing fractions best represents the value of ( ) ?  
5/ 10 
51 100 1/ loo 
5/ 1000 
He has to give a reason for his answer and then when q uestioned further explain and j ustify his responses? 
First of all discuss your  answers and then be ready to explain and justify them. Think also of questions 
the group might ask you and other ways of explaining what they ask. 
0 ( )  
Mahaki and Chanal are explaining in a large sharing session a solution strategy for the 
previous problem: 
Chanal [records a numberli ne and records 1 1 1 0  at the end] . We think that it  is five 
hundredths. We think that that ' s  the end [points at 1 1 1 0] because one tenth 
can ' t  go anymore back. You can ' t  equivalent it less. So if it was two tenths 
you could have done i t . . . you wi l l  have to go decimal points with your 
fractions. 
Many students i ndicate they have a question .  
Ava 
Sandra 
Chanal 
Sandra 
Mahaki 
Jae 
Chanal 
Rangi 
Mahaki 
[prompts] Just butt in with your questions. 
Why do you think you have to do decimal points with the one tenth? 
Because with one tenth and fractions you can ' t  go half of just one tenth . . .  
So you thought it was five hundredths? 
[records the notation scheme as he explains using fractions and decimals] 
Yeah. We thought i t  was five hundredths because one tenth times ten equals 
ten hundredths and if you half that ten it will be five, five hundredths .  
I f  you half the top one you have to  half the bottom one? 
No, because we are not halving  the numbers we are halving the fraction . 
Why did you timesed it? 
So we could get the hundredths. So we could half i t .  
After further questions and challenge to the terms, formal and i nformal representations and 
the reasoning Sandra introduces an alternative explanation. 
Sandra 
Ava 
Aroha 
[marks above 1 / 1 0, 1 0%]  There ' s  also a different type of way you can do 
this. The five would have to stand for five percent. 
So what Sandra has shown you, is that one tenth or ten percent which she 
said was another way is not near half, not near five tenths. It is the first 
mark. So half of that is five one hundredths because as Chanal and Mahaki 
told you one tenth i s  equivalent to ten hundredths and half of that segment i s  
five one hundredths. Certain ly  not half of ten tenths, or i t ' s  equivalence of  
one hundred one hundredths, or  one whole as Mahaki explained so wel l .  
[asks the two boys] So why do you put it  in decimals? 
1 44 
Sandra I know, because there ' s  three different ways to basical ly explain a fraction, 
the fraction way, a decimal point way and a percentage way. That' s why he 
has picked one of them. I nstead of just doing the fraction or percentage, he' s  
picked the decimal point way, because he may think that' s actual ly his easier 
point of doi ng the fraction way. 
(Term 3 Week 10) 
Through these actions A va encouraged student agency over when and how they 
participated in the mathematical discourse. At other times she explicit ly intervened to 
faci l i tate slower exchanges. This occun?ed particularly when new mathematical topics were 
introduced or when she considered the mathematical concepts under consideration 
particularly challenging. For example, in the fol lowing vignette she participated in a group 
discussion of the problem used in the previous vignette. This time her interventions 
maintained a slower flow of productive discourse which enabled the students to focus on 
inquiry and challenge and ensured that they reflectively considered and reconsidered their 
reasoning. 
Tipani 
Ava 
Pania 
Tipani 
Mahaki 
Tipani 
Ava 
Tipani 
Ava 
Chanal 
Facilitating a slower flow of productive mathematical discourse 
[draws a numberl ine, records 0 then 9 marks and then 1 1 1 0] Here is 0 and 
1 1 1 0. 
Now take you time. Think about it .  Right any questions? 
Why are you doing those l ines? 
[records 51 1 00 at the middle mark] Because each of those lines is 
representing one tenth, I mean ten tenths. I am thinking that this one is 
meant to be 51 1 00. 
Why? 
Basically because of what you said Mahaki . 
Which was? Explai n  it i n  your own words and see if Mahaki agrees. 
That if  you times . . .  ten by ten . . .  well I am not actually that sure. I just think 
that it  i s  five one hundredths .  I don ' t  think that it  is five one thousandths. 
Wel l  what do you think Mahaki, and you other people who heard what 
Mahaki explained? Let ' s  take a look at these fractions and think about what 
Tipani and Mahaki were saying. What do you see when you look at these 
fractions . . .  what other ways can they be represented apart from that? 
[points at the 51 1 00 mark] Mahaki said that one tenth can be ten percent 
because if  you times one by ten you get ten, and you times the ten by ten 
you get one hundred. So that wi l l  be one tenth i s  l ike ten percent, so in the 
middle that wil l  be five percent there. 
1 45 
Ava 
Chanal 
Pania 
Ava 
You accept that Mahaki? So what you are saying is that that means five parts 
out of a hundred and the one tenth there means ten out of a hundred .  So what 
is this one? [Points at 5/ 1 0] 
That i s  fifty percent. 
So how? 
Yes you jump in here Chanal, if you can explain it in a different way. 
Chanal leans forward and picks up the problem sheet with the diagram of a numberl ine of 
one divided into ten segments as he says: 
Chanal 
Pania 
Chanal 
Ava 
Mahaki 
[points at the first segment on the numberline] I know what. If you go back 
to there and just pretend you shrink that down to there. There' s  a hundred 
right? So that half way mark in brackets wou ld be right there 
[points at the position it would be in  if you had a whole number l ine not just 
to one tenth and it represented 1 1 1 0] and that would be ten percent and if you 
halved that ten percent it would be five. 
Five what? 
[records 5% and 5/1 00] Five percent or five hundredth. 
Are you all convinced? Or do you want to ask some more questions? 
It is five hundredth because as Chanal said that thing there would be just l ike 
a little piece of this line . . .  But the other way i s  to go the percent way. You 
get ten percent and then half that. That' s the quickest way to explain i t .  
(Term 3 Week 10) 
Mutual engagement, intellectual agency and collaborative partnerships, scaffolded 
participation in mathematical argumentation, exploratory talk 
A va had provided the students with a predictable framework for inquiry and argumentation. 
Within this frame she had established that the students were required to take personal 
ownership of their sense-making and justification .  Through providing the students with 
ways to circumvent the need to seek permission to speak she positioned the students as 
more autonomous participants in the mathematical i nteractions .  In turn, the students 
responded by participating in the dialogue with decreased teacher assistance. They would 
analyse the mathematical reasoning then often bypass Ava to ask a question or challenge 
the reasoning. Or they would step in ahead of Ava ' s  prompting to respond to questions and 
justify or offer alternative ways of reasoning. Evident in the data is student development of 
i ntellectual agency within a cl imate of mutual engagement (Wenger, 1 998) and respect. 
A va and her students had constituted a mathematical communi ty within an intel lectual 
partnership. As Cobb (2000a) explains, when students view themselves as autonomous 
1 46 
learners in a mathematical community they are able, independently of teacher or text, to 
validate their own and others' contributions and reach consensus in the argument. 
Ava drew on her ' insider' (Bassey, 1 995 ; Jaworski, 2003) knowledge of classroom culture 
and from moment to moment selected how she positioned herself in the discourse. In this 
situation,  as the more experienced 'knower and user' of mathematics she worked within 
negotiated zones of proximal development (Vygotsky, 1 978) guiding and scaffolding the 
students in col laborative argumentation .  Andriessen (2006) explains that "when students 
col laborate in argumentation in the classroom, they are arguing to learn", activity that 
involves "elaboration, reasoning, and reflection" (p. 443 ).  Ava 's  actions in slowing the 
discourse provided intellectual space (Watson & Mason, 2005)-a platform for all 
participants to access and develop reasoning from. In this space the different students were 
positioned and repositioned to make and justify claims and counterclaims. Arguments were 
extended through revoicing (O'Connor, 2002), rephrasing, and elaborating on. I n  the 
discursive in teraction, the words and reasoning used by others were appropriated, explored, 
reworked and extended before they became a communal 'voice' (Brown & Renshaw, 
2000) .  
The collaborative construction of  a shared view was not always premised on  immediate 
consensus. Instead in this community, as in other inquiry communities reported on (e.g. ,  
Brown & Renshaw, 2000; Goos, 2004; Mercer & Wegerif, 1 999a, 1 999b; Rojas?
Drummond & Zapata, 2004), a lack of understanding, dissension and disagreement often 
acted as the catalyst for further examination and evaluation of the reasoning. The students' 
use of specific words ("because", "I think", "but why", "so if you", and "I agree") to 
negotiate and reach a common perspective were evident. These words and phrases Mercer 
and his col leagues identified as those u sed in exploratory talk. Exploratory talk as I 
explained in  Chapter 3 is a form of talk which supports proficient u se of mathematical 
practices. 
1 47 
6.5.2 TRANSFORMING INFORMAL INSCRIPTIONS TO FORMAL NOTATION 
SCHEMES 
In this final phase of the study, informal i nscriptions and formal notation schemes were 
important tools which supported a range of mathematical practices. Ava continued to place 
drafts of the inscriptions on the wal l and this provided the students with public access to a 
range of solution strategies. They regularly reviewed and discussed these independently, 
particularly when arguing about possible solution strategies, or when making comparisons, 
or seeking more proficient solution strategies. The inscription drafts on the wall were also 
regularly used to "fold back to" (McClain & Cobb, 1 998, p. 59) during argumentation or 
for reflective analysis by the groups .  Evident in the data is how they would be appropriated, 
tried out, extended and innovated on and used to scaffold more proficient forms of notation .  
In group work Ava pressed the students to use mathematical symbols and formal notation 
schemes alongside their less formal notational schemes. In the sharing sessions Ava would 
often bui ld on a group' s  informal notation scheme to develop more formal notation 
schemes. The students would also begin from informal inscriptions and then,  in search of 
ways to convince others, develop more formal ways to explain .  The students became adept 
at translating across their multiple representations i ncluding both formal and informal 
schemes to provide explanatory justification as i l lustrated in the fol lowing vignette. 
Using multiple representations to validate reasoning 
The small group have extensively explored a problem and constructed multiple 
representations of their reasoning including a formal notation scheme. They are asked for 
justification and one student explains the reasoning used by another in their group: 
Rangi [using diagrams, fraction symbols and gesturing] First she started off with 
twelve eighths which she added to six eighths which she knew was two 
whales plus two eighths, then she plussed ten eighths because she knew that 
that made three whales and one half. Then she was adding five eighths 
which made four wholes and one eighth and then she added seven eighths 
onto her four whales and one eighth which made five wholes. 
(Term 3 Week 10) 
1 48 
Abstracting and generalising notation schemes, folding and dropping back, appropriating 
inscriptions 
The press to use more proficient notation schemes supported a shift from the use of l iteral 
interpretations embedded in real world contexts toward mathematising-abstracting and 
generalising and using more formal notation schemes. Concrete situations provided a 
starting point and in  the reflective dialogue the students gradual ly  advanced from modeling 
informal schemes to the use of more formal symbolic notation. Consistent with the findings 
of Whitenack and Knipping (2002), inquiry and chal lenge also increased the press toward 
formal notation. This occuned either when a group member could not make sense of an 
inscription, or when a student introduced a more sophi sticated form which progressed al l 
the group' s  thinking. In these situations, Ava would intervene and explore with them 
formal notating patterns. However, although Ava specifi cally pressed towards the use of 
formal notation schemes, she also recognised the value of informal models to make the 
reasoning visible. She would select a group who had used informal inscriptions or return to 
an explanation which had been explained using a less proficient notation scheme as a 
means to fold or drop back to when needed, to c larify the reasoning (McClai n  & Cobb, 
1 998) .  For example, she explained when we analysed a video excerpt: I picked Rangi first 
because I knew that they would use their diagram as a model and that would really help the 
others to see how the pieces combined. 
6.5.3 INCREASING THE PRESS FOR GENERALISING REASONING 
In this final phase of study Ava had become attuned to hearing spontaneously voiced 
generalisations used to justify reasoning and now regularly bui l t  on them.  She introduced a 
requirement that students analyse, compare, and justify differences i n  efficiency and 
sophistication between explanations. To extend how the students participated i n  
constructing and exploring generalisations Ava introduced a set o f  questions which 
potential ly supported the students to generalise. She modeled their use, regularly asking 
questions l ike 'does i t  always work' and 'how can you know for sure' . She also constructed 
a wall chart of s imi lar questions and prompted the students to use them. She used open?
ended problems which had been discussed and written in the study group. The fol lowing 
1 49 
vignette i l lustrates her students working on equivalence general isations. In the rich 
discussion, Ava' s request for further backing shifts the lens from a single model of number 
combinations towards a more general model for number combinations. 
Generalising concepts of equality 
Marama, Arohia and Rangi provide a solution strategy for a problem8 which they have 
verified using other combinations in response to an expectation that they were required to 
convince their listeners. 
Arohia 
Marama 
Arohia 
Rangi 
Rongo/Hemi 
Hinemoa 
Arohia 
Ava 
Rangi 
Hone 
Rangi 
Hone 
Rangi 
Ava 
Sandra 
Hemi 
Sandra 
Ava 
Sandra 
Ava 
What we did is we kept on doing ninety and we kept on making the middle 
one . . .  
Going smaller. 
Going decreasing and we made the last l ine going up, increasing. We went 
nine hundred, ninety, and then ten .  Then we went nine hundred, eighty and 
twenty . 
It's going down in  tens and then i t ' s  going up in  tens .  
Can you prove that . . .  i t  equals up to the number like one thousand? 
Does it always work? 
Yeah, to us it  always does. But I can do it by using another number. 
That is even better. Arohia what are you saying? Can you te ll us? 
[records 1 000 = 900, 95, 5; 900, 85, 1 5 ;  900, 80, 20] I can because all you 
have to do . . .  first I go in fives and I go up in fives and then I go . . .  
Why do you do that? 
Wel l  because I know that ninety five plus five equals one hundred and then 
ninety plus ten equal s one hundred. 
So everything adds up to one hundred? So it is al l the same as one hundred? 
Then I write nine hundred all the way down.  So i f  I add them together they 
all equal one thousand. 
I know what. Just write down B and in the third place write C. Now that 
could be anything all  right? Now what are you going to do to B? Make a 
conjecture. 
You could do B take away five and C take away five. 
No C plus five 
It' s l ike what we thought. Basically you can put any number there. It does 
work with any number. 
And? 
Like B take away n ine hundred and C plus n ine hundred. You are keeping 
the same amount. 
As long as you . . .  
8 Mrs Hay goes shopping for Christmas presents. She spends $ 1 000 on three presents and she only uses 
dol l ars not cents for them but she uses al l of her money. Can you explore patterns for the different amounts 
each present might have cost? Is there any way you could work out all the combinations that are possible 
without listing every single one? 
1 50 
Sandra Are balancing. So if you go B minus seven hundred and then C plus six 
hundred it would not work out because it' s not balanced. You would have 
one hundred less. 
(Term 4 Week 1 )  
Powerful questions, abstracting and mathematising, open-ended problems as 'example 
spaces ' 
Making comparisons and searching for patterns and connections between the solution 
strategies were tools A va used to develop more generalised reasoning patterns with the 
students. Likewise, her modeling of specific questions and the wal l chart supported 
students' search for patterns and relationships across problem situations and solutions. 
Evident in the data i s  students' increased use of logical connectives (because, but, if . . .  then)  
to  generalise. As other studies have shown (e.g., Blanton & Kaput, 2003; Strom et  al . ,  
200 1 ; Zack, 1 997, 1 999) teacher questioning and the use of specific questions were 
important elements in classrooms in which students participated in argumentation and 
justified through generalised reasoning. These studies emphasised the key role of the 
teacher in facilitating the discourse. In this study Ava's  faci l i tation of sharing sessions was 
the key to how the students mathematised problem si tuations so that they became models 
for more abstract and generalised mathematical thinking. She would structure the dialogue 
so that gradually student reasoning was pressed toward abstraction.  
A va was mindful that for the students to be able to formulate, model and generalise 
solutions of one problem situation-as models for problem situations-they needed 
opportunities to make connections in patterns and relations across problem situations. At 
times she transformed existing material so that the students were required to provide 
mul tiple conjectures. These and the use of open-ended problems provided what Watson and 
Mason (2005) termed 'example spaces' . These problems offered the students many 
opportunities in their groups to suspend judgement to conceptual i se and re-conceptualise ,  
represent and re-present problem solutions in many forms as they searched for p lausible 
patterns or counterexamples. 
1 5 1 
6.5.4 SUMMARY OF THE THIRD PHASE OF THE STUDY 
The participation and communication patterns of i nquiry provided the basi s to further 
develop intellectual partnerships within the classroom community. Justification and the use 
of many inscriptions provided a platform for the development and exploration of 
generalised reasoning. Ava facil i tated increased student autonomy within student discourse. 
Mutual engagement in learning partnerships evolved within reciprocal interaction which 
created multi -directional zones of proximal development. 
6.6 SUMMARY 
This chapter has documented the journey Ava and her students made as a community of 
mathematical i nquiry was constructed. The tools  which mediated the transformation of the 
classroom' s  learning culture were described. These included the communication and 
participation framework which guided the pedagogical actions A va took to engage the 
students in increasingly proficient use of mathematical practices. 
The classroom cu lture began as what Wood and McNeal (2003) termed a conventional one. 
Through steady consistent shifts in the communication and participation patterns the 
classroom culture became strategy reporting and then an inquiry or argument culture. 
Change was initiated in the mathematical discourse patterns through specific emphasis 
placed on modeling the processes of mathematical inquiry within a safe learning 
environment. As changes in  the discourse patterns shifted toward a predominant use of 
inquiry and argumentation, so too were the mathematical practices extended. Both Ava and 
her students ' use of questions and prompts were important factors which pressed reasoning 
to higher and more complex intellectual levels .  The variations in interaction patterns in the 
different classroom cultures had differential outcomes for student learning .  Student 
autonomy, responsibility for sense-making of reasoning was most evident in the argument 
culture. Simi larly, the mathematical practices the community members participated in and 
used became interrelated tools used to justify and validate reasoning. 
1 52 
Inducting students into a culture of i nquiry and argumentation responded to both the 
pedagogical actions Ava took and the beliefs she held about doing and using mathematics. 
The findings in thi s chapter showed how each shift in the mathematical discourse and 
mathematical practices matched shifts in Ava' s beliefs toward her students engaging in 
i nquiry and argumentation. The following chapter describes a second teacher Moana who 
held a different set of beliefs about doing and using mathematics to those held by Ava. 
Description is provided of how these shaped her construction of a mathematical community 
of inquiry and the mathematical practices which evolved. 
1 53 
CHAPTER SEVEN 
LEARNING AND USING MATHEMATICAL PRACTICES IN A 
COMMUNITY OF MATHEMATICAL INQUIRY: MOANA 
The transformative practice of a learning community offers an ideal context for developing 
new understandings because the community sustains change as part of an identity of 
participation. (Wenger, 1 998, p. 2 1 5) 
7.1 INTRODUCTION 
The previous chapter described the transformation of participation and communication 
patterns which resulted in construction of a community of mathematical i nquiry. This 
chapter also documents the transformation of a c lassroom community but it  i l lustrates how 
a different set of beliefs about doi ng and learning mathematics held by the teacher shaped a 
slower, more circuitous route to developing a community of mathematical inquiry .  
Section 7 .2  outl ines Moana's beliefs about doing and using mathematics at  the beginning of 
the research. This provides an important background to understanding her subsequent 
actions. Section 7.3 describes Moana' s pedagogical actions which mediated student 
participation in communicating mathematical reasoning. Effects of the previous learning 
culture are examined and ways in which change was initiated are described. Descriptions 
are provided of the wide resources Moana drew on when transforming the interaction 
patterns from conventional to strategy reporting. 
Section 7 .4 describes the pedagogical actions Moana took to engage her students at higher 
intel lectual levels  of i nquiry. A close relationship is i l lustrated between Moana' s previous 
experiences as a mathematical learner and user and how she posi tioned student 
participation in the mathematical discourse. Section 7 .5 describes how Moana drew on her 
students' social and cultural identities to scaffold their appropriation of the mathematical 
discourse as a social language (Bakhtin, 1 984; Gee, 1 992). Participants'  shifting roles in the 
1 54 
community towards shared ownership of the di scourse and mutual engagement in  a range 
of reasoning and performative mathematical actions (Van Oers, 200 1 )  are described. 
7.2 TEACHER CASE STUDY TWO: MOANA 
In an i nterview at the start of the research study Moana outli ned her understandi ng of the 
nature of mathematics.  Al though she had secondary school mathematics qual ifications she 
explained that she disl iked mathematics and lacked confidence in  the use of mathematics 
beyond a restricted school s i tuation. She considered that she was not as clever as those 
individuals who enjoyed mathematics or appeared more confident in  their use and 
knowledge of mathematics.  Thus Moana had constructed a situated identity (Gee, 1 999) 
about herself as a mathematician . She drew on this identity to explain both her personal 
disl ike for mathematics and her difficulties in doing and using mathematics, despite her 
apparent academic success. 
Moana stated that for her mathematics was a body of knowledge and a set of rules which 
were useful tools i n  a school context for ' school mathematics' . But beyond school she said 
she was not confident about applying mathematics to real l ife problem situations. Moana 's  
bel iefs about mathematics as  a static body of knowledge are consistent with a traditional 
view which Stipek and her col leagues (200 1 ) report many teachers hold. In thi s  view, 
mathematics involves "a set of rules and procedures that are applied to yield one 
answer. . .  without necessari ly understanding what they represent" (p. 2 1 4) .  
