Visualisation for Different Mathematical Purposes

Abstract

Visualisation is often suggested as a useful heuristic for generating new ideas when one is stuck on a problem. Yet generating ideas is just one aspect of mathematical activity. Visualisation can also help students generalise mathematical discoveries and communicate mathematical ideas. This chapter describes how the nature of a person’s visualisation can change depending on the purpose for which it is used, and how this in turn influences the mathematical structure one perceives. A pair of participants used gestures to visualise maxima and minima in an antiderivative problem. Their visualisation techniques changed to encourage more local analysis when they began generalising a rule for discovering maxima and minima. Then, they developed a simple graphical visualisation tool to communicate their rule to layperson clients. The study highlights the need for students and teachers to be aware of the different mathematical purposes for which visualisation can be used, and the kinds of semiotic activity that can facilitate each case.