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The student as a learner, a knower and a facilitator: Share epistemic agency enacted as an innovative school-based mathematics pedagogy

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This study is an inquiry into my students' participation and empowerment in all aspects of their mathematics learning that aims to explore the possibility of improving their relationship with and knowledge of mathematics. It considers the learning process in general, as well as the way it shapes the knowledge that it facilitates, and the different relationships that students have with this knowledge. This study is guided by the belief that improving my students’ mathematics knowledge requires the establishment of an innovative pedagogy based on knowledge creation. I elaborate the concept of shared epistemic agency to explain the phenomenon of students taking responsibility for the advancement of their own mathematics knowledge and that of the classroom community. The concept draws on Damşa’s notion of shared epistemic agency, Scardamalia & Bereiter’s discussion of knowledge building, and Nonaka’s analysis of knowledge creation. I carried out action research that applied these principles over one academic year, requiring the participants of my classroom to blend their authority with mine as they assumed the roles normally reserved for the teacher. Using six key characteristics of shared epistemic agency that I identify in the existing research, as well as the unit of analysis, I analysed the qualitative data that was collected. I show that the shared epistemic agency that is necessary for knowledge advancement in a secondary school mathematics classroom emerged as the students participated in my innovative pedagogical environment. Moreover, I demonstrate that this agency can be reconceptualised in terms of particular kinds of student behaviour and a particular kind of learning community. I argue that the student, in such an environment, is a competent, adaptive Participant who takes up flexible positions as a learner, a knower, and a facilitator. The classroom that developed as a democratically interactive learning community sustained the emergence of the Participant.
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The student as a learner, a knower, and a facilitator:
Shared epistemic agency enacted as an innovative school-based
mathematics pedagogy
Ijeaku Iheoma Mezue
UCL Institute of Education
Doctor of Philosophy
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I, Ijeaku Iheoma Mezue, confirm that the work presented in this thesis is my own.
Where information has been derived from other sources, I confirm that this has been
indicated in the thesis.
Signed,
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Abstract
This study is an inquiry into my students' participation and empowerment in all
aspects of their mathematics learning that aims to explore the possibility of improving
their relationship with and knowledge of mathematics. It considers the learning
process in general, as well as the way it shapes the knowledge that it facilitates, and
the different relationships that students have with this knowledge. This study is
guided by the belief that improving my students mathematics knowledge requires
the establishment of an innovative pedagogy based on knowledge creation.
I elaborate the concept of shared epistemic agency to explain the phenomenon of
students taking responsibility for the advancement of their own mathematics
knowledge and that of the classroom community. The concept draws on Damşa’s
notion of shared epistemic agency, Scardamalia & Bereiter’s discussion of
knowledge building, and Nonaka’s analysis of knowledge creation.
I carried out action research that applied these principles over one academic year,
requiring the participants of my classroom to blend their authority with mine as they
assumed the roles normally reserved for the teacher. Using six key characteristics
of shared epistemic agency that I identify in the existing research, as well as the unit
of analysis, I analysed the qualitative data that was collected.
I show that the shared epistemic agency that is necessary for knowledge
advancement in a secondary school mathematics classroom emerged as the
students participated in my innovative pedagogical environment. Moreover, I
demonstrate that this agency can be reconceptualised in terms of particular kinds of
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student behaviour and a particular kind of learning community. I argue that the
student, in such an environment, is a competent, adaptive Participant who takes up
flexible positions as a learner, a knower, and a facilitator. The classroom that
developed as a democratically interactive learning community sustained the
emergence of the Participant.
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Impact Statement
I argue that the secondary school student is a competent individual who can take
responsibility for their mathematics knowledge, and that of their learning community.
In an age in which knowledge is readily available and accessible to young people, I
present the notion of the Participant who can use the resources available to them,
specifically through interaction with other Participants, to build and share
mathematics knowledge without relying on the teacher as the epistemic authority.
This Participant is identifiable in their capacity as:
A learner who controls their knowing and unknowing, who is productive of
epistemic interactions, and who is not knowledge-less. This learner has the
potential to be transformative.
A knower with epistemic authority who is relational in their response to an
unknowing, and who is interdependent with a learner.
A facilitator with process authority who can negotiate the blending of their
authority with that of other Participants to advance collective knowledge.
The new Learning Community is identifiable as a classroom community that is:
Characterised by an interactive practice in which the Participants learn
mathematics through epistemic interactions and, in which their participation
positions them as learners, knowers, and facilitators.
Productive for the creation of mathematics knowledge in a way that is
demonstrably connected with the enactment of the pedagogy and its
capacities for epistemic interaction.
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Democratic and egalitarian. It was presumed that all Participants are able to
participate, and their participation justified this presumption.
This learning community is sustained by:
Its definition of competence as participation in epistemic interaction.
The formation of a common identity and a sense of belonging as a result of
mutual participation, and the subsequent accountability of the students to the
practice of the learning community.
Of equal benefit is the notion of the teacher as an Educator, whose purpose is to
draw out from the participant their latent potential.
This study contributes to the theory and practice of mathematics education as it
presents the revitalised conception of the Participant as a challenge to the efficacy of
the current discourse on education in the UK. It sheds light on government policy
that implicitly construes students as incapable, highlights how classroom practices
such as questioning can limit students participation, and presents an alternative to
practices such as ability setting. This study presents the possibility of an alternative
student, classroom, and teacher that could transform mathematics education.
The study also presents the possibility of considering the teacher as a professional
capable of bringing about change in the classroom, in a manner that could be
transformative to educational practices from the bottom up, heeding and reiterating
the calls for reform made in both current and older research (cf. Elliott, 2011; Schon,
2008; Stenhouse, 1981).
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The benefits of this research can be realised through publication in journals,
discussions at educational forums and presentations at conferences; as well as,
more personally, through my relationship to my own teaching practice. As a senior
leader in a secondary school, I continue to challenge the presuppositions of my
colleagues, urging them to consider the student as competent, and to share authority
with them as partners in an educational journey.
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Table of Contents
Abstract…………………………………………………………………………………… 3
Impact Statement……………………………………………………………………….. 5
Acknowledgements…………………………………………………………………….. 25
1 INTRODUCTION ............................................................................................... 27
1.1 The Context of this Research Study ............................................................ 29
1.1.1 Mathematics An Important Secondary School Subject ...................... 30
1.1.2 The Secondary School ......................................................................... 32
1.1.3 Myself: The Mathematics Teacher ........................................................ 33
1.1.3.1 Getting Expectations Wrong .......................................................... 34
1.1.3.2 Students Taking Responsibility for their Mathematics Knowledge . 36
1.1.3.3 Questioning the Taken-for-Granted................................................ 37
1.1.4 The Thesis Outline. .............................................................................. 40
2 THEORETICAL FRAMEWORK ........................................................................ 43
2.1 Agency ........................................................................................................ 43
2.1.1 Human Agency ..................................................................................... 45
2.1.2 Situated Agency ................................................................................... 49
2.1.3 Agency as Epistemic ............................................................................ 54
2.1.4 Summary .............................................................................................. 56
2.2 Theories of Social Learning......................................................................... 57
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2.2.1 Metaphors for Collective Learning ........................................................ 58
2.2.2 Communities of Practice ....................................................................... 62
2.2.2.1 Learning as Doing .......................................................................... 63
2.2.2.2 Learning as Experience ................................................................. 64
2.2.2.3 Learning as Belonging ................................................................... 65
2.2.2.4 Learning as Becoming ................................................................... 66
2.2.2.5 The Notion of Community .............................................................. 67
2.2.2.6 Power Relations in Society and the Classroom as a Community ... 69
2.2.2.6.1 Power Relations in Schools ....................................................... 70
2.2.2.6.2 Power Circulating between Teacher and Student ...................... 71
2.2.3 Summary .............................................................................................. 74
2.3 Pedagogy .................................................................................................... 75
2.3.1 The Conventional Pedagogy ................................................................ 75
2.3.2 Authority in the Classroom .................................................................... 79
2.3.2.1 Shared Authority ............................................................................ 82
2.3.2.2 Positioning ..................................................................................... 85
2.3.3 Summary .............................................................................................. 87
2.4 Theoretical Framework ............................................................................... 88
2.4.1 Knowledge Building/Knowledge Creation ............................................. 89
2.4.1.1 Knowledge Creation ....................................................................... 93
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2.4.2 Shared Epistemic Agency ..................................................................... 97
2.4.3 Summary ............................................................................................ 100
2.5 Researching Innovative Pedagogies ......................................................... 108
2.5.1 Knowledge-Building Pedagogies ........................................................ 108
2.5.2 Transformative Pedagogies in England .............................................. 112
2.5.3 Summary ............................................................................................ 118
3 METHODOLOGY ............................................................................................ 119
3.1 The Pedagogy ........................................................................................... 120
3.1.1 The Stages of the Innovative Pedagogy ............................................. 123
3.2 Rationale for Action-Research Approach .................................................. 129
3.2.1 History of Action Research ................................................................. 129
3.2.2 What is Action Research? .................................................................. 133
3.3 The Research Design ............................................................................... 136
3.3.1 The Teaching Cycles .......................................................................... 136
3.3.1.1 Teaching Cycle Stage 1 ............................................................... 137
3.3.1.2 Teaching Cycle Stage 2 ............................................................... 137
3.3.1.3 Teaching Cycle Stage 3 ............................................................... 138
3.3.1.4 Teaching Cycle Stage 4 ............................................................... 139
3.3.2 The Research Cycle ........................................................................... 139
3.3.2.1 Research Cycle Stage 1 .............................................................. 141
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3.3.2.2 Research Cycle Stage 2 .............................................................. 141
3.3.2.3 Research Cycle Stage 3 .............................................................. 142
3.3.2.4 Research Cycle Stage 4 .............................................................. 142
3.3.2.5 Research Cycle Stage 5 .............................................................. 143
3.3.3 Schedule of Action Research Cycles .................................................. 143
3.3.4 Ethics .................................................................................................. 146
3.4 Enacting the Research Design .................................................................. 151
3.4.1 Action Research Cycle 1 .................................................................... 152
3.4.1.1 Selecting Participants ................................................................... 153
3.4.1.2 Selecting Teacher Participants .................................................... 155
3.4.1.3 The Quality of Mathematics Knowledge ....................................... 159
3.4.2 Data Collection Methods..................................................................... 164
3.4.2.1 Observation .................................................................................. 164
3.4.2.1.1 Classroom Layout and Decisions about Video Focus .............. 165
3.4.2.1.2 Field Notes ............................................................................... 169
3.4.2.2 Interviews ..................................................................................... 170
3.4.3 Reflecting on Action Research Cycle 1............................................... 173
3.4.3.1 Reflecting On the Pedagogy ........................................................ 173
3.4.3.2 Reflecting on the Data Collection ................................................. 174
3.4.4 Action Research Cycle 2 .................................................................... 177
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3.4.4.1 Selecting Teacher Participants .................................................... 179
3.4.5 Data Collection Methods..................................................................... 181
3.4.5.1 Observations ................................................................................ 181
3.4.5.1.1 Video Recordings ..................................................................... 182
3.4.5.2 Student Interviews........................................................................ 185
4 ANALYTICAL METHODS ............................................................................... 189
4.1 The Unit of Analysis .................................................................................. 189
4.1.1 Intentions ............................................................................................ 195
4.1.2 Knowledge Building ............................................................................ 197
4.1.2.1 Extension ..................................................................................... 198
4.1.2.2 Explication .................................................................................... 198
4.1.2.3 Expertise ...................................................................................... 199
4.1.2.4 Mutual Relations .......................................................................... 200
4.1.3 New Knowledge .................................................................................. 203
4.2 Episode Selection ..................................................................................... 206
4.2.1 The Selection Processes .................................................................... 206
4.2.1.1 Completing the Summary Sheet .................................................. 208
4.2.1.2 Barriers to Episode Selection ....................................................... 213
4.2.2 Transcribing an Episode ..................................................................... 217
4.2.2.1 Explaining the Extract Heading .................................................... 217
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4.2.2.2 Coding the Transcript ................................................................... 218
5 FINDINGS ....................................................................................................... 220
5.1 Elaborating on the Unit of Analysis ........................................................... 222
5.1.1 Intentions ............................................................................................ 222
5.1.2 Knowledge Building ............................................................................ 229
5.1.2.1 Modes of Extension ...................................................................... 230
5.1.2.1.1 Questions ................................................................................. 231
5.1.2.1.2 Seeks Affirmation ..................................................................... 234
5.1.2.1.3 Requests .................................................................................. 236
5.1.2.1.4 Challenges ............................................................................... 240
5.1.2.1.5 Articulates Unknowing.............................................................. 243
5.1.2.1.6 Summary .................................................................................. 246
5.1.2.2 Modes of Explication .................................................................... 247
5.1.2.2.1 Clarifies .................................................................................... 247
5.1.2.2.2 Affirms ...................................................................................... 249
5.1.2.2.3 Tells ......................................................................................... 250
5.1.2.2.4 Explicates Unknowing .............................................................. 252
5.1.2.2.5 Summary .................................................................................. 254
5.1.2.3 Modes of Expertise ...................................................................... 255
5.1.2.3.1 Controls .................................................................................... 255
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5.1.2.3.2 Checks ..................................................................................... 256
5.1.2.3.3 Manages .................................................................................. 259
5.1.2.3.4 Summary .................................................................................. 263
5.1.2.4 Mutual Relations .......................................................................... 264
5.1.2.4.1 Mutual Relations as Contextual or Non-contextual .................. 264
5.1.2.4.2 Mutual Relations as Conducive or Non-Conducive for the
Advancement of Knowledge...................................................................... 267
5.1.3 New Knowledge .................................................................................. 268
5.1.3.1 Acknowledging the resolution of an episode ................................ 269
5.1.3.2 Building New Knowledge as Dimensions of Appeal. .................... 270
5.1.3.2.1 Appeal to Conceptual Knowledge ............................................ 271
5.1.3.2.2 Appeal to a Knower .................................................................. 272
5.1.3.2.3 Appeal to Procedural Knowledge ............................................. 276
5.1.3.2.4 Appeal to a Knower and Procedural Knowledge ...................... 280
5.2 Elaborating on Participants’ Interactions ................................................... 284
5.2.1 Positioning .......................................................................................... 284
5.2.1.1 Positioning as a Learner .............................................................. 291
5.2.1.2 Positioning as a Knower ............................................................... 291
5.2.1.3 Positioning as a Facilitator ........................................................... 292
5.2.1.4 The Learner and the Knower as Productive Agents ..................... 293
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5.2.2 Process Authority in Interaction .......................................................... 298
5.2.2.1 Blending of Process Authority ...................................................... 298
5.2.2.2 Control of Social Behaviour (Freedom of Dialogical and Physical
Interaction) ................................................................................................... 301
5.2.2.3 The Position of the Learner as Authority ...................................... 308
5.2.3 Epistemic Authority in Interaction ....................................................... 309
5.2.3.1 Knowledge as a Prerequisite for Extension .................................. 310
5.2.3.2 Disregarding Presumed Ability Labels ......................................... 311
5.2.3.3 Individual and Community Knowledge and Responsibility ........... 314
5.3 Summary ................................................................................................... 318
6 DISCUSSION .................................................................................................. 320
6.1 Answering Research Question 1 ............................................................... 321
6.1.1 Theme 1: The Concept of “Student as a Participant” .......................... 321
6.1.1.1 The Participant as a Learner ........................................................ 323
6.1.1.1.1 A Learner Can Take Control of their Knowing and Unknowing 324
6.1.1.1.2 A Learner as Productive of Epistemic Interactions ................... 326