In th is initial interview Moana outl ined how she considered mathematics t o  be: a valuable 
commodity in society. But she added that she considered that there were many: barriers to 
access that knowledge for many of her diverse group of students. To enable them to access 
the mathematical knowledge she explained that she u sed particular pedagogy practices 
which she considered best suited her students' learning needs and learning styles. In her 
description of her predominantly Maori and Pasifika students she said that they were : more 
practical, hands-on visual learners and so she said that she always used concrete and 
manipulative material when teaching mathematics. Thi s  label Moana had attached to her 
1 55 
students conforms to one which Anthony and Walshaw (2007) contend is often used to 
describe Maori and Pasifika students-ki naesthetic learners. However, when teachers 
attach the label to Maori and Pasifika students they make assumptions based on what they 
perceive their learning needs to be. A lton-Lee (2003) maintains that this often results in 
negative outcomes for these specific students. 
Moana explained that she combined a physical hands-on approach with: a lot of routines 
and structure. Writing on the lines, from left to right, numbers keeping them nice and tidy 
in their boxes . . .  lots of practice . . .  lots of hands on . . .  then dragging it all back, putting it on 
the board . . .  practice over and over. She described how she was the one who did most of the 
talking in her mathematics lessons because she considered that her role was to instruct 
students on how to use the various mathematical procedures. She believed that if they 
l istened careful ly and then practised the procedures she showed them they should learn 
what was required. Her description of the c lassroom context she had constructed is 
consistent with what Wood and McNeal (2003) describe as a traditional or conventional 
classroom. The way in which Moana described her use of mathematics talk as a tool which 
focused on transmitting knowledge, corresponds with descriptions of traditional c lassrooms 
where teacher talk predominates (Mehan, 1 979; Wood, 1 998) .  Moreover, the stance she 
took as personal ly responsible for her students to learn specific rules and procedures 
corresponds to that taken by many teachers (Stigler & Hiebert, 1 999). Stigler and Hiebert 
describe these teachers as positioning themselves as external authority figures in 
conventional classrooms. 
Moana had recently been a participant i n  professional development in the New Zealand 
Numeracy Development Project (Ministry of Education, 2004a). She described how she 
had adapted the project ' s  knowledge and strategy activities so that they were always at a 
concrete and manipulative level. She justified this by referring back to her need to maintain 
a teaching style which best met the learning needs of her students . Although a central tenet 
of the NDP is to develop student use and explanation of a range of strategies, Moana 
explained that she had focused on the use of concrete materials .  She said her students learnt 
through 'doing' and that they found explaining difficult. Moana's adaptation of the NDP 
1 56 
material meant that she could continue to teach in a way which conformed to her own 
beliefs about the nature of mathematics and what she perceived were the learning needs of 
her students. 
7.3 CHANGING THE INTERACTION NORMS TOWARDS A 
COMMUNITY OF MATHEMATICAL INQUIRY 
Moana had been a quiet member of the study group when the communication and 
participation framework was discussed and explored at the end of Term One. However, at 
the combined study group of senior and middle school teachers in the second term Moana 
participated actively in discussions. As she watched the senior c lassrooms' video records 
she probed and questioned the teachers about how they had initiated changes to their 
classroom communication and participation patterns. She questioned the effect the changes 
were having on their students. Then she used their descriptions of how they had 
restructured their classrooms to establish changes she wanted to make. In discussion with 
me she stated that her immediate focus was on developing student abi lity to construct and 
present mathematical explanations .  In order to achieve this she described a need to change 
the ways in which the students interacted and talked together i n  mathematics. 
7.3. 1 THE INITIAL START TO CHANGE THE COMMUNICATION AND 
PARTICIPATION PATTERNS 
Moana introduced her expectations for col laborative behaviour by using a laminated chart 
obtained from another teacher. She told the students that they would be working in small 
groups and then she read the chart to the students as she displayed it prominently on the 
wal l :  so when you are working and you have a person in your group who can 't manage and 
cope, they need to go and they need to Look at the solution pathway and the participation 
norms. Each lesson in the first week of the research fol lowed a simi lar pattern. After 
? students were placed in groups or before large group sessions began Moana read aloud 
sections of the chart. However, she did not explore or discuss what the chart meant with her 
students. In this initial stage she discussed what l istening meant but did not extend 
1 57 
description beyond the physical act of listening. For example, when she asked the students 
what they needed to do to listen, a student  said :  use taringa [ears]. To this, she responded: 
yes listen and that ended the discussion. 
At the start of the first research phase Moana used the problems which had been devised i n  
the study group (see Appendix G) to support development of  early algebraic reasoning. 
Consi stently each lesson began with the students organised i nto smal l groups. Then Moana 
read them the problem, provided them with concrete material and then i nstructed them to : 
work together and find an answer. Moana walked around the groups and regularly used 
comments l ike :  come on participate people, what did I say about active participation ? As 
the students worked i n  these groups observational data revealed a predominant use of 
disputational or cumulative talk-a form of talk Mercer (2000) describes as not conducive 
to developing shared thinking. 
Small group activity was fol lowed by l arge group sharing sessions .  In these sessions the 
students constantly i nterjected and made negative comments to each other. Moana, in turn, 
asked single answer questions and then validated the answers she wanted by recording 
them on  the whiteboard. 
Moan a' s actions i l lustrate the considerable challenges Ball and Lam pert ( 1 999) contend 
teachers face in constructing i nquiry environments. Her behaviour I in terpreted as 
indicating that she had limited vision of what teaching and learning in such an environment 
meant. Her students were unable to understand the roles they were now being asked to take. 
As Hufferd-Ackles and her col leagues (2004) describe, many teachers have had no personal 
knowledge or experience in learning, or teaching, in i nquiry environments. Moana showed 
her novice approach i n  her use of her col league' s  chart. Her i nexperience was also evident 
when she assumed that tel l ing students to l isten and participate in groups was sufficient to 
establish the social in teraction norms she required. She had put i n  place edges of the 
changes required but her core pedagogical practices remained largely  u nchanged. 
1 58 
7.3.2 CONSTITUTING SHARED MATHEMATICAL TALK 
In informal discussion duri ng the first week of the study Moana indicated her need to 
further consider how she could change the interaction patterns .  Our col laborative review of 
events in this first week and examination of classroom video recordings led to a closer 
examination of the communication and participation framework. Smaller, more incremental 
steps (see Appendix K) in the communication and participation patterns were planned to 
guide the change. 
In the first instance to remediate how the students interacted, Moana noted the need for an 
immediate focus on increasing student cooperation in l istening and talking. So that she 
could carefu l ly  guide and shape how the students participated in mathematical activity 
Moana returned to teaching mathematics in a large group. At this stage she had seen how 
the students quickly lost focus of the discussion and became disruptive in the smal l groups. 
Not using small groups was thus an intermediary step; she eventually wanted to develop 
student abil ity to work in them. 
Her immediate action as she led whole c lass mathematical activity and discussion was to 
ask the students to talk together in pairs . For example, in one of these early lessons Moana 
made an array of two sets of eight counters then directed the students to consider: If I 've got 
one set of eight there and another set of eight, how many sets of eight have I got? I don 't 
want you to answer that. I want you to talk. I want you to turn and talk to each other. 
Fol lowing their discussion she asked one member to report what the other had explained. In 
another instance, she provided a problem and required that together they make a conjecture 
and discuss their reasoning but she emphasised their need to both be able to report back. At 
other times, to keep the focus on l istening and sense-making she asked one member of the 
pair to explain their reasoning and then asked the other to outline and model with materials 
what their partner said .  
Through dai ly  discussion and our i nformal analysis of how different students participated 
in the mathematical talk Moana began to more confidently outl ine her requirements for 
col laborative activ ity. Initially she had depended on instructions and strategies she saw 
1 59 
modeled by other study group members when she viewed video records of their classrooms 
but now she tentatively explored and developed her own repertoire of talk. She used thi s to 
establish with the students what she meant by active participation and col lective 
engagement i n  mathematical discourse . Each lesson began with an outline of her 
requirements for active participation and she closely monitored how the students worked 
together. In the fol lowing vignette examples are provided of how she adapted her 
instructions specifical ly  to what she saw happening during shared mathematical activity. 
Shaping ways to talk and actively listen to mathematical reasoning 
Moana observed that the students had begun to record without discussion. She intervened 
and told them: 
Moana Before you pick up a pen or touch the paper you need to discuss it first and I 
want you to show me how you can work together listening and bui lding on 
each other' s thinking with those questions. 
(Term 2 Week 6) 
After Moana had explained a problem and provided concrete material she stopped them and 
directed: 
Moana I want you to talk to each other before you even touch the sticks. Lots of 
talking and l istening and I might ask you what someone said, not you, so 
you need to discuss things please and make sense of what someone else 
says. Listen careful ly to each other. I want you to discuss what is happening 
in your patterns of two . . .  and how many you have. 
(Term 2 Week 6) 
Conflicts in changing beliefs, reshaping the rules of talk. 
The classroom culture and the interactions i n  it had previously been shaped by the beliefs 
Moana had about the nature of mathematics and what it meant to learn mathematics.  These 
beliefs remained robust throughout the first part of this study. Researchers (e.g . ,  Boaler, 
2002a; Lampert, 200 1 )  argue that students in schools learn particular mathematical 
dispositions through the 'school mathematics '  they experience. Beliefs that mathematics 
ideas are predetermined and unarguable are fostered when prior learning experiences 
emphasise scripted construction of roles, rules and procedures (Hamm & Perry, 2002) .  This 
had been Moana ' s  prior experiences and she in turn had constructed a culture in which her 
students were developing simi lar dispositions and beliefs. Moana i l lustrated this  view in a 
1 60 
reflection, recorded after a lesson, as to why the students had difficulties engaging in  the 
talk: I have always told them, not asked them to talk together. I am used to doing the 
talking and they are used to me talking too. 
Although Moana discussed the changes she wanted to make to classroom communication 
and participation patterns with me and the other study group members she was ambivalent 
about them. She was aware of the contradictions she was causing in her students' beliefs 
about what constituted a mathematics lesson. She noted as she reviewed a video record of a 
lesson observation at the end of the second week: I really wanted these children to do this 
but they really couldn 't see any sense in it at all . . .  it was more of the mechanical or 
cosmetic workings on how I wanted them to go about talking and listening. Researchers 
(e.g. ,  Ball & Lampert, 1 999; Mercer, 2000; Wells, 1 999) report that students may be 
chal lenged by the different rol es and scripts they are required to take when shifting from 
conventional to inquiry classrooms . Conflicts and contradictions i n  the students' bel iefs 
were expected given the former sociocultural norms which had prevai led including the 
external attribution of authority to the teacher and more able or domi nant students 
Contradictions existed also because although Moana had taken small incremental steps 
towards change, she retained the central position as the authority in the classroom. When 
she viewed a video excerpt of a c lassroom observation she critically described her 
pedagogical actions as: book style participation norms . . .  going through the cosmetics . . .  but 
not really going into it . . .  not really actually unearthing it . . .  yes you must validate 
everybody 's answers but at the end of the day just give me the answer. Hamm and Perry 
(2002) maintain that teachers relinquishing authority to the c lassroom community, stepping 
away "from a central role as classroom leader to allow true public discourse about 
mathematical ideas" (p. 1 36),  comprises a major chal lenge . Moana illustrated the on-going 
difficulties she had in adopting a more facil itative approach although she remained 
critically aware of her i ntentions to do so. 
1 6 1  
7.3.3 CONSTRUCTING MORE INCLUSIVE SHARING OF THE TALK IN THE 
COMMUNITY 
In the second week of data col lection Moana voiced her concern that the girls were often 
passive participants in the discussions. She described their lack of confidence to talk and 
al igned thi s with her own experiences as a school student. She outlined her observations of 
the difficulties the Maori and Pasifika girls in particular had when required to speak to, or 
question, boys. A decision was made to initial ly put girls and boys in separate pairs rather 
than in mixed pairs and monitor the outcome. At times Moana was even more specific and 
paired particular girls, for example Pasi fika girls together. She explained that she wanted al l 
students to develop a mathematical voice and identity and so she was exploring ways to 
increase how speci fic students engaged in the mathematical tasks . 
Moana read and di scussed with the study group a research article ' which described the 
pedagogical strategies a teacher used with diverse learners and as a result she became more 
mindful that an environment which supported her students as risk-takers was needed. 
Engaging in the discourse of inquiry was such a new experience for Moana and her 
students that many were understandably hesitant. It was not unusual for a less confident 
student to stop mid-sentence or i ndicate that they wanted to ask a question, but withdraw 
when asked to speak and say "I forgot". At other times, a student would make a conjecture 
but then sit si lently when questioned or shake their head when asked by Moana to respond 
to further questioning. To establish a caring environment Moana talked with the students 
about inclusion, support, and collegiality. She c losely monitored less confident students '  
participation, actively supporting them to participate appropriately .  For example, when a 
Pasifika girl responded quietly to a question, she interceded and told her: you don 't have to 
whisper. You can talk because you want to make sure that you are heard. She participated 
in discussions and provided specific scaffolding for less confident students, eliciting and 
extending their mathematical responses. For example, Moana asked Tere: What are we 
1 White, D .  Y. (2003).  Promotin g  productive mathematical classroom d i scourse with diverse students. The 
Journal of Mathematical Behaviour, 22( 1 ) , 37-53. 
1 62 
looking at here? When Tere responded with: ones Moan a probed further: one group of . .  
and thus drew the more extended answer of: oh . . .  twenty, one group of twenty. 
At the end of the first month Moana observed that the patrs were able to more 
col laboratively construct and examine their explanations and so she began to vary the 
number of students working together. She tentatively placed boys and girls together but 
conti nued to closely monitor Pasifika and Maori girl s '  participation. As she gradual ly 
increased group sizes she affi rmed their ski l l s  of working together and introduced and 
talked about the notion that they were all members of one whanau (fami ly). She drew on  
their home experiences and together they explored how family members supported each 
other. She emphasi sed that in  such groupings different levels of expertise exist but tasks are 
accomplished through the cooperative ski l l s  of al l members . She described herself as a 
whanau member who took the lead i n  some si tuations and in  other situations depended on 
other whanau member' s expert ise.  
Moana' s i ntroduction of the whanau concept indicated that a gradual shift in  positioning 
had occurred in  the classroom culture .  Moana had begun to move from the posi tion of total 
authori ty-the teacher in control of the discourse-to become a participant in ,  and 
facil i tator of the mathematical discourse. This shift in  positioning aligned to changes in  her 
expectations and obl igations of the students. Previously her requests for explanation had 
used 'me '  and 'I '  with l ittle reference to other participants : Tell m e  about yours . . .  Speak up 
Rata. I like the way you are thinking but I need to hear you. Now in an observable shift 
Moana positioned herself as a participant and l istener i l lustrating this in  her request: can 
you show us what four groups of four look like because we want to think of other ways than 
the adding on, the skip counting don 't we ? 
Changing beliefs, changing roles, mediating different forms of participation in the 
discourse 
Cobb and Hodge (2002) explain that often teachers and researchers attribute poor 
mathematical ach ievement to specific attributes the students themselves lack. Simi l arly, 
1 63 
Moana attributed her own problems with mathematics to qual ities she lacked stati ng in an 
i nterview before the research began : I didn 't have the ideas that other kids had so I just 
never said anything. I just thought they were all brighter than me. In the same i nterview 
she described her students ' low achievement levels attributing these to their lack of prior 
experiences at home, their lack of i nterest and engagement in mathematics, and their 
passive approach to learni ng. In using a "deficit-model approach" (Civil & Planas, 2004, p. 
7) that attributed the problems in constructi ng mathematical understandings, hers i ncluded, 
to i ndividual traits she u ndervalued the role of the social learning context. Shifti ng  
classroom communication  patterns and awarding i ncreased focus on the social context was 
understandably the site of many tensions :  at some point I am thinking is this maths ? Is this 
going to work? Trusting that someone has the best interests of these children in mind with 
such an emphasis on talking and them hearing what they say and I wonder about if they get 
confused by what they say to each other. 
Moana' s actions in explicitly scaffolding specific groups of studen ts were designed to assist 
all students to access the shared di scourse. Many researchers (e.g., Civi l & Planas, 2004; 
Khisty & Chval ,  2002; Planas & Gorgori6, 2004; White, 2003) have i l lustrated successful 
outcomes for diverse learners when teachers developed classroom cultures which mediated 
student participation in the mathematical discourse . The discourse Moana wanted the 
students to access Bakhti n ( 1 98 1 )  and Gee ( 1 992) term a social language. She had begun to 
recognise that the students needed to learn more than mathematical knowledge and a set of 
procedures-they were also required to learn ways of talking, l istening, acting and 
i nteracting  appropriately  when working with mathematical ideas . 
Moana' s focus on different  forms of grouping aimed to reposi tio n  less confident students to 
participate . She wanted to i ncrease their participation and change how they saw 
themselves-their roles and identities as mathematical doers and users. Changing the 
organisational structures of who worked with who caused an immediate shift in  a l l  
participants' roles. Moana noted the :  changes evident in girl culture. The girls are starting 
to say something. Just like testing out what they can say. Who participates and how i n  
mathematics classrooms i s  influenced b y  the organisational structures i n  them and the 
1 64 
memberships these create. As Civil and Planas (2004) explain, the "the internal ization of 
certain roles, derived from these memberships, certain ly has many impli cations for 
learning" (p. 1 1 ) . This was evident in this study. 
Moana had previously maintained a distance between herself and the students. Within the 
first month she made an observable shift to include herself in the community. Her actions 
reveal the value she placed on developing affective relationships. Drawing on the concept 
of the whanau indicated that she wanted to build reciprocity-mutual respect which 
empowered al l members of the community. According to MacFarlane (2004) whanau "is 
often defined as the notion of a group sharing an association, based on things such as 
kinship, common locality, and common i nterests" (p. 64 ) .  Integrated within the whanau 
concept are specified forms of behaviour. These include care and concern within 
col laborative support but also "assertive communication" (p. 78) which links to appropriate 
ways to voice thinking. Moana' s careful scaffolding of each student 's  participation laid  the 
foundations for them to assertively communicate their reasoning duri ng mathematical 
activ ity. Moreover, I interpreted her introducing the whanau concept as evidence that she 
too had begun to develop her own pathway and a more assertive voice in constructing an 
inquiry community. 
7.3.4 LEARNING TO MAKE MATHEMATICAL EXPLANATIONS 
As described previously, Moana had adapted the Numeracy Development Project (Ministry 
of Education, 2004a) to better match the pedagogical beliefs she held. As a result, the 
students had learnt a different form of mathematics than that intended-one i n  which the 
use of concrete materials,  and recording and practis ing rules and procedures prevai led. 
Initial classroom observations revealed the students' immediate response when given a 
problem was to compute an answer. Moana wanted to address this behaviour but she was 
concerned that her students' growing mathematical in terest and confidence be maintained. 
In the first instance she redesigned a set of problems and used a family of television cartoon 
characters which she considered would better engage student attention. Using these she 
addressed their persistent answer seeking behaviour, directly outli ning her expectations: it 's 
1 65 
not about the answer. It 's about how you solve it. You need to be talking about it. Discuss 
it, what you are doing and then what you are doing all the way. 
Moana was aware that the students had many difficulties explaining their reasoning ful ly .  
In accord with a revised section of the communication and participation framework (see 
Appendix K) Moana explicitly scaffolded ways to extend their explanations. In their pairs 
she emphasised need for extensive exploration and examination of each of the sequential 
steps. Through these actions the students began to recognise what needed to be included in 
an explanation to meet their audience 's  need. They were also learning how to respond to 
questioning with clarification of their reasoning. 
In sharing sessions Moana closely attended to their verbal explanations and stepped in  to 
scaffold those who needed support and to address students' use of everyday language and 
short utterances. The fol lowing vignette i l lustrates how Moana revoiced to name solution 
strategies or to press the students to explain their actions and solution strategies. 
Scaffolding mathematical explanations 
Aporo uses counters to model how his group solved the problem2 . 
Aporo Two, four, six, eight, ten, twelve. 
Moana So that ' s  cal led skip counting because you are skipping across the numbers. 
Tere We kept adding like two more. We counted in twos .  
Moana Counting in twos. Yes that is skip counting. 
(Term 2 Week8) 
Faa J ays out an array of three rows of three counters to explain a solution for the problem3 . 
Faa 
Moan a 
Faa 
Moan a 
Tui 
We went three times three equals nine people. 
But there might be people in here who are not sure . . .  
Yes. Because three plus three equals six and plus another three equals nine. 
Did you see how he solved that and explained it? What he said? 
He said three plus three equals six plus another three equals nine or three 
times three equals nine. 
(Term 2 Week8) 
2 Mrs Dotty has baked some muffins. She puts them on the bench to cool in two rows. Each one of the s i x  
Dotty chi ldren sneaks in and takes two of them so w h e n  Mrs Dotty returns they have all gone. H o w  many 
muffins did Mrs Dotty bake? 
