Approaches to Qualitative Research in Mathematics Education (eBook)

Preface 6
References 11
Contents 14
Part I: Grounded Theory Methodology 17
Chapter 1: Grounded Theory Methods 18
1.1 The Development of “Grounded Theory” 18
1.1.1 Overview of Research Processes 19
1.2 Place of Literature Review in Grounded Theory 20
1.3 Data Analysis: Open Coding 21
1.4 Memoing 22
1.4.1 Writing Memos and Using Diagrams 22
1.4.2 Using Computer Programs 23
1.5 Intermediate Coding and the Use of a Coding Paradigm 24
1.5.1 Heuristic Concepts 25
1.5.2 Coding for Process 25
1.6 Delimiting the Study 27
1.6.1 Theoretical Sampling and Saturation 27
1.6.2 Core Category 28
1.7 Theoretical Integration 28
1.7.1 Sorting Memos 29
1.7.2 Validating the Theory 30
1.8 Interpretive Frameworks 30
1.8.1 Pragmatism 31
1.8.2 Corbin and Strauss Circa 2008: Pragmatism and Symbolic Interactionism 32
1.8.3 Constructivist Grounded Theory 33
1.8.4 Situational Analysis 33
1.9 End Comment 35
References 35
Chapter 2: To See the Wood for the Trees: The Development of Theory from Empirical Interview Data Using Grounded Theory 37
2.1 Background and Focus of the Study 38
2.2 Realization of the Study 39
2.3 Theoretical Sensitivity and Sensitizing Concepts 42
2.4 Interdependence of Data Collection, Analysis, and Development of Theory 42
2.5 Data Analysis 47
2.5.1 Open Coding 48
2.5.2 Axial Coding 49
2.5.3 Exemplary Illustration of Open and Axial Coding Using Memos and Diagrams 50
2.5.4 Selective Coding 58
2.6 Going Beyond Grounded Theory 59
2.7 Conclusion 60
References 60
Part II: Approaches to Reconstructing Argumentation 63
Chapter 3: Methods for Reconstructing Processes of Argumentation and Participation in Primary Mathematics Classroom Interaction 64
3.1 Introduction 65
3.2 The Concepts of Argumentation and Participation 65
3.2.1 The Example “Thirteen Pearls” 67
3.2.1.1 The Transcript 67
3.2.1.2 Analyses of the Scene 68
The Analysis of Argumentation 68
The Analysis of Participation 70
3.2.2 The Example of “Mister X” 74
3.2.2.1 The Transcript 74
3.2.2.2 The Analyses of the Scene 75
The Analysis of Argumentation 75
The Analysis of Participation 78
3.2.3 Comparison of the Results of the Analyses of the Two Scenes 78
3.3 Some Theoretical Remarks 81
3.3.1 Further Research on the “Production-Design” in Mathematics Classes 82
3.3.2 More Complexly Structured Argumentations 82
Appendix: Transcripts and Rules of Transcription 83
Thirteen Pearls 83
Mister X 84
Rules of Transcription 85
References 86
Chapter 4: Reconstructing Argumentation Structures: A Perspective on Proving Processes in Secondary Mathematics Classroom Interactions 88
4.1 Introduction 88
4.2 The Importance of Understanding Proving Practices in the Classroom 90
4.3 Approaches to Describing Arguments 92
4.3.1 The Inadequacy of Logical Analysis for Reconstructing Proving Processes in Classrooms 92
4.3.2 Toulmin’s Functional Model of Argument 93
4.3.3 Local and Global Arguments 95
4.4 A Method for Reconstructing Arguments in Classrooms 96
4.4.1 Reconstructing the Sequencing and Meaning of Classroom Talk 97
4.4.1.1 Layout of Episodes 97
4.4.1.2 Turn by Turn Analyses 98
4.4.2 Analyzing Arguments and Argumentation Structures 99
4.4.2.1 Functional Reconstruction of the Argumentation 99
4.4.2.2 Reconstructing the Argumentation Structure of Proving Processes in Class 102
4.4.3 Comparing Global Argumentation Structures 104
4.4.3.1 Source-Structure 105
4.4.3.2 Spiral-Structure 106
4.4.3.3 Comparing Source-and-Spiral Argumentation Structures 109
4.5 Conclusion 111
References 112
Part III: Ideal Type Construction 115
Chapter 5: Empirically Grounded Building of Ideal Types. A Methodical Principle of Constructing Theory in the Interpretative Research in Mathematics Education 116
5.1 Introduction 116
5.2 Theories and Their Significance 117
5.3 The Notion of Theory in Interpretative Mathematics Education Research 119
5.4 Theory-Developing Research 120
5.5 Looking Back: The Roots of the Ideal Type Concept 122
