Mathematical Knowledge in Teaching (eBook)

Contents 5
Contributors 7
1 Introduction: Mathematical Knowledge in Teaching 9
Background: The Topic and the Book 9
Introduction to Section 1 10
Introduction to Section 2 11
Introduction to Section 3 12
Critical Discussion and Synthesis 13
References 13
Part I Conceptualising Mathematical Knowledge in Teaching 14
2 Conceptualising Teachers' Mathematical Knowledge in Teaching 15
Introduction 15
Shulman's Conceptualisation 16
Fennema and Franke's Conceptualisation 19
The Mathematics Teaching and Learning to Teach Project (MTLT) and the Learning Mathematics for Teaching Project (LMT): A Practice-Based Framework of Teachers' Mathematical Knowledge for Teaching 21
The Knowledge Quartet 24
Synthesis 26
Implications and Limitations 28
References 30
3 Knowing and Identity: A Situated Theory of Mathematics Knowledge in Teaching 32
The Problem of Mathematics Teacher Knowledge 33
A Case Study from Primary Mathematics: Alexandra's Knowledge of the Multiplication and Division of Fractions 34
Is this just an Issue for Primary Teaching? 39
The Contribution of Situated Theories: What Does This Mean for Teacher Knowledge? 40
Implications for the Practices of Teaching, Teacher Education and Development 43
References 45
4 Changed Views on Mathematical Knowledge in the Course of Didactical Theory Development: Independent Corpus of Scientific Knowledge or Result of Social Constructions? 48
Introduction 48
The 'Stoffdidaktik' Elaboration of Mathematical Knowledge as an Essential Factor Influencing Teaching and Learning Processes 49
The Synchronization Between the Dynamics of Knowledge Development and the Processes of Teaching and Learning 51
Mathematics Education Research and Mathematical Teaching-Learning-Practice as Independent Institutional Systems 56
Mathematical Knowledge in Teaching: A Case Illustrating the Epistemology-based Interaction View on Teaching Learning Processes 60
References 67
5 Teaching Mathematics as the Contextual Application of Mathematical Modes of Enquiry 70
One of Bill's Experiences 70
One of Anne's Experiences 70
Introduction 71
The Roles of Mathematical Modes of Enquiry in Teaching 71
Mathematical Modes of Enquiry 73
An Artificial Teacher Activity 75
Stimulus 1: Problems About Inverse Proportion 77
Stimulus 2: The Day's Newspaper 80
Discussion 81
Moving Forward 84
Conclusion 85
References 85
Appendix 87
Exercise 12 87
6 Conceptualising Mathematical Knowledge in Teaching 88
Subject Knowledge Differentiated 88
Subject Knowledge Contextualised 91
Subject Knowledge Interactivated 93
Subject Knowledge Mathematised 95
Reconceptualising Subject Knowledge in Teaching 98
References 100
Part II Understanding the Cultural Context of Mathematical Knowledge in Teaching 102
7 The Cultural Location of Teachers' Mathematical Knowledge: Another Hidden Variable in Mathematics Education Research? 103
Introduction 103
Mathematical Knowledge in Teaching: A Culturally-Located Model 104
The Project 107
Pauline 109
Eva 112
Discussion 115
Pauline 115
Eva 117
Conclusion 119
References 120
8 How Educational Systems and Cultures Mediate Teacher Knowledge: `Listening' in English, FrenchINTbreak and German Classrooms
Introduction 123
Listening to and 'Hearing' Students 125
Teacher Knowledge, Pedagogic Practice and Classroom Environments 127
The Study 128
Mathematics Classroom Environment 129
Teacher Knowledge and Listening to Pupils 131
Content Knowledge for Teaching 131
'Listening Knowledge' in/for Teaching 133
Discussion and Conclusions 137
References 138
9 Modelling Teaching in Mathematics Teacher Education and the Constitution of Mathematics for Teaching 142
Introduction 142
Mathematics Teacher Education in Post Apartheid South Africa 143
Studying Mathematics and Teaching in Mathematics Teacher Education 145
Reading 'What' in the Constitution of Mathematics in and for Teaching 146
Reading 'How' in the Constitution of Mathematics in and for Teaching 152
Three Cases of Mathematics Teacher Education 152
Case 1: Teaching and Learning Mathematical Reasoning 153
Case 2: Algebra Content and Pedagogy 153
Case 3: Reflecting on Mathematics Teaching 158