3 Mrs Dotty has a car which has her driving seat and then three rows of three seats i n  the back. How many 
Dotty chi ldren can she fit in her car? 
1 66 
Teacher dilemmas, changing scripts to focus on the reasoning, teacher revoicing 
In i nquiry c lassrooms the teacher role is a complex one with many challenges (Yackel,  
1 995) .  Moana described these in her reflections of her lesson video: as a teacher I am put 
in a position of 'sculpting ' an outcome without the proper tools. The long silences and I 
feel quite lost on when to jump in and when to let the children struggle. These children are 
just not used to me letting them struggle. In informal discussions with me Moana outl ined 
her personal conflict at allowing her students to struggle or be confused. S imi larly, she 
described problems she had scaffolding student development of viable explanations while 
also making sense of them herself. She also described the challenge of knowing when to 
question and challenge, when to insert her own ideas, and when to lead discussion back to 
the mathematical understandings under consideration. Simi lar dilemmas have been 
described by other researchers (Chazan & Bal l ,  1 999 ; Lobato et al . ,  2005 ; Schwan Smith, 
2000) when teachers shift the communication and participation patterns towards inquiry. 
Previously the students had experienced making mathematical explanations through 
calculating objects arithmetically in an instrumental manner (Skemp, 1 986). Constructing 
conceptual explanations posed immediate and on-going di fficulties for many students. 
Some students continued to interpret explaining as providing procedural steps; others had 
difficulties knowing what detai ls were required for sense-making and what they could 
assume as taken-as-shared. These difficulties are simi lar to those other researchers have 
described as students learn to explain and justify their reasoning in i nquiry classrooms 
(Cobb, 1 995 ; Cobb et al . ,  1 993; Kazemi & Stipek, 200 1 ;  Yackel ,  1 995 ;  Yackel & Cobb, 
1 996). 
To transform how the students had interacted previously in mathematical activity required 
that Moan a directly address their previous scripts (Gal l imore & Goldenberg, 1 993 ). These 
involved establ ishing a range of socio-cultural norms identified by many researchers (e.g. ,  
B lunk, 1 998;  Lampert, 200 1 ;  Sull ivan et al . ,  2002) as supporting rich mathematical activity 
and discussion. The small i ntermediate steps added to the communication and participation 
framework mediated a gradual shift in the communication and i nteraction patterns .  Moana 
1 67 
used specific pedagogical practices to establ ish the foundations for peer collaboration 
(Forman & McPhai l ,  1 993) .  These included use of what O'Connor and Michaels ( 1 996) 
term revoicing-an interactional strategy used to socialise students into mathematical 
situations. Moana' s revoicing subtly repositioned students to extend their explanations. 
7.3.5 LEARNING HOW TO QUESTION TO MAKE SENSE OF 
MATHEMATICAL EXPLANATIONS 
Owing to the previous univocal patterns of discourse (Knuth & Peressini , 200 1 ) used in the 
classroom the students were inexperienced at l istening to, and making sense of, each 
others' explanations. Observing their novice behaviour in video records of classroom 
observations Moana noted their need to learn to talk and to listen to each other. She stated 
that she planned to address a former interaction pattern in which interjections were an 
accepted norm. Observations had provided evidence that the interjections maintained a 
focus on conect answers, affected self-esteem negatively, and detracted from other students 
having time or space to actively listen, thi nk, and examine the reasoning. 
In the first i nstance, Moana began to halt mathematical explanations at specific points in 
the large group discussions. She directed the students to take time to think about what had 
been explained, and then to ask questions. To focus students on responding appropriately to 
enoneous reasoning she required that they ask questions and cause the explainer to rethink. 
For example, when an enoneous explanation was presented she halted discussion, withheld 
her own evaluation and asked the students to frame a question:  Who has got a really good 
question they can ask Wiremu to make him rethink. I like the way you are thinking about 
the question. Would you like some help with your question ? Moana drew on a set of 
questions (see Appendix E) which had been developed in the study group context and used 
these as models for the students, of how to question to extend explanatory reasoning. She 
would l isten c losely to explanations and regul arly halt  the explainer to provide space for the 
other students to ask questions, if none were forthcoming she would often ask questions 
herself as i l lustrated in the fol lowing vignettes .  These show how Moana adopted a number 
of different roles as she participated in discussions. These i ncluded her directly modeling 
asking questions, scaffolding student questioning and shaping how and what questions 
1 68 
were asked, or acting as an observer and directing attention to student models of active 
listening and questioning. 
Questioning mathematical explanations 
Moana as a participant in a discussion l istening to an explanation asks a question. 
Moana What have you actually done there Mahine? 
Mahine I have plussed 1 0  onto 47. 
Moana So you have added 1 0  onto 47? Are there any questions?  Questions like 
where did you get the ten from? 
(Term 2 Week 7) 
Moana stops an explainer and asks the l isteners. 
Moana Do you want to ask a question? You need to start with what. 
(Term 2 Week 8) 
Moana halts an explanation and directs student attention to an example of active listening 
Moana Donald is  real ly  l i stening. He is  actually l istening and watching what has 
been taking place . He is not only l istening to the person, he is watching 
when they write things down. His questions are real ly specific to what the 
person is doing. 
(Term 2 Week 10) 
Active listening, questioning to provide space for rethinking 
In the i nteraction patterns previously established Moana had assumed that the students were 
learning through l istening. This was evident in a statement she recorded after a lesson when 
she observed the outcomes of the new interaction patterns :  good to see different thinking 
coming through, blows me away because before they never had a chance to explore . . .  / just 
thought that they understood by me talking all the time. Now she had begun the gradual 
process of inducting students into an inquiry culture which placed value on active l isten ing 
and questioning. Within this changed learning climate there was a discernible shift i n  how 
the students l istened to each other with i ncreased respect. 
Evident in the data is how errors had become learning tools ,  valued as a means to examine 
and analyse reasoning, rather than cause loss of self esteem .  Moana' s approach to errors as 
a way to widen discussion and questioning is simi lar to the approach used by teachers i n  
1 69 
Kazemi and Stipek' s  (200 1 )  and White ' s  (2003) studies. The previous focus of the 
classroom on provision of answers to predetermined solutions had established a learning 
culture in  which knowing mathematics was "associated with certainty: knowing it, with 
being able to get the right answer, quickly" (Lam pert, 1 990b ) .  Moan a wanted to address 
this pattern and so she emphasised a need for the students to stop, think, and reconsider the 
reasoning as she explained: I was reinforcing rethinking because calling out has been the 
prevalent means but they need to learn to hold back and do some thinking first. Moana' s 
actions scaffolded the students to use a more 'mindfu l '  approach to listening and using 
specific questions to understand the mathematical reasoning. As Lampert i l lustrated 
"teaching is not only about teaching what is conventionally called content. It is also 
teaching students what a lesson i s  and how to participate in it" (p. 34). 
7.3.6 SUMMARY OF THE FIRST PHASE OF THE STUDY 
In this section I have outli ned the many hurdles Moana encountered as she laid the 
foundations of the interaction norms of an inquiry community. Moana had appeared to 
support the use of the communication and participation framework to plan out shifts in the 
classroom community ' s  i nteraction patterns but the novice status of both her and her 
students in the inquiry environment meant that initially they lacked knowledge and 
experience of the many roles they were required to take in this culture. Many researchers 
(e.g. , Ball & Lam pert, 1 999; Franke & Kazemi, 200 l ;  Huferd-Ackles et al . ,  2004; Mercer, 
2000; Rittenhouse, 1 998;  Sherin, 2002b; Wel ls ,  1 999) report similar chal lenges when 
teachers shift their c lassroom culture from a conventional to an inquiry classroom. In  an 
i nterview at  the end of  the research Moana described what th is  first phase was like for her. 
She explained the effect of the shifts i n  c lassroom communication and participation  
patterns :  I was slowly coming awake . . .  i n  those initial stages thinking and I suppose when 
you are a teacher and you do professional development and you think I will try this . . .  then I 
started seeing . . .  and I was more impressed not by their maths but by their talking, how they 
were talking. Then I didn 't feel so harsh on myself for focusing on the participation norms 
because they were actually talking. So I just started to settle down to the maths. The 
reconstruction of the communication and participation framework into smal ler more 
incremental steps supported a gradual change in the classroom i nteraction patterns .  This i n  
1 70 
turn appeared to provide Moana with confidence to conti nue making shifts in the learning 
culture . 
The students now actively engaged in construction and examination of conceptual 
explanations-an important shift because according to Cobb and his col leagues ( 1 997) 
being able to provide explanatory reasoning i s  an important precursor for supporting 
development of explanatory justification and argumentation. Moreover, the close attention 
Moana had placed on active listening, questioning and rethinking shi fted the students 
towards sense-maki ng within zones of proximal development (Forman & McPhail, 1 993) .  
They had begun to view their reasoning from the perspective of others which potentially 
provided an important foundation for future mutual engagement (Wenger, 1 998) in 
collaborative activity. The classroom culture Moana had consti tuted had shifted toward 
what Wood and McNeal (2003) define as a strategy reporting discussion context .  
7.4 FURTHER DEVELOPING THE COMMUNICATION AND 
PARTICIPATION PATTERNS OF A COMMUNITY OF 
MATHEMATICAL INQUIRY 
Moana' s pedagogical actions m the first phase had focused on establishing the socio?
cultural norms which supported the students to provide mathematical explanations. Our 
joint discussion of the students' current communication and participation patterns prompted 
us to establish the next steps on a trajectory towards developing reasoned col lective 
discourse . Moana considered that the students were ready to learn how to engage in 
mathematical inquiry and argumentation to justify their reasoning. However, she voiced her 
key concern that the students retain their growing mathematical confidence explaining that 
she thought that: this was risky stuff moving them, upping the ante; I 'll be hanging out there 
as much as they will be hanging out there but they are ready. Together, we careful ly  
analysed the communication and participation framework and constructed smal l specific  
steps (see Appendix K) to  support the shift towards student use of  mathematical i nquiry 
and argumentation. 
1 7 1  
7.4. 1 COLLECTIVELY CONSTRUCTING AND MAKING MATHEMATICAL 
EXPLANATIONS 
In the first i nstance Moana focused on further developing how the students worked 
collaboratively. She restructured the dai ly lesson format so that she was no longer leading 
mathematical activi ty from a central position. Now in each lesson, after an initial short 
di scussion the students worked in smal l heterogeneous problem solving groups and then 
returned to the larger group situation to conclude with a sharing session. 
Moana's previous focus had been to develop individual student capacity to actively li sten, 
explain their reasoning and make sense of the reasoning of others. Now Moana wanted to 
press group behaviour towards increased col laborative interaction. She stated that she 
wanted the students to di scuss, negotiate, and construct a collective solution strategy. In 
shaping their interactions Moana emphasised their personal responsibility to engage and 
understand the reasoning used by other members. She established a pattern where the 
students began their small group activity with a mathematical problem which they were 
directed to read and think about individually, then discuss, interpret and together negotiate 
a solution strategy. Moana gave each group one sheet of paper and pen to use. She moved 
from group to group l istening to their talk and only intervened to ensure all i ndividuals 
contributed and could explain the developing reasoning. She explicitly establi shed with 
them that they cou ld only bring questions or problems to her if the whole group agreed that 
they required assistance. When a group member requested help she discussed with the 
group their prior actions, drawing from them ways they might solve their problems 
together. She explained that she wanted to assist the students to recognise and use their 
col lective strengths to engage more autonomous ly  in mathematical activity. 
Although Moana required that the students develop a joint explanation for the larger 
sharing session she encouraged them to explore and d iscuss a range of ideas, then select the 
one they agreed they could all understand and explain .  She guided their negotiation and 
selection of a shared strategy solution through establishing a set of ground rules (see 
Appendix J) for talk loosely based on those developed by Mercer (2000) but pertinent to 
this group of students. When she observed that they had developed a shared explanation she 
1 72 
asked them to predict which sections their audience might find difficult, discuss and 
explore questions they might be asked, and rehearse ways they might respond. She said that 
she wanted to maintain their confidence but also to increase their shared understanding and 
prepare them to respond appropriately  to questions. The following vignette i l lustrates how 
Moana facil i tated discussion which supported the students to question and probe an 
explanation for sense-making but at the same time ensured that ownership of the reasoning 
remained with the explainer. 
Scaffolding exploratory talk to explore an explanation 
Moana joins a small group and is a l istening participant. Anaru is making an explanation 
for the problem4. 
Anaru 
Moan a 
Anaru 
Moan a 
Wiremu 
Anaru 
Donald 
Anaru 
I drew six cakes and my way was halving it .  
Can you show us? 
[draws four squares and divides each one in  half] And then I halved it .  
Has anybody got any questions they want to ask Anaru? 
Why did you cut them in halves? 
Oh because they said that there were four people and she was to half it so 
they have equal amounts. 
What did you do next? 
Each person took three parts and there was one whole cake left and I halved 
it and then I halved another half and they took one each. That ' s  four pieces 
they took. 
Aporo watches closely and moves his hands as he counts the pieces. 
Aporo 
Donald 
Anaru 
How could she [referring to Mrs. Dotty] half it again? 
If it was al l gone? 
Yeah. There was one whole cake left and then I halved it into fourths. 
The l istening students count the halves and then look at Anaru questioning. 
Moan a 
Donald 
Anaru 
Rona 
Moan a 
Do you all agree? You have l istened to her but remember you have to agree. 
[points at the last two cakes] Can you tel l  us what you said in that bit? 
I said that I halved it and each person took three . 
I know. So you are saying they got a whole and a half. There are two cakes 
left that are halves so they got a whole cake and a half. 
Do you agree Anaru? Is that what you mean? 
4 Mrs Dotty had just baked 6 chocolate cakes and suddenly she heard a knock on the door. Guess what? 4 
friends had come to visit i n  their flash car! They were real l y  hungry so she c u t  up the cakes and they shared 
them between them. She didn't eat any because she had already had lunch. If they each ate the same amount 
how much did each person eat? What fraction of the six. cakes did each person eat? 
1 73 
Anaru No because I just found out that there was halves of a cake. I said i n  quarters 
but I meant in halves. 
Hone So if she cut al l  six cakes i nto halves and gave them three halves each that 
means one and a half or three halves each eh? 
(Term 3 Week 2) 
Facilitating the talk, increased student autonomy, gaining consensus through exploratory 
talk 
The pedagogical actions Moana took to establ ish collaborative group skil ls are simi lar to 
those used by Lampert (2002).  The direct attention Moana gave to the group processes, as 
she stepped in and out of a range of roles, meant that her students learnt that both social and 
academic  outcomes result from group work. 
Evident in the data is a shift in positioning of all the members of the classroom. Moana 
analysing her actions in a video excerpt of a lesson observation recorded: more korero 
(discussion) less teacher talk and I am really moving into a facilitating role. Her more 
facil itative approach could be attributed to her direct attention to group processes. Like 
other researchers (e .g . ,  Rojas-Drumrnond & Mercer, 2003;  Rojas-Drummond & Zapata, 
2004; Wel ls, 1 999) who explicitly scaffolded a talk-format with diverse students, this 
resulted in  a progressive shift toward increased exploratory talk. 
The changes evident in  Moana' s c lassroom resonate with those Hufferd-Ackles and her 
col leagues (2004) maintain are important in the growth of "a math-talk learning 
community" (p. 87) .  These researchers described the developmental growth of a classroom 
community i n  four dimensions. These included questioning; explaining mathematical 
thinking; the source of mathematical ideas; and responsibi l i ty for learni ng. As Moana' s 
students gained greater agency, i ncreasingly they initiated questioning. Similarly, their 
mathematical thinking became an important source for mathematical discussion. Moana 
noted the confidence the students showed in explaining, e laborating on, and defending their 
reasoning when she recorded after a l esson : Anaru shows confidence and she has a definite 
sense of trust about her way. 
1 74 
The c lassroom observations reveal that explanations had become mathematical arguments 
as a direct result of Moana' s expectation-the negotiation of a collective view. Through the 
close examination of mathematical explanations the criteria for what the students 
considered acceptable as mathematical explanations was establ ished. Not only were 
explanations required to be experiential ly real and relevant but the l isteners also expected 
elaboration or an alternative explanation. 
7.4.2 PROVIDING A SAFE RISK-TAKING ENVIRONMENT TO SUPPORT 
INTELLECTUAL GROWTH 
Taking intel lectual space to rethink reasoning had become an established practice in the 
first research phase and when Moana and I discussed the most recent observational data we 
noted that student talk had increased significantly. But Moana was concerned that the 
students: haven 't fully got to the aspect of valuing what the other person is saying because 
they are so busy gushing. They are so excited about making sense themselves. But that will 
happen, well it is happening I guess because when you go back and look at the videos of 
them in the beginning . . .  now actually you see them really on track. Together, we planned 
adjustments to the ground rules for talk discussed in the previous section and the pathway 
Moana had p lanned using the communication and participation framework. We wanted to 
extend how the students used the concept of 'rethinki ng' to provide them with cognitive 
space-time to shape responses to conjectures, respond to questions, or examine the 
reasoning of others. Moana introduced this shift by asking the students in their sharing 
sessions to use 'rethink time' to develop more questions to gain clari ty. This action 
reinforced that they needed to be constantly thinking through every mathematical action 
and questioning unti l they had complete understanding. She emphasised that 'rethink time' 
provided a way to work from confusion to understanding. For example, as she watched 
students struggling to develop an explanation she said to the group I can see you are 
confused. Me too, that 's all right we can take some time . . .  rethink about it. It 's good to take 
some risks with our thinking sometimes. In this statement and others l ike it, Moana modeled 
that she too was a learner and at times needed to persist with question ing her own thinking 
to comprehend explanations. 
1 75 
Moana' s validation of the acceptability of confusion as important to mathematical sense?
maki ng caused a further shift in the ways in which erroneous thinking was considered in 
the community. Erroneous reasoning emerged most often when the students were using the 
more specialised di scourse of mathematical language or when they over-general ised 
number properties. Moana drew on these as valuable learning tools ,  using them to explore 
alternative ideas, support rethinking and reformulation of conjectures, or to refine the use of 
mathematical language. At the same time, she remained respectful of student reasoning. For 
example, in the fol lowing vignette Moana faci l i tates discussion, providing opportunities for 
reconsideration and clarification of reasoning but she maintains the self-esteem of the 
explainer and ensures that the ownership of the reasoning remains with the explainer. 
Rethinking and reconsidering to clarify reasoning 
During sharing Moana records 3 x 4 and 4 x 3. Donald provides another explanation for the 
problem5 using counters and an array. 
Donald 
Moan a 
Aporo 
Donald 
Moan a 
Jim 
Moan a 
Donald 
Moan a 
Donald 
Three times three equals six and another group of three groups of three 
equals six and then you just put them together and they equal twelve. 
Are there any questions? Do you agree with that? I will just write it down 
[records 3 x 3 and 3 x 3 ] .  
Oh he has done too much, l ike too much threes. 
Oh true I forgot to say different. I said it wrong. 
That 's  okay. So can somebody explain? I l ike the way you are thinking 
about it. 
Three groups of four. 
[points at the first recording of 3 x 4 and 4 x 3]  We are just looking at this 
and trying to make some sense out of it and comparing it with this one 
here. [points at the recording of 3 x 3 and 3 x 3]. There i s  a disagreement 
happening. So we need to look at these and we need to sort it  out. What I 
need you to do i s  look at these again and think careful ly about what they 
look l ike. Donald can you please put three groups of three and three groups 
of three . . .  show us what three groups of three and three groups of three look 
like please [points to the pattern] .  Here we' ve got three groups of three and 
another three groups of three. 
Oh because at first I said three plus but I accidentally said times. I got mixed 
up. 
That ' s  fine because now you are making it c lear. So you have got three plus 
three and three plus three. 
Yes I got mixed up first. 
5 Mrs Dotty baked some muffms. She only has a very small oven. So she fits them i n  the oven by putting 
them in four rows with three muffins in each row. How many muffi ns has she baked? 
1 76 
Moana [records 3 x 3 = 9 then 3 + 3 + 3 = 9] So you have mixed the times with the 
plus? 
(Term 3 Week 2) 
Moana continued to emphasise need for communal support, placing responsibil ity back 
with a group to c larify reasoning including when they used e1Toneous reasoning. When she 
observed that a student had difficulties, she would require that another group member step 
in and support them. She always affirmed thei r self esteem and ri sk taking. This  was 
i l lustrated when she told the students : Caliph has taken a wonderful risk. She is out there 
and she wants some help. 
Through our discussions Moana was aware that often students reveal their misconceptions 
through extended mathematical dialogue. She introduced specifical ly designed problems 
that required the students to devise ways to explain and c larify mathematical situations for a 
character in them who was confused or had erroneous reasoning. The problems (see 
Appendix I) provided the students with public opportunity to examine and explore possible 
partial understandings the students themselves held. The students had the opportunity to 
work from a point of confusion or an erroneous position through to sense-making as valid 
mathematical problem solving activity. 
Respectful interaction, extending independent student engagement with mathematical 
reasoning, partial understandings and interthinking in shared communicative space 
Weingrad ( 1 998)  maintains that intel lectual risk-taking requires an environment which i s  
respectful o f  student reasoning. The careful ly crafted interaction patterns Moana was 
building are those which many researchers (e.g. ,  Boater, 2006a, 2006b; Martin et al . ,  2005 ; 
Pourdavood et al . ,  2005) identify as important for diverse students if they are to develop 
intel lectual autonomy. Moreover, Moana was enacting specific expectations which were 
designed to press the students toward independent engagement with mathematical ideas. 