5.6 Ideal Type Construction: Method of Everyday Understanding 126
5.7 Empirically Based Ideal Type Construction: A New Beginning 131
5.8 Ideal Type Construction in Research of Mathematics Education 132
5.8.1 Ideal Type Construction by Idealizing of Prototypes 132
5.8.2 Ideal Type Construction: Principle of Factual Theory Construction 134
5.8.3 A Model of Polar Ideal-Type Construction 135
5.8.4 Construction of Epistemic Action Types 137
5.8.5 Construction of Production Types 138
5.8.6 Ideal Type Construction with Ideal Types 140
5.9 Summary and Conclusion 142
References 143
Chapter 6: How Ideal Type Construction Can Be Achieved: An Example 147
6.1 Introduction 147
6.2 Methodological and Theoretical Considerations 148
6.3 Example: Constructing Ideal Types of Epistemic Processes 150
6.3.1 Step 1: Re-constructing the Cases Illustrated by an Epistemic Process as a Case for Ideal Type Construction 150
6.3.1.1 Approaching the Empirical Case with Peirce’s Sign Concept 150
6.3.1.2 The Epistemic Actions of Gathering and Connecting Mathematical Meanings 152
6.3.1.3 The Epistemic Action of Structure-Seeing 155
6.3.1.4 Representing the Course of the Epistemic Process 156
6.3.1.5 A Pictograph Representing the Phase Structure 158
6.3.2 Step 2: Grouping the Cases 158
6.3.3 Step 3: Building Ideal Types 160
6.3.4 Step 4: Building Theoretical Knowledge 161
6.4 What Can Be Learnt from This Example? 162
Appendix: Transcription Key 163
References 163
Part IV: Semiotic Research 165
Chapter 7: The Question of Method in a Vygotskian Semiotic Approach 166
7.1 Introduction 167
7.2 Method as the Central Problem of Scientific Inquiry 168
7.3 A Vygotskian Semiotic Approach 170
7.3.1 Knowledge 171
7.3.2 Learning 173
7.4 The Methodology of Our Semiotic Approach 174
7.5 Multi-Semiotic Analysis: An Example Concerning Pattern Generalization 178
7.5.1 Words-Gesture Combinations in the Production of a Factual Generalization 179
7.5.2 Words, Gesture and Rhythm: Refining the Generalization 184
7.6 Concluding Remarks 187
References 189
Part V: A Theory on Abstraction and Its Methodology 192
Chapter 8: The Nested Epistemic Actions Model for Abstraction in Context: Theory as Methodological Tool and Methodological Tool as Theory 193
8.1 Theory 194
8.2 The AiC Methodology 197
8.2.1 Design for Abstraction 198
8.2.2 A Priori Analysis 198
8.2.3 Data Collection and Preliminary Analysis 200
8.2.4 Need 201
8.2.5 Analysis According to the RBC-Model 202
8.2.6 Consolidation 205
8.2.7 Who Is Constructing? 206
8.3 A Focus Group in a Classroom as an Example 206
8.3.1 Design for Abstraction 207
8.3.2 A Priori Analysis 209
8.3.3 Data Collection and Preliminary Analysis 211
8.3.4 Analysis According to the RBC-Model, Including Need and Consolidation 211
8.3.4.1 Episode 1: Constructing EP 211
8.3.4.2 Episode 2: Partially Constructing ECE and ESS 213
8.3.4.3 Episode 3: The differences’ Task (Task 2) 216
8.3.5 Additional Methodological Comments 218
8.3.5.1 Knowledge Construction and Social Interaction 218
8.3.5.2 Tools as Contextual Elements in Knowledge Construction 218
8.3.5.3 Revision of the Instructional Design on the Basis of RBC Analysis 219
8.4 The Relationship of Theory and Methodology in AiC 219
References 223
Part VI: Networking of Theories 226
Chapter 9: Advancing Research by Means of the Networking of Theories 227
9.1 Introduction 227
9.1.1 The Evolution of Networking 228
9.1.2 Why Networking? 228
9.2 Language for Networking 229
9.2.1 The Semiosphere 229
9.2.2 The Essence of Networking and Its Limits 229
9.3 Methodological Considerations 230
9.3.1 Networking Strategies: The Spectrum of Networking Theories 230
9.3.2 Cross-Methodologies for Networking 231
9.4 Different Cases of Networking 232
9.5 Methodological Reflections: Difficulties That Accompany the Networking 234
9.6 Benefits of Networking: Advancing Research by Means of the Networking of Theories 235
References 235
Chapter 10: A Cross-Methodology for the Networking of Theories: The General Epistemic Need (GEN) as a New Concept at the Boundary of Two Theories 239
