New Directions for Situated Cognition in Mathematics Education (eBook)

Anne Watson is Reader in Mathematics Education at the University of Oxford. Before that she taught for many years in maintained secondary schools which served socially diverse areas. She now works in teacher education, and her work with students, and in local schools, and her research is characterised by a concern for social justice through education. In particular, the nature of mathematics classrooms and adolescents’ relationships within them are a central concern. She has published numerous books, articles and papers for both professional and academic audiences and is often asked to talk to national and international audiences of researchers and practitioners. Peter Winbourne currently lectures in mathematics education at the London South Bank University. After a long career teaching mathematics in inner city multicultural maintained schools, his passion for mathematics and social equity took him into teacher education, preparing people to work in similar schools. His interests developed from specific focus on the uses of new technologies to support learning to answering difficult questions such as why people should bother to learn at all. In answering these questions he has developed his understanding and ideas of theories of situated cognition, seeing these as illuminating the experiences of individuals as they become the people they are going to be.

Contents 6
Contributing Authors 8
Chapter 1 Introduction 13
1. INTRODUCTION 13
2. MATHEMATICS 14
3. KNOWLEDGE 15
4. SITUATED PERSPECTIVES: POWER AND LIMITATION 16
5. WHY SITUATED PERSPECTIVES? 18
6. THIS BOOK 20
REFERENCES 23
Chapter 2 School Mathematics As A Developmental Activity1 25
1. INTRODUCTION 25
2. IMPLICIT EPISTEMOLOGY: WHAT DOES IT MEAN ‘ TO BE DOING MATHEMATICS’? 27
3. LEARNING CLOSE TO PRACTICAL CONTEXTS AND SITUATIONS 30
4. LEARNING IN THE SCHOOL CONTEXT 32
4.1 Utilitarian vs. epistemic attitude to the world and to language 33
4.2 Unreflected, or not consciously developed vs. planned and conscious procedure 35
4.3 Learning as a relation of operations, tasks and the object of cognitive activity 37
5. PERFORMANCE IN THE SITUATION VS. DEVELOPMENT 39
REFERENCES2 41
Chapter 3 Participating In What? Using Situated Cognition Theory To Illuminate Differences In Classroom Practices 43
1. INTRODUCTION 43
2. OUR POSITION ON SITUATED COGNITION 47
3. CASE STUDIES 48
3.1 Case 1: Norma 49
3.2 Case 2: Roisin 51
3.3 Case 3: Susan 52
4. ANALYSIS OF CLASSROOM SEQUENCES AS LOCAL COMMUNITIES OF PRACTICE 55
4.1 How do students seem to be acting in relation to mathematics? What kind of participants do they seem to be within the lesson? 56
4.2 What developing mathematical competence is publicly recognised, and how? 57
4.3 Do learners appear to be working purposefully together on mathematics? With what purpose? 57