6.1.1.1.3 A Learner is not Knowledge-Less ............................................ 327
6.1.1.2 The Participant as a Knower ........................................................ 329
6.1.1.2.1 A Knower has Epistemic Authority ........................................... 330
6.1.1.2.2 A Knower is Interdependent on a Learner and Relational ........ 331
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6.1.1.3 The Participant as a Facilitator ..................................................... 335
6.1.1.3.1 Process Authority Facilitates Learning ..................................... 336
6.1.1.3.2 Process Authority as Negotiated .............................................. 336
6.1.1.3.3 Summary .................................................................................. 337
6.1.2 Theme 2: The Concept of a “Learning Community” ............................ 340
6.1.2.1.1 The Interactive Classroom Practice ......................................... 341
6.1.2.1.2 A Productive Community.......................................................... 346
6.1.2.1.3 Democratic Participation .......................................................... 348
6.1.2.2 Summary ...................................................................................... 349
6.2 Answering Research Question Two .......................................................... 351
6.2.1 Competence ....................................................................................... 352
6.2.1.1 Competence as a Knower ............................................................ 354
6.2.1.2 Competence as a Learner ............................................................ 355
6.2.1.3 Competence as a Facilitator ........................................................ 356
6.2.1.4 Competence as Productive Interaction ........................................ 357
6.2.2 Accountability ..................................................................................... 359
6.2.3 Sustaining the Community .................................................................. 362
6.2.4 Summary ............................................................................................ 369
6.3 Reflecting on the Action Research ............................................................ 371
6.3.1 My Role as a Participant ..................................................................... 372
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6.3.1.1 My Position as a Learner ............................................................. 373
6.3.1.2 My Position as a Knower .............................................................. 376
6.3.1.3 My Position as a Facilitator .......................................................... 378
6.3.2 The Pedagogy of Trust ....................................................................... 379
6.3.2.1 The Student, a Participant ............................................................ 380
6.3.2.2 The Teacher, an Educator ........................................................... 381
6.3.2.3 The Classroom, a Learning Community ....................................... 383
6.4 Summary ................................................................................................... 385
7 CONCLUSION ................................................................................................. 387
7.1 A Contribution to the Field of Mathematics Education ............................... 391
7.1.1 A Contribution The Participant and The Educator ........................... 392
7.1.1.1 A Challenge to Educational Policy ............................................... 396
7.1.1.2 A Challenge to Educational Practice ............................................ 398
7.1.2 A Contribution The Innovative Pedagogy ........................................ 401
7.1.2.1 The Learning Community as a Challenge to the Educational Policy
of Mastery ..................................................................................................... 404
7.1.2.2 The Pedagogy as a Challenge to Educational Practice of Ability
Differentiation ............................................................................................... 406
7.1.2.3 The Pedagogy as Empowerment ................................................. 408
7.2 A Contribution to Theory ........................................................................... 410
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7.2.1 Shared Epistemic Agency is a Manifestation of Who the Participants
Become ........................................................................................................... 411
7.2.2 Shared Epistemic Agency is the Practice of a Type of Learning
Community ...................................................................................................... 413
7.3 A Contribution as a Teacher-Researcher .................................................. 415
7.3.1 A Particular Action Research Methodology ........................................ 417
7.3.2 An Authentic and Empowering Methodology ...................................... 420
7.3.2.1 A Participatory Methodology ........................................................ 422
7.4 Limitations and Suggestions for Future Research..................................... 424
BIBLIOGRAPHY….….….….….….….….….….….….….….….….….….….….….….429
Appendices
Appendix 1 Principles of knowledge building……………………………………….458
Appendix 2 Overview of actions indicative of Damşa et al’s (2010) Shared
Epistemic Agency……………………………………….………………………………..460
Appendix 3 Parent and student consent form……………………………………….462
Appendix 4 Limitation of having three teacher participants An extract from field
notes……………………………………….………………………………………………465
Appendix 5 Interview 1 questions…………………………………………………….467
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Appendix 6 Interview 2 questions……………………………………….……………468
Appendix 7 Elaboration of intentions across all Episodes…………………………469
Appendix 8 A summary sheet inclusive of modes………………………………….473
Appendix 9 Frequency of modes table………………………………………………474
Appendix 10 Jayzee and Daniel Extract from field notes………………………..475
Extracts
Field notes extract 3.1 Teacher participants knowledge of GCSE questions……160
Field notes extract 3.2 Screenshot of worked examples…………………………..162
Transcript extract 3.3 Transcript of audio recording during the planning
session………………………………………….…………………………………………176
Transcript extract 3.4 From transcript of interview with Adam…………………….187
Extract 4.1 Unit of analysis Parts 1 and 2 Episode 19…………………………193
Extract 4.2 Unit of analysis New knowledge Episode 9……………………….201
Extract 4.3 Example of a deselected Episode………………………………………216
Extract 5.1 Intentions (Ext, Dialogic interaction, Identified) Episode 1…………223
Extract 5.2 Intentions (Exp, Dialogic/Physical interaction, Assumed) Episode
9……………………………………….……………………………………….…………..224
Extract 5.3 Intentions (Exp, Dialogical/Physical interaction, Assumed) Episode
19……………………………………….………………………………………………….227
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Extract 5.4 Extension (Questions to student Participant) Episode 18……….…231
Extract 5.5 Extension (Questions to teacher Participant) Episode 29………….232
Extract 5.6 Modes of Extension (Seeks affirmation) Episode 10……………….235
Extract 5.7 Modes of Extension (Requests) Episode 3 …………………………238
Extract 5.8 Modes of Extension (Challenge A) Episode 2……………………….241
Extract 5.9 Modes of Extension (Challenge B) Episode 14…………………….242
Extract 5.10 Modes of Extension (Articulates unknowing) Episode 6…………243
Extract 5.11 Modes of Explication (Clarifies knowledge) Episode 23…………248
Extract 5.12 Modes of Explication (Tells) Episode 6……………………………250
Extract 5.13 Modes of Explication (Explicates) Episode 1……………………..252
Extract 5.14 Modes of Expertise (Checks) Episode 25…………………………257
Extract 5.15 Modes of Expertise (Manages) Recording 7………………………260
Extract 5.16 Appeal to a knower Episode 6………………………………………273
Extract 5.17 Appeal to procedural knowledge Episode 2……………………….277
Extract 5.18 Appeal to a knower and procedural knowledge Episode 30…….280
Extract 5.19 Positioning Episode 2……………………………………………….286
Extract 5.20 Learner/Knower as productive agent Episode 23………………..294
Extract 5.21 (Control of social behaviour) Episode 10………………………….302
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Extract 5.22 Community, individual knowledge and responsibility Episode
6……………………………………….………………………………………………….314
Field notes extract 6.2 Sustaining accountability………………………………….365
Figures
Figure 1.1 Edexcel Mathematics GCSE Summer 2017 Higher Paper 1, Question
5……………………………………….……………………………………………………..28
Figure 1.2 Solving linear equations in ‘border-line’ GCSE Mathematics
classes……………………………………….……………………………………….……..35
Figure 2.1 Relating Emirbayer & Mische’s (1998) situational agency to Bandura’s
(2001) human agency……………………………………….…………………………….53
Figure 2.2 Paavola & Hakkarainen’s (2011) three metaphors of learning………...60
Figure 2.3 Components of a social theory of learning (Wenger, 1998, p. 5)
……………………………………….……………………………………….……………..63
Figure 2.4 Process/Epistemic dimensions of authority vs. teacher/student authority
distribution……………………………………….………………………………………....83
Figure 2.5 The Knowledge Creation Spiral (Umemoto, 2002, p. 464)………….….95
Figure 2.6 Theoretical background interconnection and relevance of concepts,
notions and perspectives.………………………………………..…………………….102
Figure 2.7 Pedagogic principle 1.……………………………………….………….105
Figure 2.8 Pedagogic principle 2.……………………………………….………… 106
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Figure 2.9 Pedagogic principle 3.……………………………………….………… 106
Figure 2.10 Pedagogic principle 4.……………………………………….……… ..107
Figure 2.11 Pedagogic principle 5.……………………………………….…………107
Figure 3.1 Stages of the innovative pedagogy.…………………………………….123
Figure 3.2 Teaching cycles and research cycle interplay.…………………………140
Figure 3.3 ICT Classroom layout with seating positions (A Ώ) and possible
camera positions (C1 C4) .……………………………………….…………………..166
Figure 3.4 Layout of teaching cycle 7 classroom.………………………………….183
Figure 4.1 The three parts of an Episode.……………………………………….….190
Figure 4.2 Intention = 1, knowledge building & new knowledge ≥ 1……………..192
Figure 4.3 The process of an Episode, the unit of analysis……………………….205
Figure 4.4 The summary sheet of Episode 9, extract 4.2.………………………...208
Figure 5.1 The unit of analysis.……………………………………….……………...221
Figure 5.2 Appeal to conceptual knowledge in Episode 9.………………………..272
Figure 5.3 Dimensions of Appeal in all Episodes ………………………………….279
Figure 6.1 The multi-faceted student as a Participant.……………………………..322
Figure 6.2 The features of the learner position.…………………………………….324
Figure 6.3 The modes of extension that point to a learner in control…………….325
Figure 6.4 The features of the knower position.…………………………………….330
Figure 6.5 The features of the facilitator position.…………………………………..335
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Figure 6.6 The features of the Participant as a learner, a knower or a
facilitator.……………………………………….………………………………………….339
Figure 6.7 The features of the Learning Community.………………………………341
Figure 6.8 Interplay of participation and reification.………………………………..360
Figure 6.9 The Participant in the Learning Community.……………………………363
Figure 7.1 My action research cycles inclusive of teaching cycles.………………417
Photographs
Photograph 3.1 Camera positions for the ICT suite recordings.…………………167
Photograph 3.2 Camera positions for the teaching cycle 7 classroom.…………183
Photograph 5.1 Intentions (Ext, Dialogical Interaction/Reification, Identified)….226
Photograph 5.2 Modes of Extension (Requests).…………………….……………238
Photograph 5.3 Disregarding presumed ability labels.…………………………….313
Tables
Table 2.1 Attitudes and practices of the banking model of education (Freire, 1970,
p. 164).…………………….…………………….…………………….……………………67
Table 3.1 The innovative pedagogy Learning, principles and characteristics of
shared epistemic agency.…………………………….………………………………….128
Table 3.2 Actual research design schedule of action research cycles…………...144
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Table 3.3 Participants selected pseudonyms.……………………………………...154
Table 3.4 Teaching schedule for teaching cycle 1-4.………………………………158
Table 3.5 Interview timetable for action-research cycle 1.…………………………171
Table 3.6 Planned and actual data collection.…………………………….………...174
Table 3.7 Teaching schedule for teaching cycle 5-7.………………………………181
Table 4.1 The thirty-six Episodes identified across the research.………………...212
Table 5.1 Elaboration of Intentions.……………………………..……………………228
Table 5.2 Classifying a mode of Extension.………………………………………....230
Table 5.3 Modes of Extension summary.……………………………...……………246
Table 5.4 Modes of Explication summary.……………………………..……………254
Table 5.5 Modes of Expertise summary.……………………………………………263
Table 5.6 Classification of mutual relations.………………………………………..268
Table 6.1 Participants GCSE outcomes.……………………………...……………368
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Acknowledgments
I owe a depth of gratitude to many people for their contribution to the successful
completion of this PhD.
My deepest thanks go to the students who over the years contributed to my
experiences as a mathematics teacher: especially my Year 10 mathematics class,
whose actions, as they sought to learn mathematics, form the basis of this study. To
them I own a huge debt.
I am immensely grateful to my supervisors Professor Candia Morgan and Dr Cathy
Smith for patiently sharing their knowledge with me and challenging my thinking.
You both made my incoherence coherent as a thesis. Thank you very much.
My thanks and gratitude go to Tony Hartney, CBE, the CEO of the school that
sponsored the majority of my PhD study; that support and faith was a source of
motivation. I also acknowledge, appreciatively, Carlene Graham who, as the Head
of Mathematics Faculty, gave me the freedom to change my classroom pedagogy. I
appreciate my professional colleagues, past and present, interactions with whom
contributed to my experience as a teaching professional, and the contribution and
thoughts Nicholas Mroczkowski made to my final draft.
Finally, I acknowledge the love, wisdom, and affection of my beautiful parents
Augustine Onyeka and Eunice Nneka, who started me on this quest for knowledge. I
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am blessed to have the love and encouragement of my intelligent children; Chisom,
Uzoma, Chiamaka, Chienyem, and Chuba, you all were and continue to be my
greatest inspiration. But most importantly, I am specially blessed with the love and
care of my husband Onyekwelu, whose quiet presence and attention to other areas
of our life ensured that I had the time, space and the peace of mind I needed for the
six years it took to complete this study. I am grateful to you, Ezenwakaenyi 1 of
Ideani.
To God be the Glory.
27
1 INTRODUCTION
Socio-cultural theories of learning (e.g. Boaler & Greeno, 2000; Lave & Wenger,
1991; Rogoff et al., 1998; Wenger, 1998) suggest that the way in which mathematics
learning occurs that is the specific practices involved in the learning process,
shape and define the knowledge produced, as well as the different relationships
students have with this knowledge and the uses they make of it.
Boaler’s research suggests that students who participate in negotiating and
interrogating mathematics as they learn it are more able to use apply its principles in
situations that require such practices when compared with students who learn
mathematics by working through exercises from a textbook (2002a). Boaler argues
that students knowledge is applied in situations outside the classroom in a way that
is dependent on the situation within the classroom, given that knowledge is co-
constituted by how the learning occurred. My study, in this spirit, starts from the
position that improving the mathematics knowledge of my students and their
relationship with the subject requires a change in their current learning practices.
For the first fifteen years of my teaching career, my classroom pedagogy was similar
to that of the mathematic teachers I had encountered, including those with whom I
taught. Summarily, my role in the classroom, as the teacher, was to transfer my
mathematics knowledge to the students; the students role was to listen to me and
internalise this information; this was the core of the learning process. At the start of
the lesson, I would explain a topic to the students and work out a select number of
examples on the board; the students then had the opportunity to ask questions.
Following questions, the students would try out similar problems relevant to the topic.
28
At this point, I would supervise the classroom, checking students work and
answering questions that may arise; finally, a summary of our learning (or my
teaching) occurred at the end of the lesson.
Due to my years of experience as a mathematics teacher, I became responsible for
teaching the students who appeared to have a particular difficulty with learning
mathematics. Over time I came to believe that my classroom pedagogy, combined
with the nature of the students apparent difficulties, contributed to reinforcing in the
students minds the conviction that they were not good at mathematics. The
anecdotal evidence that formed the basis of my belief was what I viewed as the
students increasing reliance on my mathematics knowledge. This reliance
manifested as a reluctance to take chances in the classroom, answer questions, or
engage with the reasoning behind mathematical principles.
This reliance is evident when students confront a mathematics question to which the
mathematical concepts to be applied are not explicit (see figure 1.1).
Figure 1.1 Edexcel Mathematics GCSE Summer 2017 Higher Paper 1, Question 5
29
While my students could grasp composite mathematical concepts, such as
Pythagoras theorem, they often struggled to make sense of what is required by
questions such as these. They resorted to approaches that are not mathematically
rational, such as manipulating the numbers using any of the four arithmetical
operations, or using formulae that relate to irrelevant information in the question,
such as by calculating the area of the triangle. With a significant proportion of the
reformed GCSE Mathematics Higher Paper requiring the application of mathematical
knowledge in a variety of routine and non-routine problems with increasing
sophistication, including breaking down problems into a series of simpler steps and
persevering in seeking solutions (National Curriculum in England, 2021, paragraph
2), my students needed to develop this particular skill in order to achieve higher
grades or even to pass mathematics at GCSE. Their learning practice needed to
change.
1.1 The Context of this Research Study
Mathematics, especially at the secondary level, is an important subject both to
schools and to students own futures. Three government policies Key Stage
testing, first undertaken in 1991; the transfer of responsibility for school inspections
to the Office for Standards in Education (Ofsted) in 1993; and the introduction of
school performance tables under the governments choice agenda in 1992 drew
attention to mathematics performance in schools. These policies have led school
stakeholders, including local education authorities, to pressure mathematics faculties
and teachers to raise their students performance in the subject. The fall in Englands
30
position in the international ranking of students performance in English, Mathematics
and Science in the OECD PISA survey of Great Britain (Department for Education,
2010) led to the 2014 wholesale reform of the national curriculum, testing, and the
performance indicators for school league tables. The government sought to emulate
the more successful education systems of Finland, Singapore, and Shanghai (e.g.
Department for Education & Truss, 2014). These policies and reforms continue to
impact the culture and structure of secondary school mathematics departments as
students performance in mathematics becomes increasingly crucial for schools
survival.
1.1.1 Mathematics An Important Secondary School Subject
In England, all 15-to-16-year-olds sit the General Certificate of Secondary Education
(GCSE) at the end of their compulsory secondary school education. Children start
secondary school having sat standardised English Reading and Mathematics tests at
age 11, the end of primary school. The difference between the two test scores (the
progress measure), statistically calculated by the Department for Education, is used
to determine students progress in mathematics in secondary schools.
In their bid to quantify what is happening in schools and give parents more
information and power regarding the choice of schools, successful governments in
England have introduced and continuously improved test-based school
accountability measures (Leckie & Goldstein, 2019). In 2016, As part of this
measure, the government introduced the floor target Progress 8 (DfE, 2019),
replacing the previous floor target that judged a given school on the percentage of its
31
students who achieved five C+ GCSE grades, including English and Mathematics.
This new target measured individual students progress in eight subjects from
primary school national tests to their GCSE examinations. A schools Progress 8
score is the average of their students scores presented with a 95% confidence
interval. While this floor target is calculated based on students performance across
eight subjects, English and Mathematics are double-counted. A Progress 8 score
ranges from -1.0 to +1.0; a score below -0.5 indicates failure to achieve the minimum
standard expected by the government, and a score of +0.5 or above indicates that
the students in the school are progressing above the expected level. These
measures further contribute to the emphasis placed on students performance in the
particular subject of mathematics that informs teaching practices in schools and
mathematics classrooms.
These accountability measures rank schools on students attainment and progress in
GCSE examinations. Before the COVID-19 pandemic began in early 2020, these
rankings, in the form of league tables, were published in the national press, and
validated a schools reputation both in the local and national contexts. As school
funding follows pupils, these accountability measures and accompanying league
tables are a form of consequential accountability (Hanushek and Raymond, 2005),
which assigns consequences to institutions that fail to meet expectations. Parental
choice determines pupil intake; schools that produce positive results, therefore,
become oversubscribed, while failing schools struggle to meet their intake quota.
Poor performance also triggers an inspection by Ofsted that results in an
Outstanding, Good, or Requiring Improvement rating; all schools are required to
make public the full documentation of these inspections. Publicity acts as a further
32
aspect of the control and policing of school performance, and legitimises government
policy.
For students, these qualifications act as a threshold for accessing post-16 education
and employment. Secondary school mathematics and its study has historically
conferred positive status on students who perform well in it. It is a gatekeeper to
entry into elite universities (P. Davies & Ercolani, 2019) and a higher earning power
leading to economic stability (Levine & Zimmerman, 1995).
1.1.2 The Secondary School
This study took place in an inner-city London state school that enjoys a measure of
popularity in the local community. The school has a cohort of 1300 mixed-gender,
culturally-diverse pupils ranging in age from 11 to 16 years old. The school has
enjoyed increasingly strong examination results, with a Progress 8 score of 0.3. In
its last school inspection, Ofsted graded the school as Outstanding. The
mathematics faculty is in a block of twelve classrooms, one of which is an ICT suite.
My classroom, where the research took place, is the ICT suite. The mathematics
faculty designed the Year 10 curriculum map with the intention of having students
progress through the GCSE mathematics content over two years; thus, they would
typically complete the program by the end of the Year 10, having started at the
beginning of Year 9.