Goos (2004) i l lustrated how a teacher in her study enacted similar expectations related to 
"sense-making, ownership, self-monitoring and justification" (p. 283 ) which made possible 
a later press toward extending more autonomous engagement with mathematical ideas. 
1 77 
The way in which erroneous thinking became considered as a valued teaching and learning 
tool reveals the significant shift which had occurred in the culture of the classroom 
community. Moana modeled norms which reinforced that rethinking and persevering to 
work through confusion was a sound mathematical learning practice (Boaler, 2006a). No 
longer were errors the cause of negative situations and loss of self-esteem . Instead, they had 
become catalysts for further problem solving or "springboards for inquiry" (Borasi ,  1 994, 
p. 1 69) .  
Evident in the data is  how the different partial understandings held by participants in the 
dialogue contributed to the continual reconstitution of a shared zone of proximal 
development. These results are similar to those of other researchers when students are 
specifical ly scaffolded to engage with the reasoning of others (e .g . ,  Brown,  2005; Brown & 
Renshaw, 2004; Goos et al . ,  1 999; Mercer, 2000). In Moana' s c lassroom the variable 
contributions pul led the participants i nto i nterthinking within a shared communicative 
space. Moana noted the shifts she had made as a partic ipant and faci l itator as she viewed a 
video excerpt and recorded: I was involved in clarification and whole class joint shared 
understanding, revoicing and talking about multiplication. I always thought maths was so 
straight forward but now it 's changing rapidly in my head. She had noted her own shift 
towards considering mathematics as an ever-expanding body of knowledge, rather than a 
limited set of mathematical facts and procedures (Stigler & Hiebert, 1 999).  
7.4.3 POSITIONING STUDENTS TO PARTICIPATE IN THE CLASSROOM 
COMMUNITY 
Moana continued to monitor closely how different students i n  the classroom participated in  
interactions. She viewed and discussed video excerpts of  the Maori and Pasifika girls '  
emerging communication and participation patterns .  She closely monitored how less 
confident or less able students managed within the heterogeneous grouping. Moana 
specifically focused on strategies to build these students' mathematical confidence. She 
actively posit ioned specific students. For example, as she l istened to a group discussion she 
1 78 
overheard a quiet comment by a less able student and responded by directing the other 
students to l isten to him. She then said: good thinking Tama, something to get you all 
going. At another time she li stened to the exchange of ideas then commented loudly: wow 
Teremoana see how you have made them think when you said that? Now they are using 
your thinking. She regularly halted expl anations as they were being shared to draw attention 
to how a low achieving or unconfident students' reasoning had contributed. The fol lowing 
vignettes i l lustrate how Moana li stened careful ly  as small groups interacted and then when 
required stepped in and positioned specific students to participate and contribute their 
reasonmg. 
Positioning the students to access and own the mathematics talk 
Moana observes Anaru nodding her head in agreement so she questions Anaru directly: 
Moan a 
Beau 
You are saying yeah . . .  what are you saying yeah to? 
She's  just been going . . .  making up suggestions . . .  dumb ones . . .  she ' s  just. . .  
Moana without responding to Beau repositions Anaru as a valued group member: 
Moana Well  it' s not dumb ones. I have been l i stening and she ' s  making you think 
because she is using the problem and making sense of it. She knows that you 
have to l isten to each group member, l i sten to their thinking and make sense 
of what they think [turns and asks Anaru] So, what was your way? Can you 
explain it  please? 
(Term 3 Week 8) 
A group is discussing partitioning a l ine segment i nto fractional pieces. Moana quietly 
questions a passive on-looker to ensure that he is accessing the reasoning. 
Moana 
Beau 
Caliph 
Do you understand it Haitokena? Come on people talk about it. 
So that one in the middle is half. 
[points at mid-point] Yeah you can just l ike draw a l ine through there. 
Moana positions Haitokena so he can sense-make and then provide an explanation. 
Moana But we [turns to i nclude Haitokena] are asking what are each of these bits? 
What i s  the fraction of those bits? Then Haitokena will be able to explain 
them to you. 
(Term 3 Week 8) 
1 79 
Changing student status, shifting beliefs, changing roles 
Through direct pedagogical actions Moana repositioned the low achievers, the passive and 
shy students, and Maori and Pasifika girls from their previous social and academic status .  
She assigned competence (Boaler, 2006b)  to  them through expl icitly drawing other 
students' attention to the intel lectual value of their reasoning. Moana recognised how her 
own actions and expectations for different students had shifted when she commented to me 
informal ly :  I have realised I did have low expectations because why would I be so excited 
when they say something that is relevant to what is happening. I thought I had high 
expectations for my class; well I had high expectations about certain people . . .  so at first 
when someone would say something and I thought did you just say that? Now I am hearing 
what they all say and its relative and it 's linking in . . .  so when I hear them talk out loud I 
help them to be heard. 
Moana regularly reviewed previous video records and discussed the shifts she observed in  
the students' communication and participation patterns.  She voiced specific concern about 
the tensions and contradictions the shifts in interaction patterns had caused for the Maori 
and Pasifika girls .  She included the changes in her expectations of them, and their own 
expectations they now held, of their role in the classroom mathematical community. She 
voiced her observations of how they had changed their view of what 'doing' mathematics 
meant: when you go back and look at the videos of them working in the groups at the 
beginning they were just pretending to be working in those groups. They were just 
parroting the words . . .  but now you see them really on track. But then as they got hooked in 
I can see the changes, it 's the girls who are really changing like Anaru. I have seen 
Anaru 's behaviour . . .  well it is . . .  it is coming to an uneasy uncomfortable place. She is taking 
herself seriously now in maths. She 's a Pacific Island girl and she has come to {1 cross road 
and she is making a choice. (Term 3 Week 8) 
Within the sociocultural approach of this study, to understand Moana' s perspective requires 
explicating the relationships between her actions and the "cultural , insti tutional and 
historical situations in which this action occurs" (Wertsch et al . ,  1 995 ,  p. 1 1 ) . To explain  
the many in ternal tensions i ndividuals i n  a community contend with Forman and Ansel l  
1 80 
(200 1 )  use the notion of "multiple voices" (p. 1 1 5) .  These multiple voices match the many 
communities individuals belong to, but also incorporate a historical dimension in which 
"memories of the past and anticipation of the future affect l ife in the present" (p. 1 1 8) .  
Moana, as  the old-timer (Lave & Wenger, 1 99 1 )  was required to  induct the students i nto 
the mathematical community but in doing this a historical voice inc luded her own 
memories of her learning experiences in mathematics as a Maori gir l .  Her expectations for 
the possible future in mathematics of the Maori and Pasifika girls coloured her present 
interactions.  Forman and Ansell drawing on social cultural theories explain this "discursive 
mechanism by which the past and future are drawn into the present as prolepsis" (p. 1 1 8 ) ;  a 
form of anticipation of the existence of something before it actually does or happens .  
Moana's statements and interactions with the Maori and Pasifika girls can be understood as 
part product of her own experiences and also her future expectations for them. 
7.4.4 PROVIDING EXPLANATORY JUSTIFICATION FOR MATHEMATICAL 
REASONING 
Through the first half of the year the participation structures Moana made available to the 
students emphasised a need for collective agreement in the construction of explanatory 
reasoning. Through discussion with me Moana real ised that when the focus of classrooms 
is toward teaching students to work together and develop collective consensus there is 
potential that they interpret this as always needing to be in agreement (Mercer, 2000) .  
Providing justification or convincing others was a key feature of a part of the 
communication and participation framework Moana aimed to enact. As an important 
incremental step, Moana establ i shed that her students needed to learn 'poli te' ways to 
disagree and challenge. She placed an immediate focus on requiring that l i stening students 
voiced agreement or disagreement with conjectures. She regularly halted explanations and 
positioned students to take a stance. For example, she told them: at some point you are 
going to have an opinion about it. You are going to agree with it or you are going to 
disagree. But she ensured that they knew that they needed valid reason to support their 
stance directing the students to: think about what they are saying. Make sense of it. If you 
don 't agree say so but say why. If there is anything you don 't agree with, or you would like 
them to explain further, or you would like to question, say so. But don 't forget that you 
1 8 1  
have to have reasons. Remember it is up to you to understand. The emphasis she placed on 
their need to j ustify the stance they took reinforced their responsibi lity to actively l isten and 
sense-make and provided a p latform to shift the discourse from questioning and examining 
explanations toward questioning for justification. 
In the study group Moana had examined the questions and prompts for justification 
described by Wood and McNeal (2003) and viewed examples in video records from her 
col leagues ' classrooms. As she analysed video excerpts of her classroom observations she 
stated : a real need to move, shift from the surface questions or practising how to . . .  to take a 
good look up close and personal, shift thinking to challenging, justifying, validating, 
creating other possibilities. To do this, before mathematical activity began she verbally 
emphasised the need for questions and chal lenge: I want you people asking 
questions . . .  throughout ask questions. Why did you come to that decision ? Or why did you 
use those numbers ? Or can you show me why you did that? Or if you say that, can you 
prove that that really works ? Or can you convince me that this one works the best? 
Through our study group discussions Moana knew that explanatory justification often 
required more than one form of explanation and so she prompted the students to consider 
development of multiple ways to validate their reasoning: sometimes . . .  remember yesterday 
we had like three strategies, three different strategies, all the same. You all came out with 
the same solutions but you did the three different ways and sometimes you need that to 
convince somebody. 
Moana used problems (see Appendix H)  purposely developed in the study group which had 
as key component the requirement that the students c losely examine their col lective 
explanations and construct multiple ways to val idate their reasoning and convince others . 
Similarly, she required that all group members be able to explain and justify the col lective 
strategy and provide support if  their explainer had difficulties responding to questions or 
challenge. She also strengthened how they responded to argumentation through requiring 
the small groups to examine and explore questions they could be asked when explain ing to 
the large group: not only are you asked to justify and explain but you are also thinking 
about what possible questions you might be asked and how you are going to go about 
1 82 
answering those questions. As i l lustrated in the fol lowing v ignette Moana actively 
participated in  group discussions facil itating how the students could respond to chal lenge 
and as a result the students examined, extended and validated their reasoning 
autonomously. 
Facilitating justification to validate reasoning 
Moana has listened as Mikaere explained and drew, and segmented a set of li nes i nto 
halves; eighths and then 1 3  pieces, and then recorded 1 31 1 3 for the problem6. 
Moan a 
Beau 
Wire mu 
Moan a 
Hone 
Wire mu 
Mikaere 
Wiremu 
Can you all understand that? Now what you people need to do . . .  other 
people are going to ask you questions about why you split these sections i nto 
these fractions .  What are you going to say? So what questions do you think 
other people are going to ask you? 
They will ask why . . .  
How did you come up with that? 
Okay. So they will come up with . . .  how did you come up with that idea? 
Why did you use those fractions? You need to clarify exactly what you were 
thinking at the time. Think of different ways to answer. 
What fractions do you have? 
Why did you put thirteen thirteenths? 
Because i t ' s  thirteen l ines. 
No. You mean thirteen bits . But why is i t  thirteen bits? How do we say? 
6 This little guy from outer space is in your classroom . He is l istening to 
Ann ie, Wade, Ruby and Justin arguing about sharing a big bar of chocolate. 
Annie says that you can only share the bar of chocolate by di viding it into halves or quarters 
Wade says he knows one more way of sharing the bar of chocolate 
Ruby and Justin say that they knows lots of ways of sharing the bar of chocolate and they can find a pattern as 
well 
The little guy from out of space is really interested i n  what they say so they s tart to explain all the different 
ways to him. 
What do you think they say? I n  your group work out a clear explanation that you think they gave. 
He needs lots of convincing so how many different ways can you use to prove what you think they were 
thinking. Make sure that you use fractions as one way to show h i m  because he l i kes using n umbers. Can you 
find some patterns to explain? 
1 83 
Faa 
Hone 
Mikaere 
Wire mu 
Hone 
Because i t  was a whole chocolate bar. If i t  was halves you have two bits. But 
you cut his one whole bar into thirteen pieces and you sti l l  have the one bar 
but thirteen thirteenths now, j ust l ittler bits than the halves. 
But they might ask if  they equal the whole. 
They all equal one whole chocolate bar just the half bits are bigger but sti l l  
one whole and we can show as  many bits but sti l l  the same. But  there is  
another pattern. The bigger the number goes the smal ler the bi t  goes. 
Cut it into halves that 's the biggest. 
But if the top and bottom are the same then you just have one whole, doesn ' t  
matter what they are if they are the same. 
(Term 3 Week 8) 
Scaffolding inquiry and justification, student voice 
Reviewing the video records of lesson observations provided insight i nto how Moana 
careful ly scaffolded a shift in the classroom discourse from the need for explanatory 
reasoning to explanatory justification. The pedagogical strategies she used are reported as 
important by other researchers (e.g. ,  Kazemi & Stipek, 200 1 ;  Lampert et al . ,  1 996; 
Lampert, 200 1 ; Nathan & Knuth, 2003) .  The use of problematic mathematical activity 
coupled with Moana's  structuring of the di scourse and requirement that the students take a 
stance positioned them to develop logical ways to refute or support reasoning. Ball ( 1 99 1 ,  
1 993)  described a similar outcome i n  her c lassroom-increased engagement i n  discipl ine 
specific inquiry and debate was matched with growth in  student agency. The use of inquiry 
and debate supported development of joint zones of proximal development. The s imi lar 
levels of reasoning but different pieces of understandi ngs held by the i ndividuals gave them 
experience in accommodating a range of perspectives and experiencing transactive dialogue 
(Azmitia & Crowley, 200 1 ) . 
Consi stently over the duration of the research Moana voiced confl icts related to the 
pedagogical role and actions she needed to take. When considering i ntroducing a press for 
justification she noted: There have been significant changes for me. I started off thinking 
should I write those questions up so they could just pick and just answer them. I am glad 
that I didn 't do that because it would have replaced their voice. It would have just replaced 
their voice with a card that they read off and that would have been something else because 
they do have their own language and they have to learn that it is okay in certain places and 
1 84 
not in others. I deduced from her comment that she had developed her own sense of trust in 
her actions in constructing an inquiry community. Moreover, she recognised that her 
students were constructing a mathematical discourse, as another speech genre or social 
language, along side their individual voice (Bakhtin ,  1 994). Her actions made possible 
opportunities for them to appropriate and explore, extend, expand and transform the 
language of inquiry i nto their own words and thoughts. Other researchers have indicated the 
importance of these actions for diverse learners (e.g. ,  Gee, 1 999; Gee & Clinton, 2000; 
Moschkovich, 2002b ;  Wertsch, 1 99 1  ) .  
7.4.5 EXPLORING RELATIONSHIPS AND PATTERN SEEKING 
Through the specifical ly designed problems and the search for multiple forms of 
justification the students began to tentatively di scuss numerical patterns they observed. 
Moana had participated in discussions in the study group of general isations students may 
use but she had given them li ttle attention at the initial phases of the study. Now, the need 
to provide multiple levels of explanatory j ustification led to increased student recognition 
and voicing of numerical patterns. Moana began to use these with the students, often as 
position statements used to explore and extend numerical connections and patterns. In the 
vignette Moana extends discussion of a student-voiced observation to press the students to 
explore and examine patterns they observed in fractional numbers. 
Shifting reasoning from justifying to pattern-seeking and exploring 
The students in groups have discussed and explored the statement Aporo made "the bigger 
the denominator the smal ler the bit". Moana began the large group discussion by 
positioning Aporo and his group to validate their conjecture. 
Aporo [uses two segmented l ines with their fraction equivalents recorded as 
symbols] Because that number is big [ 1 3/ 1 3] and this number is l i ttle [5/5] 
and you can tel l  the pieces because the five ones are bigger and the thirteen 
ones are smaller. 
Moana Questions? What have you got there on your .  . .  what is your fraction there 
Wire mu? 
Wiremu/Pita Four out of four, four quarters. 
Moana Okay. In comparison to five out of five, five fifths which bit is bigger? But  
what happens if  you have thirteen thirteenths? Is  what Aporo said true? Pita 
you j ust said 'different' why? Why? Why do you think it was different? 
1 85 
Rona 
Moan a 
Rona 
Aporo 
Moan a 
Rona 
I know why [points at segmented li nes and notation for 5/5 and 4/4] .  That 
one [4/4] is a bit smal ler that that one [5/5] .  
S o  what are you saying Rona? 
Like that' s . . .  just a l ittle number [4/4] and this is a kind of a little one [5/5 ] .  
But that one ' s  got l ike the bigger piece [4/4] and this one has got the l itt le 
pieces [5/5 ] .  
Because you have got to cut this into five pieces and you have to cut that 
into four pieces so smal ler number, bigger piece. 
So if I had a chocolate bar and I said to you that you can have five fifths or 
thirteen thirteenths or twenty twentieths . . .  
That wil l  be the same. 
Beau They are all the same, because it' s just smaller but sti l l  one whole piece. 
Moana So how does that work? Can anyone see the pattern? Is there any ru le we 
can use? 
Beau Yeah. If the top and bottom are the same then you just have one whole 
doesn ' t  matter what they are if they are the same. 
(Term 3 Week 8) 
Pattern seeking and exploration, a collective zpd, importance of validating conjectures 
Evident in the classroom observational data is how a shift from explaining to justifying 
increased the community l i stener-ship. Moana acknowledged the value of student 
contribution to progress col lective reasoning noting after the lesson: this session was 
interesting because it put me in the position of a sponge board or sponge board/spring 
board. I am revoicing, not in a negative or condescending way, what they are making sense 
of I am not taking their words and changing them, but adding to a shared understanding 
for them and me. It helps me view things from their perspective. Lerman (200 1 )  describes 
how increased participation in discourse and reasoning practices pulls all participants i nto a 
shared zpd. This was evident i n  this classroom. 
Moana ' s  press on the students to validate conjectures scaffolded potential development of 
more generalised reasoning. Blanton and Kaput (2003, 2005) and Carpenter and his  
col leagues (2004b) note the importance of teachers using student validation of conjectures 
as a tool to mediate generali sed reasoning. 
1 86 
7.4.6 USING MATHEMATICAL LANGUAGE 
Although the students had developed an increased repertoire of questions to inquire and 
chal lenge they often sti l l  used short utterances and informal colloquial language to explain 
or respond to questions. We agreed that they needed richer ways to share their reasoning 
but Moana voiced concern that a push toward more extended responses might cause loss of 
confidence and withdrawal from participating. A research article7 mediated Moana ' s  next 
shifts as she used the teacher' s actions in the article to map out her next steps. She 
increased her explicit models of mathematical talk, actively participating and describing the 
mathematical actions using informal , then formal descriptions . She l istened carefu l ly  to 
their explanations and then revoiced and extended what they said using multiple layers of 
meaning. For example, after a student described cutti ng a chocolate bar into: six bits Moana 
responded with: yes six bits, sixths and six of them, six equivalent pieces all the same size, 
six sixths of the one whole chocolate. Then she asked : is there anyone else who can model 
another equivalent fraction ? Good Rona for taking a risk like this. Just go ahead and 
construct another fraction which is the same, equivalent. Gradual ly Moana' s phrasing of 
questions and responses were appropriated. The students used Moana' s models, often 
rephrasing and using terms and concepts she had previously introduced. 
Increasing and extending fluency in mathematical discourse 
The discourse and communication patterns had been appropriate for their former si tuated 
classroom context (Gee & Clinton ,  2000; Moschkovich, 2003 ; Nasir et al . ,  2006) .  Moana 
mode led her actions on Khisty and Chval ' s (2002) description of teachers who inducted 
students i nto more fluent forms of mathematical talk through use of specific models of rich 
multiple layers of mathematical words and statements. Moana' s actions were designed 
intentionally to shift students from using colloquial talk which had been accepted 
previously when the answer was the focus  but which now l imited how they explained and 
justified their reasoning. Meaney and Irwin (2003)  describe how informal and imprecise 
7 Khisty, L. L., & Chval, K. B. (2002) .  Pedagogic discourse and equity in mathematics:  
When teachers' talk matters. Mathematics Education Research Journal, 14(3) ,  1 54- 1 68 .  
1 87 
use of a mathematical di scourse eventual ly restricts students' mathematical reasonmg. 
Simi larly, Latu (2005) emphasises the importance of extending student understanding of 
mathematical concepts beyond the exact context in which they are learnt. Moana 
recognised student growth of communicative competence (McCrone, 2005) when she 
informal ly commented: when we come to maths it 's like these little antenna go up. They go 
right we are in maths what 's the language and they start to think about the language, the 
strategies, like they always talk about what strategies are you using or why. Like when I 
was watching them working in a group the other day and one of them said 'just prove that 
then ' and it was said so naturally, just part of what they say all the time. 
