Phillip Kent and Phil
Ramsden
Imperial College, London, United Kingdom.
Emails: p.kent@ic.ac.uk, p.ramsden@ic.ac.uk
It's probably safe to assume that nearly everyone at this workshop would answer "yes" to the question "can computer algebra systems contribute anything important to mathematical learning?" There may be less agreement about why this is so, or about what is the best way to make this contribution as valuable as possible.
Given any technological tool, one should ask whether there are ways of using it that make some kinds of learning easier, or more satisfactory, or perhaps even possible for the first time. We show, with the aid of two contrasting examples, how we see computer algebra systems contributing to that part of mathematical learning which is concerned with experiment, conjecture and the construction of fresh concepts. We make the case that software of this kind presents students with a unique opportunity to make and test multiple computational representations of their mathematical ideas.
We will describe and demonstrate two pieces of work in progress, based on the Mathematica system. Both are what we call "mathematical microworlds": computational environments that represent particular mathematical domains, designed expressly for the purpose of learning about those domains. The first example is concerned with the beginning study of ordinary differential equations: here we have designed a range of learner activities around a single tool (a Mathematica function) that embodies the central mathematical concept of "direction field". In the second example, we have designed a kit of specialised tools for the study of the applications of group theory to molecular symmetry.
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