This study focuses on one mathematics class of students who were, at the beginning
of the research, just commencing Year 10 studies. I chose to focus on this class as,
at the time, I only taught one Year 10 class and one Year 11 class; I did not select
33
the latter class for two reasons. Firstly, Year 11 is a shortened year, as the GCSE
examinations start in May, and the students are no longer in lessons; secondly, Year
11 was planned by the mathematics faculty as a revision year, and the students
would have already completed the curriculum content. I wanted the study to take
place over a sufficient amount of time, and involve students learning new content.
The students in this study were 14 to 15 years old and in their fourth year of
secondary education (that is Year 10), having commenced a programme of study
that culminates in a series of external examinations across May and June the
following academic year. The students were loosely assigned to mathematics class
groups based on assumptions about their mathematical abilities, as is conventional
(Boaler, 2014). This perception was based on students performance in the internal
mathematics examinations that took place at the end of the previous academic year;
the mathematics faculty considered my class as of a lower-middle ability.
1.1.3 Myself: The Mathematics Teacher
From the beginning of my teaching journey, I have been aware of the potential of
alternative approaches to secondary school education, having spent my formative
years educated in another continent. As a Postgraduate Certificate of Education
(PGCE) trainee at the UCL Institute of Education (IOE) in London, I was acutely
aware of the fact that the form of UK education system was not universal, and this
contributed to my initial endorsement of alternative pedagogy. Attending to these and
other personal motivations and assumptions is a significant aspect of becoming a
reflexive qualitative researcher; it is crucial to be faithful to the influence of my
34
positionality on the research process and findings. Jane Miller referred to telling
ones story as part of the research process, as the autobiography of the question”
(Miller, 1995, p. 23); she argued that it is a powerful validation of our experiences
and their potential for rethinking teaching. To this end, in reflecting on the journey
that led me to this study, I highlight two further motivations that changed my thinking
as a mathematics teacher and made me consider adapting my pedagogical methods
in the classroom.
1.1.3.1 Getting Expectations Wrong
In January 2008, certain events caused me to rethink the traditional pedagogical
approach to mathematics that I was implementing, and particularly to question its
assumptions regarding the role of the learner. It started with a student who took her
GCSE mathematics examination twenty months early, in November 2007, at the
start of Year 10.
Kaome (real name withheld) was one of the students in what was then my Year 10
mathematics class. My class was a border-line class; mathematics teachers use
this term to refer to groups of students whom the faculty considers to have difficulty
learning mathematics, but who, with academic support, could achieve a pass grade
C in the GCSE examination at the end of their secondary education (June 2009 for
Kaome). Achieving a grade C was of great importance to schools, given the
presiding governments accountability measures, which were based on the
percentage of students who achieved a grade C and above in subjects (see section
1.1.1). Kaomes academic profile hitherto was based on her performance in the
35
primary school Mathematics Standardised Assessment Tests (Year 6 SATs) that
positioned her on entry to secondary school as of average abilityhaving achieved
the national expected level (Gibbs, 2011) and her performance in the Year 9 SATs
that positioned her as of border-line ability having achieved the national expected
level (Department for Children, Schools and Families, 2009)]. Based on our schools
internal statistics, 52% of students who achieved a level 5 in their Year 9 SATs
achieved a C grade [or above] at the GCSE level Kaome achieved a level 5 in her
Year 9 SATs.
Being border-line, Kaome was availed of only a limited field of mathematical
concepts; she was perceived by the pedagogical authorities as lacking the cognitive
capacities required to engage with higher-level concepts. Moreover, the teaching
procedures even prevented Kaome and other border-line students from being able
to explore such concepts on their own. For instance, when covering the topic of
linear equations such as 2x + 7 = 15, students in border-line classes were only
exposed to methods informed by what is happening to x?-style flow diagrams such
as that shown in figure 1.2 below.
Figure 1.2 Solving Linear Equations in Border-line GCSE Mathematics Classes
This method cannot be applied to equations such as 2x + 7 = 3x + 11, which have
variables on both sides of the equal sign.
36
In January 2008, Kaome achieved a grade B in GCSE Mathematics. Her parents
had sent her to a Saturday school in preparation for the November 2007 GCSE
examinations. In personal communications, her mother informed me that the
Saturday school had expected Kaome to achieve an A grade; thus, their view was
that she had underachieved, while we (myself and the mathematics faculty) believed
that she had over-achieved. It came down to a difference in expectations.
After achieving a B grade, Kaome moved to the higher-ability mathematics class; the
faculty no longer considered her to be a border-line student, but now assumed that
she was capable of reckoning with more advanced material. Due to a subsequent
change in self-perception, she herself behaved like such a student who achieves A
grades in both GCSE Mathematics and Statistics. Three years later, she went on to
study medicine at university.
As a mathematics teacher, I had judged Kaome wrongly; I had relied on statistical
information to limit my expectations of my students, including Kaome. In doing so, I
justified to myself the restriction of the mathematics learning that I made accessible
to them. As a consequence of my experience with Kaome, I decided to change this
approach.
1.1.3.2 Students Taking Responsibility for their Mathematics Knowledge
As Head of Faculty, in November 2009 I decided to give all students in Year 11 the
opportunity to enter their GCSE examinations eight months early. As a result of this
decision, the school achieved its best GCSE Mathematics results to date, with 84%
of the cohort achieving a grade C or better by the end of Year 11. In the following
37
year, the mathematics faculty allowed all students in any secondary-level year group
to enter GCSE Mathematics at a time of their choosing within the broader timeline of
secondary study. Expectations for achievement became the responsibility of the
individual students themselves; expectations became an index of students beliefs
about themselves and their own agency, and were no longer limited by teachers or
based on past examination performance.
What became immediately noticeable to myself and my fellow faculty members was
the change that took place in students participation in their learning once they had
decided to sit their GCSE examinations. The students took responsibility for what
they did not know and sought to know; they became more tenacious and creative in
their desire for knowledge, and supported each others learning. Over the next four
years, the faculty achieved figures ranging between 79% and 84% of students
achieving a grade C or above. More students referred to themselves as good at
maths, and, upon receiving their results in the January of the academic year, it
became common for students to register to take the next set of GCSE examinations
to achieve a better grade. In September 2013, however, the government began to
penalise schools for early entry examinations, and our faculty stopped offering this
opportunity to students.
1.1.3.3 Questioning the Taken-for-Granted
Having observed how early entry for GCSE examination challenged the taken-for-
granted relationships between assessment procedures and student performance, I
challenged myself to look further beyond my current thinking. Part of the learner
38
discourse that I had initially internalised tended to link certain coordinates, such as
presumed ability, ethnicity, gender, and economic profile to students’ mathematics
achievement (cf Boaler et al., 2011). Subsequent independent research led me to
discover that, beyond what I had seen as fact or simply assumed, other factors such
as students perceptions of gender and ability can impede progress, especially
during group work (Pozzi et al., 1993). I observed that the differential performance
of ethnic minority groups is partly explained by other factors such as their attendance
of lower-performing schools (Kingdon & Cassen, 2010); and that the teachers
attitude towards characteristics such as ethnicity (positive or prejudicial) can have a
significant impact on students participation and achievement in mathematics
learning (Boaler et al., 2011). The literature confirmed what I had come to realise:
that my perception of my students influenced my behaviour towards them, and,
therefore, their experiences in my classroom. I decided to attempt to bring my
actions in line with my expectations of the students.
I started by changing how I expected the students in my classroom to learn
mathematics; I sought to restore the motivation I noticed in my students when they
were able to take responsibility for the timing of their entry into GCSE Mathematics.
Above all, I wanted them to make more decisions about what they wanted to learn
and how they learnt it. Two years before the commencement of my doctoral
research, I began pursuing this aim by giving the students in my mathematics
classes the opportunity to choose the sequence in which we would learn the topics in
the curriculum; I also gave them new responsibilities, asking each to prepare a
mathematics topic and teach it to their peers, with the hope of bolstering their
confidence in their abilities. While I sincerely believe that they had a positive effect
39
on my students’ participation, given their informal nature, I could not effectively
analyse the impact of these new measures. This research study was undertaken in
order to discover, with analytical clarity, the most effective means for improving the
conditions of my students engagement with mathematics.
Thus, this study aims to empower students to actively participate in all aspects of
their mathematics learning in order to improve their relationship with the subject and
their grasp of it. Most concretely, I am concerned to discover more effective forms of
pedagogy that encourage students to apply their knowledge rationally to solve
problems in the secondary school mathematics classroom; and, ultimately, that will
improve their performance in GCSE examinations. To this end, I explore how
shared epistemic agency is developed and sustained in mathematics classrooms.
Shared epistemic agency, discussed in full in chapter 4, is the central concept that I
have developed and used to describe and analyse students participation in learning
environments for the creation of knowledge. I propose that students with this type of
agency are actively engaged in their learning, taking responsibility for what they
know and do not know and acting to further their own and their peers knowledge; if
this agency is able to be sustained over a period of time, I hold that it is a powerful
facilitator of the advancement of the collective knowledge of all the students in the
classroom.
40
1.1.4 The Thesis Outline.
I have organised this study across six further chapters. In chapter 2, in which I
develop my theoretical framework, I review literature on the key concepts of agency,
social learning theories, pedagogy and the existing constructs of knowledge building,
knowledge creation and shared epistemic agency that inform the design of the study.
This review leads me to conclude that my elaborated idea of shared epistemic
agency, which embodies the six essential characteristics of the ideal learner that I
have extracted from the extant literature on education theory, was the particular kind
of agency required to improve the participation of the students in my classroom. An
innovative pedagogy that could support the development of this agency was also
needed. Students with shared epistemic agency: intentionally act to resolve a
mathematics unknowing, they seek to extend their knowledge, they explicate
knowledge to each other, they take control of the learning process and as a result,
they create knowledge new to them. The review revealed characterisations of
shared epistemic agency in short-term classroom projects, outside a high-stakes
assessment system, but these were important differences to my classroom setting.
The following research questions then emanated
1. What are the indicators of shared epistemic agency in the mathematics
classroom?
2. What sustains the emergence of shared epistemic agency in the mathematics
classroom?
Chapter 3 presents the qualitative action-research methodology I employed
throughout the study, and the specific research design that it informed, which was
41
developed to fit my particular aims. In this chapter, I explain the innovative
pedagogy that is at the heart of this study, address the ethical issues in relation to
the intervention and explain the methods of data collection.
In the chapter 4, which concerns my analytical methods, I present an original unit of
analysis: an Episode of shared epistemic agency that exemplifies the objects of
interest; that is, the interplay of the six characteristics mentioned above. An Episode
is a snapshot of students purposeful interactions to resolve an unknowing, hence
produce knowledge new to the students. Focusing the analysis on Episodes thus
allows me to select relevant moments from the hours of data.
In chapter 5, I present the findings from these Episodes. To facilitate answering the
research questions and meet the aims of the study, in the first section of this chapter,
I used my analysis of episodes to present a more detailed description of how the
characteristics of shared epistemic agency manifested in the classroom as the
students enacted the innovative pedagogy. In the second section, I elaborate on
what was unique about students’ epistemic interaction and I present findings that
highlight how student positionings and authority impacted on the way they advanced
mathematics knowledge in the classroom.
Subsequently, chapter 6 contains a discussion of the two themes that emanate from
these findings in responds to the research questions. I critique the idea of shared
epistemic agency as an encapsulation of the six characteristics and I propose a
more holistic view of the construct. The chapter also puts forward a
conceptualisation of the student and the mathematics classroom that emerged from
42
the study and it reflects on the action research process. This reflection focusses on
my role as a participants and the innovative pedagogy.
In the seventh and final chapter, I outline the contributions this study makes to the
field of mathematics education, and I present a challenge to current educational
policy and classroom practice. In my contribution to theory, I present my extension
to the existing construct of shared epistemic agency and I indicate the extent to
which this study has fulfilled its aims of participation and empowerment. My final
contribution as a teacher researcher identifies the value of action research as a
meta-methodology. I note the limitations of the research study and end with a call for
teachers to become researchers in a bid to improve the profession.
43
2 THEORETICAL FRAMEWORK
This chapter, organised into five sections, reviews the key concepts and constructs
that inform this studys design. The first section addresses the concept of student
agency, which includes the narrower concept of epistemic agency, and outlines the
learning theories pertinent to its forms. The second section focuses on social
theories of learning, in particular addressing the knowledge-creation metaphor of
learning and Wengers (1998) communities of practice. The third section considers
the conventional pedagogy that this study seeks to transform in order to achieve its
aims, and discusses a picture of authority that is useful for describing pedagogy in
general and teachers and students participation in it. The fourth section lays out the
twin theories of knowledge building and shared epistemic agency that underpin this
study. In contrast, the fifth section examines other studies that have worked on
transforming pedagogies, especially within the context of mathematics education,
supporting my claim that what these studies lack is a focus on an innovative
pedagogy such as I am developing that supports everyday practice in the
mathematics classroom.
2.1 Agency
This section discusses three approaches to agency that have informed the approach
that I develop and utilise in this study. They are:
Bandura’s individualistic and calculative perspective on human agency as the
capacity of individual human beings to make choices and to act on these
choices in a way that makes a difference in their lives (Martin, 2004, p. 135).
44
This perspective opposed the tradition of behaviourism that viewed human
behaviour as determined mechanistically by environmental stimuli.
Emirbeyer and Mische’s situated agency (Emirbayer & Mische, 1998, p.
963)), which, drawing on the work of influential 20th-century social
philosophers George Herbert Mead, Hans Joas, and John Deweys, views of
agency as a rational and evaluative capacity. In their view, individuals (actors)
can respond to changing environments by continually reconstructing their
view of the past as they attempt to understand the conditionings of the
emergent present, and use this subsequent understanding as the basis upon
which to shape and control their future responses. The inherited conception
of a “deliberative attitude” (Mead, 1932, p. 76) represents actors as able to
actively constitute their environment by selectively controlling their responses
to emergent situations and structural factors such as race, culture, gender,
and poverty that otherwise constrain their agency.
Scardamalia’s epistemic agency, which identifies the academic sphere as a
locus of the knowledge-building practice of learning, and which connects this
practice with the general capacity of the human being (Bereiter &
Scardamalia, 1998). Epistemic agency refers to the amount of individual or
collective control people have over the whole range of components of
knowledge building (Scardamalia & Bereiter, 2006, p. 106). The word
epistemic” itself, from Ancient Greek epìstamai (“to know”), means “relating
to knowledge and knowing.
These three approaches to agency are all underpinned by an attention to the social
and relational qualities of agency, though the first two have slightly different
45
backgrounds and assumptions from each other. While this study draws on ideas
from both Bandura and Emirbayar and Mische’s theories, Scardamalia’s work on
epistemic agency is the primary influence.
2.1.1 Human Agency
This section starts with the work of Albert Bandura the locus classicus of a
discussion of agency to which a considerable majority of researchers in the social
sciences have referred since its initial dissemination. In a gesture that helped to
make him one of the most influential psychologists in modern history, he challenged
the then-predominant behaviourist perspective, positing his Social Cognitive
Theory of learning and development. Bandura dealt with human behaviour and
agency in terms of a triadic framework of reciprocity among environmental variables,
behaviours, and personal factors such as cognition (Bandura, 1999, p. 156) He later
extended this theory to address how people seek to exercise control over their lives
by means of the self-regulation of their actions and thoughts (Bandura, 1986). He
claimed that much of human behaviour is performed not only to accommodate the
preferences of others, but is also “motivated and regulated by internal standards and
self-evaluative reactions to [one’s] own actions” (Bandura, 1986, p. 20). Moreover,
he argues for construing agency as emergent and interactive, claiming that thoughts
emerge from neurological processes initiated and sustained by social interactions.
From this socio-cognitive perspective, he identifies four moments of human agency
that determine the influence of thought on human actions: intentionality
(distinguished from the ‘intentionality’ that is discussed by earlier psychologists
46
Brentano (Fréchette, 2013) and Husserl (Husserl et al., 2019)) , and which
continues to be used as a term in cognitive science and philosophy of mind),
forethought, self-reactiveness, and self-reflectiveness (Bandura, 2001). Agency, in
the first place, can be understood as a characteristic of whosoever carries out their
actions intentionally; people are agentic if their actions are intentional. Intentions
themselves are understood as the proactive commitment to bringing about a desired
outcome. Furthermore, successful outcomes that are brought about accidentally,
even with intention, are not viewed as agentic, given the separation of intention from
the decisive action or event. On the other hand, a successfully intentional action may
confer agency on a person even if it does not succeed in bringing about the desired
outcome. The critical feature of individual agency is the power to generate actions for
a given purpose, regardless of whether the outcome of such actions is of benefit or
not, or whether it produces the intended consequences. Student A asking an
adjacent student, B, for help with a mathematics question is evidence of a students
intention to solve a mathematics problem. The agency emerges in the activity of
asking for help, and is present regardless of the outcome of the request that is,
whether or not help is eventually received or whether such help in fact leads to a
correct solution.
Forethought extends agency temporally beyond the present moment of intentionality,
connecting it with forward-directed planning (Bandura, 2001, p. 7). People anticipate
future consequences of their actions and select current actions to bring about future
success. An anticipated future success cannot be a source of current motivation and
action (i.e. an intention) since it does not exist. However, when individuals represent
the consequences of their intended actions cognitively in the present, they become a
47
source of present self-guidance, motivation, and behavioural regulation in
anticipation of a projected goal and future outcome. Individuals exercise agency by
acting to shape the present to meet a desired future. In this sense, they transcend
the constraints of the present. Following through with the previous example, student
A asks the questions of student B because they feel that student B’s response would
help them solve the mathematics problem, a goal which it is in their interest to
achieve. The decision to ask the question requires a degree of forward-planning. In
Bandura’s terms, forethought is the capacity of student A to be motivated to
persevere with seeking to answer the question, as student A can imagine the future
benefits that will accrue if they can solve the mathematics problem (p. 7)
Self-reactiveness as a feature of human agency is the ability of the individual to
motivate and self-regulate themselves to execute intended actions for a desired
outcome. It includes all the sub-functions of self-regulation that link thought to action,
such as self-monitoring, self-guidance, and self-correction. Self-reactiveness is an
important element for the achievement of intended actions. Thus, in our example,
student A is not only a planner and a forethinker; they can also change how they
behave in order to encourage student B to give them the answer to their question or
to answer further questions. This could involve such strategies as, for example, not
giving in to frustration if student B is too slow to respond.