Mathematics for Teaching Across Cases of Mathematics Teacher Education 160
In Conclusion 161
References 162
10 Audit and Evaluation of Pedagogy: Towards a Cultural-Historical Perspective 164
Introduction 164
Accounting for the Dialectic of Audit 165
What Is the Purpose of Audit and Evaluation of Teachers Knowledge? 170
What Kinds of Knowledge ‘Should’ (Mathematics) Teachers ‘Have’ for – or ‘Display’ in – Teaching? 171
How Can We Audit/Assess Teacher Knowledge: What Tools/Technologies Do We Have? 172
Conclusion 177
Discussion: Towards a Collective Subject 178
References 179
11 The Cultural Dimension of Teachers' Mathematical Knowledge 182
A Case for Considering Culture in Research on Teachers' Mathematical Knowledge 182
The Interplay Between the Cultural Context and Mathematical Knowledge for/in Teaching 185
The Cultural Embedding of Mathematical Knowledge in Teaching in the Context of National Educational Systems 185
The Cultural Embedding of Mathematical Knowledge for Teaching in the Context of Diverse Teacher Education Programmes 187
The Embedding of Mathematical Knowledge for Teaching in a ''Knowledge Economy'' Culture 188
Implications for Teacher Education 189
References 192
Part III Building Mathematical Knowledge in Teaching by Means of Theorised Tools 195
12 The Knowledge Quartet as an Organising Framework for Developing and Deepening Teachers' Mathematics Knowledge 196
Introduction 196
Rationale 196
Developing the Knowledge Quartet 197
Context and Purpose of the Research 197
Method 198
Conceptualising the Knowledge Quartet 200
Foundation 201
Transformation 201
Connection 202
Contingency 202
The Knowledge Quartet and Mathematics Teaching Development 203
Development in Conceptions of Mathematics Teaching 204
Development of Content Knowledge 207
Conclusion 210
References 212
13 Learning to Teach Mathematics Using Lesson Study 214
Introduction 214
Enhancement of Teaching Through Lesson Study 214
Lesson Study Appraised 216
The Role of Knowledgeable Other(s) 217
The Dublin Study 218
Overview of the Lesson Study Elective Course 219
Data Analysis 220
'Doing' Lesson Study 221
Preparing the Lessons: Cycle One 221
Research Lessons: Cycle Two 222
Learning Takes Time 222
'Doing' Lesson Study: Cycle Three 222
'Doing' Mathematics 223
'Being' in the Lesson Study Elective Community of Practice 224
Discussion 225
Identity in Terms of Learning to Teach Mathematics 225
The Case of Brd 225
Descriptive Synopsis of Bríd’s Lesson 226
Learning from Teaching 226
Mathematics Teaching and Matters of Interpretation 228
Findings 228
Lesson Study as a Tool for Developing Mathematical Knowledge in Teaching 229
References 230
14 Using Theories to Build Kindergarten Teachers' Mathematical Knowledge for Teaching 232
Introduction 232
Combining Theories of Teacher Knowledge with Theories of Mathematics Knowledge 233
Dimensions of Knowledge for Teaching 233
Concept Image-Concept Definition (CICD) 234
The Combined Framework 236
Setting 237
Research Segments 238
Building Kindergarten Teachers' SCK Regarding Concept Images and Concept Definitions of Triangles 238
Differentiating Between SCK and KCT 240
Building Kindergarten Teachers' KCT Regarding Concept Definitions and Concept Images of Triangles 242
Kindergarten Children's Knowledge of Pentagons 245
Summing Up and Looking Ahead 247
References 249
15 Teachers' Stories of Mathematical Subject Knowledge: Accounting for the Unexpected 252
Introduction 252
Teachers' Mathematical Knowledge 253
Testing Subject Knowledge 256
Personalised Diagnostic Maps of Subject Knowledge 258
Narrative Accounts -- The Impetus of 'Troubles' 261
Lorna 262
Charlene 265
Comparison, Contrast and Limitations 268
Conclusion and Discussion 269
References 271
16 Building Mathematical Knowledge in Teaching by Means of Theorised Tools 273
Introduction 273
Theorised Tools from Teachers Knowledge: KQ, SMK & PCK, MKT
The CICD -- A Theorised Tool from Mathematics Knowledge 278
The Role of the Researcher/Instructor/Teacher Educator in Building Mathematical Knowledge in Teaching 280
The Role of the Mathsmaps in Building Mathematical Knowledge and PCK 282
Final Remarks 283
References 285
17 Conclusion 288
Author Index 291
Subject Index 296