7.4.7 SUMMARY OF THE SECOND PHASE OF THE STUDY 
Threaded through this research study is the influence of Moana' s own past experiences and 
beliefs she had constructed about doing and learni ng mathematics. Her observations of the 
positive outcomes which emerged for her students were key factors which convinced her to 
continue the shift in communication and participation patterns towards inquiry and 
argumentation. My collaborative support, study group activity, the communication and 
participation framework, research articles and video observations supported each tentative 
step. These occurred in conjunction with Moana' s careful analysis of their effect on the 
students' self-esteem and the growing confidence in doing and using mathematics. 
Influencing how Moana scaffolded interactions were the presence of multiple voices, past 
and future, but also a present ' new voice' Moana had constructed as she guided 
development of the sociocultural norms of an i nquiry community. 
Moana's many pedagogical actions to scaffold student engagement in more proficient 
mathematical practices parallel those described by other researchers who studied teachers 
working with diverse learners (e.g. ,  Boaler, 2006b ;  Khisty & Chval, 2002; White, 2003) .  
Through these careful ly considered actions Moana laid the foundations for a community of 
inquiry. Increasingly, the students participated in  exploratory talk (Mercer, 2000) as they 
explained and justified their reasoning in classroom interaction patterns which had begun to 
resemble more closely what Wood and McNeal ( 2003)  defined as an i nquiry culture. The 
1 88 
students had increased agency within a more balanced intel lectual partnership (Amit & 
Fried, 2005).  
7.5 TAKING OWNERSHIP OF MATHEMATICAL PRACTICES IN 
A COMMUNITY OF MATHEMATICAL INQUIRY 
The gradual development of a community of mathematical inquiry had been a long, steady, 
change-process. At this point Moana noted her facil i tative role and that the students: expect 
everybody to make sense of what they are saying . . .  they ask lots more questions all the time 
of each other, they just expect that they have to justify what they are thinking and that they 
can use words and other ideas to back up what they are saying. Participation in 
communication of mathematical reasoning had become an integral part of what it meant to 
'know and do' mathematics in the classroom community. Moana outl ined to me that she 
considered being able to participate in mathematical di scourse a fundamental right of her 
students. She metaphorical ly li nked the way in which Maori are privi leged when they know 
how and when to speak on a Marae to her faci l i tative role in mathematics lessons : they all 
have the right to have that privilege. I am making sure that these children all know that 
they have got the right to talk and be heard in maths. Moana wanted to sustain  the press on 
mathematical i nquiry and argumentation but also ensure ownership of it was vested in  the 
contributions and reasoning of all community members. 
7.5 . 1  USING MODELS OF CULTURAL CONTEXTS TO SCAFFOLD STUDENT 
ENGAGEMENT IN INQUIRY AND ARGUMENTATION 
Whilst maintain ing an expectation that individual students engage actively i n  i nquiry and 
argumentation Moana directed attention to their responsibility to each other. She re-vi sited 
notions of the whanau (family) and emphasised the strengths inherent in being a member. 
She made direct l inks to the students' famil y  context, for example emphasising that family 
members take different  roles to make a Cook Is land haircutting ceremony successful .  When 
she observed groups working together she drew attention to their col laborative actions, 
paral lel ing these with the actions of a Kapa Haka group (Maori cultural group) who i nduct, 
support and challenge members unti l  they ach ieve simi lar levels of expertise. She drew 
attention to similarities in the role of the tuakana (elder brother or sister or cousin in a 
1 89 
whanau), l inking these to specific i ndividuals she saw taking leadership and actively 
supporting group members to challenge thinking or to promote development of col lective 
reasoning. This was i l lustrated when Moan a told the class : There 's really interesting korero 
(talk) going on. I really spent most of my time with this group because they were having 
problems and arguments and Wiremu was really good . . .  you were really good in that 
position Wiremu, you were helping your group and you weren 't giving out the answers and 
that 's really good but you were pushing them to think. Yes you had everyone talking about 
and discussing how they were going to sort out the ideas. You were challenging and other 
people were following your lead so the arguing was really kapai (good). 
The students appropriated Moana' s view of col lective responsibi l ity. They drew attention to 
their rights or responsibil ities and readily stated their lack of understanding or need for 
support. They expected other group members to actively l isten and engage in the reasoning 
and when explaining reasoning they would stop, wait, and then ask for questions or 
chal lenge. If other group members appeared inattentive they challenged their behaviour as 
i llustrated when a student saw another scribble on the recording sheet, she said: don 't, man, 
you listen and ask or you aren 't even learning, man. On another occasion when a student 
responded to a conjecture with a disputational comment another student responded with: 
don 't dis her, man, when she is taking a risk. Their recognition of their collective 
responsibi l i ty extended to their provision of explanatory justification in the larger sharing 
group. This is i l lustrated in  the fol lowing vignette when a col lective explanation is provided 
by group members stepping in and out to support each other. They then provide 
justification for their selection of the strategy solution explai ning that it was the one they al l 
understood and could explain .  
Collective provision of explanatory justification 
Anaru explains a solution strategy for 36 + 47 as her group clusters around tracking closely 
and quietly discussing her actions. She draws a numberli ne, records 47,  draws a l ine from 
the 47 to 57, records + 1 0  above it, draws a l ine from 57 to 77, records +20. 
Hone 
Anaru 
Wire mu 
What are you doing? 
Oh. I am adding . .  . I  am adding up my numbers. 
How is yours different from Mahine 's?  
1 90 
Anaru 
Beau 
Anaru 
Wiremu 
Anaru 
Moan a 
Anaru 
Hemi 
Different numbers . 
What different numbers? 
Like the 20 and the 1 0. 
Why have you only got one ten ?  
[points at the 20] There are two tens. 
[steps in to clarify further] What did you do with the first ten? 
I added . . .  oh . . .  I added it. 
[a group member extends the explanation] She added it to the 57. Plus 
another 20 were the two tens. That gave her 77. 
Anaru records +3, 80 on the line, completes the solution recording +3 and 83 as group 
members Jim and Hone verbal ise her actions. 
Jim 
Hone 
Donald 
Anaru 
Jim 
Eighty . . .  
Then you have got three so eighty three. 
Why did you want to do it that way? 
Because it was the easiest way. Yes for me and for Alicia to understand. 
[poi nts at a previous explanation] Because it was a bit of a shorter way from 
there too. 
(Term 4 Week 3) 
Inquiry culture, ethnic socialisation, the role of groups 
From the on-going interview data and reflective notes Moana made, I i nterpreted that she 
had constructed a c learer view of inquiry c lassrooms and what the learning in them looked 
l ike . Observational data reveals that she confidently drew on her situated knowledge of Te 
reo and Tikanga Maori and used her understandi ng of the "ethnic socialisation" 
(MacFarlane, 2004, p. 30) of her students . Her knowledge of their background supported 
their understanding  of the changes she enacted in the interaction patterns. Within the 
sociocultural perspective of this study, groups were used as i nstructional agents. Moana 
guided the group members' appropriation and use of mathematical understandings of those 
more knowledgeable to advance collective understanding and access to the mathematical 
discourse. Within the Maori and Pacifi ca dimension Moana drew on, she positioned 
students to consider the more knowledgeable students not as individuals but rather as a 
knowledge component of a whanau (a collective) .  In turn, the students responded with an 
increased sense of interdependence as whanau members . 
1 9 1  
7.5.2 FURTHER DEVELOPING STUDENT AGENCY OF THE 
MATHEMATICAL DISCOURSE 
Moana observed the students' growth in col lective responsibility but she noted that in her 
presence the students behaved less autonomously and sti l l  looked to her to lead. Whilst  she 
had addressed the former pattern of interjecti ng by requiring that students put their hands 
up Moana recognised that the students needed to contribute more to the management of the 
flow of discussion. After our analysis and discussion of a video record of a classroom 
observation she stated: opportunities here exist for children to ask and dispute and so I 
need to let the children guide their own questioning and discussion more. Moana 
introduced the use of koosh bal ls8 to scaffold student management of the inquiry and 
debate. These were placed in the middle of the discussion circle; the students picked them 
up indicating a question or challenge. The following vignette i l lustrates how the koosh bal l 
mediated the discourse, slowing discussion down and making questioners more 
accountable. Although picking up the ball indicated a question or challenge, the self esteem 
of the explainer was protected and they were provided with cognitive space to take time to 
think, and then respond . 
Using the koosh ball as a tool to facilitate explanatory justification 
Hone explains  as he notates a col lective addition strategy for 236, 2 1 9, 22 1 ,  and 2 1 4 :  
Hone 
Aroha 
Hone 
Aroha 
Moan a 
Hone 
We added 236 and 2 1 4 which equals 450. 
[picks up a koosh ball then points at 2 1 9  and 22 1 ]  Why didn ' t  you add those 
two? 
Because it tidies up these numbers . Because there is a six and four which 
equals ten. 
But why didn ' t  you use the other ones? 
[steps in to press for further clarification and justification] But why didn ' t  
you just add the 236  and 2 1 9? 
Because then it wil l  be harder. . .  because i t  won ' t  . . .  l ike not equal . . .  it won ' t  
l ike be tidied up. 
(Term 4 Week 3) 
8 A soft ball that d i d  not roll but which fitted into the students' hand 
1 92 
Intellectual partnerships, student agency and autonomy, teacher facilitator 
The students were provided with a predictable structure to engage in inquiry and 
argumentation and the introduction of the koosh ball i ncreased their agency to question and 
inquire. An intel lectual partnership had been established and Moana noted the faci l itative 
role she now held as she analysed a video record: I am more guiding and refining the 
explanatory process . . .  or making the understanding available to everyone. I am identifying 
children who still have misconceptions, or those who are not fully convinced, or those who 
know but want to know why. Martin et al. (2005) i l lustrated that both the culture and 
structure of classroom communities positively influences student beliefs and attitudes 
toward mathematics and provides diverse "students with opportunities to gain deeper 
mathematical dispositions" (p. 1 9) .  
Discussion driven by student talk and contribution positioned Moana as much a participant 
in the discussion as she was the faci l i tator of it. Moana was drawn into what Lerman (200 1 )  
describes as a "symbolic space" (p. 1 03 ). Within this zpd, she too was required to engage 
with and work to understand the perspectives of all other participants . She acknowledged 
her role when she recorded the following reflection after a lesson : Making sense of the 
process as much as they are. Sometimes my questions are to support me making sense as 
much as they are to show the kids what sorts of questions they need to be asking to make 
sense. In further discussion Moana explained that she needed to question to sense-make and 
then she used the students' ideas to progress col lective understandings. As Martin and 
col leagues (2005) describe "the centrality of students' voices provides teachers with 
opportunities to develop and modify their instruction" (p . 1 9) .  
Interaction patterns had shifted and the predominant form of  talk was exploratory (Mercer, 
2000). Mathematical discussions were extended as the students negotiated and renegotiated 
their reasoning to develop a shared perspective founded i n  the variable contributions of 
different members . At t imes, their partial understandings caused dissension, doubt and 
confusion but they now persistently explored, explai ned, and argued until all members 
understood and were convinced. These findings are simi lar to those reported by a number 
of researchers (e.g. ,  Azmitia & Crowley, 200 1 ;  Brown & Renshaw, 2004; Goos et a l . ,  
1 93 
1 999; Goos, 2004) when students of equal levels of understanding work collaboratively in 
zones of proximal development. 
7.5.3 PROBLEM SOLVING AND A SHIFT TOWARDS GENERALISING 
Moana introduced the use of a series of problems specifical ly constructed to support the 
students making connections from day to day and across problem situations .  Previously 
Moana had careful ly  controlled how the students worked with problems but our discussion 
of video records of c lassroom observations prompted her to reconsider; now she gave the 
problems directly to the groups and asked them to conjecture solution strategies and then 
use their 'rethink time' to pattern-seek. To support them, Moana modeled questions they 
could use to compare, evaluate, and make connections between the conjectures. She told 
the students: in your group there are going to be different strategies and different ways and 
you are going to pool all your combined collective knowledge to solve these problems. You 
need to be sharing the knowledge. You need to be asking questions. Questions like why did 
you use that number, why did you do that, what strategy are you using, is it more efficient 
than that one, which is the most efficient way, is my way more efficient than yours, why is 
it, why isn 't it, is that the easiest way to understand it? 
Moana' s requirement that the students examine and compare solution strategies provided 
them with opportunities to construct mathematical relationships-relationships which 
extended mathematical reasoning beyond the context of the immediate problem. A problem 
solution became applicable across problems as Moana worked with the students to identify 
and connect patterns and regularities. They were pressed to connect with their prior 
understandings and use this to construct new ways of reasoning. For example, after an 
explanation and before the next one Moana told the students: what is happening here is that 
you are comparing the two strategies in your head. Okay? Then you might be able to try 
those strategies out. Or add them to the knowledge that you already have, the maths 
knowledge that you already have. So that 's why this is really importan t, because you are 
really learning from each other . . .  you just add to your existing base of knowledge. Think 
about those strategies. Think about your strategy and how you would apply them . . .  and 
which one you liked . . .  and which one you thought worked best . . .  which one you understand. 
1 94 
Not only were students pressed to make connections to their own understanding, Moana 
also required them to analyse, compare, and justify differences in the efficiency or 
sophistication of solution strategies . Through the shift Moana enacted, the students were 
positioned to analyse and compare solution strategies and justify their stance. The vignette 
i l l ustrates how Moana' s press led to pattern seeking and provision of explanations which 
included generalised number properties. 
Validating mathematical reasoning autonomously 
Aroha explains a solution strategy for adding of 43, 23, 1 3 , 3. She records 43, 23, 1 3 , 3 and 
then 3 x 4 = 1 2  
Aroha 
Kea 
Aroha 
Donald 
Hone 
Donald 
I am adding forty three, twenty three, thirteen and three, so three times four 
equals twelve. 
Why are you trying to do that with those numbers? Where did you get the 
four? 
[points at the 3 digit on the four numbers] These threes, the four threes.  
[another member of the group] All  she is doing i s  l ike making it shorter by 
l ike doing four times three. 
[the third group member] Because there are only the tens left. 
Three times four equals twelve and she got that off all the threes like the 
forty three, twenty three, thirteen and three. So she is just l ike adding the 
threes al l up and that equals twelve. 
After an alternative explanation is provided Moana posi tions the students to compare 
strategy solutions. 
Moan a 
Faalinga 
Moan a 
Danny 
Moana 
Anaru 
Did they use the same strategy as Anaru? 
I don ' t  think so. 
You don' t  think so? So they haven ' t  used the same strategy? [Moana 
observes that Danny nodded his head when she asked if they had used the 
same strategy so she turns to him] Argue with them Danny.  
They have used the same strategy because they have both added the ten and 
the twenty and the forty and they have both added the threes together. 
Wel l  done you guys. They al l  added the threes together. You prove that you 
did the same thing Anaru . You show what you did with the threes. 
[draws an array of four sets of three] Three plus three plus three plus three. 
[points at each set of three] Or four times three or three times four [points 
across the top of the array and down the side of the array] . 
(Term 4 Week 4) 
1 95 
Rethink time as an example space, interthinking 
The classroom community was now one in which the students had learnt to suspend 
judgment in order to examine and explore a range of possible solution strategy steps in  
what Watson and Mason (2005 ) term 'example ' spaces. The problem contexts and the 
careful attention paid to identi fying relationships, regularities, and patterns, beyond the 
immediate surface features of problem situations were tools which mediated a shift towards 
general ising. Moreover, as they analysed and di scussed the conjectures their use of 
exploratory talk (Mercer, 2000) was evident in the words they used (because, why, so, but 
and forms of ei ther agreement or disagreement). Mercer outl ines the importance of these 
words if students are interthinking within zones of proximal development. 
7.5.4 JUSTIFYING EXPLANATORY REASONING THROUGH INSCRIPTIONS 
Moana used the communication and participation framework as a reflective tool to analyse 
student engagement in proficient mathematical practices. In our discussion she outli ned 
how the communication and participation framework and the questions and prompts 
framework developed in the study group : helps if I am going to plan a focus. I say this is 
the mathematical practice I am going to focus on. So what do I need to do to help the 
children engage better in it ? What prompts do I need to use and what prompts could I help 
them use because I look on myself as a total facilitator, facilitating them engaging in 
mathematical practices. She then described the limited attention she attributed previously 
to student notation and how she planned an explicit focus on them in this phase. 
Moana establ ished a requirement that the students i l lustrate their reasoning using a range of 
symbolic schemes, i nc luding i nformal notational schemes, diagrams and pictures. She 
modeled the use of different notational schemes, notating as students explained, or asking 
students to specifical ly  justify their reasoning using their i nscriptions . As a result, the 
fol lowing vignette i l lustrates how the symbol ic schemes became sense-making tools which 
mediated students' analysis, j ustification and validation of their reasoni ng. 
1 96 
Notating to validate mathematical reasoning and clarifying terms 
Donald is explaining a solution strategy for 63 - 26 using an empty number-l ine 
Donald 
Mahine 
Donald 
Caliph 
Mahine 
Donald 
Mahine 
Donald 
Mahine 
Donald 
First it  was 26 so I took away 20 and that equals 43. 
Hang on . Can you hear what you are saying? 26 take away 20? 
63.  I mean . 63 take away 20 equals 43. Take away 3 .  That equals 40? 
You tidied that up then? 
[points at the beginning of the numberl i ne] Why did you start there? I t ' s  take 
away? 
It doesn ' t  matter if I add up or subtract but if i t ' s  easier for you to see I wi l l  
take away, so take away twenty, take away three. 
But why three now? 
Because it i s  easier and it takes you to the nearest ten. Do you understand 
that way so far? Okay then I take the last three so 37. Got it? If you don' t  
wel l  just ask.  
So where did the . . .  oh yes I see the other three was from the s ix .  
Because I was subtracting twenty six so first it  was the twenty then the 
three . . .  
Caliph/Mahine [ in unison add] And then the other three. 
(Term 4 Week 3) 
Inscriptions as public reasoning tools, informal notation schemes 
The explicit focus placed on inscriptions pressed the students to examine and analyse 
whether their i nformal notation scheme represented their reasoning. During this  dialogue 
explanatory steps were en larged or modified and other notation forms devised in 
preparation for provision of explanatory justification as the inscriptions became valued as 
public reasoning tools .  Sawyer (2006) i l lustrated that different representational forms 
students use are important in that they make mathematical reasoning public and accessible 
to an audience. As Cobb (2002) has described previously, in Moana' s classroom the 
obligation to make available multiple explanations of reasoning influenced how students 
learnt and used representations. C larification of explanations and the need to further j ustify 
were powerful reasons to re-present and recast notation schemes. In addition, many of the 
invented notation schemes close ly  approximated more standard procedures while all the 
time retaining a conceptual sense. 
1 97 
7.5.5 SUMMARY OF THE THIRD PHASE OF THE STUDY 
The enacted participation and communication patterns had established an i nquiry 
community which supported student engagement in a range of mathematical practices. 
Learning and using mathematical practices developed steadily as the students gradually 
gained access to productive mathematical discourse. Inquiry and the need for multiple 
forms of justification led to increased focus on the need for engagement in generalising and 
notating reasoning. 
All the roles i n  the classroom had shifted significantly .  Moana outl ined the view she held of 
her role toward the end of the research explaining: I had that big debate; are you a teacher, 
are you afacilitator, but now I see that as long as they are learning I don 't care what I am. 
My interpretation was that she had confidence in the intel lectual community as a site which 
provided the students with many opportunities for participation in  rich learning situations. 
She had constructed a new set of bel iefs about doing and learning mathematics which 
valued mathematical communication and mutual engagement in collective reasoning. 
Moana had drawn on her own knowledge of the ethnic socialisation of her students to 
accommodate the social and cultural practices the students were accustomed to in  their 
homes and communities, and those being constituted in the c lassroom context. 
7.6 SUMMARY 
This chapter has mapped out the gradual, often circuitous and challenging journey Moana 
and her students made as the foundations of a community of mathematical i nquiry were 
constructed. How the use of the communication and partic ipation framework, the 
framework of questions and prompts, research articles, video records, study group activities 
and on-going discussion with me mediated a shift in  beliefs and transformation of the 
sociocultural and mathematical norms of the c lassroom are i l lustrated. 
The i nitial c lassroom culture was described as a conventional one. Shifting the interaction 
patterns towards a strategy reporting one (Wood & McNeal .  2003)  caused many confl icts in  
Moana' s and her students' beliefs about doing and using mathematics . A new set of roles 
1 98 
and scripts were needed for all participants. The communication and participation 
framework was reconstructed as small, incremental steps and used to provide foundations 
for a collaborative community. Constructing the norms of a strategy reporti ng classroom 
community took a full three months as the students learnt to actively engage in explaining, 
questioning and mathematically sense-making. Student abi lity to make conceptual 
explanations provided an important foundation from which other mathematical practices 
were learnt and used more proficiently.  Questioning and challenge were important factors 
in deepening conceptual understanding and extending the col lective reasoned di scourse. 
Moana used specific pedagogical practices to i nduct the Maori and Pasifika students and 
girls to actively participate in col lective reasoning. In turn, these students' positive response 
to changes in the classroom context and mathematical activity prompted shifts in Moana' s 
bel iefs and supported her continued enactment of changes in the interaction patterns .  
Evidence was provided of  the multiple voices (Forman & Ansel l ,  200 1 )  which operated i n  
the classrooms and which shaped the carefully measured pedagogical actions Moana took 
to press student engagement in inquiry and argumentation. At the conclusion of the study 
there had been significant changes in student autonomy and agency and this held potential 
for continued growth in the efficient use of interrelated mathematical practices. Moana had 
established her own voice and i n  doing so had constructed an intellectual community which 
placed value on the central ity of student voice. 
