Combinatorics and Reasoning Representing, Justifying and Building Isomorphisms
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Carolyn A. Maher, Arthur B. Powell, Elizabeth B. Uptegrove
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Carolyn A. Maher, Arthur B. Powell, Elizabeth B. Uptegrove
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Combinatorics and Reasoning Representing, Justifying and Building Isomorphisms
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Springer-Verlag
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9789400706156
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1
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CHF 85.30
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Schulpädagogik, Didaktik, Methodik
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English
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226
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Wasserzeichen/DRM
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PC/MAC/eReader/Tablet
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PDF
Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms is based on the accomplishments of a cohort group of learners from first grade through high school and beyond, concentrating on their work on a set of combinatorics tasks. By studying these students, the editors gain insight into the foundations of proof building, the tools and environments necessary to make connections, activities to extend and generalize combinatoric learning, and even explore implications of this learning on the undergraduate level. This volume underscores the power of attending to basic ideas in building arguments; it shows the importance of providing opportunities for the co-construction of knowledge by groups of learners; and it demonstrates the value of careful construction of appropriate tasks. Moreover, it documents how reasoning that takes the form of proof evolves with young children and discusses the conditions for supporting student reasoning.
Preface
7
Acknowledgements
8
Contents
9
Introduction
11
Contributors
15
Introduction, Background, and Methodology
16
The Longitudinal Study
17
Methodology
23
Foundations of Proof Building (1989–1996)
29
Representations as Tools for Building Arguments
30
Towers: Schemes, Strategies, and Arguments
39
Building an Inductive Argument
56
Making Pizzas: Reasoning by Cases and by Recursion
69
Block Towers: From Concrete Objects to Conceptual Imagination
83
Making Connections, Extending, and Generalizing ( 1997– 2000)
97
Responding to Ankur’s Challenge: Co- construction of Argument Leading to Proof
98
Block Towers: Co-construction of Proof
105
Representations and Connections
113
Pizzas, Towers, and Binomials
129
Representations and Standard Notation
140
So Let’s Prove It!
152
Extending the Study, Conclusions, and Implications
162
“Doing Mathematics” from the Learners’ Perspectives
163
Adults Reasoning Combinatorially
176
Comparing the Problem Solving of College Students with Longitudinal Study Students
189
Closing Observations
205
Appendix A Combinatorics Problems
209
Appendix B Counting and Combinatorics Dissertations from the Longitudinal Study
217
References
218
Index
223
