Software to Think With: The Case of Algebra

  • Judah L. Schwartz
Conference paper
Part of the NATO ASI Series book series (volume 107)


This paper describes a series of ‘intellectual mirror’ software environments for the teaching and learning of algebra. These environments are distinguished by the fact that they allow users to manipulate the symbolic representation of functions symbolically and the graphical representation of functions graphically, thus leading to a deeper understanding of the concept of function both as process and as entity.


mathematics education algebra visualizing algebra functions function analyzer function supposer symbol manipulation 


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Note and References

  1. 1.
    Schwartz, J.L., Intellectual Mirrors: A Step in the Direction of Making Schools Knowledge-Making Places, In: Harvard Educational Review, Vol. 59, No. 1, February 1989.Google Scholar
  2. 2.
    This is not the place to document the parlous state of secondary mathematics teaching and learning the US (or elsewhere). Nor is this the place to talk about the mathematical and pedagogic flaws with current curricula in algebra, and thus the need for a new curriculum.Google Scholar
  3. 3.
    There is a particular kind of ‘number recipe’ that requires special mention, i.e. the constant function. The number recipe F(x)= 1 is a recipe that produces the value 1 no matter what value of x you supply it with. This number recipe is different from and should be distinguished from the number 1.Google Scholar
  4. 4.
    The VISUALIZING ALGEBRA series of software environments was developed at Education Development Center by Judah L. Schwartz and Michal Yerushalmy under the direction of Wayne Harvey. The series is published by Sunburst Communications, Pleasantville, NY 10570 to whom inquiries may be addressed. The series consists of The Function Analyzer - published 1989 The Function Supposer - published 1990 The Function Comparator(functions of one variable) - in pressGoogle Scholar
  5. The Function Comparator(functions of two variables) - in pressGoogle Scholar
  6. In addition, the following software environments also embody the approach to algebra described in this paper.Google Scholar
  7. un*SOLVING Linear Quadratic Equations Inequalities, by Judah L. Schwartz, Sunburst Communications, 1990.Google Scholar
  8. The Function Family Register, by Judah L. Schwartz, Sunburst Communications, 1990.Google Scholar
  9. The Algebraic Proposer, by Judah L. Schwartz, True BASIC, West Lebanon, NH 03784, published 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Judah L. Schwartz
    • 1
    • 2
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Harvard Graduate School of EducationCambridgeUSA

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