Mathematics Education Library

Approaches to Algebra

Perspectives for Research and Teaching

Editors: Bednarz, N., Kieran, C., Lee, L. (Eds.)

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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.

Table of contents (21 chapters)

  • Approaches to Algebra: Perspectives for Research and Teaching

    Bednarz, Nadine (et al.)

    Pages 3-12

  • From Euclid to Descartes: Algebra and its Relation to Geometry

    Charbonneau, Louis

    Pages 15-37

  • The Roles of Geometry and Arithmetic in the Development of Algebra: Historical Remarks from a Didactic Perspective

    Radford, Louis

    Pages 39-53

  • The Role of Problems and Problem Solving in the Development of Algebra

    Rojano, Teresa

    Pages 55-62

  • Expressing Generality and Roots of Algebra

    Mason, John

    Pages 65-86

Buy this book

eBook $79.99
price for USA (gross)
  • ISBN 978-94-009-1732-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $269.00
price for USA
  • ISBN 978-0-7923-4145-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $102.00
price for USA
  • ISBN 978-0-7923-4168-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Approaches to Algebra
Book Subtitle
Perspectives for Research and Teaching
Editors
  • N. Bednarz
  • C. Kieran
  • L. Lee
Series Title
Mathematics Education Library
Series Volume
18
Copyright
1996
Publisher
Springer Netherlands
Copyright Holder
Kluwer Academic Publishers
eBook ISBN
978-94-009-1732-3
DOI
10.1007/978-94-009-1732-3
Hardcover ISBN
978-0-7923-4145-1
Softcover ISBN
978-0-7923-4168-0
Series ISSN
0924-4921
Edition Number
1
Number of Pages
XVI, 348
Topics