Having solved the problem with the help of student B, student A can also look back
and decide on whether their course of action was the right one. This attests to
Banduras final feature of agency, self-reflectiveness: the capacity to understand and
be aware of ones thoughts and actions and to evaluate their adequacy. In this
48
metacognitive activity, individuals judge the validity of their predictions against the
anticipated outcome of their actions. They consider external effects, such as the
impact of other peoples actions, established practices and beliefs, and the
anticipated impact of these factors on their future success. Peoples beliefs in their
capacity to exercise control over their own functioning and over environmental
events constitute the final frontier of human agency (p. 10). People act because they
believe they can produce effects with their actions. The strength of ones belief in this
ability correlates positively with the effort invested in actions.
Bandura’s social cognitive theory also recognises the necessity of collective agency
in the precipitation of positive effects; indeed, it is clearly the case that individuals
work with others to bring about what they cannot accomplish independently. A key
ingredient of collective agency is the belief, mutually held by the individuals that
make up a group, in their collective power to bring about the desired results;
Bandura refers to this as the belief of collective efficacy , noting that it consists in
the group members knowledge, intentions, skills, and the interactive, coordinated,
and synergistic flexibles of their transactions (p. 14), which together determine the
groups attainments.
Although Banduras view of human agency is interactive and relational, it still
emphasises the capacities of the individual, even as it recognises collective agency.
This individualist view, though proffered in the distinctive context of modern
psychology, can be traced back to the conception of agency as personal autonomy
leading to individual empowerment and emancipation that was articulated by
Immanuel Kant in the 18th century (Biesta & Tedder, 2006, p. 4). An emphasis on the
49
empowerment of the individual student, who can follow a course of action to meet a
desired outcome, and persevere and reflect on the achievement of the outcome for
future purposes, is relevant to this study. However, students in mainstream
education do not learn in isolation, and this study would be limited if it did not
progress beyond the individual perspective alone. Schools are institutions with
social structures such as rules and regulations, traditional teaching practices,
curriculum maps, and school-wide assessments. Educational policy that includes, for
instance, the GCSE curriculum also has bearing on the agency of students. Both
social structures and educational policies impact students and their agency in
emergent classroom situations. They impose competing views of how students
should engage with learning and constrain the actions they may want to take to
produce an outcome, or cause students to re-evaluate their thoughts, habits, and
beliefs about consequential outcomes. Since this study seeks to challenge the
received views of students as passive and constrained, it requires as a framework a
conception of human agency that follows Bandura’s – in other words, one which
considers the subject to be emergent, dynamic, and interactive while also
mitigating the individualist emphasis of the latter’s theories, in order to account for
the distributed nature of the social and policy-led pressures that weigh on the
students’ agency.
2.1.2 Situated Agency
In order to do justice to this interplay, I turn to Emirbayer and Misches sociological
conception of agency as situated (Emirbayer & Mische, 1998, p. 963). As noted
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above, Emirbayer and Mische drew on the work George Herbert Mead; they were
also influenced by Hans Joas, and John Dewey’s work, situating them within the
tradition known as American Pragmatism. Pragmatism rejects the mind-matter and
rational-normative dichotomies, offering a theory of knowledge that takes as its point
of departure the interactions and transactions that take place in nature itself
understood as “a moving whole of interacting parts” (Dewey in Biesta, 2014, p. 36).
On a pragmatic view, the experiences of living organisms cannot be separated in
thought from their implication in an environment; organisms interactively adapt to
their living circumstances, and are constituted by their attunement to ever-changing
environmental conditions. Emirbayer and Mische (1998) characterise their approach
as relational pragmatics’, due, on the one hand, to their allegiance with
contemporary and classical pragmatism, and, on the other, to their conception of
agency as intrinsically relational and social (p. 973). Their view of agency focuses
on actors and their engagement (and disengagement) with the different contexts and
environments that constitute their flexible yet structured social universes.
Emirbayer and Mische argue that a conception of agency should neither be limited to
considerations of the individual pursuit of interests and needs (as in the Kantian
tradition), nor to a view of human actions as totally constrained within cultural and
structural contexts (as, for example, in structuralist anthropology (p. 974). Thus, they
seek to reconceptualise agency in order to account for the historical and temporal
nature of human experience, and to demonstrate how this temporality interacts with
structural contexts informed by the past and oriented towards the present and the
future. In their view, human actions, through an interplay of habit, imagination, and
judgement, reproduce and can also transform the contextual determinations to which
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they respond. Individuals can orient themselves towards the past, present, or future
at any point in time, and change their orientation as they see fit. Applied to our case,
the consequence is that students can and do change their relationships with each
other and with their contexts.
According to Emirbayer and Mische, agency has three dimensions. In the first
dimension, the iterational element, individuals can change the dogmatic schemes of
action that have developed over time in a society (p. 976). Their agency lies in the
capacity for selecting, deciding, locating, and recognising which actions to change,
or else contemplating whether to reproduce existing schemas of experience,
activities, expectations of others, or situations developed in the past. In other words,
it involves participants knowing what to do with existing knowledge and practices.
The second dimension is the projective element, on which agency is conceptualised
as the ability of individuals to reconfigure their current actions to create a desired
future. This dimension, that draws parallels to Bandura’s forethought and self-
reactiveness, is the creative-reconstructive dimension of agency, where existing
cultural practices do not constrain agents’ actions, but rather, constitute challenges
to which they can respond. They are able to invent new thoughts and actions to bring
about a desired future, and do not have to repeat existing actions and established
practices; they can develop new responses to problems. They use current
knowledge to move beyond themselves and decide where they are now, where they
want to be, and how to get there from where they are in the present (p. 984). The
third dimension, the practicalevaluative element, views agency as the capacity of
individuals to exercise contextual judgements. That is, prudent, intelligent, and
practical decisions concerning which actions to perform in order to address
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problematic situations. Here, agency lies in agents’ ability to read the present
situation and make decisions in real time that may challenge a given state of affairs.
This element sees participants increasing in their capacity to bring about change
where the consequence of their actions cannot be structured or controlled. In effect,
Emirbayer and Mische posit that human agency should be conceptualised as a
temporal embedded process of social engagement informed by the past (in its
habitual aspect), but oriented towards the future (as a capacity to imagine alternative
possibilities) and towards the present (as the capacity to contextualize past habits
and future projects within the contingency of the moment) (p. 963).
Emirbayer and Mische stress that these three dimensions of agency are analytical
distinctions, and that all three can be identified in various degrees within any
empirical instance of action. In Figure 2.1 below, I relate these three dimensions of
agency with the moments of Bandura’s analysis. As an individual proactively
commits to bringing about a future action (intentionality), sets in place a course of
action to bring about a future result (forethought), motivates themselves to see their
plans through (self-reactiveness), and reflects on the adequacy of their actions (self-
reflectiveness), this individual’s thoughts and actions are seen to be able to
transform or reproduce their structural environment, and can be informed by past
habits, oriented towards an imagined future, or based on present judgments.
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Figure 2.1 Relating Emirbayer & Mische’s (1998) Situational Agency to Bandura’s Human
Agency (2001)
These two perspectives on agency can be used to analyse the character of student
engagement in a school classroom community. Banduras view is relevant to the
extent that it elaborates on the features that underpin students actions as they
strive to produce a desired outcome. His theory offers insights into how students can
work interdependently with others to bring about outcomes that they cannot deliver
independently. Emirbayer and Mische, on the other hand, contribute the insight that
students can bring about desired outcomes by making ad-hoc decisions in the
present that could transform the restricted structures in which they are acting. In the
context of this study, these decisions might involve deviating from existing habits
relating to how students should act in the classroom and finding new ways to
develop mathematical knowledge, or indeed simply retaining good habits and
traditions. The following section explores how students decisions and actions can
lead to knowledge.
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2.1.3 Agency as Epistemic
Scardamalia (2002) argued that the notion of responsibility links human agency, as
defined by Bandura (2001), to knowledge, which is the central focus of
Scardamalias conception of epistemic agency. Knowledge arises from choices for
which the agent is responsible (Reed, 2001). To know, individuals or collectives
need to be in control of their actions and have the ability to determine how to apply
their will towards concrete forms of action. Individuals or collectives that take
responsibility for their learning are aware of what they know or do not know and act
on this awareness to advance their knowledge.
The idea that active engagement by participants is required for them to learn or
construct knowledge has its roots in Vygotskys social constructivism (Bereiter, 2002;
Scardamalia, 2002; Valsiner & Veer, 2000). Constructivism is a philosophical and
psychological perspective on learning that contends that individuals construct or form
much of what they learn through their actions and interactions in the world (Packer &
Goicoechea, 2000). The sociological applications of constructivism, which
emphasise the influence of the social environment on learning, have driven
contemporary discussions of agency, its meaning, and its expression in educational
environments.
Marlene Scardamalia, a psychologist and educational researcher who is considered
one of the pioneers of computer-supported collaborative learning, put forward the
notion of epistemic agency (2002) in the context of knowledge-building pedagogy
(Scardamalia & Bereiter, 2006). A self-described deep constructivist (Scardamalia,
2014), she distinguished between shallow constructivist methods such as guided
55
discovery (Brown & Campione, 1994), in which teachers plan what the students are
to discover, and deep constructivist methods, in which the highest-level capacities
such as planning and the evaluation of learning which, in our age, are typically
accorded only to the teacher are handed over to the students. Students not only
construct their understanding but the whole space of invention, operating as a
professional knowledge environment (Scardamalia, 2014, 2:20mins). Emerging from
the context of this new learning environment, Scardamalia presents her notion of
epistemic agency as the metacognitive ability concerning goal-setting, motivation,
evaluation, and long-range planning (Scardamalia, 2002, p. 79). In her view,
students with epistemic agency assume responsibilities typically left to teachers,
and, pace Bandura, these students can have collective metacognitive abilities that
are different from the mere combination of individual ones. Collectively, students who
take responsibility for their learning, form ideas, relate them to others’ ideas, and
agree upon an ideal compromise. It is the collective contribution of students that
results in and sustains the collective knowledge advancement.
Scardamalia did not provide a clear theoretical account of the concept of epistemic
agency, nor describe how it can be identified in an educational setting. I consider
the idea of epistemic agency to emerge from her work on collective cognitive
responsibility (Scardamalia, 2002). Collective cognitive responsibility exists in groups
such as medical teams that carry out knowledge-based work. These groups exhibit
qualities such as flexibility, continued learning, collaboration, and rational thinking.
Though each member has a specific duty and/or area of expertise, roles are not
necessarily fixed. When problems arise, team members can take over from each
other without relying on a higher level of authority. The groups success is
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distributed across all of the individuals, rather than attributed solely to the leader. In
addition to the more tangible and practical aspects, individuals within these teams all
take cognitive responsibility to acquire the knowledge that their activities require and
ensure that everyone is adequately knowledgeable. In teams with collective cognitive
responsibility, the individuals and the teams are more productive and innovative than
those without such responsibility (Scardamalia, 2002). A classroom in which
students develop epistemic agency exhibits the characteristics of collective cognitive
responsibility. These classrooms act as a community, with collective contributions
creating new knowledge and advancing collective knowledge.
2.1.4 Summary
This section has outlined three conceptualisations of agency that emanate from
three related perspectives on knowledge, learning, and human development. The
first perspective, from which Bandura’s (2001) conception of human agency
emanates, is Social Cognitivism. This perspective views learning as a reciprocal triad
of personal factors, environmental variables, and behaviour. What is in students
minds (thoughts, beliefs) and the teachers expectations (rules, procedures)
influence students actions and the outcome of these actions. The second
perspective is Pragmatism, the perspective of Dewey (1900) and Mead (1932) that
represents knowing as based on ones experiences in one’s environment; this
informs the relational pragmatist viewpoint of Emirbayer & Mische (1998) that
represents the relations between ends and means as pre-eminently dynamic, and as
unfolding and ongoing processes (Emirbayer, 1997). This view recognises that each
57
student experiences the world uniquely and can react to this experience
idiosyncratically as the situation changes for them. Finally, the third perspective of
deep Constructivism (Bereiter, 2002; Scardamalia, 2002) argues in favour of
students taking responsibility for what they know and do not know and creating
knowledge from this process. These three perspectives of agency are compatible,
and dovetail in the notion of epistemic agency, on which taking responsibility for what
one knows or does not know transforms individual-situational agency into a new form
of agency related to knowledge.
I hold the position that students have the capacity to change and adapt to an
innovative pedagogy. While I recognise the agency of the individual students and
that of students as a collective as they respond to their classroom learning
environment and its pre-existing structures, I lean towards the notion of deep
constructivism, appreciating that students can create knowledge as they take
responsibility for their learning in a secondary school mathematics classroom. To
supplement this perspective, I require a theory that reconciles the social character of
learning with this interest in classroom practice.
2.2 Theories of Social Learning
Epistemic agency, as Scardamalia defines it, is, in the classroom context, a quality
that sustains the creation of new knowledge by the collective contributions of
students who take responsibility for their learning. Having established this, I can
identify one goal of this study to be the designing of a pedagogy that supports
students in the development of such agency. This innovative pedagogy, elaborated
58
upon in chapter 5, restructures the classroom as an environment in which students
can learn as a community. To this end, it draws upon Sfard’s (1998) two metaphors
of learning, the knowledge-creation metaphor (Paavola et al., 2006), and the social
perspective of learning (Wenger, 1998); each of these connects the pedagogical
environment with a notion of the community of practice. In the section that follows, I
will review these theories to the extent that they underwrite the development of my
own theoretical construct. This review will include an elaboration of the notion of
community and power relation.
2.2.1 Metaphors for Collective Learning
Metaphors for learning respond to questions such as who the subject of learning is,
the kind of knowledge they should learn, and how they learn it. They reveal certain
essential features of learning by asking us to consider it in terms of other behavioural
practices. In her article On Two Metaphors for Learning and the Dangers of
Choosing Just One (1998), the mathematics educator Anna Sfard proposed two
primary ways of thinking about how learning occurs: the acquisition and the
participation metaphors. The acquisition metaphor depicts knowledge as the
capacity of an individual mind, and learning as a process whereby the individual is
guided in assimilating or constructing pre-given knowledge. Sfard’s participation
metaphor, on the other hand, focuses on knowing rather than knowledge.
Knowledge does not exist in individuals minds or in the world, but is situated in the
cultural practices of a community (Lave & Wenger, 1991; Rogoff et al., 1998;
Wenger, 1998). Learning occurs as individuals participate in and are inculcated into
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the forms of life that constitute their community. Sfard’s presentation of the
participation metaphor does not seek to inspire changes in pedagogical practice;
rather, her focus is on mastering existing practices. However, thinking her
participation metaphor together with the notion of a community of practice as
discussed in section 2.2.2, it is clear that participants could, through active
negotiation, develop a practice that is both historical and dynamic (Wenger, 1998, p.
53).
Indeed, Paavola, Lipponen, & Hakkarainen (2006) suggest an approach that relies
upon but goes beyond the two metaphors mentioned above, highlighting the capacity
for advancing collective knowledge. Their metaphor, that of knowledge creation(p.
536), addresses the possibility of innovative learning activities for the creation of
knowledge; taking it seriously requires a theory or model of learning that clearly
emphasises innovation in relation to learning and knowledge. The knowledge-
creation view of learning is connected with the theories of knowledge-building
(Scardamalia & Bereiter, 2010) and knowledge creation (Nonaka, 1991) that I
discuss in section 4.1 in order to examine what is vital in knowledge communities
and innovations in learning, and, ultimately, in order to suggest new approaches to
pedagogy.
This knowledge-creation approach to learning explicitly builds upon Sfards (1998)
two metaphors for learning. The acquisition metaphor represents the monological
view of human cognition and activity, according to which important events happen
exclusively within the human mind. In contrast, the participation metaphor
emphasises a dialogical view of human cognition, whereby important events such
as learning occur as the individual interacts with culture, other people, and the
60
surrounding environment. Finally, the knowledge-creation metaphor corresponds to
a “trialogical model (see Figure 2.2); emphasis is placed on the way individuals
collaboratively develop shared knowledge objects and artefacts (Paavola et al.,
2006, p. 539). In innovative knowledge communities based on the third model,
learning occurs during collaborative practices that create shared objects of
knowledge.
Figure 2.2 The Three Metaphors of Learning (Paavola & Hakkarainen, 2011, p. 535 - 557)
This proposed innovative pedagogy stands in contrast with a conventional pedagogy
(see section 2.3), which relies on the acquisition metaphor, considering the teacher
to be in sole control of the transmission of knowledge, and rendering students as
passive receivers of this knowledge, having no other role than to store the
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information received from the teacher. The participation metaphor suggests a
pedagogy in which the students are not passive but are required to take an active
role in their learning, and points to the idea of students learning as a community.
Indeed, in most pedagogies that uphold this metaphor, such as the community of
learners model (Brown & Campione, 1996; Rogoff et al., 1998), the classroom is
organised as a community with the students working together, all serving as
resources for each other and guided by the teachers leadership.
The knowledge-creation metaphor allows for a further departure from this model,
allowing me to describe the classroom and pedagogy that I propose in this study, as
it examines learning in terms of the social structures it creates and the existing
processes of collaboration that support innovation and knowledge advancement.
