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Approaches to Algebra

Perspectives for Research and Teaching

  • Book
  • © 1996

Overview

Editors:
  1. Nadine Bernarz

    1. CIRADE, Université du Québec á Montréal, Montréal, Canada

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  2. Carolyn Kieran

    1. Département de Mathématiques, Université du Québec à Montréal, Montréal, Canada

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    You can also search for this editor in PubMed   Google Scholar

  3. Lesley Lee

    1. Département de Mathématiques, Université du Québec à Montréal, Montréal, Canada

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    You can also search for this editor in PubMed   Google Scholar

Part of the book series: Mathematics Education Library (MELI, volume 18)

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-xv
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  2. Introduction

    1. Front Matter

      Pages 1-1
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    2. Approaches to Algebra: Perspectives for Research and Teaching

      • Nadine Bednarz, Carolyn Kieran, Lesley Lee
      Pages 3-12

  4. A Generalization Perspective on the Introduction of Algebra

    1. Front Matter

      Pages 63-63
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    2. Expressing Generality and Roots of Algebra

      • John Mason
      Pages 65-86

    3. An Initiation into Algebraic Culture through Generalization Activities

      • Lesley Lee
      Pages 87-106

    4. Some Reflections on Teaching Algebra through Generalization

      • Luis Radford
      Pages 107-111

  5. A Problem-Solving Perspective on the Introduction of Algebra

    1. Front Matter

      Pages 113-113
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    2. Emergence and Development of Algebra as a Problem-Solving Tool: Continuities and Discontinuities with Arithmetic

      • Nadine Bednarz, Bernadette Janvier
      Pages 115-136

    3. Developing Algebraic Aspects of Problem Solving within a Spreadsheet Environment

      • Teresa Rojano
      Pages 137-145

    4. Rough or Smooth? The Transition from Arithmetic to Algebra in Problem Solving

      • David Wheeler
      Pages 147-149

    5. Algebraic Thought and the Role of a Manipulable Symbolic Language

      • Alan Bell
      Pages 151-154

    6. Placement and Function of Problems in Algebraic Treatises from Diophantus to Viète

      • Louis Charbonneau, Jacques Lefebvre
      Pages 155-165

    7. Problem-Solving Approaches to Algebra: Two Aspects

      • Alan Bell
      Pages 167-185

    8. “When is a Problem?”: Questions from History and Classroom Practice in Algebra

      • John Mason
      Pages 187-193

  6. A Modeling Perspective on the Introduction of Algebra

    1. Front Matter

      Pages 195-195
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Keywords

About this book

In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magnitudes are exactly measured by a certain unit. The theory of proportions does not have access to such operations. It cannot be seen as an "arithmetic" of ratios. Even if Euclidean geometry is done in a highly theoretical context, its axioms are essentially semantic. This is contrary to Mahoney's second characteristic. This cannot be said of the theory of proportions, which is less semantic. Only synthetic proofs are considered rigorous in Greek geometry. Arithmetic reasoning is also synthetic, going from the known to the unknown. Finally, analysis is an approach to geometrical problems that has some algebraic characteristics and involves a method for solving problems that is different from the arithmetical approach. 3. GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th century, Euclid's Elements was considered a model of a mathematical theory. This may be one reason why geometry was used by algebraists as a tool to demonstrate the accuracy of rules otherwise given as numerical algorithms. It may also be that geometry was one way to represent general reasoning without involving specific magnitudes. To go a bit deeper into this, here are three geometric proofs of algebraic rules, the frrst by Al-Khwarizmi, the other two by Cardano.

Editors and Affiliations

  • CIRADE, Université du Québec á Montréal, Montréal, Canada

    Nadine Bernarz

  • Département de Mathématiques, Université du Québec à Montréal, Montréal, Canada

    Carolyn Kieran, Lesley Lee

Bibliographic Information

  • Book Title: Approaches to Algebra

  • Book Subtitle: Perspectives for Research and Teaching

  • Editors: Nadine Bernarz, Carolyn Kieran, Lesley Lee

  • Series Title: Mathematics Education Library

  • DOI: https://doi.org/10.1007/978-94-009-1732-3

  • Publisher: Springer Dordrecht

  • eBook Packages: Springer Book Archive

  • Copyright Information: Kluwer Academic Publishers 1996

  • Hardcover ISBN: 978-0-7923-4145-1Published: 30 June 1996

  • Softcover ISBN: 978-0-7923-4168-0Published: 30 June 1996

  • eBook ISBN: 978-94-009-1732-3Published: 06 December 2012

  • Series ISSN: 0924-4921

  • Series E-ISSN: 2214-983X

  • Edition Number: 1

  • Number of Pages: XVI, 348

  • Topics: Mathematics Education, History of Mathematical Sciences

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