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Combinatorics and Reasoning

Representing, Justifying and Building Isomorphisms

  • Book
  • © 2011

Overview

Editors:
  1. Carolyn A. Maher

    1. Graduate School of Education, Rutgers University, New Brunswick, USA

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    You can also search for this editor in PubMed   Google Scholar

  2. Arthur B. Powell

    1. Department of Urban Education, Rutgers University, Newark, USA

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    You can also search for this editor in PubMed   Google Scholar

  3. Elizabeth B. Uptegrove

    1. Department of Mathematical Sciences, Felician College, Rutherford, USA

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    You can also search for this editor in PubMed   Google Scholar

  • Unique Longitudinal research over 20 years
  • Presents compelling evidence of the advantages to using combinatorics in mathematics education
  • Extensive discussion of mathematical tasks

Part of the book series: Mathematics Education Library (MELI, volume 47)

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Table of contents (17 chapters)

  1. Front Matter

    Pages I-XVII
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  2. Introduction, Background, and Methodology

    1. Front Matter

      Pages 1-1
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    2. The Longitudinal Study

      • Carolyn A. Maher
      Pages 3-8

    3. Methodology

      • Carolyn A. Maher, Elizabeth B. Uptegrove
      Pages 9-14

  3. Foundations of Proof Building (1989–1996)

    1. Front Matter

      Pages 15-15
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    2. Representations as Tools for Building Arguments

      • Carolyn A. Maher, Dina Yankelewitz
      Pages 17-25

    3. Towers: Schemes, Strategies, and Arguments

      • Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz
      Pages 27-43

    4. Building an Inductive Argument

      • Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz
      Pages 45-57

    5. Making Pizzas: Reasoning by Cases and by Recursion

      • Carolyn A. Maher, Manjit K. Sran, Dina Yankelewitz
      Pages 59-72

    6. Block Towers: From Concrete Objects to Conceptual Imagination

      • Robert Speiser
      Pages 73-86

  4. Making Connections, Extending, and Generalizing (1997–2000)

    1. Front Matter

      Pages 87-87
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    2. Responding to Ankur’s Challenge: Co-construction of Argument Leading to Proof

      • Carolyn A. Maher, Ethel M. Muter
      Pages 89-95

    3. Block Towers: Co-construction of Proof

      • Lynn D. Tarlow, Elizabeth B. Uptegrove
      Pages 97-104

    4. Representations and Connections

      • Ethel M. Muter, Elizabeth B. Uptegrove
      Pages 105-120

    5. Pizzas, Towers, and Binomials

      • Lynn D. Tarlow
      Pages 121-131

    6. Representations and Standard Notation

      • Elizabeth B. Uptegrove
      Pages 133-144

    7. So Let’s Prove It!

      • Arthur B. Powell
      Pages 145-154

  5. Extending the Study, Conclusions, and Implications

    1. Front Matter

      Pages 155-155
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    2. “Doing Mathematics” from the Learners’ Perspectives

      • John M. Francisco
      Pages 157-169

    3. Adults Reasoning Combinatorially

      • Barbara Glass
      Pages 171-183
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Keywords

About this book

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.

Editors and Affiliations

  • Graduate School of Education, Rutgers University, New Brunswick, USA

    Carolyn A. Maher

  • Department of Urban Education, Rutgers University, Newark, USA

    Arthur B. Powell

  • Department of Mathematical Sciences, Felician College, Rutherford, USA

    Elizabeth B. Uptegrove

Bibliographic Information

  • Book Title: Combinatorics and Reasoning

  • Book Subtitle: Representing, Justifying and Building Isomorphisms

  • Editors: Carolyn A. Maher, Arthur B. Powell, Elizabeth B. Uptegrove

  • Series Title: Mathematics Education Library

  • DOI: https://doi.org/10.1007/978-94-007-0615-6

  • Publisher: Springer Dordrecht

  • eBook Packages: Humanities, Social Sciences and Law, Education (R0)

  • Copyright Information: Springer Netherlands 2011

  • Hardcover ISBN: 978-94-007-0614-9Published: 14 January 2011

  • Softcover ISBN: 978-94-007-3495-1Published: 25 February 2013

  • eBook ISBN: 978-94-007-0615-6Published: 25 November 2010

  • Series ISSN: 0924-4921

  • Series E-ISSN: 2214-983X

  • Edition Number: 1

  • Number of Pages: XVIII, 226

  • Topics: Mathematics Education, Combinatorics

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