This pedagogy views learning as a social process while still recognising the
competencies and initiatives of the individuals that make up the community. It
focuses on the process of innovation that occurs as people interact, rather than on
the contents of individual minds, and brings the dynamics between individuals and
environmental structures for creating new knowledge to the forefront. The
individuals initiative feeds the communal effort to innovate, while the social
environment feeds the individuals initiative and cognitive development. Constructing
shared objects of knowledge requires more than dialogue; it requires the interaction
of individuals contributions and collective contributions in a community learning
environment. The proposed pedagogy will focus on students both individually and
collectively taking responsibility for their own knowledge creation; the knowledge-
creation” metaphor underpins this pedagogy.
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2.2.2 Communities of Practice
My proposal for a new pedagogy based on knowledge creation also requires a
sufficiently dynamic conception of the community in which learning takes place. My
thinking about community draws on ideas of communities of practice in the work of
Etienne Wenger, wherein community relations are of mutual benefit to participants in
achieving their shared goals and advancing their mathematical knowledge. In
communities of practice, learning is not an individual experience, but rather a social
phenomenon that occurs as individuals engage in activities that are essential to the
community. Thus, knowledge is competent participation; knowing is the ability to
participate in the community’s endeavours, and learning involves the transition
towards such competence, changing who a person is. Figure 2.3 below shows the
components that characterise participation in Wengers social theory of learning and
knowing, and I will discuss them in turn.
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Figure 2.3 Components of a social theory of learning (Wenger, 1998, p. 5)
2.2.2.1 Learning as Doing
A practice is a way of doing things developed over time by participants of a
community of practice to fulfil their purpose of coming together. In a mathematics
classroom viewed as a community of practice, the classroom participants, that is, the
students and teacher, through their engagement (their doing) over time, develop
ways of communicating and behaving that fulfil their aim of learning mathematics.
These modalities of communication and behaviour could include students’
knowledge of how to communicate with each other and with the teacher, or of how
they access homework and receive feedback. These tacit and explicit classroom
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practices, negotiated over time, include actions and reifications (Wenger, 1998) that
are unique to the participants of that classroom. The term negotiation intends to
convey a flavour of continuous interaction, of gradual achievement, and of give and
take (p. 53), emphasising that the participants practice is a production of their
individuality of who they are as individuals and who they become as a community.
2.2.2.2 Learning as Experience
Our experience gives meaning to our participation in activity. Wenger used the
concept of the “negotiation of meaning to “characterize the way we as individuals
experience the world we are in and how we experience our engagement in it as
meaningful” (p. 53). For example, consider the case of students who attend
mathematics classes. As they engage in their learning, their activities develop into
patterns of action. It is the development of these patterns all over again, lesson after
lesson, that constitutes the experience learning mathematics of what mathematics
means to them. The term negotiation is used in the sense of continuous interaction,
of the continuous development of meaning through the interactions of participants
with their practices.
Reification is a connected term that, in this study, functions to explain the role of
material objects in the community of practice. The term refers to the capacity for
abstract, distributed, complex ideas to achieve reality as material objects as they
assume central functions within a practice. Thus, the curriculum is a reification, as is
a lesson plan or a tick in a student’s exercise book. In the classroom, reifications are
products of students experiences. It is the experience gained through their
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participation that gives meaning to what they do. A tick in the book, for instance,
conveys to the students that they are correct, knowledgeable, and have given the
right answer or solved the problem. A tick is authentic to the students because of the
meaning it projects. The meaning of the tick is based on their experience of being in
a classroom. Thus, when students say give me a tick or my work was ticked or
shall I tick it?, the tick itself is only representative; it is the experience of its
meaning, the idea of which it is a reification, that is really circulating. Summarily,
reification is the process of giving form to our experiences by producing objects that
congeal this experience into ‘thingness’” (p. 58). The actions and reifications that
form the practice of a mathematics classroom give this practice meaning that is,
make visible what learning mathematics is to the participants.
2.2.2.3 Learning as Belonging
The participants of the mathematics classroom negotiate what constitutes
competence in the practice of learning mathematics. Competence reflects the
actions and reifications that define belonging, that is, being a classroom community
member (Wenger, 1998). The community determines competence; it is what the
community recognises as competence that defines competence in their community.
In some mathematics classrooms, competence is accorded to students who
frequently answer questions or put their hands up, or else who complete the set work
quickly. Wenger emphasised, however, that belonging to a classroom community
requires more than competence alone; it also requires experience of participation.
Experience of participation includes mutuality of engagement, establishing
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relationships with other participants, engaging with them, and responding to their
actions and reifications. Accountability for the other participants includes doing what
is required to learn in ways acceptable to the community and its participants.
Competent members of a community show their belonging by participating in the
practices of the community.
2.2.2.4 Learning as Becoming
As individuals participate in the pedagogy of a mathematics classroom, they build an
identity that emerges from the negotiation of what it means to be a member of the
classroom community and to engage in its practice of learning mathematics. A
participant’s identity in the classroom is who they become as a member of the
classroom, how they influence the community practice, and how it influences their
participation. As participants of a mathematics classroom engage in the practice of
learning mathematics, other participants develop relations with them that reify them
based on their competence; they are viewing, for example, as good at algebra, at
explaining, or at showing their working. Identity involves how we experience our
participation and how others project their reifications of our participation on us.
Identity can be defined, then, as a layering of events of participation and reification
by which our experiences and its social interpretations inform each other (p. 151).
A mathematics classroom operating as a community of practice can benefit from the
mutual relations inherent in any community with a common purpose. The purpose of
advancing their collective mathematics knowledge directs students participation and
practice. As they participate, they negotiate this practice and develop competence
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as mathematics learners. Their competent participation and participation
experience make them belong to the classroom community; as such, participants
develop identities and influence the identities of other members of the classroom
community. While these ways of viewing learning in a community complement the
aims of this study, Wenger’s trajectories of participation (Wenger, 1998, p. 153) that
legitimises unequal forms of membership in a community of practice is at odds with
this study’s view of community. In addition, Wenger has been criticised for his
benevolent view of community and for not considering the detrimental impact
power/knowledge relations can have on the members of a community (Creese,
2009; Paechter, 2003; Tusting, 2005).
2.2.2.5 The Notion of Community
The notion of community I propose for my mathematics classroom, to meet the aims
of this study, draws from the work of the British philosopher John Macmurray. He
views community as a mode of unity informed by relationships of the individuals
intrinsic worth (McIntosh, 2015, p. 14). These communities are not created or
sustained by force but emerge voluntarily and are sustained through friendship. He
argued in his book Conditions of Freedom (Macmurray, 1950), that what
differentiates these communities from society in general is that they are constitutive
of equality and freedom (p.73-74). Akin to friendship, where there is equality of
consideration and value, each member of the community has equal value, and their
voice counts equally. This does not imply that the individuals are not different in
terms of their natural disposition or their capabilities, rather in these communities, the
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relations between members overrides these differences. The individuals are free to
be their authentic self’s and express their uniqueness. In essence, equality and
freedom are mutually inclusive, they are conditional of each other. Being equal
means one can act in accordance with their nature and freedom of expression is
made possible amongst equals. Macmurray’s view of community suggests human
relations in which the individual and the community are interdependent, “we enter
into personal relations with others because it is through them that we can be and
become ourselves” (Fielding, 2012, p. 685). The learning is learning to live as a
community, both the teacher and the students voluntarily take responsibility for
advancement of each other’s mathematics knowledge and avoid exercising their
freedom in a way that will limit the freedom or the voice of others (McIntosh, 2007, p.
75).
Though Macmurray was calling for education to focus on human fulfilment rather
than personal gain, and did not give illustrations of the freedom he described, I can
extend this mode of community to a mathematics classroom. The relationships
between students and teachers and or between students and students do not
depend on their individual functions, that is how they benefit each other, or individual
achievement, rather, it is about reciprocal caring for how each other feel in the
classroom as they learn together as equals.
I propose a democratic community where the relations of equality and freedom, that
exist between students and with the teacher, include participation with a democratic
stance (Vinterek, 2010); a classroom where the students trust and respect each
other, and have the freedom to take control of how they learn and what they learn,
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they exhibit a willingness to listen to others, to speak up and a willingness to give
voice to their own thoughts (p. 373). This proposed classroom community
contrasts with the relationships of power that exist in society. Classrooms are
microcosms of society, as such, if allowed, hegemony and relations of power, can
impacts on the relations between teachers and students and between students and
students.
2.2.2.6 Power Relations in Society and the Classroom as a Community
Relationships of power exist in all human interactions and structure human
behaviour (Foucault, 1978, p. 96). As individuals are constantly interacting, power is
constantly at play in these interactions. Foucault put forward a productive view of
power as both positive and negative. He analysed it as something that is capillary,
and circulates with individuals as vehicles of power, “not something that is acquired,
seized, or shared, something that one holds on to or allows to slip away” (p.94). In
effect we all exercise power and are subjected to power by others. In Foucaults
view, power exists only when it is put into action. “In effect, what defines a
relationship of power is that it is a mode of action which does not act directly and
immediately on others. Instead, it acts upon their actions: an action upon an action,
on existing actions or on those which may arise in the present or the future”
(Foucault, 1982, p. 789). That is to say, the exercise of power directs the conduct of
others, it opens possible actions or outcomes, that can be harmful or productive. It
also implies a degree of freedom or the possibilities of resistance from others
(Foucault, 1980, p. 780), otherwise, there would be no need to direct their conduct.
70
While all individuals or collectives are implicated in power relations, that does not
mean that all have equal power. Foucault also posited that power circulates as
knowledge and is visible in discourse and discursive practices, such as in the
discursive practices of a mathematics class. With this conceptualisation of power in
mind, in a classroom community, where teachers and students relate with each other
relations of power are at play and could have a positive or negative impact on
individuals and the community. This power that circulates will result from the
innovative classroom pedagogy, the discourse of schooling that ascribes knowledge
hence power to the teacher and from power ascribed to constructs such as race,
gender, class, and socioeconomic factors that act to marginalise individuals in
society. An awareness of the workings and source of power are important for this
study, if I seek a democratic classroom community as described in the previous
section.
2.2.2.6.1 Power Relations in Schools
Some sociologists have claimed that school are structured, designed and organised
to mirror the divisions, ranks and hierarchies’ existent in society (Giroux, 2011;
Giroux & Penna, 1979). The interconnection between ideology, pedagogy and the
curriculum acts as a tool to socialises students into society (Bernstein, 2009; Giroux
& Penna, 1979). Bourdieu argues that cultural reproduction occurs in schools by
normalising what constitutes as knowledge and truth (Bourdieu, 1990). He posits that
schools subtly reproduce the power relations that exist in society through mediating
the dominant culture that tacitly confirms what being educated means. Michael Apple
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(2004, pp. 2930) describes schools as agents of cultural and economic
reproduction, maintaining the inequity of society. Hence factors such as race,
gender, disability, sexual orientation, socioeconomic status, immigration that
disenfranchise sections of our society from participating equitably and
democratically (Fraser, 2012; Fraser & Sunkara, 2019; Wallace et al., 2022) can be
mirrored in school and in classroom as students and teachers relate with one
another.
Educational research (Boaler et al., 2000; Boaler & Greeno, 2000; Gore, 1995;
Hargreaves et al., 2021; Smith, 2014; Solomon, 2009c) show that, beyond the
inherent discursive practice of school, the power relations at play in society are
evident in the mathematics classroom and act to exclude students from full
participation. Class, culture and gender caused teachers to position students, in the
mathematics classroom, as competent or not competent thereby, restricting
student’s access to mathematics knowledge and impacting student’s self-belief in
their ability to participate in mathematics (Solomon, 2007, 2009c). This positioning
acts to limit access to good teaching for low-attaining students (Hargreaves et al.,
2021), and limits girls take up of A level mathematics (Smith, 2014).
2.2.2.6.2 Power Circulating between Teacher and Student
The discursive practice of school generally places the teacher by virtue of their
knowledge in a position of social dominance in the student teacher relationship,
referred to by Bernstein as “control of the social base” , (Bernstein, 2000, p. 30).
During student to student interactions, student can copy this teacher attribute and
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power as social dominance can circulate as mathematics knowledge limiting other
students contributions and mathematic meaning making (Langer-Osuna, 2017).
Though the reason why the students mimic the teachers behaviour could be related
to broader institutional norms that focus on competition and comparison rather that
community learning (Barron, 2000, p. 432). Understanding the power-relations at
work in the classroom is essential if this research seeks to achieve its aims.
As previously stated, power circulates as knowledge, knowledge which Foucault
posits is arbitrary. He argued that knowledge is a product of power relations
asserting what truth is constructed and kept in place through strategies such as
discourse that support and affirm it and exclude and counter other discourses
(Foucault, 1978, pp. 100101). Power operates in the processing of information that
selects what is being labelled as fact, that is, in what the curriculum and teachers
allow to be circulated in the classroom. This fact becomes the dominant discourse
and other dominated discourses are excluded. Knowledge within schools and in
society is carried out and kept in place through a wide range of strategies that affirm
and support it such as practice, institutions and hegemony, where those who are
dominated by others such as students, take on board the values and ideologies of
the dominant teachers in schools and accept them as their own: this leads to
students accepting their position within the hierarchy as natural or for their own good.
This internalisation of the dominant discourse by the dominated is the capillary form
of power. In this sense, discourses, truth, power and knowledge are intricately
linked. This interconnection may give an explanation why the power relations that
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exist between teachers and students are pervasive such that my attempts to change
this dynamic in the classroom may be resisted by the students this study aims to
empower.
I find Foucault’s work on discourses useful in helping me think about how I know what
I know; under what circumstances the information is produced, where it comes from
and whose interest it serves. Thus, the discourses of teaching and pedagogy do not
hold universal truth but are constructed and held in place by the practices of schooling,
it is thus possible to think differently about practices and to trace how what we in
schools accept as ‘true’ is kept in its privileged position.
Consequently, discourses can be seen as a means of resistance as well as a means
of oppression. Discourse transmits and produces power; it reinforces it, but also
undermines and exposes it, renders it fragile and makes it possible to thwart it”.
(Foucault, 1978, pp. 100101). Though transforming the pedagogy is an act of
resistance it is equally an exercise of power because both the students and the
teacher have the freedom to effect change. However, for change to be sustained,
the students have to feel that it is purposeful, I have to make the logic of the
innovative pedagogy clear to the students, and the aims of the study decipherable
(pp. 9495). To improve students’ relationship with and learning of mathematics, it is
possible for the students and I to interact on a basis of mutual authority and
competence. In exercising power, we can direct each other’s conduct towards
respect and trust and through enacting an innovative pedagogy build a democratic
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community empowering the students to take responsibility for what and of how
they learn mathematics.
2.2.3 Summary
At the beginning of this section, I framed a goal of this study, which is to develop a
pedagogy that would support students achievement of epistemic agency. To
develop responsibility for their learning requires a pedagogy in which learning is a
social endeavour. Thinking about this pedagogy begins with a decision about the
metaphor of learning used to describe who the subject of learning is, the kind of
knowledge learners should learn, and how learners learn in the pedagogy. The
knowledge-creation metaphor (cf. Paavola et al., 2006) provides a way of
conceptualising learning in terms of innovative communities of knowledge that does
not exclude learning as acquisition or learning as participation; instead, it
emphasises how individuals collectively participate to acquire shared knowledge
objects and artefacts. This metaphor of learning is of interest to this study as it
prepares the context in which epistemic agency can develop and gives form to the
goals of the innovative pedagogy that I am developing.
In the second part of this section, I outlined Wenger’s social learning theory that
discusses how learning can occur in a classroom that operates as a community of
practice, I highlighted the notion of community that will support the aims of this study,
considered the power relations at work in society and the relationships of
participation that this study’s innovative pedagogy aims to develop. However, while
the theory outlined four ways of learning in a community learning as doing, learning
as experience, learning as belonging, and learning as becoming it focuses on
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knowing rather than knowledge. This social learning theory that focuses of students
participating in established practices, therefore, could be viewed as being at odds
with a study that focuses on mathematics knowledge and innovative forms of
learning within a classroom community. Thus, I hope to draw on the ideas of learning
through participating in a community from Wenger (1998), while also moving beyond
them by means of the ideas of collective learning from Paavola et al. (2006) in the
design of my innovative pedagogy.
2.3 Pedagogy
This section focuses on aspects of pedagogy that will influence this study’s
innovative pedagogy design. The first sub-section will describe the conventional
pedagogy alluded to in section 2.2.1 above. The following two subsections will
introduce the constructs of authority and positioning. These two constructs show
how the pedagogy can impact the students experience of and relationship with
mathematics in a classroom.
2.3.1 The Conventional Pedagogy
The notion of conventional pedagogy that I introduce here has its roots in my own
experience (see section 1.1.3.2), as well as in Paulo Freire’s critique of what he
describes in his seminal book, Pedagogy of the Oppressed, as the banking concept
of education(1970, p. 72). In this conception, education takes the form of
depositing. The teacher, as the depositor, narrates knowledge to the student who
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acting as depositories, mechanically receives, memorises, and repeats the
information. I consider this teacher-student relationship akin to the acquisition
metaphor of learning introduced in section 2.2.1 above. In the banking model,
according to which there is an asymmetrical relationship between the teacher and
the students, the teacher controls the subject knowledge and its learning as outlined
in table 2.1 below.