The fol lowing chapter draws together this chapter and Chapter Six. The different pathways 
the two teachers took on their journeys to develop mathematical i nquiry communities are 
discussed and the similarities and differences in the pedagogical practices they used are 
elaborated on. The contributions this research has made to the research field and the 
l imitations and implications of this study are examined. A concluding section confirms that 
communities of mathematical inquiry can be constructed and these provide opportunit ies 
for students to partic ipate in rich collective reasoning practices. 
1 99 
CHAPTER EIGHT 
CONCLUSIONS AND IMPLICATIONS 
As students engage in classroom practices, and in mathematical practices, they develop 
knowledge and they develop a relationship with that knowledge. Their mathematical 
identity includes the knowledge they possess as well as the ways in which students hold the 
knowledge, the ways in which they use the knowledge and the accompanying mathematical 
beliefs and work practices that interact with their knowing. (Boater, 2003b, p. 1 6- 1 7) 
8.1 INTRODUCTION 
The intention of this thesis was to examine how teachers construct communities of 
mathematical i nquiry in which the participants collectively engage in the use of rich 
mathematical practices. The l iterature review examined the differential outcomes which 
result from the different patterns of participation and communication in mathematics 
classrooms. An important thread maintained in the review of the l i terature was the 
significance of constructing i ntel lectual learning communities in which students learn to 
participate in ,  and use, reasoned mathematical actions and di scourse. 
The complexities and chal lenges teachers encounter in developing col lective inquiry and 
argumentation within the classroom context was considered. As a result, the use of design 
research to simultaneously support and examine the pedagogical actions teachers take to 
engage students in proficient mathematical practices in communities of inquiry was 
proposed. A key element of the design research was a communication and participation 
framework which was used by the teachers to map out possible pathways they could take to 
construct the interaction patterns of i nquiry communities. 
Detailed descriptions were presented of the pedagogical actions two case study teachers 
took to constitute the sociocultural and mathematical norms which supported student 
engagement in reasoned communal mathematical i nquiry and argumentation. Whi lst the 
transformation of communication and participation patterns was a gradual process, the 
200 
impact on student engagement in mathematical practices was significant. Significant 
changes were revealed in both classrooms as the teachers enacted progressive shifts in the 
sociocultural and mathematical norms which validated collective mqmry and 
argumentation as learning tools. Higher levels of student involvement in the mathematical 
dialogue resulted in increased intel lectual agency and higher intel lectual levels of 
verbalised reasoni ng. 
This research focused on one key question: 
How do teachers develop a community of mathematical inquiry that supports 
student use of effective mathematical practices ? 
Section 8.2 summarises the different pathways the two teachers took to construct 
mathematical i nquiry communities. Key features of the constitution of the sociocultural and 
mathematical norms and how these resulted i n  the emergence and evolution of 
mathematical practices are explained. Section 8 .3 describes the pedagogical practices the 
teachers used to support their students' participation i n  collective reasoned communication 
within i nquiry communities. Section 8 .4 examines and describes the cri tical features of the 
tools which mediated the teacher development of i nquiry communities and its discourse. 
Section 8 .5 presents the contributions this research has made to the research field. I n  
Section 8 .6 and 8.7 the l imitations and implications of  this study are examined alongside 
suggestions for further research. Section 8 . 8  provides a final conclusion to this research . 
8.2 THE PATHWAYS TO DEVELOPING COMMUNITIES OF 
MATHEMATICAL INQUIRY 
This research documented the journey two teachers took as  they developed their classroom 
participation and communication patterns as part of their goal to establish student 
engagement in effective mathematical practices in c lassroom inquiry communities. Within 
the research study both teachers participated in the same core professional development 
activities and u sed simi lar tools to mediate change but their development of mathematical 
20 1 
learning communities took different pathways. These may be attributed to the differing 
intermediary goal points on their trajectories and the "dis-coordination and resulting 
conflicts" (Brown & Cole, 2002, p .  230) the teachers experienced. 
The way in which each teacher considered the discip line of mathematics was of central 
importance to how they instituted and maintained changes in their pedagogical practices 
and their students' learni ng practices. At the start of the research Ava readi ly adopted a 
view of mathematics as a discipline of "humanistic enquiry, rather than of certainty and 
objective truth" (Goos et al . ,  2004, p. 1 1 2) .  Her classroom culture reflected small shifts 
towards i nquiry. This provided her with a foundation to further develop and refine a 
learning context founded in mathematical i nquiry and argumentation. Moana, in contrast, 
viewed mathematics as a discipline of absolute certainty rather than humanistic inquiry. 
The conventional learning climate she had constructed supported her use of prescriptive 
teaching practices, her central position of authority, and a predominant use of univocal 
discourse. As a result, changing the classroom context and learning a new set of roles and 
responsibil ities for a mathematical inquiry community presented Moana and her students 
with many contradictions and conflicts. 
It was evident that the role the teachers adopted in the classroom communities shaped their 
response to initiating changes in the communication and participation patterns .  At the 
beginning of the study Ava readily repositioned herself as a participant i n  the discourse. 
She modeled the process of inquiry, sense-making and self-monitoring, and emphasised the 
community's shared authorship of the mathematical reasoning. Moana' s pathway to 
establishing a learning partnership with her students required significant reshaping of 
expectations and obligations in the mathematics community and took a lengthy period of 
time to real ise. To achieve the shift Moana drew on her social and cultural understandings 
of a whanau (fami ly  grouping). She emphasised the students' responsibility for their own 
mathematical learning but also the need for col laborative support and responsibil ity for the 
learning of others as well as respectful l istening and assertive communication. The 
teachers' observations that changes in the learning culture resulted i n  positive learning 
202 
outcomes for students, were powerful tools  which reshaped their subsequent 
understandings of teaching and learning mathematics. 
The constitution of a safe supportive learn ing environment-one which promoted 
inte llectual risk-taki ng-was a critical component in the formation of the inquiry 
communities. Both teachers used a range of strategies to attend to their students ' affective 
needs, including direct di scussion of the need for col legiality and inclusion, risk-taking, and 
the repositioning of themselves and their students as risk-takers . The existi ng classroom 
culture within Ava's classroom supported her reconstitution of the interactional norms 
through direct discussion of her expectations and the students' obligations. Moana engaged 
in a lengthier process, in i tially needing to address some of her students ' attitudes and 
behaviour during mathematical activi ty. She did thi s  by closely engineering learning 
partnerships, specifical ly  placing the Maori , Pasifika and female students in supportive 
pairs . When using larger groups she monitored the interactions and engendered learning 
competence through specific posi tioning of the shy, low achieving students, and the girls .  
She used her knowledge of the students' ethnic sociali sation to provide models of the 
sociocultural norms she wanted enacted. 
It was evident in both classrooms that an emphasis on independence, not dependence, was a 
key feature in  the students' growth in mathematical agency. 
8.2.1 SCAFFOLDING STUDENT COMMUNICATION AND PARTICIPATION 
IN MATHEMATICAL PRACTICES 
Within the classrooms, communal construction and examination of mathematical 
explanations were an important precursor for supporti ng  the development of explanatory 
j ustification and general isation. The level of student engagement in productive discourse 
was the key factor which shifted the focus of mathematical reasoning past mathematical 
explanations to student communication and participation in many interrelated mathematical 
practices. 
203 
Initial ly,  the most common form of talk used in  both classrooms were those which Mercer 
(2000) terms cumulative or di sputational. To change these unproductive forms of talk Ava 
immediately scaffolded coll aborative small group construction and examination of 
mathematical explanations. In a larger group setting, she facilitated questioning and 
clarification of conceptual explanations, providing 'think-time' so that explanations and 
errors became reflective tools .  Moana took a more gradual route with her in itial focus 
concerned with addressing students' negative interjections and persi stent attention to 
answers rather than sense-making. Her use of ground rules for talk establ ished clear 
boundaries. Through smal l ,  i ncremental steps she gradual ly established active li stening,  
questioning, explaining and rethinking mathematical reasoning. The deliberate focus both 
teachers placed on student analysis of their reasoning and the reasoning of others provided 
the foundations for developing the discourse of inquiry and argumentation. 
It was evident in this study that enacting the norms which supported inquiry and 
argumentation caused all participants on-going conflicts and contradictions .  The teachers 
acknowledged their own novice status in a mathematics environment which used inquiry 
and argumentation. They also expressed concern at what they perceived to be a lack of fit 
between the cultural and social norms of their students and the requirement that they 
engage in the mathematical discourse of inquiry and argumentation. The li terature (e .g. ,  
Andriessen, 2006; Mercer, 2000; Wells, 1 999) recognises that many students hold contrary 
views on argumentation and the teachers expressed s imi lar views. They also recognised 
that mathematical i nquiry and argumentation as a specifi c  speech genre was not currently 
part of their students' repertoire of cul tural practices and therefore developing it required 
careful attention. Of significance in achieving this was the explicit attention they gave to 
discussing and exploring with their students' their attitudes towards mathematical 
argumentation .  
Almost immediately, us ing the communication and participation framework to map out a 
pathway, Ava scaffolded the foundations for what Wood and McNeal (20030 term an 
i nquiry or argument culture .  She scaffolded the use of questions and prompts and asked that 
the students construct multiple explanations, examine these closely, and rehearse possible 
204 
responses to questions or challenge. She pressed her students-revoicing and positioning 
them to take a stance to agree or disagree-but required that they mathematically j ustify 
their position. She provided models of mathematicians' use of 'maths arguing' as a tool to 
progress col lective reasoning. These actions shaped the form and content of the discourse 
the students used to justify and validate their reasoning and provided them with a 
predictable framework for i nquiry and argumentation. 
In contrast, Moana maintained a lengthy focus on developing mathematical explanations 
and in her classroom the interaction patterns steadily shifted from a conventional classroom 
culture to what Wood and McNeal (2003) term a strategy reporting c lassroom. When 
Moana observed evidence of student  abi lity to construct, explain ,  clarify and elaborate on 
their i ndividual and col lective mathematical reasoning she increased her expectation that 
they take a stance, explain and justify agreement or disagreement. Drawing on her 
observations of their growth in mathematical confidence and increased use of productive 
mathematical discourse, she further scaffolded the use of inquiry, all the time considering 
the students' home and school social and cultural contexts. The focus of questions shifted 
from questioning for additional information or clarification, to questioning for justification .  
This gradual shift took more than half the school year before the classroom culture could be 
described as an inquiry or argument context. However, i n  the final research phase it was 
evident that both Ava and Moana and their students readily accepted the need for extended 
mathematical discourse and exploratory talk. The students used mathematical i nquiry and 
argumentation, during dialogic discourse, to examine and explore the perspectives of 
others' and ultimately achieve consensus. 
In this study, clear evidence is provided of the difficulties both teachers had attending to 
and developing general i sed reasoning in  their c lassrooms. It seemed that neither had 
previously abstracted the fundamental numerical patterns and structures of numbers or 
operational rules, nor had they considered exploring these with their students. I n  the first 
phase of the study the teachers often did not appear to 'hear' student voiced intuitive 
generali sations. However, an i ncrease in inquiry and argumentation supported them 
attending to, and building on, the students' observations of patterns and relationships. The 
205 
earlier use of inquiry and argumentation i n  Ava's classroom explains the earlier shift to the 
examination and use of general ised reasoning and why it was only in the latter stages of the 
research that this occurred in Moana' s classroom. Although the increase in shared 
classroom talk prompted the teachers to afford explicit attention to developing generalised 
reasoning in the classroom communities other tools were also of importance. These 
included student provision of multiple ways to justify and val idate reasoning, the use of 
numerical patterns to validate reasoning, the use of a specific set of questions and prompts 
for generalisations, position statements, the use of open-ended problems and a requirement 
that the students analyse and compare solution strategies for efficiency and sophistication. 
Importantly, the students' i ncreased participation m mathematical reasoning at higher 
intel lectual levels was the prompt which caused the teachers to continue to press for i nquiry 
and argumentation. In turn, their increased expectations provided the students with a 
p latform to develop explanatory justification, general ised reasoning, the construction of a 
range of inscriptions to validate the reasoning, and a more defined use of mathematical 
language. Clearly apparent in this study were the differential outcomes which emerged as 
the frequency and complexity in the questioning and chal lenge used by the teachers and 
students i ncreased. Higher levels of complexity in articulated reasoning were achieved 
earlier in Ava' s classroom but both c lassroom communities at the conclusion of the 
research readily used interaction patterns most often premised i n  i nquiry and argument and 
the use of exploratory talk. 
This section has described the pedagogical practices the case study teachers took to 
develop productive mathematical discourse which supported the development of 
mathematical explanations, explanatory justification, and generalised reasoning. However, 
it would be reasonable to suggest that the more difficult task which confronted the teachers 
was knowing which actions not to take-that is reducing cognitive chal lenge when 
introduci ng problems or doing the mathematical reasoning and talking for the students. 
Allowing the students' time to wrestle with confusion and erroneous thinking in what 
appeared messy and inefficient ways before attaining sense-making was chal lenging. Their 
complex role required 'on the spot' decision-making of pedagogical practices which best 
206 
facil i tated cognitive and social opportunities in which the students come to know and use 
mathematical practices-practices which focus on not only "the learning of mathematics, 
but the doing of mathematics-the actions in which users of mathematics (as learners and 
problem solvers) engage" (Boaler, 2003b, p. 1 6). 
8.3 SUPPORTING STUDENTS TO BECOME MEMBERS OF 
COMMUNITIES MATHEMATICAL INQUIRY 
It was evident in this study that constituting mathematical inquiry communities conflicted 
with how the students had previously viewed mathematics-beliefs formed through their 
previous experiences in more prescriptive learning environments. The two case study 
teachers used a range of pedagogical roles and practices to shift the sociocultural and 
mathematical norms so that their students came to know and do mathematics as a process 
of inquiry and argumentation. The extended time and the pedagogical roles the teachers 
assumed to enact sociocultural norms which supported shared constitution of mathematical 
norms were important .  Of significance were the changes in their pedagogical roles from a 
predominant use of univocal discourse to one in  which they facilitated dialogic discourse . 
This  supported a shift in the students' role as new identities were created . As the teachers 
adopted a range of varying roles their students gradual ly assumed mathematical agency 
within classroom communities premised on i ntel lectual partnerships. 
Important actions the teachers took to develop their students' mathematical agency 
included the expectation for the students to take ownership for communal responsibi l ity for 
sense-making during mathematical activity. Activities, for example included the 
expectation of a group recording of the reasoning, and a flexible approach to pairing and 
grouping with consideration for the social and cultural context of the students. The Maori 
and Pasifika dimensions of the students were drawn on to establish key aspects of the 
whanau (family and collective) concept. These included assertive communication, the value 
of diversity and multiple perspectives, valuing effort over abi lity, assigning competence to 
individuals, and positioning the more knowledgeable as valued knowledge sources within 
the col lective. 
207 
The explicit framing of classroom interaction patterns so that al l  participants engage in the 
reasoning activity supported student ' interthinki ng' within shared communicative space. 
Scaffolding to consider mathematical reasoning within multiple perspectives occuJTed 
through the guidance offered by the teachers. This included requiring the groups to consider 
questions they might be asked, or sections of their explanations others might find difficult; 
and developing multiple ways to explain,  elaborate, justify, and val idate mathematical 
reasoning. Through the need to engage with group members' different perspectives, 
variable contributions, and partial understandings, multiple zones of proximal development 
evolved . In turn, in order to negotiate shared perspectives the students encountered 
mathematical situations which required transactive exploration and speculation?
mathematical activity which often closely approximated those used by competent users of 
mathematics. 
The teachers used many tools to mediate student communication and participation m 
mathematical activity. Of particular consequence was the provision of a predictable 
framework for strategy reporting, i nquiry and argument. The teachers directly modeled 
ways to explain and justify mathematical reasoning. They ensured a gradual shift in the use 
of specific questions and prompts for the different mathematical practices so that increased 
levels of intellectual reasoning resulted. Consistent teacher revoicing, reshaping, and 
extending student use of informal terms and concepts whi le maintaining focus on the 
mathematical content, provided the students with access to a mathematical register and to 
knowledge of how to participate in mathematical discourse. The teachers ensured that the 
students developed understanding of inscriptions as sense-making tools to explain ,  
e laborate, and validate mathematical arguments through direct modeling. Other tool s  
i ncluded the problematic and  open-ended tasks used during mathematical activity. A 
fundamental feature in the design of the problems focused on key mathematical content 
while at the same time emphasising enactment of specific  participation and communication 
goals .  Tasks were frequently supported with direct guidance to the students on how they 
were to engage in and communicate their mathematical reasoning. 
208 
Evident in thi s research is the significance of sustained t ime which supported student 
learning of key reasoning practices. Provision of 'time and space ' was apparent i n  
mathematics lessons in which examination of mathematical reasoning extended across both 
a lesson to lessons and a problem to problems. Partial understandings and misconceptions 
common to the student group were identified and extensively explored and as a result errors 
became valued tools used within the community to grapple with complex ideas and develop 
deeper conceptual understandings. Opportunities for sustained examination supported 
analysis and exploration of commonalities and differences and identification of 
sophistication or proficiency in solution strategies. 'Extended time and space' was also a 
tool the teachers used to mediate student appropriation of the communication and 
participation patterns of inquiry. They explicitly facili tated ' time and space' to support the 
students to explore, experiment with, examjne, extend and innovate on models of the 
questions and prompts. Both teachers readi ly accepted and provided opportunities for the 
students to copy, practi se, try out, and experiment with ways to use discipline specific 
dialogue. S imi larly, they ensured space for the students to use, extend, and innovate on the 
inscriptions. 
8.4 SUPPORTING TEACHERS TO CONSTRUCT COMMUNITIES 
OF MATHEMATICAL INQUIRY 
In the teachers ' construction of communities of mathematical inquiry there was no single 
unique factor which supported them although some influences proved particularly effective. 
The communication and participation framework (see 5 .4. 1 )  was a significant tool which 
threaded through every stage of the project. Initially, it  was used as a tool to map out shared 
possible pathways the teachers could u se to transform the communication and participation 
patterns in their c lassrooms. As the project progressed it served as a reflective tool ,  used by 
the teachers to analyse the development of i nquiry communities and plan their next focus. It 
was a flexible scaffold which over the duration of the study supported backward and 
forward movement, the renegotiation of contexts and the development of different or more 
detailed pathways for i ndividual teachers. 
209 
The framework of questions and prompts for mathematical practices (see 5 .4. 1 )  was an 
additional tool which augmented the communication and participation framework. The 
framework focused the teachers' attention on the need to consider how the questions and 
prompts they and their students used, shaped mathematical activity and dialogue. It served 
as a more explicit reflective tool to help their analysis of c lassroom interactions and plan 
their next focus. It shaped their viewing of classroom interactions on video records and 
provided the teachers with important evidence that higher levels of complexity in questions 
and chal lenge were matched with higher levels of articulated mathematical reasoning. 
Reali sation of the positi ve student mathematical learning outcomes was a powerful factor in 
the shift of teachers' pedagogical practices. The video classroom records and transcripts 
provided c lear evidence of how shifts in classroom interaction patterns advanced the 
students' learning and social goals .  In the study group context maintaining a focus on 
identifying and analysing the communication and participation patterns, and questions and 
prompts, assisted identification of how these actions supported student engagement in 
mathematical practices, the emerging use of different mathematical practices, and the 
actions the teacher took to enhance their student 's  engagement in more proficient 
mathematical practices. 
As i l lustrated in the previous section, of prime importance in the construction of c lassroom 
communities of mathematical i nquiry were the opportun ities the teachers gave the students 
to learn how to participate in collective inquiry and argumentation. Simi larly for the 
teachers, participating in dialogic i nquiry was a key factor in the transformation of their 
beliefs and attitudes towards knowing and doing mathematics within communities of 
mathematical i nquiry. The teachers engaged in dialogic i nquiry on many levels i n  a range 
of contexts, i ncluding formal staff meetings, study group meeti ngs, and less formal 
discussion with their colleagues and me. Research articles, a DVD, and videos which 
i l lustrated aspects of i nquiry classrooms and their sociocul tural and mathematical norms 
were powerful tools. These provided opportunities for the teachers to reflect on the habitual 
patterns of interaction they used in their mathematics lessons .  They suggested possibilities 
for change and formed the basis for objective discussions .  
2 1 0  
It was evident in this study that for the teachers the study group was an important learning 
site.  Within this context the teachers developed deeper collegial relationships as they eo?
constructed through lengthy conversations the reshaping of their c lassroom mathematical 
communities. Although the primary focus of this group was the negotiation and 
renegotiation of the communication and participation patterns towards inquiry, the study 
group was responsive to other needs. These included extensive discussions of the tensions 
and contradictions the teachers encountered as they scaffolded i nquiry and argumentation 
in their c lassrooms and exploration of solutions to these . The teachers used the study group 
context to review and reflect on their own history as learners and users of mathematics and 
reconcile thi s with the changes they were implementing. Another activity which evolved as 
an on-going need was the collaborative construction and exploration of mathematical 
problems. These problems gave the teachers' opportunities to examine and explore the 
small pieces of mathematical knowledge, used in mathematical activity, in detai l .  
