Ebook: Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms

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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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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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.

Content:
Front Matter....Pages I-XVII
Front Matter....Pages 1-1
The Longitudinal Study....Pages 3-8
Methodology....Pages 9-14
Front Matter....Pages 15-15
Representations as Tools for Building Arguments....Pages 17-25
Towers: Schemes, Strategies, and Arguments....Pages 27-43
Building an Inductive Argument....Pages 45-57
Making Pizzas: Reasoning by Cases and by Recursion....Pages 59-72
Block Towers: From Concrete Objects to Conceptual Imagination....Pages 73-86
Front Matter....Pages 87-87
Responding to Ankur’s Challenge: Co-construction of Argument Leading to Proof....Pages 89-95
Block Towers: Co-construction of Proof....Pages 97-104
Representations and Connections....Pages 105-120
Pizzas, Towers, and Binomials....Pages 121-131
Representations and Standard Notation....Pages 133-144
So Let’s Prove It!....Pages 145-154
Front Matter....Pages 155-155
“Doing Mathematics” from the Learners’ Perspectives....Pages 157-169
Adults Reasoning Combinatorially....Pages 171-183
Comparing the Problem Solving of College Students with Longitudinal Study Students....Pages 185-200
Closing Observations....Pages 201-204
Back Matter....Pages 205-224

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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.

Content:
Front Matter....Pages I-XVII
Front Matter....Pages 1-1
The Longitudinal Study....Pages 3-8
Methodology....Pages 9-14
Front Matter....Pages 15-15
Representations as Tools for Building Arguments....Pages 17-25
Towers: Schemes, Strategies, and Arguments....Pages 27-43
Building an Inductive Argument....Pages 45-57
Making Pizzas: Reasoning by Cases and by Recursion....Pages 59-72
Block Towers: From Concrete Objects to Conceptual Imagination....Pages 73-86
Front Matter....Pages 87-87
Responding to Ankur’s Challenge: Co-construction of Argument Leading to Proof....Pages 89-95
Block Towers: Co-construction of Proof....Pages 97-104
Representations and Connections....Pages 105-120
Pizzas, Towers, and Binomials....Pages 121-131
Representations and Standard Notation....Pages 133-144
So Let’s Prove It!....Pages 145-154
Front Matter....Pages 155-155
“Doing Mathematics” from the Learners’ Perspectives....Pages 157-169
Adults Reasoning Combinatorially....Pages 171-183
Comparing the Problem Solving of College Students with Longitudinal Study Students....Pages 185-200
Closing Observations....Pages 201-204
Back Matter....Pages 205-224
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