4.4 What are the shared values and ways of behaving in relation to mathematics: language, habits, tool- use? 58
4.5 Does active participation of students and teacher in mathematics constitute the lesson? 59
4.6 Do students and teacher appear to be engaged in the same mathematical activity? What is the activity? 60
5. ANALYSIS OF THE AFFORDANCES, CONSTRAINTS AND ATTUNEMENTS OF MATHEMATICAL ACTIVITY IN EACH SITUATION 61
6. INFLUENCE OF STUDENTS’ CONTRIBUTIONS 64
6.1 Susan’s class on ‘Equations’ 65
7. CONCLUSION 67
ACKNOWLEDGEMENTS 68
REFERENCES 68
Chapter 4 Social Identities As Learners And Teachers Of Mathematics 70
The situated nature of roles and relations in mathematics classrooms. 70
1. INTRODUCTION 70
2. PUPILS’ SOCIAL IDENTITIES AS LEARNERS OF MATHEMATICS 71
3. LEARNERS’ SOCIAL IDENTITIES 72
4. FROM IDENTITIES TO RELATIONS 75
5. CHANGING CLASSROOM RELATIONS 77
6. CLASSROOM RELATIONS 79
6.1 Parallel calculation chains 81
6.2 Solver and recorder 82
6.3 Clue problems 83
7. DISCUSSION 86
8. CONCLUSION 87
REFERENCES 88
Chapter 5 Looking For Learning In Practice: How Can This Inform Teaching 90
1. INTRODUCTION 90
2. IDENTITY-CHANGING COMMUNITIES OF PRACTICE 94
3. A TEACHING MOMENT 96
3.1 Methodology: the teaching moment 97
3.2 Three stories including the teaching moment 98
4. A LEARNING EXPERIENCE 105
4.1 Methodology - the biology story 106
5. DISCUSSION: WHAT DOES THIS HAVE TO SAY ABOUT TEACHING AND LEARNING? 108
5.1 The learning of school students 108
5.2 The conceptualisation of teaching 110
ACKNOWLEDGEMENTS 112
REFERENCES 112
Chapter 6 Are Mathematical Abstractions Situated? 114
1. INTRODUCTION 114
1.1 Empiricist views on abstraction 115
1.2 Situation and context 117
1.3 Contextual views of abstraction 119
2. THE STUDY, THE TASKS AND PROTOCOL DATA 122
2.1 Protocol data 124
3. ABSTRACTION: MEDIATION, PEOPLE AND TASKS 128
3.1 Mediation 128
3.2 People 131
3.3 Tasks 134
4. CONCLUSIONS 136
REFERENCES 136
Chapter 7 ‘ We Do It A Different Way At My School’ 139
Mathematics homework as a site for tension and conflict 139
1. INTRODUCTION 139
2. RYAN, HIS MOTHER AND HIS HOMEWORK 143
2.1 The practice of homework 150
2.2 Ryan’s school and home identities 152
2.3 Ryan’s mother and mathematics 153
2.4 Tensions and conflicts during the homework event 154
3. DISCUSSION 156
ACKNOWLEDGEMENTS 160
REFERENCES 160
Chapter 8 Situated Intuition And Activity Theory Fill The Gap 162
The cases of integers and two-digit subtraction algorithms 162
1. INTRODUCTION 162
2. IMPLEMENTING THE INSTRUCTIONAL METHOD 165
2.1 The teaching of integers 166
2.2 The teaching of subtraction of two-digit numbers 174
2.3 Comparison of the 2-digit subtraction and the integer- operations cases 180
3. SITUATED INTUITION AND AUTHENTIC CLASSROOM LEARNING 182
3.1 Finally, what can we say about situated cognition and activity theory 184
REFERENCES 185
Chapter 9 The Role Of Artefacts In Mathematical Thinking: A Situated Learning Perspective 188