Examination of  the problems also supported the teachers t o  anticipate the erroneous 
thinlGng which might emerge, or the possible strategy solutions, as well as the patterns and 
relationships inherent in the problems . The communally constructed problems were valued 
tools, used in the classrooms to advance both the mathematical content agenda and the 
social agenda. 
In mediating change, ' time and space' was as important for the teachers as it  was for their 
students.  Time supported each teacher to develop their own pathway and approach in  
changing the interaction patterns in their c lassrooms. Space provided them with 
opportunities to draw on their own situated knowledge, reflect on it, and engage in dialogic 
i nquiry about it with their peers and the researcher. Through talGng ' time and space' they 
were able to construct their own perspective of what i nquiry communities and the 
mathematical discourse and mathematical activities i n  them might look l ike. Moreover, i t  
provided the room for exploration,  while sustaining teacher agency. 
2 1 1 
8.5 LIMITATIONS 
While the research contributes new knowledge to the discipline at a variety of levels, any 
research has its l imitations. The results of this research are based on empirical analysis of a 
small sample of teachers and students, in one school, i n  one urban area of a city. Given the 
small sample the generalisability of the findings for teachers in the context of different 
c lassroom setti ngs in New Zealand may be l imi ted. However, the explicit outl i ne of the 
participation and communication framework, the framework of questions and prompts for 
mathematical practices, and the clear descriptions of the teachers' pedagogical practices, 
allows others to trial a simi lar study. 
Because of the complex nature of schools and classroom practices, i nterpretation of the 
results in this study can only provide an emerging understanding of the pedagogical 
practices teachers use to enact the communication and participation patterns of i nquiry 
communities. Although triangulation methods were used, consideration needs to be given 
to the possibility of bias in the results of this research. The presented findings are based on 
one researcher' s i nterpretation of data from audio records and notes from study group 
meetings, classroom video records, interviews, teacher reflections and field notes. Other 
interpretations are possible, although the interpretations are strengthened by the use of a 
wide range of data sources, the use of a grounded approach in the search for confirming and 
disconfirming evidence, and the prolonged engagement with the teachers as eo-researchers. 
Consideration needs to be given to the impact of the research process on the school and 
classrooms. Included in the impact is the level of i ntrusion the presence of a researcher/ 
university lecturer caused in the school and classrooms. Thought also needs to be given to 
the intrusive effect the v ideo recording had on the classroom context and the behaviour of 
al l participants. This issue-the way in which the presence of another person and video 
recording in classrooms influences change-was discussed earlier in Chapter Five and the 
steps to minimise the disturbance explained. These included the col laborative relationship 
the researcher established with the teachers, the role of the researcher as participant 
2 1 2  
observer, and the discussion and practice runs which famil iarised the students to the use of 
v ideo capture before the start of the study. 
Although there was evidence of student resistance and conflict as the interaction patterns 
shifted, this was not a focus of the study. Nor did the research design encompass 
exploration or examination of individual student ' s mathematical learning, or include 
exploration of the i ndividual student' s views and attitudes. The parents' and other 
community members' views were also not examined. These are key factors which are 
regularly considered in  conjunction with community of learners approaches. Furthermore, 
despite the extensive time in the school the research did not col lect data related to 
continued development of mathematical communities of i nquiry, or the use and refi nement 
of mathematical practices within the community. It is acknowledged that the processes 
i nvolved in teachers constituting communities of mathematical inquiry are on-going. 
8.6 IMPLICATIONS AND FURTHER RESEARCH 
Possibilities for different ways of thinking about teachi ng and learning mathematics in New 
Zealand primary classrooms are suggested by this study. Evidence within this study 
suggests that it  is the classroom teacher who makes possible the development of the 
mathematical discourse of i nquiry in learning communities .  This study appears to be the 
first of its kind in New Zealand in that it  simultaneously supports teachers to constitute 
intellectual learning communities in mathematics classrooms and at the same time explores 
the pedagogical practices teachers use to develop them. It is important to extend this base of 
knowledge beyond the urban, low deci le, primary school in this study. Understanding needs 
to be extended to how mathematical i nquiry communities can be developed in other age 
and year levels, types of schools, deciles, locations, and with different ethnic groupings.  
I t  was evident i n  this study that the teachers held mathematical content and pedagogical 
knowledge grounded in more traditional forms of mathematics teaching and learning. 
Initially, they lacked experience in ,  and understanding of the pedagogical knowledge 
required to enact and maintain the shared discourse of mathematical i nquiry and 
2 1 3  
argumentation, communal learning partnerships, small group interaction patterns ,  the 
development of exploratory talk, and the sociocultural and mathematical norms of i nquiry 
communities. Additional research needs to incorporate understandings developed i n  this 
study of the pedagogical practices teachers use to constitute the mathematical di scourse of 
mathematical i nquiry communities. In particular, it  also needs to extend the examination of 
the ways teachers directly scaffold exploratory talk. The design component and the 
communication and participation framework which mapped out shifts in the interaction 
patterns in this study need to be further tri al led with different teachers in different contexts. 
Although teacher attitudes and beliefs were not intended as a focus of this study, their effect 
on the constitution of the mathematical discourse of inquiry and argumentation became a 
significant factor. Further research i s  needed in New Zealand to explore how the past 
experiences of teachers in learning and using mathematics i nfluences their current 
pedagogical practices. In particular, exploration is required of the specific attitudes and 
bel iefs teachers hold towards the value and use of mathematical inquiry and argumentation. 
Specific interventions like those used in this study need further trial l ing to analyse how, 
when, and which factors are significant in changing attitudes. 
Within this study the different pathways the teachers took to construct inquiry communities 
resulted in part from their situated knowledge of the ethnic socialisation of their students. 
Further research is required which draws on Maori and Pasifika dimensions to explore 
optimal means to scaffold Maori and Pasifika students' participation in inquiry and 
argumentation . 
The professional development of the teachers was not a focus of this study, but the study 
group activities in which the teachers participated had an important influence on the 
outcome of the research .  Important factors included the year-long sustained focus, access to 
relevant research material, regular objective collegial discussions of video records, 
evaluation of the interaction patterns ,  analysis of the questions and prompts on the video 
records, the examination of mathematical content and the experience of being in a 
community of learners. Simi larly, through directi ng  attention on the examination and 
2 14 
analysis of critical incidents of students engagi ng in and using mathematical practices on  
the video records the teachers developed deeper understanding of  the nature of  
mathematical practices. This finding suggests implications for the ways in which the 
professional development of teachers rrtight be enhanced. 
Within this study the findings provide clear evidence of the positive outcomes for student 
reasoning when students engage in interrelated social practices. However, it was evident i n  
this research that initial ly  the teachers consistently focused on  teaching mathematical 
content knowledge (or specific solution strategies) and they demonstrated little awareness 
of how mathematical practices could support the learning. It was only when direct attention 
was focused on scaffolding the emergence and use of mathematical practices that the 
teachers began to understand and explore what they were and recognise their value. The 
communication and participation framework and the development of the framework of 
questions and prompts were significant in progressing the teachers' understanding of 
mathematical practices as individual and interrelated social practices. Both tools need to be 
further trialled with a range of different teachers, schools, ethnic groups and across 
different age groups to grow the knowledge base of how teachers can be supported to 
develop richer understandings of how to construct and maintain communities of 
mathematical inquiry. 
8.7 CONCLUDING WORDS 
The intention of this research was to explore and exarrtine how teachers developed their 
mathematical c lassrooms to embrace i nquiry communities in which the students come to 
know and use mathematical practices within reasoned communal dialogue. Mathematical 
practices in this study were shown to be col lective and interrelated social practices which 
emerged and evolved within the classroom mathematical discourse. The study revealed 
findings that add to the research knowledge of the relationship between patterns of 
participation and communication in mathematics classroom and differential outcomes for 
student engagement in mathematical practices. 
2 1 5  
The rich data generated in this collaborative design study provided evidence that 
communities of mathematical inquiry can be constructed despite complex and difficult 
challenges. In particular, the research reported in this study resonates with and extends the 
current body of li terature that seeks to understand the many factors involved in the 
construction of such classroom communities. It documents in detai l a range of pedagogical 
practices which support the constitution of mathematical inquiry communities, the resultant 
changes and effects of shifts in communication and participation patterns on the 
participants ' roles and attitudes towards doing and knowing mathematics, and the 
mathematical practices which emerge and are used. It has extended understanding of these 
practices as initiation into social practices and meaning, within a New Zealand context and 
in primary (elementary) school classrooms. 
Of particular importance for New Zealand teachers i s  that the findings of this research 
provide possible models for ways teachers can draw on and use their Maori and Pasifika 
students ' ethnic socialisation to consti tute mathematical inquiry communities which align 
with the indigenous patterns of learning of these students. It i s  in that sense that the findings 
might act as a springboard for significant educational change. 
2 1 6  
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243 
APPENDICES 
APPENDIX A :  TEACHER INFORMATION SHEET AND CONSENT FORM 
My name is Bobbie Hunter and I am currently doing Doctoral research focusing on 
understanding mathematical practices and how those practices are developed by teachers 
and learned by students. I am writing to invite you to participate in collaborative research 
during Terms 2, 3 and 4 of this year. At the end of last year (2003) the management staff of 
Tumeke School expressed an interest to take part in the Numeracy Practices and Change 
Project funding by the Ministry of Education Teaching and Learning Research Initiative. 
The research on Mathematical Practices which I hope to undertake in your school is part of 
this larger project which aims to investigate equitable effects of numeracy and factors 
associated with sustained reform numeracy practices. 
Mathematical practices refers to those activities and modes of thinking that successful 
mathematical learners and users actually do. To investigate the development and 
enhancement of effective mathematical practices this study wil l  involve a group of teachers 
from your school in a professional development programme directly li nked to your own 
classroom. 
All Year 4-8 teachers at the school who participated in the New Zealand Numeracy Project 
wi ll be invited to participate in the study. The professional development programme wil l  
involve you as eo-researchers trial l ing activities, and evaluating your teaching in relation to 
mathematical practices. A particu lar focus wil l  be on the communicative and interactive 
nature of the learning environment. To faci l i tate this inquiry you wi l l  be involved in 
reflective practices such as classroom observations (sometimes with audio/video records) 
and journal writing. Permission with regard to audio/video recording wi l l  be sought from 
both parents and chi ldren in  your class. The focus of the study is on the teaching strategies, 
and thus it wi l l  be possible to organize the recording devices to avoid those students who 
do not consent to participate. 
The project is comprised of four phases: 
1 .  Term 1 :  Professional development wil l  focus on the New Zealand Numeracy 
Framework and Teaching Model . 
2 .  Term 2 :  Over a five week period teachers in each of the senior syndicates wil l  
work within a collaborative partnership to develop and trial strategies re lated to 
student mathematical practices. 
3 .  Term 3 :  Bui lding on the work in  Term 2, a second Numeracy unit wil l  be  trialed 
focusing on key numeracy ideas and communication patterns. I wi l l  case study 3 
teachers (self-identified) as they use an explanatory framework to examine their 
own and each others mathematical teaching practices related to mathematical 
practices. Video, audio and journal material wil l  be used in reflective observations .  
4. Term 4: The teachers and researcher as  a group wi l l  refine the explanatory 
framework and evaluate its usefulness as a professional development tool for 
teachers. In addition, we wil l  reflect and report on the process of teacher change 
and changes in student mathematical practices and learn ing. 
244 
The time involved in the professional development meetings for you will  be no more than 
five school days out of classrooms. The proposed professional development i s  in accord 
with the school strategic plan for numeracy focus. Your teacher release time wil l  be fu l ly 
funded by the Teaching and Learning Research Initiative Project: Numeracy Practices and 
Change. 
All project data will be stored in a secure location, with no public access and used only for 
this research and any publications arising from this research .  After completion of five years, 
all data pertaining to this study wil l  be destroyed in a secure manner. All efforts wi l l  be 
taken to maximize confidentiality and anonymity for participants. The school name and 
names of all participants will be assigned pseudonyms to maintain their anonymity. Near 
the end of the study a summary wil l  be presented to you to verify accuracy, and fol lowing 
any necessary adjustments, a final summary wi l l  be provided to the school and teachers 
i nvolved. 
Please note you have the fol lowing rights in response to my request for you to participate in 
this study. 
? decline to participate ; 
? decline to answer any particular question; 
? withdraw from the study at any point; 
? ask any questions about the study at any time during participation; 
? provide information on  the understanding that your name will not be used unless 
you give permission to the researcher; 
? be given access to a summary of the project findings when it is concluded . 
If you have further questions about this project you are welcome to discuss them with me 
personally :  
Bobbie Hunter: Massey University (Albany), Department of Technology, Science and 
Mathematics Education. Phone: (09) 4 1 40800 Extension 9873. Emai l .  
R.Hunter@massey.ac .nz; or contact either of my supervisors (eo-directors of the Numeracy 
Practices and Change Project) at Massey University (Palmerston North), Col lege of 
Education 
Associate Professor Glenda Anthony: Department of Technology, Science and 
Mathematics Education. Phone: (06) 350 5799 Extension 8600. Emai l .  
G.J .Anthony@massey.ac .nz 
Dr. Margaret Walshaw: Col lege of Education . Department of Technology, Science and 
Mathematics Education. Phone: (06) 350 5799 Extension 8782. Emai l .  
M.A.Walshaw@massey.ac .nz 
This project has been reviewed and approved by the M assey Universi ty Human Ethics 
Committee, ALB Protocol NO/NO (insert protocol number). If you have any concerns 
about the conduct of this research,  please contact Associate Professor Kerry P Chamberlain, 
Chair, Massey University Campus Human Ethics Committee : A lbany, telephone 09 443 
9700 x9078, emai l K.Chamberlain@massey.ac .nz .  
245 
CONSENT FORM: TEACHER PARTICIPANTS 
THIS CONSENT FORM WILL BE HELD FOR A PERIOD OF FIVE (5) YEARS 
I have read the Information Sheet and have had the detai ls  of the study explained to me. 
My questions have been answered to my satisfaction, and I understand that I may ask 
further questions at any time. 
I agree to participate in this study under the conditions set out in the Information Sheet. 
I understand that professional development will involve discussion of my own and other 
teacher' s classroom practice and I agree to keep descriptions of specific classroom epi sodes 
confidential. 
Signature: Date: 
Full Name - printed 
246 
APPENDIX B :  STUDENT INFORMATION SHEET AND CONSENT FORM 
My name is Bobbie Hunter and I am currently doing Doctoral research focusi ng on 
understanding how mathematical practices are developed by your teacher and learned by 
you . Your teacher is one of several teachers in the school taking part in this research study 
during Terms 2 and 3 of thi s year. 
As part of the research we wil l  need to make some classroom observations and therefore I 
am writing to ask your permission for you to be audio or video recorded as part of your 
teacher' s record of their practice. The focus of the recordings will be on your teachers ' 
teaching strategies and so at no time wil l  you be focused on or audio or video recorded for 
any length of time. The recordings would be of usual mathematics lessons and so you 
would not need to do anything special for the cameras or tape-recorder. In addition, your 
teacher may want to take a copy of some of your written work to help with their recordings. 
All data recordings will be stored in  a secure location, with no public access and used only 
for th is  research. In order to maintain anonymity the school name and name of a l l  
participants wil l  be assigned pseudonyms in any publications arising from this research.  At 
the end of the year, a summary of the study wil l  be provided to the school and made 
available for you to read. 
Please note you have the following rights in response to my request for you to participate i n  
this study. 
? decline to participate; 
? decline to answer any particular question; 
? withdraw from the study at any point; 
? ask any questions about the study at any time during participation; 
? provide information on the understanding that your name wi l l  not be used unless 
you give permission to the researcher; 
? be given access to a summary of the project findings when it i s  concluded ; 
? have the right to ask for the audio/video tape to be turned off at any time during the 
observations ;  
? have the right to not allow copies of your written work to be taken. 
If  you have further questions about this project you are welcome to discuss them with me 
personal ly :  
Bobbie Hunter: Massey University. Albany. College of Education .  Department of 
Technology, Science and Mathematics Education. Phone: (09) 4 1 40800 Extension 9873.  
Emai l .  R.Hunter@massey.ac .nz 
Or contact either of my supervi sors : 
? Associate Professor Glenda Anthony:  
College of Education. Department of 
Education. Phone: (06) 350 
G.J .Anthony@ massey.ac .nz 
247 
Massey University. Palmerston North .  
Technology, Science and Mathematics 
5799 Extension 8600. Email .  
? Dr. Margaret Walshaw: Massey University. Palmerston North .  Department of 
Technology, Science and Mathematics Education. Phone: (06) 350 5799 Extension 
8782. Emai l .  M.A.Wal shaw@massev .ac . nz  
This  project has been reviewed and approved by  the Massey University Human Ethics 
Committee, ALB Protocol NO/NO (insert protocol number). If you have any concerns 
about the conduct of this research, please contact Associate Professor Kerry P Chamberlain, 
Chair, Massey University Campus Human Ethics Committee: Albany ,  telephone 09 443 
9700 x9078, emai l K.Chamberlain@massey.ac .nz .  
CONSENT FORM: STUDENT PARTICIPANTS 
THIS CONSENT FORM WILL BE HELD FOR A PERIOD OF FIVE (5) YEARS 
I have read the Information Sheet and have had the detai ls of the study explained to me. 
My questions have been answered to my sati sfaction, and I understand that I may ask 
further questions at any time. 
I agree/do not agree to be audio taped during mathematical lessons. 
I agree/do not agree to be video taped during mathematical lessons. 
I agree to copies of my written work being col lected. 
I agree to participate in this study under the conditions set out in the Information Sheet. 
Signature: Date: 
Full Name - printed 
248 
APPENDIX C:  BOARD OF TRUSTEES INFORMATION SHEET AND CONSENT 
FORM 
To the Chairperson 
Board of Trustees 
Tumeke School 
Dear Sir/Madam 
My name is Bobbie Hunter and I am currently doing Doctoral research focusing on 
understanding mathematical practices and how those practices are developed by teachers 
and learned by students. I am writi ng to request permission to undertake col laborative 
research with teachers in your school during Terms 2, 3 and 4 of this year. At the end of 
last year (2003) the management staff of Tumeke School expressed an interest to take part 
in the Numeracy Practices and Change Project funding by the Ministry of Education 
Teaching and Learning Research Initiative. The research on Mathematical Practices which 
I hope to undertake in your school is part of this larger project which aims to investigate 
equitable effects of numeracy and factors associated with sustained reform numeracy 
practices. 
Mathematical practices refers to those activities and modes of thin!Ung that successful 
mathematical learners and users actually do. To investigate the development and 
enhancement of effective mathematical practices this study wil l  involve a group of teachers 
in a professional development programme directly li nked to the teachers' own c lassrooms. 
All Year 4-8 teachers at the school who participated in the New Zealand Numeracy Project 
will be invited to participate in the study. The professional development programme wi l l  
involve teachers a s  eo-researchers trial l ing activities, and evaluating their teaching in  
relation to mathematical practices. A particular focus wi l l  be  on  the communicative and 
interactive nature of the learning environment.  To faci l itate this i nquiry teachers wi l l  reflect 
on teaching practices through observations (sometimes with audio/video records) and 
journal wri ting. Permission with regard to audio/video recording wil l  be sought from both 
parents and children i n  each teacher' s class. The focus of the study is on teaching strategies, 
and thus it wi l l  be possible to organise the recording devices to avoid those students who do 
not consent to participate. 
The project is comprised of four phases: 
5. Term 1 :  Professional development wi l l  focus on the New Zealand Numeracy 
Framework and Teaching Model .  
6. Term 2 :  Over a five week period teachers i n  each of the senior syndicates wil l  
work within a col laborative partnership to develop and trial strategies related to 
student mathematical practices. 
7. Term 3 :  Building on the work in Term 2, a second Numeracy unit wi l l  be trialed 
focusing on key numeracy ideas and communication patterns .  I wi l l  case study 3 
teachers (self-identified) as they use an explanatory framework to examine their 
249 
own and each others mathematical teaching practices related to mathematical 
practices .  Video, audio and journal material wi l l  be used in reflective observations. 
8 .  Term 4 :  The teachers and researcher as a group wil l  refine the explanatory 
framework and evaluate its usefu lness as a professional development tool for 
teachers. In addition, we wil l  reflect and report on the process of teacher change 
and changes in student mathematical practices and learning. 
The time involved in the professional development meetings for teacher participants wil l  be 
no more than five school days out of c lassrooms. The proposed professional development is 
in accord with the school strategic plan for numeracy focus. Teacher release time wil l  be 
ful ly  funded by the Teaching and Learning Research Initiative Project: Numeracy 
Practices and Change. 
All project data wil l  be stored in a secure location, with no public access and used only for 
this research and any publ ications arising from this research. After completion of five years, 
all data pertaining to this study wil l  be destroyed in a secure manner. All efforts wil l  be 
taken to maximize confidentiality and anonymity for participants. The school name and 
names of all participants wil l  be assigned pseudonyms to maintain their anonymity. Near 
the end of the study a summary wil l  be presented to the teachers to verify accuracy, and 
fol lowing any necessary adjustments, a final summary will be provided to the school and 
teachers involved. 
