Combinatorics and Reasoning

Representing, Justifying and Building Isomorphisms

By: Carolyn A. Maher (Editor), Arthur B. Powell (Editor), Elizabeth B. Uptegrove (Editor)

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Hardcover | 17 September 2010

At a Glance

Hardcover
246 Pages

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23.39 x 15.6 x 1.6

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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.

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You Can Find This Book In

Non-Fiction Mathematics Combinatorics & Graph Theory Groups & Group Theory Philosophy Education Teaching of a Specific Subject

Introduction, Background, and Methodology
The Longitudinal Study	p. 3
Methodology	p. 9
Foundations of Proof Building (1989-1996)
Representations as Tools for Building Arguments	p. 17
Towers: Schemes, Strategies, and Arguments	p. 27
Building an Inductive Argument	p. 45
Making Pizzas: Reasoning by Cases and by Recursion	p. 59
Block Towers: From Concrete Objects to Conceptual Imagination	p. 73
Making Connections, Extending, and Generalizing (1997-2000)
Responding to Ankur's Challenge: Co-construction of Argument Leading to Proof	p. 89
Block Towers: Co-construction of Proof	p. 97
Representations and Connections	p. 105
Pizzas, Towers, and Binomials	p. 121
Representations and Standard Notation	p. 133
So Let's Prove It!	p. 145
Extending the Study, Conclusions, and Implications
"Doing Mathematics" from the Learners' Perspectives	p. 157
Adults Reasoning Combinatorially	p. 171
Comparing the Problem Solving of College Students with Longitudinal Study Students	p. 185
Closing Observations	p. 201
Combinatorics Problems	p. 205
Counting and Combinatorics Dissertations from the Longitudinal Study	p. 213
References	p. 215
Index	p. 221
Table of Contents provided by Ingram. All Rights Reserved.

Combinatorics and Reasoning

Representing, Justifying and Building Isomorphisms

At a Glance

Hardcover

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