1. INTRODUCTION 188
2. RATIONALE FOR THE THEORETICAL OPTIONS AND THE FOCUS OF ANALYSIS 189
2.1 Why bring activity theory into a situated perspective? 189
2.2 Why look deeper into the situated role of artefacts? 190
3. CONCEPTS IN ACTION IN THE ANALYSIS OF THE STUDY 190
3.1 Activity in activity theory 191
3.2 Artefacts in activity theory 192
3.3 Activity from a situated perspective 193
3.4 Structuring resources and artefacts 195
4. THE STUDY OF THE ARDINAS’ PRACTICE 196
5. WHAT EMERGED FROM THE DATA ANALYSIS OF THE ARDINAS’ PRACTICE 199
5.1 First encounter: what is in a table and what does it tell us? 200
5.2 Second encounter: the calculator as artefact what does it tell us?
6. ARTEFACTS AND RESOURCES: HOW THE TECHNOLOGY OF THE PRACTICE PRODUCES A SHARED REPERTOIRE 208
REFERENCES 211
Chapter 10 Exploring Connections Between Tacit Knowing And Situated Learning Perspectives In The Context Of Mathematics Education21 214
1. INTRODUCTION 214
2. EXPLORING CONNECTIONS BETWEEN SCHOOL MATHEMATICAL PRACTICES AND OTHER SOCIO- CULTURAL ‘ MATHEMATICAL’ PRACTICES 218
3. TACIT KNOWLEDGE (OR KNOWING) AND VARIATIONS 224
4. CONNECTING TACIT MATHEMATICAL KNOWING, SITUATED PERSPECTIVES AND SCHOOL MATHEMATICS PRACTICES 227
4.1 What might a tacit-explicit dimension of school mathematics practice consist of ? 228
4.2 How does tacit knowledge (or knowing) manifest ‘ psychologically’ in mathematical activities? 234
4.3 Characterising the site of learning for the jangadeiros 234
4.4 What pedagogical implications result from connecting tacit mathematical knowing, situated perspectives and school mathematics practices? 236
5. CONCLUSION 237
REFERENCES 238
Chapter 11 Cognition And Institutional Setting 241
Undergraduates’ understandings of the derivative 241
1. INTRODUCTION 241
2. THEORETICAL FRAMEWORK OF THE STUDY 242
3. THE RESEARCH 245
4. INSTITUTIONAL CONTEXTS OF THE DEPARTMENTS 247
4.1 The Mechanical Engineering department 247
4.2 The Mathematics department 248
5. RESULTS 249
5.1 Student tests 249
5.2 The calculus modules 256
5.3 Lecturers’ views regarding their teaching practices 256
6. INSTITUTIONAL SETTINGS: STUDENTS AND LECTURERS 259
6.1 Calculus modules and students’ developing conceptions 259
6.2 But why did ‘what you teach’ differ? 260
6.3 Students’ situated developing conceptions 262
6.4 How do institutional settings influence lecturers and students? 264
REFERENCES 266
Chapter 12 School Practices With The Mathematical Notion Of Tangent Line 268
1. INTRODUCTION 268
2. RESEARCH FRAMEWORK 270
3. MATHEMATICS, SCHOOL MATHEMATICS AND LOCAL COMMUNITIES OF PRACTICE 272
4. SCHOOL PRACTICES AND SCHOOL MATHEMATICS 274
4.1 The technical design classroom 276
4.2 The highway system project classroom 281
5. THE MATHEMATICAL EXPERIENCES SHARED IN THE CLASSROOMS OBSERVED 286
5.1 The shared ways of behaving, language, habits, values and tool- use. 287
5.2 The developing mathematical competences, as recognized within the lessons observed 287
5.3 The common direction of learning: functioning mathematically, across school classrooms 289
6. SOME IMPLICATIONS FOR TEACHING 290
ACKNOWLEDGEMENTS 291
REFERENCES 291
Chapter 13 Learning Mathematically As Social Practice In A Workplace Setting 293
1. INTRODUCTION 293
2. THEORETICAL FRAMEWORK 294
3. RESEARCH METHODOLOGY 295
4. FACTORY PROCESSES 296
5. STUDENT ACTIVITIES 296
6. THE MATHEMATICAL DIMENSIONS OF THE WORK PRACTICES ON THE FACTORY FLOOR 299
7. DATA ANALYSIS 301
7.1 Adult mentors’ retellings of (mathematical) performance 301
7.2 Focus group interview with students 302
8. DISCUSSION 303
9. CONCLUSIONS 305
REFERENCES 306
Chapter 14 Analysing Concepts Of Community Of Practice 308
1. INTRODUCTION 308
2. DISTINGUISHING AMONG THE CONCEPTS OF COMMUNITIES OF PRACTICE 310
2.1 A characterisation of CPT1 311
2.2 A characterisation of CPT2 312
2.3 Examples in mathematics education 313
3. COMPARING AND CONTRASTING CTP1 AND CPT2 317
3.1 Community of practice: concepts of learning, identity and boundary 317
3.2 Problems arising from these comparisons 322
4. DEVELOPING THE CONCEPTS OF COMMUNITY OF PRACTICE 326
4.1 Sociology of structures 327
4.2 Discursive practice 328
4.3 Final remarks 329
ACKNOWLEDGEMENTS 332
REFERENCES 332
Chapter 15 ‘ No Way Is Can’t’: A Situated Account Of One Woman’s Uses And Experiences Of Mathematics 334
1. INTRODUCTION 1.1 One by One 334
1.2 Placing 335
1.3 Measurement 335
2. WRITING THIS CHAPTER 335
3. STARTING FROM ZERO 337
4. THE BLOCK OF FLATS 340
4.1 Alison’s version 340
4.2 Sandra’s version 341
4.3 What’s the situation? 342
5. PARTICIPATION AND PERSISTENCE 343
5.1 Peter’s view 343
6. LEARNING, FORGETTING AND MATHEMATICS 348
7. THE STORIES WE TELL AND DON’T TELL 351
8. CONCLUSION 354
REFERENCES 355
Acknowledgements 357
Index of Authors 358
Index 362