Title: Computer Algebra Systems and Mathematics Education: Resolving the Disharmony -or- Why You Cant Have Your Cake and Eat it

Author: Martyn Quigley* (Brunei, martyn@mail.ubd.edu.bn)

Affiliation: University Brunei Darussalam, Brunei

Abstract: Lurking just beneath the current enthusiasm for ICT in our schools is a fundamental disharmony. A teacher of middle age certainly completed all of their mathematics education without any ICT. Since the curriculum now is essentially the same as it was 30 years ago (at least in mathematics), why is it suddenly so necessary for our children to learn it with ICT?

In this paper I shall use the solution a problem in mathematics to demonstrate as clearly as possible those steps in the solution which can be done by a computer algebra system (CAS) and those steps which cannot. Clearly, if some steps in a solution cannot be done with a CAS, they must be done by the human. Generalising, this results in a distinction between "machine possible" steps and "machine impossible" steps.

This distinction leads us to consider two issues. Firstly, just because the machine possible steps can be done with a CAS, should they? The way forward in this respect is not so easy and will require some hard decisions to be made. Arguments will be put forward to support the contention that either we use a CAS or we don't, there is no half-way house, and that this decision is ultimately a philosophical one. Secondly, we need to consider where the "interesting" mathematics lies, in the machine possible steps or the machine impossible steps? An answer to this controversial question is given which it is hoped will guide us in developing 21st century curricula in mathematics.