INTRODUCTION

ICMI [1984] poses the question “What is the mathematics underlying symbolic mathematical systems”? The aim of this paper is to give some answers to this question, and also to address the following question that ICMI does not directly answer: “How does this mathematics relate to current curricula”, which could be re-phrased as “What aspects of current curricula are rendered obsolete, or drastically changed by symbolic mathematical systems”. It should be emphasised that this paper does not address the question “How should algebra systems be used to teach existing mathematics in the same way”, though that is a very important question.

ELEMENTARY CALCULATIONS

Symbolic mathematical systems are capable of a variety of essentially trivial calculations. An obvious example is the multiplication of polynomials. The algorithm for doing this is taught at school, and there is little doubt that any competent student knows how to multiply polynomials. He may make a mistake while doing so, but that would be an accident, and he would recognise the mistake if it were pointed out to him. This does not mean that the student could actually do the calculations. They may well be too long for him, either in terms of time or in terms of the probability of there being an error.

Either the student or the experienced mathematician may wish to use a computer algebra system to multiply polynomials. Andrews [1979] used one to multiply four polynomials together to verify a 752-term identity.