Mathematics as a Constructive Activity: Learners Generating Examples

by Watson; Anne

• ISBN13: 9780805843446
• ISBN10: 0805843442
• Format: Nonspecific Binding
• Copyright: 2005-05-17
• Publisher: Routledge

Summary

This book explains and demonstrates the teaching strategy of asking learners to construct their own examples of mathematical objects. The authors show that the creation of examples can involve transforming and reorganising knowledge and that, although this is usually done by authors and teachers, if the responsibility for making examples is transferred to learners, their knowledge structures can be developed and extended.

Table of Contents

Preface	ix
Introduction to Exemplification in Mathematics	1	(9)
What Is an Example?	3	(3)
On Whose Shoulders Are We Standing?	6	(3)
Summary	9	(1)
Learner-Generated Examples in Classrooms	10	(23)
Teacher-Initiated, Learner-Generated Examples	12	(13)
Learners Using Examples, Counterexamples, and Extreme Examples	25	(5)
Initial Theorizing: Shifting Responsibility	30	(2)
Summary	32	(1)
From Examples to Example Spaces	33	(26)
A Difference of 2	34	(5)
Inter-Rootal Distances	39	(12)
How Do These Themes Relate to Our Work With Other Groups?	51	(6)
Summary	57	(2)
The Development of Learners' Example Spaces	59	(33)
Reminder: What Is an Example Space?	59	(1)
Metaphors	60	(2)
Central Examples	62	(3)
Nonexamples and Counterexamples	65	(5)
Promoting Development	70	(20)
Summary	90	(2)
Pedagogical Tools for Developing Example Spaces	92	(41)
Kinds of Examples	92	(11)
Encountering Important Examples	103	(7)
Constructing New Objects for Oneself	110	(20)
Being Asked to Give Examples: The Larder Metaphor	130	(2)
Summary	132	(1)
Strategies for Prompting and Using Learner-Generated Examples	133	(27)
Case Studies	133	(17)
Summary of Strategies	150	(7)
Focus on Action	157	(1)
Practice and Fluency	158	(1)
Summary	159	(1)
Mathematics as a Constructive Activity	160	(37)
Constructive Constraints	161	(7)
Building Confidence	168	(8)
Learning as Construction	176	(7)
Gathering Threads	183	(8)
Constructed Outcomes	191	(4)
Epilogue: Constructing Tasks	195	(2)
Appendix A: Some Historical Remarks on Teaching by Examples	197	(13)
Early Practices and Implicit Theories	198	(2)
Explicit Theories and Implicit Practices	200	(4)
The Inductive Method	204	(2)
Toward the More Active Learner	206	(2)
Summary	208	(2)
Appendix B: Suggestions About Some of the Tasks	210	(3)
References	213	(10)
Author Index	223	(4)
Subject Index	227