The Teacher
The Students
teaches
are taught
knows everything
know nothing
thinks
are thought about
talks
listen meekly
disciplines
are disciplined
chooses and enforces their choice
Comply
acts
have the illusion of acting through
the action of the teacher
Chooses the program content
(who are not consulted) adapt to it
Confuses the authority of knowledge with his
or her professional authority, which he or she
sets in opposition to the freedom of the student
is the subject of the learning process.
are mere objects
Table 2.1 Attitudes and practices of the banking model of education. Quoted from
Pedagogy of the Oppressed (Freire, 1970, p. 73)
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Freire argues that this oppressive pedagogy prevents students agency from being
creative and transformative. He called for an equitable pedagogy based on inquiry in
which knowledge emerges only through invention and re-invention, through the
restless, impatient continuing, hopeful inquiry human beings pursue in the world, with
the world, and with each other (Freire, 1970, p. 53). Jacques Rancière, an
influential French philosopher who continues to engage with social issues, also
criticised the prevalent pedagogy of 1980s France that positioned students as of
unequal intelligence to the teacher. He called for an emancipatory pedagogy in
which the intelligence of students is recognised and not stultified by what he termed
the explanation logic (Bingham et al., 2010, p. 3). He posited that schools
presuppose students to be ignorant, and present knowledge as needing to be
explained by teachers; instead of making students’ intelligence equal to that of the
teacher, this explanation perpetuates the myths that further explanation is needed,
that students are unable to learn without the explanation of the teacher, and that,
therefore, they are always of unequal intelligence. Both Rancière and Freire called
for a pedagogy of equality, where the polarised view of teacher as knowledgeable
and in control, and the students as ignorant and powerless, is replaced by a
pedagogy in which students and teachers share authority in the classroom and learn
alongside each other.
Although each of these critiques of education had as their contexts different parts of
the world and moments in history, I see similarities between the banking model
observed by Freire, the inequality of intelligence described by Rancière, and the
pedagogy experienced by students in most parts of my school. This pedagogy is
clearly based on an unequal relationship between the students and the teacher
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similar to that outlined in table 2.1 above; I will refer to this as the ‘conventional
pedagogy’, and argue with Boylan (2010), Pratt & Kelly (2007), and Wright et al.
(2020) that this is typical of learning mathematics across England.
Critical mathematics education research, such as that of Gutstein (2006) and Wright
(2017), has drawn inspiration from the work of Freire (1970), and has developed
mathematics pedagogies with social justice commitments that help students to
understand the communities they live in and the ways inequality is contested and
produced in the world. This study does not seek to assume a critical perspective on
society, though there is an overlap with critical mathematics education in the fact of
this study’s desire for equality in the authority relations between students and
teachers.
An important difference in my aims here, compared with Freire’s and Rancière’s, is
that, while these thinkers aimed at overhauling society to achieve equality and social
justice, this study aims to achieve equality in the humbler context of the mathematics
classroom, and aims above all at improving the student’s relationship with the
subject in order to better facilitate their learning.
In the summary of the previous section, I mentioned that the innovative pedagogy
based on a knowledge-creation metaphor of learning would aim to have students
taking responsibility for their mathematics learning. Taking responsibility for learning
requires a pedagogy in which learning is a collective community endeavour, and in
which students participate in their learning actively; achieving this state of affairs is
an aim of this study. In the conventional pedagogy, wherein the teacher has sole
authority, these relations of authority can constrain students abilities to engage with
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mathematical ideas and reflect on their learning. In extreme cases, it interferes with
their ability to obtain mathematical insights and solve problems in the first place (cf.
Amit & Fried, 2005; Brubaker, 2012; Schultz & Oyler, 2006). Thus, in order to avoid
these pitfalls, I turn to a consideration of the phenomenon of classroom authority
itself in the preparation of my own pedagogy.
2.3.2 Authority in the Classroom
In an educational context, authority can be defined as a social relationship where
some people are granted the legitimacy to lead, and others agree to follow (Pace &
Hemmings, 2007). It is distinguishable from the form of power, which connotes
subjugation of one individual to anothers will by some form of coercion (see section
2.2.2.6). Instead, authority involves a relation of obedience and voluntary
submission that is quasi-reciprocal rather than coerced. Authority operates in
situations in which a person or group, fulfilling some purpose, project, or need,
requires guidance or direction from a source outside himself [sic) or itself (Benne,
1970, p. 392). Those who lead and those who submit are both relevant to
determining the claims to the legitimacy of the authority. Both can determine the
extent to which the need for guidance is fulfilled and change the relationship
accordingly. Authority requires legitimate claims to competence; otherwise, it
becomes a power relationship that involves coercion, a pattern of over and for, rather
than with (McNay in Brubaker, 2012).
Authority in education appeals to a value system or normative order that students
uphold with their teacher, giving sense to their relationship. Authority cannot be
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disassociated from the idea of freedom as the students are free to acknowledge the
legitimacy of the teachers authority (Perry et al., 2008). If the students are coerced
to accept the teachers authority, the latter cannot claim their authority as legitimate.
The students have the freedom to reject or resist the teachers authority, but do not
do so as they recognise its legitimacy (Goodman, 2010; Peters, 2015). This could
be evidenced in a classroom in which students often moan about the relevance of a
particular mathematics topic to their lives “Miss, how will this help me in real life?” –
but nevertheless capitulate to the curriculum requirements stressed by the teacher,
knowing of the future benefits of a good mathematics grade. Having said this, in
classrooms where the students exercise their freedom to reject or resist teachers
authority, they could expose themselves to negative consequences, and hence
coercion, regardless of whether a given teacher has legitimate claims to competence
(Hargreaves, 2017).
As stated in the previous section, in a conventional pedagogy, the teacher is the sole
authority. Relevant to this study is the analysis of teacher authority as two
interwoven but distinct dimensions of content authority and process authority
(Oyler, 1996a). These dimensions of authority originated in Peters' (1966) view of
the teacher as both an authority and “in authority (p. 239-240).
The content dimension of authority refers to one who is validated as a knower and
viewed as the legitimate possessor of knowledge (i.e., of content). A teacher is an
authority carrying out their role as a teacher to teach their subject content. This
content authority is referred to in this study as epistemic authority (Hargreaves et
al., 2018). The use of the term epistemic as opposed to content is in keeping with
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Solomon’s (2009a) use of the word epistemic that views mathematics knowledge as
open to negotiation and knowers as creative negotiators of mathematics knowledge.
Epistemic authority is attributed to the teacher by the definition of the teacher role.
It presupposes that the teacher has studied to attain the subject knowledge, and is
therefore employed by the school. However, the teacher has to demonstrate and
establish this authority in the classroom for it to be legitimised by the students
(Hargreaves et al., 2018; Wagner & Herbel-Eisenmann, 2014).
On the other hand, a teacher has process authority due to an aspect of the
prevailing culture: how the knowledge is taught in the classroom in a given society.
This process dimension of authority, synonymous with being ‘in’-authority (Peters,
1966), is best understood in terms of the notion of framing (Bernstein, 2000).
Framing relates to how knowledge is communicated and the nature of the relations
that go along with it. It relates to who is in control of selecting the knowledge to be
communicated, the how of learning, its sequencing, its pacing, the instructional
criteria, the control of the social base, the regulative criteria, and the dominant values
of the society that make the communication of knowledge possible (Bernstein, 2000,
p. 37). When the teacher is in control, such as in a conventional pedagogy, the
framing is said to be strong. The teacher has authority over the processes of how
the knowledge is communicated to the students. Theoretically, where the students
are in control, the framing could be said to be weak; it is important to understand that
this is not an evaluation of quality, but of the potency of individuals’ relations to the
determination of practice.
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As I view students active participation in all aspects of their learning as necessary
for the aims of this study to improve student’s relationships and learning of
mathematics this study requires a move away from the conventional pedagogy in
which authority is in solely the teachers possession in order to achieve its aim.
Instead, it calls for a shared authority pedagogy, where the students participate in all
aspects of their learning. As Oyler (1996a) notes, this is a more significant move
than it would seem: Sharing authority then is much more than offering activity
choices; rather it requires that teachers and students develop and negotiate a
common destination or agenda (p. 23).
2.3.2.1 Shared Authority
The process and epistemic dimensions of authority are not the only ways to construe
authority in an educational setting. Various authors identify a range of types of
authority (Amit & Fried, 2005; Pace & Hemmings, 2007; Solomon, 2009b; Wagner &
Herbel-Eisenmann, 2014). From my reading of Solomon (2009c), I would argue that
the process/epistemic distinction points to what one has authority over, while the
notion of shared authority addresses whether/how participants distribute authority
amongst themselves. Shared authority, also referred to as revised authority (Amit
& Fried, 2005, p. 151), is the authority characterised by co-participation that involves
both the students and the teacher; in this case, the legitimacy of either the students
or the teachers authority comes from mutual interdependency where those involved,
such as the teacher and the student, are continually learning and reaching beyond
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their present relationship to a relationship that supports independence while
acknowledging differences in knowledge, skill and status (Benne, 1970, p. 401).
Figure 2.4 Process/Epistemic dimensions of authority vs teacher/student’s authority
distribution.
In classrooms with revised authority, students, and the teacher, through their
participation, can negotiate how process and epistemic authority is shared (see
figure 2.4). The revised authority shifts the focus of authority to negotiation and
consent, and renders the relationships upon which authority supervenes as dynamic
and fluid (Amit & Fried, 2005). The students do not blindly expect the teacher to be
the expert, but see expertise in themselves and in each other (Brubaker, 2012).
Epistemic authority refers to who is viewed as legitimately knowledgeable and
process authority refers to how the knowledge is taught in the class. However,
teachers also have other relationships to knowledge that support their authority.
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Shulman (1986, 2013) coined the term pedagogical content knowledge (PCK) to
emphasise that a discussion of one’s knowledge of a subject is not sufficient to
explain what is necessary for teaching. He suggests a trichotomy of categories of
content knowledge: subject matter content knowledge, pedagogical content
knowledge, and curricular knowledge. In this study, epistemic authority refers to
subject matter content knowledge, and the notion of process authority subsumes
pedagogical content knowledge and curricular knowledge. As the term suggests,
subject matter content knowledge refers to the structure and amount of subject
knowledge in the teachers mind. Pedagogic content knowledge refers to the
generic principles of classroom organisation and management, the most useful
representations of ideas that make them comprehensible to students
preconceptions, and common misconceptions that students bring with them to
topics. Curricular knowledge refers to the full range of topics required for the subject;
it includes the sequence of topics, instruction material, and assessment
requirements. From a mathematics perspective, pedagogical content knowledge and
curricular knowledge can be conceptualised as mathematics knowledge for teaching
(MKT) (J. Silverman & Thompson, 2008); in other words, as what is necessary for
successfully teaching mathematics.
It is prudent to assume that teachers and students can share content authority and
process authority in a pedagogy in which both students and teachers participate
equally in a classroom community. However, as the teacher and students negotiate
their practice (see section 2.2.1) to advance their mathematics knowledge, their
mutual relations of interdependence would recognise that some aspects of process
authority such as mathematics knowledge for teaching will best reside with the
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teacher. Due to their training and experience in the profession, the teacher is more
likely to possess knowledge such as the exam board requirements that influence the
questions students practice in class.
2.3.2.2 Positioning
In her book chapter Doing Undergraduate Mathematics: Questions of Knowledge
and Authority”, Solomon (2009a) discussed how students are positioned in
mathematics learning communities by their perceptions of authority. The positioning
of students in the classroom can result from how the pedagogy distributes authority
between the teacher and students in a mathematics classroom. Davies and Harré
(1990) described positioning as the discursive process in which speech and action
are used to arrange people in social structures by locating them in conversations as
observably and subjectively coherent participants in jointly produced storylines
(discourses) (p. 48). Storylines (discourses) are the ongoing repertoires that are
already shared culturally or they can be invented as participants interact (Herbel-
Eisenmann et al., 2015, p. 188). Interactions are communications, dialogue, or
actions that occur among people, either face to face or through other media.
Interaction occurs in a mathematics lesson between participants, whether between
teacher and student or student and student. As participants interact, they assign
positions for themselves and others participating in the interaction.
Positioning constrains what one may meaningfully say or do. With every position
comes a connected discourse. In this way, positioning may diminish the domain of
what one does out of the possibilities of what one can do (Harré & Slocum, 2003, p.
86
106). There are many positions available for the students and the teacher formed by
their interactions in the discourse of schooling. A teacher standing in front of the
class positions themselves as in authority (process authority) and consequently
positions the students as subject to such authority. This positioning of the teacher
constrains them to control the students behaviour, while it expects the students to
behave in a certain way, such as sitting quietly. Subsequently handing the
whiteboard pen to the students, the teacher is able to position the student as an
authority (epistemic authority) in a way determined by the particular context; having
been so positioned by the teacher, the student is expected to answer correctly. In
this sense, people are positioned through interaction with others, and this positioning
tracks these interactions (Davies & Harré, 1990). Positions are responsive to
context, and participants’ relations to them are dynamic, as one can occupy more
than one position and shift between positions.
To position someone is to establish what their duties and rights are, and to
determine what they are obliged and allowed or not obliged and not allowed to do
(Harré & Moghaddam, 2003; Harré & Slocum, 2003). A participants rights constitute
what others must do for them, and their duties constitute what they must do for
others. Having been positioned, either interactionally by others or reflexively by
themselves, a person sees the world from the vantage point of that position (Davies
& Harré, 1990, p. 6). The position gives meaning to the participants and others
speech, writing, and actions (Harré & Moghaddam, 2003). The meaning of a
position is influenced by and influences the past, present, and future of the
participants interactions and participation; thus, in an educational setting such as a
classroom in which the teachers are in authority, the conventional positioning of
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students can include or exclude them from participating in mathematics learning
(Solomon, 2007, 2009c). Positions are defeasible (Harré & Slocum, 2003) and can
be disputed over time or in the moment. This study aims to develop a pedagogy that
challenges the teacher-student discourse that positions the teacher as
knowledgeable and the students as not knowledgeable.
A useful distinction for my thinking is that between position and roles. In contrast to
flexible and situation-specific positions, roles in interactions are static, though long-
term positions approximate the status of a role (Harré, 2012). The static nature of a
role can be understood when considering its close relationship with the function of a
job. A role, like a job, represents a set of constraints and requirements that is
rather pervasive in someones life (Harre & Slocum, 2003, p. 104). Teacher is a
fixed role in a school, while the teacher themselves can, through their interactions,
be positioned temporarily or lastingly as an authority or otherwise in different
situations, dependent on the discourse. This study, by proposing a pedagogy that it
takes to be innovative, follows the heels of other research that has tried to change
mathematics classroom pedagogy in England by challenging existing authority
relations. The pedagogy and its discourses determine the location of authority, as
well as the roles and positions available to its subjects in the maths classroom.
2.3.3 Summary
This section examines the pedagogy critiqued by Paulo Freire and Jacques Rancière
and its similarities to the conventional pedagogy experienced by students in many
present day classrooms in England. The proposed innovative pedagogy will seek to
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facilitate co-participation and interdependence between students and teachers
(Benne, 1970), as against the established forms. Students and teachers sharing
authority in the classroom will learn from each other and negotiate how best to use
their different skills and experiences in mathematics learning.
In the first three sections of this chapter, I have described aspects of agency relevant
to the aims of this study, emphasising the usefulness of the deep constructivist
notion of epistemic agency, according to which students take responsibility for their
knowledge. I have also discussed the knowledge-creation metaphor, which
represents learning as both an individual and collective endeavour; this metaphor
prepares the way for the possibility of a dynamic pedagogy, where learning occurs
as students interact, rather than where knowledge is merely transmitted into their
passive minds by a teacher, as described in section 2.2. Wengers social learning
theory allowed me to examine how learning can occur through students participation
in a mathematics classroom. In this third section, I developed the notion of authority
in the context of mathematics pedagogy. In the following two sections, I will begin to
argue for the notions and concepts that I rely upon in working to achieve the aims of
this study, first constructing the theoretical framework.
2.4 Theoretical Framework
The aim of this study is for the students in my mathematics classroom to actively
participate in all aspects of their learning, and to thereby improve their relationship
with and their learning of mathematics. To achieve this, existing constructs that have
achieved similar aims to mine will be considered in order to help develop the
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theoretical argument that will underpin this study. The two focal ideas of knowledge
building and shared epistemic agency will be introduced in this section as they build
upon the previously discussed notions of agency in particular epistemic agency
that made visible the possibility of students taking responsibility for their learning
introduced in section 2.1. The theory of communities of practice reflects this study’s
interest in the classroom as a learning community and its possibilities for changing
participants relationship with mathematics; however, the community of practice
alone cannot account for the acquisition of knowledge, such as mathematics
knowledge, and has been supplemented with social learning theories.
2.4.1 Knowledge Building/Knowledge Creation
This section discusses in further depth Scardamalia and Bereiters conceptualisation
of knowledge building and Nonaka’s contemporary account of knowledge creation.
The concept of knowledge building is helpful for this study as it illuminates students
engagement with knowledge to the extent that it is useful to all classroom
participants. It goes beyond the weak constructivist notion of learners active
construction of knowledge to include the two characteristics of intentionality and
community knowledge (Scardamalia & Bereiter, 2010) addressed in sections 2.1 and
2.2 respectively. From a weak constructivist perspective, learning is personal and
occurs unconsciously through engagement in activity. By contrast, the deep
constructivist perspective of knowledge building considers students as intentionally
producing purposeful and valuable knowledge; it furthermore concerns the creation
of knowledge in the form of conceptual artefacts for the benefit and advancement of
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the community. Although individual learning could occur in the process, it is not the
ultimate goal of the activity; the primary goal is to solve problems, develop new
thoughts and ideas, and advance community knowledge.
Understanding knowledge building requires a prior understanding of conceptual
artefacts (Bereiter, 2002, p. 64) and their role in collaborative knowledge building.