10.1 Introduction 239
10.1.1 Abstraction in Context 240
10.1.2 The Theory of Interest-Dense Situations 241
10.1.3 General Description of the Cross-Methodology of Networking the Two Theories 242
10.2 An Illustrative Example: Investigating the General Epistemic Need 243
10.2.1 The Task and Its Setting 244
10.2.2 Beginning a Cross-Analysis 245
10.2.3 Separate Analysis from the AiC-View 247
10.2.4 A Re-Analysis from the IDS-View and Its Results 249
10.3 Methodological Reflections About the Networking Process 253
Transcription Key 254
References 255
Part VII: Multi-Level-Analysis 257
Chapter 11: Understanding Learning Across Lessons in Classroom Communities: A Multi-leveled Analytic Approach 258
11.1 A Conceptual Framework for Analyzing the Generation of Common Ground 259
11.1.1 Core Constructs 260
11.1.2 The Reproduction and Alteration of a Common Ground: An Illustrative Exchange 261
11.1.3 Analyzing Common Ground at Collective and Individual Levels 264
11.1.3.1 The Collective Level 264
11.1.3.2 The Individual Level 266
Microgenesis 268
Ontogenesis 270
Sociogenesis 271
The Interplay Between Micro-, Onto-, and Sociogenetic Developments in Collective Activities 273
11.1.3.3 A Final Note on Collective and Individual Activity 274
11.2 An Illustration of Empirical Techniques: The Learning Mathematics Through Representations Project 275
11.2.1 Empirical Techniques Used to Inform Design Choices for the LMR Lesson Sequence 275
11.2.1.1 Preliminaries 275
11.2.1.2 Empirical Techniques and Design Choices 277
Interview Studies 277
Tutorial Studies 282
Classroom Studies 285
Preliminary Classroom Studies 285
LMR Classroom Studies: Preliminary Lessons and Their Iterative Refinement 286
Support for Use of the Lesson Sequence with New Teachers 287
Efficacy study 287
Student Assessment Instrument 287
Student Assessments and Growth 288
11.2.2 The Complete Lesson Series 288
11.2.2.1 Supports for a Common Ground of Talk and Action with Shifting Lesson Topics 290
Ordering of Lesson Topic 291
Definitions and Principles 291
Cuisenaire™ Rods (C-Rods) 294
Recurrent Lesson Structure 295
11.2.3 Empirical Techniques Used to Analyze the Reproduction and Alteration of a Common Ground of Talk and Action in a Classroom Community 296
11.2.3.1 Empirical Techniques: Data Collection 296
Video Records (of Lessons) 297
Assessment of Integers and Fractions Knowledge 297
After Class Interviews 298
Sociogram 298
Teacher Interviews 298
11.2.3.2 Empirical Techniques: Data Reduction and Analytic Approach 299
Collective Level: A Focus on Emergent Norms in the Classroom Community 301
Sociomathematical Norm #1. Use Definitions to Support Your Ideas: Especially When Explaining Thinking or Justifying Reasoning 302
Sociomathematical Norm #2. When Definitions Are Used, Connect Them to a Particular Problem Context and/or to Other Definitions 304
Individual Level: A Focus on the Microgenesis, Ontogenesis, and Sociogenesis of Form-Function Relations 306
Microgenesis: Empirical Techniques and the Unit Interval 307
Ontogenesis: Empirical Techniques and the Unit Interval 308
After Class Interviews: Ravena’s After Class Interview and Shifts in Thinking Through the Lesson 309
Student Assessment Instrument to Document Longer-Term Ontogenetic Shifts: Pre-, Interim-, Post-, and Final Assessments 310
Sociogenesis: Empirical Techniques and the Unit Interval 313
Documenting Shifts in the Distributions of Form-Function Relations 314
Explaining Shifts in the Distributions of Form-Function Relations 316
11.3 Final Thoughts and Next Steps 318
References 321
Part VIII: Mixed Methods 324
Chapter 12: The Combination of Qualitative and Quantitative Research Methods in Mathematics Education: A “Mixed Methods” Study on the Development of the Professional Knowledge of Teachers 325
12.1 Introduction 326
12.2 “Mixed Methods”: Challenging the Qualitative-­Quantitative Divide in Social and Educational Research 327
12.3 The Dispute About “Quan” and “Qual” and Mixed Methods in Research on Mathematics Education 329
