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Chapter 1 Introduction. 13 

1.1 Overview.. 13 

1.2 Education as a design science. 14 

1.3 Systemisation of design. 14 

1.4 Design as a method, an object and an outcome of study. 16 

1.5 The Demonstrator Domain: Learning About Number Sequences. 17 

1.6 The research setting. 18 

1.7 Process and methods. 19 

1.8 Aims of this study. 20 

1.9 Structure of the Thesis. 21 

Chapter 2 TEME as A Design Science of Learning. 24 

2.1 Introduction. 24 

2.2 What is a Science of Design, and what value does it bring to TEME?. 25 

2.3 Further Challenges for a Design Science of TEME. 34 

2.4 Summary and Conclusions. 35 

Chapter 3 Aims and Method of Enquiry. 39 

3.1 Introduction. 39 

3.2 Notes on Aim 1. 41 

3.3 Notes on Aim 2. 42 

3.4 Notes on Aim 3. 43 

3.5 Method of Enquiry. 43 

3.6 Conclusions. 44 

Chapter 4 A Pattern-Based Approach to Design Research in TEME.. 45 

4.1 Introduction: In Search of an Epistemic Infrastructure. 45 

4.2 A Possible Path from Experience to Knowledge. 49 

4.3 The Narrative Turn in Mathematics Education Research. 52 

4.4 Design Narratives. 53 

4.5 Design Patterns. 57 

4.6 Bringing it all Together: Attaching the Representations to the Research Cycle. 65 

4.7 “Yes, but is it methodological?”. 67 

4.8 Conclusions. 69 

Chapter 5 Methodological Framework, Methods and Settings. 70 

5.1 Introduction: a Methodological Framework. 70 

5.2 Experimental Setting. 72 

5.3 Collection and Management of Data. 79 

5.4 From Data to Design Narratives. 86 

5.5 From Design Narratives to Design Patterns. 88 

5.6 Summary and Conclusions. 94 

Chapter 6 The Demonstrator Problem Domain. 96 

6.1 Overview.. 96 

6.2 Number patterns, sequences and mathematical structure. 96 

6.3 Representational systems and mathematical conceptualization. 111 

6.4 Conclusions. 121 

Chapter 7 Design Narratives. 122 

7.1 Introduction. 122 

7.2 Basic Sequence Activities. 123 

7.3 Guess My Robot 133 

7.4 Convergence and Divergence. 149 

7.5 Technological Infrastructure. 158 

7.6 Conclusions. 167 

Chapter 8 Design Patterns. 171 

8.1 Introduction. 171 

8.2 Mathematical Game Pieces. 176 

8.3 Soft scaffolding. 181 

8.4 Narrative spaces. 185 

8.5 Objects to talk with. 189 

8.6 Guess My X.. 192 

8.7 Transparent Stream.. 197 

8.8 Task-in-a-box. 202 

8.9 Thumbnails. 206 

8.10 Conclusions. 207 

Chapter 9 Summary and Conclusions. 209 

9.1 Overview.. 209 

9.2 Findings related to Aim 1. 210 

9.3 Findings related to Aim 2. 211 

9.4 Findings related to Aim 3. 213 

9.5 Limitations of this study. 216 

9.6 Open Questions and Future Research. 218 

9.7 Final comments. 221 

References. 222 

Appendix I Final design of activities and the tools to support them.. 256 

Appendix II Examples of data sources and methods of analysis. 279 

Appendix III Data sources for the design narratives in Chapter 7. 286 

Appendix IV Published Papers
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