Conceptual artefacts are abstract knowledge objects (e.g., ideas, theories,
algorithms) that can be realised in some material form, typically through discussion
or physical construction. Logical relations exist between conceptual artefacts; for
example, one conceptual artefact could justify another, and be derived from yet
another. Artefacts can be criticised, tested, and improved. Bereiter and Scardamalia
claim that in order for conceptual artefacts to be treated as objects of new knowledge
and credited as evidence of knowledge, they must: i) be of value to people other
than the individual; ii) have value that endures beyond the moment in which it is
conceived; iii) apply beyond the situation that gave rise to them; and iv) display
evidence of a modicum of creativity in their production (Bereiter & Scardamalia,
2011, p. 3). For example, consider a situation in which an individual, through
experience as a decorator, develops a good sense of symmetry. For Bereiter and
Scardamalia, the individual has acquired knowledge, not built it. If the individual
produces a short video that shows how images are reflected from one side to
another, the individual would be said to have created an artefact. This artefact,
though not conceptual, would enable others to access and acquire the tacit
knowledge and skills that the individual has. For the artefact to be termed
conceptual, the individual would have to produce a mental theory that explains how
the symmetric image is produced. This theory is a conceptual artefact, and it can be
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treated as knowledge that is represented in the video, which therefore fulfils the
criteria above. Developing the theory that supports the conceptual artefact is the
process of knowledge building. When students build knowledge, they are actively
engaged, as a community, to create conceptual artefacts. This collective approach to
creation shares and advances the knowledge of the community.
Knowledge building therefore consists in the continuous collective production of
improved forms of ideas (conceptual artefacts) that contribute to the advancement of
knowledge in a community (Bereiter, 2002). It challenges learners to go beyond
individual capabilities and to collaborate, with whom they share a common epistemic
goal. Bereiter (2002) and Scardamalia & Bereiter (2014) derived knowledge building
from an epistemological outlook that treats ideas as entities in their own right,
independent of the mental states of individuals. In classrooms organised around
knowledge-building pedagogy, individual students are recognised for their
contributions to collective knowledge advancement rather than for what is in their
minds. In these classrooms, students find respect and acceptance as contributors
in knowledge creation (Scardamalia & Bereiter, 2006).
Thus, on the basis of their theory of knowledge building, Scardamalia and Bereiter
proposed a pedagogy that encourages an individual to intentionally execute higher
level cognitive processes on their own, without depending on their teacher, within a
classroom community that further sustains knowledge advancement by providing
opportunities for student-to-student feedback. The pedagogy is based on twelve
principles (see Appendix 1) which deviate from currently prescribed procedures (Lai
& Campbell, 2018; Scardamalia, 2002). Six of these principles align with the aims of
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this study and the innovative pedagogy I propose. The other are less relevant to
secondary school mathematics pedagogy that follows the GCSE curriculum. I will
here outline the principles that align with this study, and subsequently synthesise
them with other active theories in order to produce my own characterisation of
shared epistemic agency in a knowledge-building pedagogy. The relevant principles
are:
Community knowledge, collective responsibility that encapsulate the
aim of knowledge-building pedagogy to produce knowledge that is
useful to and usable by the participants of a classroom community (see
section 2.2.2 on communities of practice).
Epistemic agency (see section 2.1.3), which is essential for supporting
the collective efforts of knowledge advancement beyond the individual
performance of tasks.
The collective improvement of ideas (see section 2.3.2.1). There are no
final truths; learners view every idea as having the potential to be
improved. The improvement of ideas comes from the students as they
seek to reconcile conflicting conceptions. There is the continual
application of a make it better heuristic (Scardamalia & Bereiter,
2014, p. 400).
Knowledge-building discourses for the improvement of ideas (see
section 2.2.1). Bereiter (2002) argues that classroom discourse should
mimic professional science discourse. It should, in other words, be
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cooperative and more concerned with creatively advancing the
collective knowledge beyond what is currently known.
The democratising of knowledge that is a result of such discourses
(see section 2.3.2.1). In a classroom based on the knowledge-building
paradigm, all participants are deemed legitimate contributors to
collective knowledge.
The use of authoritative information, such as multimedia resources, in
these classrooms. In my classroom, this involves the careful use of
such things as MathsWatch, textbooks, and other media in order to
construct coherent knowledge from diverse representations.
The following section will discuss the concept of knowledge creation”, not to be
confused with the knowledge-creation metaphor for learning described in section
2.2.1.
2.4.1.1 Knowledge Creation
Though distinct from knowledge building, Nonakas (1991) concept of knowledge
creation relates to the former in its focus on the ways in which a community can
create new knowledge from within, through active engagement; this concept is useful
for the secondary mathematics classroom in which students need to make
mathematics knowledge and their problem-solving strategies explicit to each other.
The distinction between knowledge building and knowledge creation is due to the
different disciplinary commitments of the associated theorists: knowledge building
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was developed in the context of education, while knowledge creation is a dynamic
that was initially identified in the context of the corporate organisation.
Nonaka’s concept of knowledge creation is germane to the aims of this study to the
extent that it recognises the value of knowledge as both explicit and tacit, placing an
emphasis on the process by which personal knowledge is made available to others.
Explicit knowledge is easy to articulate, while tacit knowledge is personal, hard to
formalise, and challenging to communicate to others; it consists of mental models,
beliefs, and perspectives (Nonaka, 1991). This concept can explain how, in the
mathematics classroom, knowledge can be tacit or procedural, and students may
find it difficult to articulate their reasoning and justify their solutions to problems; or
else the knowledge can be explicit, in which case students will typically find it easy to
communicate their thinking. Both types of knowledge are of value, and Nonaka’s
theory further reveals the process by which the two interact in a spiral of knowledge
(p. 97) to generate innovations; that is, to create new knowledge. This presents the
interaction between students as a process of knowledge creation
The knowledge spiral, which depicts the iterative transformation and sharing of
knowledge from the level of the individual to that of the organisation, and even
among organisations, is grounded in four complementary knowledge-creation stages
that operate between individuals and groups in an organisation (Figure 2.5).
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Figure 2.5 The Knowledge Creation Spiral. Source: (Umemoto, 2002, p. 464)
The first stage involves the transmission of tacit knowledge from individual to group
due to the sharing of experiences in the activity socialisation. It is essential to
develop trust between individuals at this stage, as close interaction and collaboration
are necessary for the effective sharing of the explicit knowledge over time. In the
second stage, tacit knowledge is transformed into explicit knowledge through
externalisation. In this stage, the tacit knowledge of a socialised group is made
explicit through discourses, metaphors, diagrams, and concepts that is, through
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artefacts. Thus, during externalisation, knowledge can be exchanged by means of
what Nonaka refers to as a “metaphors, analogies and models” (p. 99), which is
broadly analogous to Bereiter’s conceptual artefact. In the third stage, the new
explicit knowledge aggrandises itself through its combination with existing explicit
knowledge, and is subsequently distributed throughout the organisation. In the fourth
stage, explicit knowledge is transformed back into tacit knowledge, through
internalisation, and begins to inform the practices of individuals. This implicit
knowledge is then itself socialised, beginning the cycle anew.
Bereiter (2002) was critical of Nonaka’s knowledge spiral on four counts, noting its
exclusion of creativity, understanding, knowledge work, and collaborative knowledge
building (Bereiter, 2002, pp. 175177). He argued that as the model does not
distinguish between “knowledge involved in productive work and knowledge that is a
product of productive work (Bereiter, 2002, pp. 177178), it cannot promote learning
that will contribute to a community’s ability to create knowledge. He noted that the
knowledge spiral could be a carrier of ritual and tradition, as it presupposes shared
implicit understanding but does not necessitate understanding at the individual level.
The individual does not become what he referred to as “a fully functioning member of
a knowledge society” (Bereiter, 2002, p. 173). However, I argue that Nonaka’s
perspective on knowledge can contribute its thinking to structure the mathematics
pedagogy that I seek to develop in this study. Supplementing her picture of the
transformation of knowledge with capacities for discussion and shared problem
solving evades Bereiter’s critiques and contributes individuals tacit knowledge to
community knowledge. In this sense, student-to-student explication of mathematical
knowledge fulfils the criteria of new knowledge, and I argue that it qualifies as
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knowledge building. Despite Bereiter’s criticism, other authors such as Paavola et
al., (2006), whom I discussed in section 2.2, and Damşa et al., (2010) whose work I
describe in the next section, have also combined these two models.
In summary, knowledge building and knowledge creation orient the design of a
pedagogy that focuses on the individuals engagement with knowledge for
community benefit. Individuals can be seen to benefit from the pool of knowledge
within the community from which they can draw. This picture of the synergy of the
individual and the community is in agreement with Wengers theory of community of
practice (see section 2.2.2), wherein he argues that through participation, benefits
such as accountability and mutual relations contribute to the advancement of a
communitys enterprise (Farnsworth et al., 2016). In the following section I describe
the kind of agency I desire the pedagogy of this study to develop in the students.
This agency is referred to as shared epistemic agency.
2.4.2 Shared Epistemic Agency
Shared epistemic agency, introduced by Damşa et al. (2010), is the central concept
of this study. It is described by these authors as an emergent construct that builds
on Scardamalia's (2002) notion of epistemic agency (see section 2.1.3), which they
used to characterise undergraduate students’ abilities to carry out complex, authentic
collaborative projects. They conceptualised shared epistemic agency to include the
notion of sharedness that presupposes intentionality (Bandura, 2001; see section
2.1.1), the collaboration between participants, the social-communicative processes
that leads to new collective knowledge (see section 2.4.1), as well as the notion of
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an established community of practice i.e. the mutual relations of participation that
support coherence in a community (Wenger, 1998, and section 2.2.2). Shared
epistemic agency describes the interdependency of partners (see section 2.3.2.1)
and the collaborative actions that do not happen when individuals work on their own.
It also draws on the knowledge-creation perspective of learning (see section 2.2.1)
that situates learning as occurring during collaborative practices that create shared
material knowledge objects.
Damşa et al.’s construct of shared epistemic agency, which lies within the
knowledge-creation perspective (see section 2.2.1), depicts a specific form of
epistemic agency (see section 2.1.3) that emerges during collaboration to create
shared knowledge objects. In this sense, the shared knowledge object is both the
outcome of the groups collaboration and the reason for the groups activity (Stahl,
2009, p. 64). Damşa et al., like Nonaka (1991), acknowledge the interaction
between explicit and implicit knowledge as of value to knowledge creation, while
arguing that shared epistemic agency goes beyond knowledge building. They
argued that knowledge building emphasises collective collaboration for the
improvement of singular ideas, whereas shared epistemic agency involves working
on more than one idea to create knowledge through the advancement and
development of complex knowledge objects (Damşa et al., 2010). These authors
posit that learning occurs as students act to give conceptual artefacts a concrete
form as material objects of shared knowledge, such as reports, essays, or software.
Shared epistemic agency can be understood as the capacity that enables
individuals, groups, or collectives to make appropriate judgments, to make plans,
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and to pursue these through purposeful action, in order to achieve the construction
of knowledge (Damşa, 2014, p. 446). In addition to sharedness, this definition
emphasises epistemic productivity and negotiation within the community. The
related notion of “temporality refers to the emergent nature of the agency in
question (p. 447); it suggests a certain kind of practice that is reflexive and iterative,
considering past practices and experiences metacognitively to solve present
problems and create plans that lead to future desired outcomes.
Shared epistemic agency is an empirical concept; in other words, it is a
conceptualisation of observable phenomena and they expressed the intentions that
materialise indicative of the agentic behaviour (Damsa et al., 2010, p. 155). The unit
of analysis, is, therefore, the group-level actions that constitute the conditions for its
emergence. These actions fall into two categories: the epistemic and the regulative.
Epistemic actions are directed towards knowledge and the creation of knowledge
objects. These include actions that serve to create awareness of the current
knowledge situation within the group (e.g., brainstorming, discussing); that create
shared understanding; that alleviate a lack of knowledge and gather information
(e.g., researching, asking, discussing); and that generate collaborative actions (e.g.,
explanations, concepts) (Damşa & Andriessen, 2012).
Regulative actions are the processes that occur at the metacognitive level and that
prepare the foundation for epistemic actions. They do not directly influence the
creation of knowledge objects, although they make their creation possible.
Regulative actions are based on the groups intentions (Bandura, 2001) to create the
knowledge object, and consist in the procedures that occur as a result of this
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intention (Emirbayer & Mische, 1998); that is, they are the result of the meta-
knowledge that the group has about the process and the progress of creating the
knowledge object that informs the actions that the group takes. These actions,
consisting of projective actions, the setting of a common goal, the creation of a plan
of action, and proactive engagement, are required for successful outcomes.
Regulative actions, such as monitoring the progress of the knowledge object and
reflecting on it, and relational actions the social aspect, i.e., validation and the
acknowledgment of individual contributions facilitate relations between individuals
and the group, making possible the maintenance of their epistemic community. An
overview of epistemic and regulative actions is offered in Appendix 5.
2.4.3 Summary
Knowledge building conceptualises a community learning environment in which
students interact with shared intentions to improve on their ideas, creating new
knowledge continuously. Shared epistemic agency is a conceptualisation of the
capacity of individuals and collectives to perform collaborative actions, bringing
together multiple ideas to create a knowledge object, which is the material realisation
of their new knowledge. To achieve the aims of this study, I consider these concepts
in the context of the capabilities of students. It is my intention to promote the
emergence of shared epistemic agency amongst the students in my mathematics
classroom, creating a learning environment in which they continuously develop new
knowledge and control their own knowledge advancement.
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Although Damşa et al. describe shared epistemic agency in terms of the epistemic
and regulative actions that, over time, lead to the creation of a knowledge object,
their empirical study reports only on undergraduate students engaged in one-off
collaborative group work to produce an authentic knowledge object such as an
instructional design project or a training and evaluation project (Damşa et al., 2010).
Their research cannot be applied without modification to a secondary mathematics
classroom, in which both participants and subject matter are considerably different
from the original objects of the study. Thus, I proceed with my own study by
apprehending and developing the notion of shared epistemic agency in this new
context; I determine that the shared epistemic agency that I want to emerge is a
quality of students that is an index of active participation in all aspects of their
learning of mathematics and an improved relationship with mathematics, which leads
to improved mathematics learning. Good GCSE grades will evidence this improved
learning in the students’ terminal secondary school examinations.
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On the strength of the theoretical background developed) in this chapter (see Figure
2.6), I can now characterise the specific kind of shared epistemic agency that I
consider appropriate for the aims of this study. Its six characteristics are given as:
a) Intention. The agency will include intentionality: the proactive commitment to
bring about a desired outcome (see section 2.1.1) that presupposes
purposefulness and will include community knowledge (cf. Bandura, 2001;
Damşa et al., 2010; Scardamalia & Bereiter, 2014).
b) Extension. The student deliberately focuses on going beyond existing
knowledge. This notion originates in the theory of knowledge building (see
section 2.4.1, first paragraph) that extends constructivism towards deep
constructivism (see section 2.1.3), in line with which students control all
aspects of learning (cf. Bereiter & Scardamalia, 2011).
c) Explication. This refers to purposeful dialogue that makes knowledge explicit
so that it can be shared (see section 2.4.1.1). Drawing on Nonakas
knowledge spiral, shared epistemic agency will acknowledge sharing personal
knowledge and the interaction between tacit and explicit knowledge that
communicates mathematics knowledge through dialogue, advancing all
students knowledge in the classroom (cf. Nonaka, 1991).
d) Expertise. Students are considered to be expert learners who set themselves
similar tasks to those typically imposed by mathematics teachers. This draws
on Damşa et al.’s notion of regulative actions (see section 2.4.2) that depict
the metaknowledge possessed by the group that allows them to manage and
monitor the advancement of the knowledge object, requiring them to not to
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rely solely on external sources such as the teacher (cf. Damşa & Andriessen,
2012).
e) Mutual Relations. In order to sustain epistemic agency, mutual relations
between individuals must be established (see section 2.2.2). The application
of my revised notion of shared epistemic agency will include a consideration
of the mutual relations that support the coherence of the community in the
project of fulfilling their common purpose of learning mathematics (cf. Wenger,
1998).
f) New Knowledge This refers to learning through collectively developing ideas
and explanations that are new to the students (see section 2.4.1) The final
object of analysis will be the new knowledge students are able to create, in
the form of a conceptual artefact that is the product of more than dialogue with
the pedagogical authority, instead combining the collective and individual
contributions of learners who are actively engaged in developing new ideas
and explanations in the context of unfamiliar mathematical concepts (cf.
Bereiter, 2002; Bereiter & Scardamalia, 2011).
The precise nature of these characteristics, in the specific context of the knowledge-
creating classroom practices that are the object of my study, will be illuminated in the
following sections. The actions and artefacts that are indicative of each of these six
characteristics will also be identified by the end of this study. Henceforth, the term
shared epistemic agency will encapsulate the six characteristics stated above. The
wider construct originating in Damşa et al. (2010) will be referred to as SEA for
differentiation. Therefore, a preliminary question that this study seeks to answer is:
What are the indicators of shared epistemic agency in the mathematics classroom?
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As previously stated, knowledge building requires a learning environment that could
support the emergence of shared epistemic agency. The innovative pedagogy I
propose draws on the concepts of knowledge building and knowledge creation to
support the emergence of shared epistemic agency. The pedagogy will be based on
the knowledge-creation metaphor of learning, according to which new knowledge is
continuously and creatively produced from within the learning community. It will
seek to reimagine the conventional teacher-student power relations by
demonstrating the interdependence of authority (see section 2.3.2.1), and by
redefining learning as a community endeavour. The pedagogy will draw on the six
key principles of knowledge building, and will notably include reflection that leads to
improvement (see section 2.5.1), as well as explicitly relying on the community
relations that support the genuine advancement of knowledge. Given my synthesis
of the previous literature performed in this chapter, I clarify the principles of the
innovative pedagogy I propose as stipulating that students are responsible for:
1. Building objects of mathematical knowledge (cf. Bereiter, 2002; Damşa et
al., 2010; Emirbayer & Mische, 1998; Reed, 2001; Scardamalia, 2002).
Figure 2.7 Pedagogic principle 1
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2. The process that makes this knowledge explicit so that it can be shared,
internalised, and used by all the classroom participants (cf. Bandura, 2001;
Damşa et al., 2010; Nonaka, 1991).