12.4 Basic Methodological Concepts of Method Integration 333
12.4.1 Combination of Methods and Techniques During Data Collection and Analysis 333
12.4.2 Integration of Methodological Approaches Within one Research Design 334
12.5 Capabilities and Functions of Mixed Methods Designs 337
12.5.1 Strengths and Challenges of Quantitative Methods 337
12.5.2 Strengths and Challenges of Qualitative Methods 340
12.5.3 Types of Mixed Methods Designs and Their Function in the Research Process 342
12.6 An Example of a Mixed Methods Research Design in Mathematics Education 344
12.6.1 Research Purpose and Mixed Methods Design of the TEDS-Telekom Study 344
12.6.2 The Quantitative Sub-Study 346
12.6.3 The Qualitative Sub-Study 349
12.6.4 Triangulation in the Mixed Methods Design: Relating Quantitative and Qualitative Findings to Each Other 351
12.7 Different Functions of Mixed Methods Designs: An Overview 357
References 359
Part IX: Qualitative Content Analysis 366
Chapter 13: Qualitative Content Analysis: Theoretical Background and Procedures 367
13.1 Methodological Background of Qualitative Content Analysis 367
13.2 Development and Definition of Content Analysis 369
13.3 Basics of Qualitative Content Analysis 371
13.3.1 Embedding of the Material Within the Communicative Context 371
13.3.2 Systematic, Rule-Bound Procedure 371
13.3.3 Categories as the Focus of Analysis 372
13.3.4 Object Reference in Place of Formal Techniques 373
13.3.5 Pilot Testing of the System of Categories and the Content Analytical Rules 373
13.3.6 Theory-Guided Character of the Analysis 374
13.3.7 Integrating Quantitative Steps of Analysis 374
13.3.8 Quality Criteria 374
13.4 Basic Procedures or Techniques of Qualitative Content Analysis 375
13.4.1 Inductive Category Formation 376
13.4.2 Deductive Category Assignment (Structuring) 378
13.5 Final Appraisal of the Qualitative Content Analysis 380
References 381
Chapter 14: A Study on Professional Competence of Future Teacher Students as an Example of a Study Using Qualitative Content Analysis 383
14.1 Introduction 383
14.2 Theoretical Framework and Research Question of the Study 384
14.3 Why Was Qualitative Content Analysis Chosen? 385
14.4 How Was Qualitative Content Analysis Used in This Study? 389
14.5 Summary 399
References 399
Part X: Triangulation and Cultural Studies 402
Chapter 15: The Contemporary Importance of Triangulation in a Post-Positivist World: Examples from the Learner’s Perspective Study 403
15.1 Introduction 404
15.2 Triangulation 406
15.3 Research as the Mobilization of Bias 409
15.3.1 Mathematics Teaching 409
15.3.2 Chinese Learners’ Paradox 410
15.3.3 National Pedagogies 410
15.3.4 Research Design Characteristics 410
15.4 Characteristic Features of the Hong Kong LPS Research Implementation 411
15.5 Triangulation and Acts of Cross-Cultural Comparison 413
15.5.1 Lesson Patterns or Lesson Structure 414
15.5.2 The Hong Kong Investigation of Lesson Structure 416
15.5.3 Contrasting the Enactment of “Lesson Events” Across Different Cultural Systems 418
15.5.3.1 The Learning Task: One Example of a Lesson Event 420
15.6 Conclusion 422
References 423
Part XI: Design Research as a Research Methodology 426
Chapter 16: An Introduction to Design-Based Research with an Example From Statistics Education 427
16.1 Theory of Design-Based Research 427
16.1.1 Purpose of the Chapter 427
16.1.2 Characterizing Design-Based Research 428
16.1.2.1 Integration of Design and Research 428
16.1.2.2 Predictive and Advisory Nature of DBR 428
16.1.2.3 The Role of Hypotheses and the Engineering Nature of DBR 429
16.1.2.4 Open and Interventionist Nature of DBR 430
16.1.2.5 Comparison of DBR with Randomized Controlled Trials (RCT) 431
16.1.2.6 Comparison of DBR with Action Research 433
16.1.2.7 Names and History of DBR 434
16.1.2.8 Theory Development in Design-Based Research 435
16.1.2.9 Summary of Key Characteristics of Design-Based Research 435
16.1.3 Hypothetical Learning Trajectory (HLT) 436