Please note you have the fol lowing rights in response to my request for your school to 
participate in this study. 
? decl ine to participate; 
? withdraw from the study at any point; 
? ask any questions about the study at any time during participation ; 
? provide information on the understanding that the participants' names wi l l  not be 
used unless you give permission to the researcher; 
? be given access to a summary of the project findings when i t  is concluded. 
If you have further questions about this project you are welcome to discuss them with me 
personally:  
Bobbie Hunter: Massey University (Albany) ,  Department of Technology, Science and 
Mathematics Education. Phone: (09) 4 1 40800 Extension 9873. Emai l .  
R.Hunter@massey.ac .nz ;  or  contact either of my supervisors (eo-directors of the Numeracy 
Practices and Change Project) at Massey University (Palmerston North) 
? Associate Professor Glenda Anthony: Department of Technology, Science and 
Mathematics Education . Phone :  (06) 350 5799 Extension 8600. Emai l .  
G .J .Anthon y@ massey.ac .nz 
? Dr. Margaret Walshaw: Department of Technology, Science and Mathematics 
Education. Phone: (06) 350 5799 Extension 8782. Emai l .  
M.A.  Walshaw@ massey.ac .nz 
250 
This  project has been reviewed and approved by the Massey University Human Ethics 
Committee, ALB Protocol NO/NO (insert protocol number). If  you have any concerns 
about the conduct of this research, please contact Associate Professor Kerry P 
Chamberlain, Chair, Massey University Campus Human Ethics Committee : Albany, 
telephone 09 443 9700 x9078,  email K.Chamberlain@massey.ac .nz. 
CONSENT FORM: BOARD OF TRUSTEES 
THIS CONSENT FORM WILL BE  HELD FOR A PERIOD OF FIVE (5) YEARS 
I have read the Information Sheet and have had the detai ls of the study explained to me. 
My questions have been answered to my satisfaction, and I understand that I may ask 
further questions at any time. 
I agree to participate i n  this study under the conditions set out in the Information Sheet.  
Signature: Date: 
Full Name - printed 
25 1 
APPENDIX D:  INFORMATION SHEET FOR PARENTS AND CAREGIVERS 
My name is  Bobbie Hunter and I am currently doing Doctoral research focusing on 
understanding mathematical practices and how those practices are developed by teachers 
and learned by students. The management staff of Tumeke School has agreed to take part in 
the Numeracy Practices and Change Project funding by the Ministry of Education 
Teaching and Learning Research Initi ative. The research on Mathematical Practices, which 
is taking part in your school,  is part of this larger project. 
To investigate the development and enhancement of effective mathematical practices this 
study wil l  involve your  child ' s  teacher working with a group of teachers at Don Buck 
Primary in a professional development programme directly linked to their own classrooms. 
The focus of the study is on the teaching strategies that support c lassroom communication 
and mathematical practices-those activities and modes of thinking that successful 
mathematical learners and users actual ly do. 
I am writi ng to formally request your permission for your chi ld to be audio or video 
recorded as part of their teacher' s record of practice. Your child ' s  involvement wil l  be no 
more than that which occurs in normal dai ly mathematics lessons. The video or audio 
recording would occur during Terms 2 and 3 during mathematic teaching involving 
numeracy work only. Sometimes a whole maths lesson wil l  be recorded, but more usual ly 
the teacher would only want to record a part of the lesson. In al l ,  t he teacher would make a 
maximum of 8 recordings (it may be necessary to have a few trials recordings so that 
students get used to the video equipment). In addition, the teachers may want to take copies 
of student ' s  written work to assi st their recordings. The teacher would review the 
recordings as part of the professional development programme, and sometimes parts of the 
recording would be shared with the other teachers working in the study as they col lectively 
develop and critique their teaching strategies.  
All data recordings wil l  be stored in  a secure location, with no public access and used only 
for this research. In order to maintain anonymity the school name and names of al l  
partic ipants wil l  be assigned pseudonyms in any publications ari sing from thi s research .  At 
the end of the year, a summary of the study will be provided to the school and made 
avai lable for you to read. 
Please note you have the fol lowing rights tn response to my request for your chi ld to 
participate in this study: 
? decl ine your chi ld ' s  participation ;  
? withdraw your child from the study at any point; 
? you may ask any questions about the study at any time during your child ' s  
participation; 
? your child provides i nformation on the understanding that your chi ld 's  name wi l l  
not  be used un less you give permission to the researcher; 
? be given access to a summary of the project findings when it i s  concluded; 
? decl ine your chi ld being video recorded; 
? dec l ine your chi ld being audio recorded; 
? decl ine to allow copies of your chi ld ' s  written material to be taken.  
252 
If you have further questions about this project you are welcome to discuss them with me 
personally: 
Bobbie Hunter: Massey Universi ty (Aibany), Department of Technology, Science and 
Mathematics Education. Phone: (09) 4 1 40800 Extension 9873. Emai l .  
R.Hunter@massey.ac .nz ;  or contact either of my supervisors (eo-directors of the Numeracy 
Practices and Change Project) at Massey University (Palmerston North), College of 
Education: 
? Associate Professor Glenda Anthony: Department of Technology, Science and 
Mathematics Education. Phone: (06) 350 5799 Extension 8600. Emai l .  
G.J .Anthonv@  massey.ac .nz 
? Dr. Margaret Walshaw: Department of Technology, Science and Mathematics  
Education. Phone: (06) 350 5799 Extension 8782. Emai l .  
M.A.Walshaw@massev .ac .nz 
This project has been reviewed and approved by the Massey University Human Ethics 
Committee, ALB Protocol NO/NO (insert protocol number). If you have any concerns 
about the conduct of this research, please contact Associate Professor Kerry P Chamberlain ,  
Chair, Massey University Campus Human Ethics Committee: Albany, telephone 09 443 
9700 x9078, emai l K.Chamberlain @massey.ac .nz. 
CONSENT FORM: PARENTS OF STUDENT PARTICIPANTS 
THIS CONSENT FORM WILL BE HELD FOR A PERIOD OF FIVE (5) YEARS 
I have read the Information Sheet and have had the detai l s  of the study explained to me. 
My questions have been answered to my satisfaction, and I understand that I may ask 
further questions at any t ime. 
I agree/do not agree to my child being audio taped during mathematical lessons. 
I agree/do not agree to my chi ld being video taped during mathematical lessons. 
I agree/do not agree to copies of my chi ld' s  written material being collected. 
I agree to my child participating in  this study under the conditions set out i n  the 
Information Sheet. 
Signature: Date: 
Full Name - printed 
253 
APPENDIX E: THE FRAMEWORK OF QUESTIONS AND PROMPTS 
Teacher_questions and prompts Student questions and prompts 
I ' m  confused? Would you tell us again what I ' m  confused can you explai n that bit. 
you thought? 
Can you use a different way to explain that? 
Okay we have got some con fused faces, any 
questions about what they d i d? I don ' t  get that, what do you mean by . . .  ? 
"' c 0 I don ' t  understand that bit can you explain it to Can I just j u mp in here and ask . . .  ?-= ? us? Explain what you did? c ? When we went. . .  what happened? c. 
? What did you do there . . .  in that bit of your ? 
-; 
:::1 
explanation? What do you mean by . . .  ? 
-
Q. 
? Does this make sense? Are t here bits that don ' t  Why did you say . . .  ? c.> c make sense? You l isteners n eed to ask questions 0 c.> 
about those bits so that they can clarify them for I don ' t  get i t .  . .  could you draw a picture of what ... ? you are think ing? ? you. (j 
? 
..:;: How did you decide this? Whose thinking did ? 
? you use to build on? 
Now that you have heard their explanation can 
you explain their strategy? 
Can you show us what you mean by . . .  ? 
But how do you know it works? Why did you . . .  
Show us . . .  Convince us . . .  So what happens if. . .  ? 
Can you persuade them that perhaps what they Are you sure it's . . .  ? 
have come up with might not be the best 
solution? Can you prove that? c 0 
?-= Can you convince them that they might need to You need to show that every bit works. eo: ? 
!5 rethink their solution? ..... But how do you know it works l i ke that. . .  "' 
::::1 . ..., Why wou Id that tell you to . . .  ? 
...... What about if you say . . .  does that sti l l  work? ""' 0 ..... Can we explore that further because I don ' t  eo: c think we are all convinced yet. Can you convince us that . . .  eo: 
-a 
? Can you j ustify why you d i d  that? Can you draw a picture of that to prove what Q.l 
? you are saying is true? c 
? Why does that work l i ke that? 
eo: What about if we . . .  can that work? ? 
So what happens if you go like that? 
So if we . . . .  
Do you agree? Do you d i s agree? Remember that 
you can agree or disagree but you must be able 
to explain mathematical l y  why. 
Is that simi lar to what you were thinking? How? 
254 
Can you see any patterns? Can you make Can you/we check this another way? 
connections between . . .  ? 
So why is it .  . .  ? 
That is a really interesting observation . .  I 
wonder if we can find out how these patterns are Is that the q uickest way? What is a quicker 
'IJ related? way? = 
.s - Remember how . . .  Think about what we were But how are you going to . . .  ? ? -? doing the other day. ? I. But if you . . .  QJ = Why? Why would you do that? What is QJ eJl 
happening? What are the patterns you can see? Because if you . . .  eJl = ? Anybody do it a different way? Is there a Is there a different way we can do that? ? 
E different way you can do it? How is this the same or d i fferent to what w e  did ? = before? ? 
'IJ How are you going to find a quicker way? = 0 In what ways is this different from our last ?.:: " Can you generalise that? Can you l i n k  a l l  the solution strategy? QJ = ideas you found in some overall way? = 0 Let ' s  work out if we can show how it always " 
? Does it al ways work? Does it work for all o f  works. = ? them? 'IJ = If you say that the . . .  I. QJ - Can you explain the difference in . . .  ? -? c. What happens if . . .  ? 
I. 
.8 Is it always true? Why does this happen? 
eJl Can we do this another way? Does that work? = 
:.;2 Can you give us another example of what you What else could we do which uses the same 0 0 are explaining? way? ...J 
Can you see how these solution strategies are 
the same? Can you see how these strategy 
solutions are different? Can you explain 
mathematically how they are the same . . .  or 
d i fferent? 
D o  you want 'think time' i n  order to revise your I don ' t  agree because . . .  
thinking? 
I agree because . . .  
eJl Does this make sense? Why, why not? = ?a 0 B ut i f  . . .  'IJ Does anyone want to comment on that? ? QJ Questions for them, but remember you are not What about if. . .  I. 
eJl telling them the answer? .s - If you say that the . . .  ? :2 Remember you need to agree or disagree but ? 
;;.. use because or if in your statement Which bits do you thi n k  we might have 
questions about? How can we answer the 
How are you making sure you can all questions? 
understand and convince use of. . .  
255 
APPENDIX F: SAMPLE TRANSCRIPT WITH TEACHER ANNOTATIONS 
Transcript Teacher annotation 
Moana: Okay we have all thought about it Participation norms 
and discussed it? Can you tell me please K Should have taken the opportunity to gauge 
what you people have discussed in your consensus 
group? Can you show me what two groups 
of four look like ? 
The chi ld laid out 
0000 
00 
Moana: Can someone ask K a question probing encouraging questions 
please . . .  that 's what K says two groups of questions are not addressing why and how 
four look like ? Yes J. 
1: Would you like to rethink that? Trying uns uccessfully to shape questions . . .  rethink 
Moana: That was good 1 that you what. . .  why? None of this is easy to get these 
suggested that. You did not say you are 
chi ldren to use. I am sure we all look lost together 
wrong . . .  but if you don 't agree with what Planning focus how, why, when 
she is showing you need to ask a question But also move them from adding and skip 
which starts with why . . .  counting 
S: Why did you do two lots of two and Actual ly covering lots of focuses maybe too many introducing of language or too much as well as 
four? understanding i n  context 
Disagreement, argument etc comparing 
A later point i n  the lesson Focus on making explanations 
Moana: Did you see how he solved that? 
Focusing closely on what is being explained and He said something ? looking at the shift from materials 
R: Oh I know I could see T P l istening closely so rather than me 
T P: He said three plus three equals six re-saying it was good to get a girl speaking up. I am 
plus another three equals nine or three watching how much talk I do so good to see who 
times three equals nine can add explanations 
W: And it wont change 
1: What did you mean by that? 
T: Mean by what? 
1: It will always be that and it won 't 
change? Trying to explore what he meant. I hoped he was 
Moana: Can you explain that? making a generalisation b u t  actual l y  I a m  n o t  sure. 
256 
APPENDIX G :  PROBLEM EXAMPLE DEVELOPED IN THE STUDY GROUP TO 
SUPPORT EARLY ALGEBRAIC REASONING 
BUILDING FENCES 
The zookeeper is bui ld ing a fence from half round posts , 
for the zebras. 
One section  
takes four 
posts. 
Two sections 
takes seven 
posts. 
Three sections 
takes ten 
posts. 
Usi ng ice block sticks continue this pattern. 
Complete the table below. 
Write the pattern you see as ordered pairs e.g.  ( 1 ,4) , (2 ,7) . . .  
Can you f ind the ru le? 
Use your rule to extend your tab le to f ind out how many fence 
posts wou ld  be needed for more sect ions of fence. 
For example:  
How many posts wou ld  be needed to bu i ld :  10 sections of fence? 
15 sections? 20 sect ions? 32 sect ions? 1 14 1  sections? 
257 
APPENDIX H :  PROBLEM EXAMPLES DEVELOPED IN THE STUDY GROUP 
WHICH REQUIRED MULTIPLE WAYS TO VALIDATE REASONING 
Mrs Dotty has made three cakes which look l ike this. She decided to 
cut them up. First she cut them into halves. Then she cut them into quarters . Then she cut 
them into eighths.  
The Dotty children come home from school where they all have been learning about 
fractions and so Mrs Dotty decides to question them about the cakes she cut up. 
Barbie Dotty says that if she cut the 3 cakes into halves she must have had 6 pieces or six 
halves or 
6/2 
Maureen Dotty says that if she cut the 3 cakes into quarters she must have had 1 2  pieces or 
twelve quarters or 1 2/4 
Gaylene Dotty says that if she cut the 3 cakes into eighths she must have had 24 pieces or 
twenty four eighths or 24/8 
Baby Dotty who is just learning about fractions gets very confused about how they could 
have the same number and size cakes and yet get all the different fractional pieces .  She 
doesn ' t  know if her si sters are right. Are all three of her sisters correct? How could you 
convince Baby Dotty? Can you work out different ways to convince her? 
Gollum says 'precious'  40 times every hour. How many times does he say 'precious' every 
day? Froddo thought that it was 1 060. Sam thought it was 860. Who was right? Convince 
us about who is  right using more than three different solution strategies. 
Listen careful ly to each member of your group and then together explore different ways to 
convince yourself and the larger group. Be ready to explain  the tricky bits you think other 
people might find difficult in your explanation of this solution strategy. 
Select one of your strategies which you think wil l  work with any numbers and explore 
whether it does. 
258 
APPENDIX I :  PROBLEM EXAMPLES DEVELOPED IN THE STUDY GROUP 
WHICH SUPPORTED EXPLORATION OF PARTIAL UNDERSTANDINGS 
Peter and Jack had a disagreement. Peter said that 5/8 of a jelly snake was bigger than 3/4 
of a jel ly snake because the numbers are bigger. 
Jack said that it was the other way around, that 3/4 of a jelly snake was bigger than 5/8 of a 
jelly snake because you are taliUng about fractions of one jelly snake. 
Who is right? When your group have decided who is correct and why you need to work out 
lots of different ways to explai n  your answer. Remember you have to convince either Peter 
or Jack . . .  and they both take a lot of convincing !  Use pictures as well as numbers in your 
explanation .  
A l ittle guy from outer space is  in  your classroom and he i s  l i steni ng to Annie, Wade, Ruby 
and Justin arguing about sharing a big bar of chocolate. 
Annie says that you can only share the bar of chocolate by dividing it  into halves or 
quarters. 
Wade says he knows one more way of sharing the bar of chocolate 
Ruby and Justin say that they knows lots of ways of sharing the bar of chocolate and they 
can find a pattern as well 
The l i ttle guy from out of space i s  real ly interested in  what they say so they start to explain 
all the different ways to him. 
What do you think they say? In  your group work out a clear explanation that you 
think each person gave and then work out who is correct. 
The little guy needs lots of convincing so how many different ways can you use to 
prove what your group thinks is correct. Make sure that you use fractions as one way 
to show him because he likes using numbers. 
Can you find and explain any patterns you find in your explanation? 
259 
APPENDIX J:  MOANA'S CHART FOR THE GROUND RULES FOR TALK 
How do we korero in our classroom? 
We make sure that we discuss things together as a whanau. We listen careful ly  and actively 
to each other. That means: 
? We ask everyone to take a turn at explaining their thinking first. 
? We think about what other questions we need to ask to understand what they are 
explain ing. 
? We ask questions 'politely'  as someone i s  explaining their thinking; we do not wait 
until they have completed their explanation. 
? We ask for reasons why. We use 'what' and 'why' questions .  
? We make sure that we are prepared to change our minds. 
? We think careful ly about what they have explained before we speak or question.  
? We work as a whanau to reach agreement. We respect other people's ideas . We 
don' t  just use our own. 
? We make sure that everyone i s  asked and supported in  the group to talk. 
? We all  take responsibi l ity for the explanation. 
? We expect challenges and enjoy explaining mathematically why we might agree or 
disagree. 
? We think about all the different ways before a decision is made about the group' s  
strategy solution. We make sure that as we 'maths argue' we use " I  think 
. . .  because . . .  but why . . .  or we use "If you say that then . . .  " 
260 
APPENDIX K:  EXAMPLES OF EXPANSIONS OF SECTIONS OF THE 
COMMUNICATION AND PARTICIPATION FRAMEWORK 
Thi nk of a strategy solution and then 
explai n it to the group. 
Listen carefully and make sense of each 
explanation step by step. 
Make a step by step explanation together. 
Make sure that everyone understands. 
Keep checking that they do. 
Take turns explaining the solution 
strategy using a representation. 
Use equipment, the story in the problem, 
a drawing or diagram or/and numbers to 
provide another way or backing for the 
exp lanation. 
Listen to each person in  your group and 
state agreement with their explanation 
OR state disagreement with their 
explanation. 
Practise talking about the bits you agree 
with and be ready to say why. 
Ask questions of each other about why 
you agree or disagree with the 
explanation. 
Pick one section of an explanation and 
provide a mathematical reason for 
agreeing with it. 
Discuss the explanation or a section of 
the explanation and talk about the bits 
that the listeners might not agree with and 
why. 
26 1 
Keep asking questions unti l every section 
of the explanation is understood. 
Be ready to state a lack of understanding 
and ask for the explanation to be 
explained in another way. 
Ask questions (what did you . . .  ) of 
sections of the explanation. 
Discuss the explanation and explore the 
bits which are more difficult to 
understand. 
Discuss the questions the listeners might 
ask about the explanation 
Provide a mathematical reason for 
disagreeing with the explanation or a 
section of the explanation. 
Think about using material or drawing 
pictures about the bit of the explanation 
that there have been a lot of questions 
about in the group. 
Ask questions of each other (why did 
you . . .  how can you say . . .  ) 
Question unti l you understand and are 
convinced. 
Explain and using different ways to 
explain until you are ALL convinced. 
APPENDIX L:  A SECTION OF THE TABLE OF DATA OF THE ACTIVITY 
SETTING IN THE CLASSROOM 
The teacher asks for collective OB : 8 9 1 0  1 1  
responsibility to the maths 
community 
Asks that all group members I 2 7 9 1 1  1 , 1 0  
understand, share, record solution 
strategies 
Asks for multiple solution strategies l b  2 9  1 0  1 , 1 0  
Asks other small group members t o  be l e  
able to continue an explanation 
Asks for collective consensus reached 2a 2 9  10  I 
on selected strategy sol ution 
Asks that all smal l group members 2b 4 9 10  3, 1 1 , 1 3  
understand each other ' s  strategy 
Asks that each member of the smal l 2c 4 1 1  
group describes a strategy 
Asks for small group collective 3 9 1 8  
development of strategy solution 
Asks for turn taking i n  explaining 3b 3,4, 1 1 
strategies in small groups 
Asks for collective analysis to predict 4 4, 9, 1 8  3,5 
questions related to solution strategy 
Asks group members to 5b 34 5 
interject/support each other in 
explanation 
Asks students to analyse and explain 34 20,2 1 
own/or grou2_ erroneous thinking 
Asks for collective support when 7 2 , 27 
student states l ack of understanding 
Uses model of how small group 7b 8, 1 1 
reworked their thinking 
Emphasises the value of working 7c 
together 
Examines with students how groups 7d 
worked collaborati vely 
Uses body language to determine 52 33 
understanding of mathematical 
thinking 
Uses think ti me/wait time to support 5 1  4 4 5 
questioning/challenging 
Provides wait time for other chi ldren 54 7, 2 1  1 6  
to ask questions 
Positions self as member of 39 
mathematics community w i th similar 
status 
Attributes mathematical 55b 1 8, 2 1 
thinki ng/explanation/generali sation to 
student 
262