Figure 2.8 Pedagogic principle 2
3. The discursive process that communicates this knowledge to the
classroom community (cf. Emirbayer & Mische, 1998; Nonaka, 1991;
Scardamalia & Bereiter, 2014).
Figure 2.9 Pedagogic principle 3
4. Maintaining the social relations and communicative processes that are
conducive to the advancement of mathematical knowledge (cf. Bandura,
2001; Damşa, 2014; Damşa et al., 2010; Wenger, 1998).
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Figure 2.10 Pedagogic principle 4
5. Reflecting on practice and making plans for the improvement of ideas and
activities (cf. Bandura, 2001; Bereiter & Scardamalia, 1998; Emirbayer &
Mische, 1998; Yang, Chen, et al., 2020).
Figure 2.11 Pedagogic principle 5
In the next section, I will investigate pedagogies that have turned control of learning
over to the students, providing a touchstone for my own suggestion of a pedagogy
that meets the aims of this study in the context of my mathematics classroom.
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2.5 Researching Innovative Pedagogies
The emergence of shared epistemic agency requires more than groups of individuals
learning collaboratively. Simply bringing students together to work on a joint task
and pooling their knowledge together is not sufficient to create new knowledge
(Barron, 2000; Scardamalia & Bereiter, 2010). As elaborated in the previous section,
it requires an established community with customary practices negotiated over time
(Damşa et al., 2010; Wenger, 1998). It necessitates an innovative pedagogy with a
purpose, namely, which goes beyond collaborative learning to include the notion of
productivity; that is, a knowledge-creating classroom. Having outlined the principles
of my innovative pedagogy, in this section I investigate knowledge-building
pedagogies and transformative mathematics pedagogies in England to inform the
design of my own.
2.5.1 Knowledge-Building Pedagogies
This section describes three pedagogies (Moss & Beatty, 2011; Yang, Chen, et al.,
2020; Yang, van Aalst, et al., 2020; Zhang et al., 2018) that are explicitly framed by
the concept of knowledge building that I described in section 2.4.1. Though online
technology, which is not a focus of my own study, heavily supports student
interaction in these pedagogies, the findings are still relevant for their analyses of the
ways in which the pedagogy was decisive in developing students participation in the
creation of new knowledge. In Moss and Beatty (2011), fourth-grade students
collaborated on an online database that provided a communal space where students
posted their ideas and read each others, engaging in critical reflective activity. In
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this way, they all contributed to the community knowledge base. The database was
entirely student-managed; the teachers voice was not present, nor were answers or
solutions provided from an external source. In other words, the students had
collective cognitive responsibility for coming up with conjectures, and solutions, and
negotiating the various approaches to mathematical problem solving.
This research illustrates how the knowledge-building principles of the
democratisation of knowledge and epistemic agency (see section 2.4.1) can further a
mathematics problem-solving and learning culture. Democratising knowledge
requires that all participants within a community are legitimate contributors to the
community knowledge and that their contributions are valued and acknowledged.
Moss and Beatty’s students working together to solve problems evidence their
epistemic agency; they supported each others suppositions and questioned when
ideas or solutions were incorrect. In this way, the community was assured that the
solutions provided were correct. In the absence of an external verifier, the students
not only verified their solutions to the problems independently, but also routinely took
responsibility for offering evidence and justification for their solutions, with the
intention being to make sure that the whole community understood the proper
solution to the problems. In this way, they took responsibility for the communitys
collective understanding. Moss and Beatty researched 8-to-9-year-olds across three
schools; the intervention took the form of a one-off addition to the existing classroom
pedagogy, which contrasts with my aim to change the overall learning experience of
secondary students for a single subject over a whole year.
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Moss and Beatty’s research, however, does bear similarities to my study; its
demographics were of a similar economic status, and a significant proportion of the
students were categorised as low-achieving. Equally, the democratisation of
knowledge and the quieting of the teachers voice are outcomes of a knowledge-
building pedagogy that resonate with this study’s aims.
Yang, van Aalst, et al., (2020) conducted research with low-achieving ninth-graders
who collaborated on an online platform. The findings from this research were similar
to those of Moss and Beatty. They illustrate how academically low-achieving
students could get involved in sustained collaborative and productive knowledge-
building discourse and inquiry (p. 1253). In addition, in a manner that is particularly
relevant to my study, the research illustrated that by engaging in reflections, students
had a better view of their contributions through the lenses of others, which led to a
more productive discourse. Reflecting on others contributions to knowledge
improvement did not lead to criticism, but became the community practice, the
classroom norm. This research focused on developing a community in which the
goal and focus of the classroom was knowledge-building collaboration that advanced
collective knowledge; reflective assessment was not based on individual attainment,
but on the progress made by the whole class.
Research by Zhang et al. (2018) sought to support student-driven inquiry within a
socially organised pedagogy. The researchers worked with two upper-primary school
classrooms on a knowledge-building initiative. The researchers sought to provide
structure to the students inquiries while still allowing the flexibility that enabled their
agency and imagination to thrive. The researchers designed an inquiry-structuring,
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timeline-based web platform, ITM (Zhang et al., 2018, p. 401), that discovered
emerging directions and interests in students interactive discourses. ITM then
formulated unfolding inquiry strands and made them visible to students to support
ongoing participation and reflection. The reflective process, facilitated by the
technological apparatus, shifted control of the inquiry from the teacher to the
students agency. While the research highlighted the value of reflection, and
knowledge building pedagogy was the established science pedagogy for a twelve-
week period, the teacher guided the students inquiry to a larger extent than is
proposed in this research. The research shed light on how to construct pedagogical
structures with students to develop a classroom community that sustains the
students ownership of their collective thinking journey to support knowledge-building
interaction, but I attempt to go further, in line with the renunciation of authority
consistent with deep constructivism.
The three studies noted above show how a knowledge-building pedagogy can lead
to the emergence of favourable characteristics in the classroom environment, such
as the democratisation of knowledge, epistemic agency, the quietening of the
teachers voice, community learning, and improved participation in learning
however, in each case, a technology platform where ideas were shared was central
to the pedagogy. In addition, the three studies took place outside of England. In the
following section, I will therefore conduct a literature review to identify further
research related to my study that has transformed pedagogies, without reliance on a
technology platform, in English secondary schools.
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2.5.2 Transformative Pedagogies in England
My literature review focuses on studies that have transformed mathematics
pedagogies in secondary schools in England in the last ten years, as this frame
bears close relevance to the context of this study (see section 1.1). I used the UCL
library search facility and put in the terms: <Any field (contains) transformative
pedagogies AND Any field is (exact) mathematics AND Any field is (exact)
England>, and I filtered for the Years: 2011-2021, Form: Articles and Book
Chapters, and Topic: including Pedagogy. Two articles from the 145 results were of
interest; the other 143 did not describe a mathematics pedagogy in England.
However, on further reading, these two were not found to be germane to the specific
aims of my study.
Ruthven et al. (2017) developed the epiSTEMe pedagogical model, which focused
on improving student engagement with mathematics and science in the first year of
secondary school education through exploratory dialogic conference. It was not
relevant to this study, as the pedagogic measures it proposed retained the privileged
position of the teacher as an authority, and it involved changing the nature of
mathematics content as opposed to improving student agency. This research, if
anything, further entrenches the roles of teachers as knowledgeable and students as
requiring continuous guidance to be knowledgeable.
The “participatory pedagogy” of Lyndon et al. (2019) focused on pedagogic
mediation and viewed the student as a social being with the capacity to construct
their knowledge in collaboration with others. Despite the similar view of the student
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in my study, the research differed in context as it focuses on nursery school children,
and was not mathematics-specific.
I altered the search term to: <Any field is (exact) pedagogy AND Any field is (exact)
mathematics classroom AND Any field is (exact) England>. I filtered for the
Years: 2011-2021, Form: Articles and Book Chapters, and Topic: including
Pedagogy, including Education & Educational Research. This produced 13 results;
of interest was the work of Hofmann & Ruthven (2018); Watson & De Geest (2014);
and Wright et al. (2020). The other 10 articles did not describe a mathematics
classroom pedagogy in England.
Watson & De Geest (2014) carried out three-year ethnographic research with three
secondary school mathematics departments in England, teaching students of a
similar socioeconomic background to that of the students in my study. The
departments sought to improve the achievement of their students. However, the
transformation did not directly focus on improving student agency. Instead, it centred
on changing classroom groupings to mixed-ability, expanding the mathematics tasks
available to students, and developing teachers confidence in their subject content
knowledge. These changes are similar to those that have been discussed in my
mathematics department and many others over the years; with this study, I propose
something more radical: a change in our beliefs about students and the historico-
cultural role assigned to them.
Hofmann & Ruthven (2018) were co-researchers on the epiSTEMe project,
alongside Ruthven et al. (2017); indeed, the limitations of the project noted above
apply to their study as well. I discuss Wright et al. (2020) below.
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Altering the search term to <Any field is (exact) mathematics pedagogy AND Any
field is (exact) student agency AND Any field is (exact) England> produced one
new result: Wright (2017).
Further manipulation of the search terms revealed Foster (2013), who is critical of
the reductionist approach to traditional mathematics pedagogy, and who and calls for
a more holistic approach to mathematics tasks; however, his article was focused on
critique, and did not put forward a pedagogy. My systematic search, therefore,
resulted in the identification of two studies that share an interest in putting forward a
pedagogy, based in an English secondary school, and focusing solely on
mathematics. These are the works of Wright (2017) and Wright et al. (2020) from
my literature review; the work of Solomon et al. (2021), which I discovered through a
search of recent articles from researchers in my bibliography, was also useful.
Wright et al. (2020) adopted a critical model of participatory action research to
transform mathematics classroom practice in a London secondary school. The
mathematics pedagogy research project they undertook was a collaboration between
Peter Wright, an academic researcher, and two secondary school mathematics
teachers, who are also co-authors. The project’s aims were twofold. The first aim
was to investigate the effect of making a progressive mathematics pedagogy visible
to students, leading to their appreciation of how to be successful mathematics
learners. Progressive pedagogy in this research referred to a problem-solving
teaching approach that was discursive, collaborative, and open-ended. The second
aim focused on developing and refining the model. Wright’s approach to
pedagogical transformation focused on developing the teachers practice.
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Wright comes from the school of critical mathematics education, also influenced by
Paolo Freire, whom I mentioned in section 2.3.1. Critical educators such as Gutstein
(2006) introduced practices that reimagine the authority relations in the classroom
and alter the mathematics teaching materials in a bid to help students to understand
the society in which they live, and recognise how inequality is contested and
produced in society. I do not advance a critical view of society; nor am I interested,
in this study, in precipitating changes in the social at large. Though my study
focuses on social justice in terms of wanting the students to be total participants in
their learning, its ultimate aim is improving exam performance to offer students
greater opportunities in life. Wright et al. indeed seek a reversal of historically
inequitable academic outcomes by making the pedagogy more visible; in this way,
their study and mine have a similar focus. However, though he argued for teachers
and students to reflect on the implicit power relations in the classroom that prevent a
relationship of trust, which would allow classroom rules to be negotiated and made
clear to students, rather than the teacher relying on their authority to control students
(Wright, 2017), Wright et al.’s transformation did not go far enough in my view. The
researchers restricted students agency to articulating the justification behind the
teachers intentions. The students did not participate in any decision-making, nor did
they initiate or direct any change within the pedagogy; this leads me to question
whether the intentions to involve students in negotiating classroom rules held the
same social learning focus of developing a practice (see learning by doing in section
2.1), as my study intends to do. The locus of the participatory action-research
practice was the relationship between the researcher and the teachers.
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In Solomon et al. (2021), the research focused on introducing Realistic Mathematics
Education (RME) to a group of low-attaining students who had not achieved the
accepted pass grade in GSCE Mathematics. The development of the RME
pedagogy is supported by “guided reintervention” that requires increased
participation on the part of the students and particular practices by the teacher, both
underpinned by a significant shift in responsibility and authority from the teacher to
the students. The teacher orchestrated whole-class mathematical discussions for a
specific goal (p. 175-6). The pedagogy positioned the students as knowledgeable
and expected them to articulate and defend their solution strategies.
The research shares similarities with this study. It sought to increase students
epistemic authority by shifting authority from the teacher to the students and
positioning them as knowers responsible for articulating their thinking and solution
strategies. However, the study was founded upon a curriculum-focused RME
theoretical base, whereas my study is driven by pupil relationships with mathematics.
I left the question of how the mathematics was to happen to the students, and our
own resources built on workbooks and exam practice.
The literature review has shown that numerous researchers in mathematics
education have sought and still seek changes to the conventional mathematics
pedagogy. Both Wright and Solomon needed longitudinal studies to embed and
research their pedagogy, and both were participatory in that they trialled new ideas
in existing cultural settings, not labs. My study, however, stands alone in seeking an
everyday pedagogy in which students take control of learning the mathematics
curriculum in a secondary school mathematics classroom in England.
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The need to change my classroom pedagogy started long before the
commencement of this doctoral study. As described in the introduction, I had begun
to consider how my actions in the classroom may constrain the students from
engaging with mathematics logically. Prior to embarking on this research, I had
started to allow the students to take greater control in the classroom and to teach
topics to each other. I also allowed them to make decisions about the sequence of
the teaching of topics. However, I knew that convincing other professionals to
change the conventional pedagogy required a systematic study. I also needed to
justify to myself the benefits of my pedagogy by rigorously collecting evidence.
I am aware that there must be other ways of designing a pedagogy that would lead
to the emergence of shared epistemic agency in a mathematics classroom. This
study’s innovative pedagogy started to develop as my classroom practice for two
years before the commencement of this study, when I had attempted to silence my
authoritative voice as teacher in the classroom so that students could find their own
ways of making sense of mathematics through their active participation. In this way,
I believed they would respond more logically to problem-solving and ultimately do
better in the GCSE terminal examinations.
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2.5.3 Summary
My pedagogy will involve the students working collaboratively in line with the
pedagogic principles I have established above (see section 2.4.3). The design of the
pedagogy will be described in fuller detail in the following section. From my
experience before this study, I found that the students act as both an epistemic
support and motivator for each other’s mathematics knowledge when the authority of
the teacher is weakened. The kind of participation that I want my students to be
engaged in will develop and change the teacher-student relationship over time. This
directs this study towards an action-research methodology that seeks to answer the
following questions:
1. What are the indicators of shared epistemic agency in the mathematics
classroom?
2. What sustains the emergence of shared epistemic agency in the mathematics
classroom?
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3 METHODOLOGY
At the end of the previous chapter, I identified the need for a study that combined
two interwoven strands: firstly, the design and enactment of an innovative pedagogy
that promotes shared epistemic agency in a school context; and secondly, data
collection and analytical methods that would enable me to answer my research
questions about what indicates and sustains shared epistemic agency. My reading
of methodology literature led me to combine these two strands under the auspices of
action research, allowing me to engage in a form of disciplined, rigorous enquiry, in
which a personal attempt is made to understand, improve and reform practice
(Ebbutt in Cohen et al., 2018, p. 345). The first section of this chapter sets out my
initial vision for what my pedagogy should achieve, informed by the literature
introduced in chapter 2. The second section reviews how action research is justified
as a research method both in general and for this specific project, and then
introduces my plan for my own cycles of action research. The third section outlines
the research design that combines the pedagogy stages that correspond to the
teaching cycles and the research cycles that outline how data is collected. The
fourth section discusses how enacting the pedagogy as part of the action-research
methodology allowed me to continuously adapt the pedagogy, its enactment, and the
design of the project to meet the aims of the study.
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3.1 The Pedagogy
This research project investigates the emergence of shared epistemic agency
amongst the students in a mathematics classroom organised around an innovative
knowledge-building pedagogy. The innovative pedagogy is based around five
principles that I have synthesised from the literature and summarised in chapter 2,
as well as being informed by practices that I personally trialled in the classroom. As
these principles stipulate a handing over of responsibility to the students, I will
henceforth refer to students as participants”, being faithful to the commitments of my
innovative pedagogy (my role as a participant will be discussed later in chapter 6).
This is to emphasis not only their responsibility but also their agency in advancing
the collective mathematics knowledge of members of the classroom. The
participants are responsible for:
1. Building objects of mathematical knowledge (cf. Bereiter, 2002; Damşa et al.,
2010; Emirbayer & Mische, 1998; Reed, 2001; Scardamalia, 2002).
My plan is to have pairs of participants take responsibility for teaching the
other members of the class a mathematics topic (these pairs are therefore
named teacher participants). They are responsible for planning and
leading the discussion and learning of a mathematics topic. They make
use of relevant information which is not supplied by myself, but discovered
independently from other sources such as mathematics websites
(MathsWatch, Corbettmaths, Maths Genie), the broader internet, or other
individuals. The knowledge objects by which they will reify their
mathematics knowledge is the PowerPoint lesson plan they are asked to
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produce for the lesson, and the answers to the mathematics questions the
participants solve during the lesson.
2. The process that makes this knowledge explicit so that it can be shared,
internalised and used by all the classroom participants (cf. Bandura, 2001;
Damşa et al., 2010; Nonaka, 1991).
My idea is that, as the teacher participants prepare their lesson plan to
teach the rest of the class (the student participants), they consider and
decide on how best to make the mathematics topic explicit so that the
student participants will be able to make sense of it. This could involve
deciding on how their exposition of the mathematics concept is structured
and how the contents of the PowerPoint lesson plan support this
exposition.
3. The discursive process that communicates the knowledge to the classroom
community (cf. Emirbayer & Mische, 1998; Nonaka, 1991; Scardamalia &
Bereiter, 2014).
I intend for the participants of the classroom to engage in discussions to
improve their knowledge of the mathematics topic being taught. Through
this discussion, tacit knowledge is explicated, and participants ask
questions and receive answers that help to clarity their knowledge. My
idea is that as I am not the mathematics authority, the participants must
find their own ways to advance their collective knowledge, including
sharing what they know and building on each other’s knowledge.