Mathematics As A Constructive Activity Learners Generating Examples

Anne Watson; John Mason0805843434, 9781410613714, 9780805843446, 9780805843439, 0805843442

Watson (Oxford U.) and Mason (Open U.) examine teaching techniques which actively engage learners, enhancing their sophistication and deepening their understanding, revealing their depths, making them accept initiative in finding and applying deeper concepts, and enriching enjoyment of mathematical structures and connections. They give examples used in classrooms, then “example spaces” specifically for learners to explore, then the tools teachers need to help learners develop those “example spaces” and strategies for prompting learners to generate them. They close with a very useful chapter for those seeking to justify expansion of programs that examines the constructive activities of mathematics, including its ability to teach how to learn, how to build evidence, and construct outcomes.

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Table of contents :
Contents……Page 8
Preface……Page 10
1 Introduction to Exemplification in Mathematics……Page 16
What Is an Example?……Page 18
On Whose Shoulders Are We Standing?……Page 21
Summary……Page 24
2 Learner-Generated Examples in Classrooms……Page 25
Teacher-Initiated, Learner-Generated Examples……Page 27
Learners Using Examples, Counterexamples, and Extreme Examples……Page 40
Initial Theorizing: Shifting Responsibility……Page 45
Summary……Page 47
3 From Examples to Example Spaces……Page 48
A Difference of 2……Page 49
Inter-Rootal Distances……Page 54
How Do These Themes Relate to Our Work With Other Groups?……Page 66
Summary……Page 72
Reminder: What Is an Example Space?……Page 74
Metaphors……Page 75
Central Examples……Page 77
Nonexamples and Counterexamples……Page 80
Promoting Development……Page 85
Summary……Page 105
Kinds of Examples……Page 107
Encountering Important Examples……Page 118
Constructing New Objects for Oneself……Page 125
Being Asked to Give Examples: The Larder Metaphor……Page 145
Summary……Page 147
Case Studies……Page 148
Summary of Strategies……Page 165
Focus on Action……Page 172
Practice and Fluency……Page 173
Summary……Page 174
7 Mathematics as a Constructive Activity……Page 175
Constructive Constraints……Page 176
Building Confidence……Page 183
Learning as Construction……Page 191
Gathering Threads……Page 198
Constructed Outcomes……Page 206
Epilogue: Constructing Tasks……Page 210
Appendix A: Some Historical Remarks on Teaching by Examples……Page 212
Early Practices and Implicit Theories……Page 213
Explicit Theories and Implicit Practices……Page 215
The Inductive Method……Page 219
Toward the More Active Learner……Page 221
Summary……Page 223
Appendix B: Suggestions About Some of the Tasks……Page 225
References……Page 228
D……Page 238
M……Page 239
U……Page 240
Z……Page 241
F……Page 242
Z……Page 243