Research Issues in the Learning and Teaching of Algebra - Vol. 4

By Sigrid Wagner; Carolyn Kieran | Go to book overview

Linking Representations in the Symbol Systems of Algebra

James J. Kaput Department of Mathematics Southeastern Massachusetts University and The Educational Technology Center Harvard Graduate School of Education

Mathematics is, among other things, a collection of languages, and languages have dual, interlocking roles: They are instruments of communication, and they are instruments of thought. In a previous paper ( Kaput, 1987b) I developed a beginning set of languages for communicating and thinking about mathematical languages, which in algebra include expressions, equations, coordinate graphs, tables of data, hybrid natural language fragments, and other notations. In this paper I will apply and extend the previously developed terminology and theoretical framework to discuss and evaluate the representational characteristics of new or potential algebra learning environments, especially those that support the linking of different algebraic representations.

Thus this paper is deliberately oriented towards future research, indeed towards the research needed to design the future. To consider research from any other perspective would contribute to the well-documented and continuing failure of school mathematics to serve students' real needs.


FOUNDATIONS

Our starting points are: (a) the notion of mental representation as the means by which an individual organizes and manages the flow of experience and (b) the notion of representation system (or symbol system) as a materially realizable, shared cultural or linguistic artifact.

Representation systems, when learned, are used by individuals to structure the creation and elaboration of their own mental representations. A mathematical representation system is used to instantiate mathematical objects, relations, and processes in material form--the physical signs produced by pen on paper, keystroke on computer, and so on. A fuller discussion of representation systems and the underlying issue of Platonism can be found elsewhere ( Kaput, 1985, 1987a, 1987b, in press).

A caveat: Our initial attention to symbol systems should not be misread as an assertion that mathematics is, and hence the curriculum should be, about symbols and syntax. On the contrary, our ultimate aim is to account for the building and expressing of mathematical meaning through the use

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