16.1.3.1 HLT in the Design Phase 437
16.1.3.2 HLT in Teaching Experiment 437
16.1.3.3 HLT in the Retrospective Analysis 438
16.1.4 Phases in DBR 439
16.1.4.1 Phase 1: Preparation and Design 439
16.1.4.2 Phase 2: Teaching Experiment 439
16.1.4.3 Retrospective Analysis 440
16.1.5 Validity and Reliability 441
16.1.5.1 Internal Validity 442
16.1.5.2 External Validity 442
16.1.5.3 Internal Reliability 443
16.1.5.4 External Reliability 443
16.2 Example of Design-Based Research 443
16.2.1 Relevance and Aim 444
16.2.2 Research Question 444
16.2.3 Orienting Framework: Diagrammatic Reasoning 445
16.2.4 Domain-Specific Framework for Action: Realistic Mathematics Education (RME) 446
16.2.5 Methods 447
16.2.6 HLT and Retrospective Analysis 450
16.2.6.1 Analysis of the First Phase of Growing a Sample 452
16.2.6.2 Analysis of the Second Phase of Growing a Sample 454
16.2.6.3 Analysis of the Third Phase of Growing a Sample 456
16.2.7 Reflection on the Example 458
16.2.8 Final Remarks 459
Appendix: Structure of a DBR Project with Illustrations 459
References 461
Chapter 17: Perspectives on Design Research: The Case of Didactical Engineering 465
17.1 Introduction 465
17.2 Didactical Engineering: An Historical Review 466
17.3 Didactical Engineering as a Research Methodology 469
17.3.1 Preliminary Analyses 470
17.3.2 Conception and a Priori Analysis 471
17.3.3 Realization, Observation and Data Collection 472
17.3.4 A Posteriori Analysis and Validation 472
17.3.5 The Nature of the Results 473
17.3.6 Didactical Engineering and Design-Based Research 474
17.4 Two Particular Examples 475
17.4.1 A Paradigmatic Example: The Extension of the Field of Numbers by G. and N. Brousseau 475
17.4.1.1 Preliminary Analyses 476
17.4.1.2 Conception and Analysis a Priori 476
17.4.1.3 Realization, Data Collection, a Posteriori Analysis, Validation and Further Outcomes 478
17.4.2 An Example of Didactical Engineering Combining the Theory of Didactical Situations with Semiotic Perspectives 480
17.4.2.1 Preliminary Analyses 480
17.4.2.2 Conception and Analysis a Priori 481
17.4.2.3 Data Collection, a Posteriori Analysis and Validation 483
17.5 Some Recent Developments of Didactical Engineering 488
17.5.1 Didactical Engineering and the Anthropological Theory of Didactics 488
17.5.2 Research and Development: Didactical Engineering of Second Generation 490
17.6 Conclusion 491
References 492
Chapter 18: Educational Design Research to Support System-Wide Instructional Improvement 495
18.1 The United States Context 497
18.2 An Orienting Vision of High-Quality Mathematics Instruction 498
18.3 Design Studies to Investigate and Support System-Wide Improvement in Mathematics Instruction 499
18.4 Developing Initial Conjectures 500
18.5 Recruiting Collaborating Educational Systems 502
18.6 Using an Interpretive Framework to Assess Designed and Implemented Improvement Strategies 503
18.7 New Positions 504
18.8 Learning Events 505
18.8.1 Intentional Learning Events 505
18.8.2 Incidental Learning Events 506
18.9 New Organizational Routines 507
18.10 New Tools 507
18.11 Summary 509
18.12 Conducting Design, Analysis and Feedback Cycles 509
18.12.1 Documenting Current Instructional Improvement Strategies 510
18.12.2 Documenting How Instructional Improvement Strategies Are Implemented 513
18.12.3 Sharing Findings and Recommendations with System Leaders 514
18.12.4 Assessing the Influence of Recommendations on Collaborating System’s Instructional Improvement Strategies 517
18.13 Testing and Revising Conjectures that Comprise a Theory of Action for System-Wide Instructional Improvement 519
18.14 Findings About the Districts’ Instructional Improvement Strategies 519
18.15 Research Literature 519
18.16 Retrospective Analyses 520
18.17 MIST’s Current Theory of Action for Instructional Improvement in Middle-Grades Mathematics 520
18.18 Conclusion 522
References 523
Part XII: Final Considerations 529
Chapter 19: Looking Back 530
References 533
Bibliography 534
Author Index 573
Subject Index 580