Introduction

Abstract

As teachers, we rightly value the ways in which our students bring meaning to the mathematical situations they encounter. There is much scope for judgment, insight and creativity in the style of mathematical work being introduced in many schools and we may aspire to encourage these qualities in the students’ learning. Yet there is still a need for an individual to reconcile her own personal mathematical understanding with the ideas and traditions which have grown out of centuries of mathematical exploration and invention (cf. Ball, 1993). Whilst students can be creative mathematicians there is still a need to be able to do everyday calculations and understand aspects of conventional mathematical thinking. We are often torn between attempting to focus on our students’ own way of seeing their mathematical endeavours, and seeing these endeavours with our own eyes, inspired perhaps by a “correct” view of mathematics. There are inevitably difficulties for us in making sense of students’ own developing understanding without using our own “expert” overview as a yardstick, especially when we pose the tasks that they are working on. Teacher descriptions of students’ learning often presuppose an adult overlay framing the mathematical ideas supposedly being addressed.

A demanding, prudent, “experimental” attitude is necessary; at every moment, step by step, one must confront what one is thinking and saying with what one is doing, with what one is. (Foucault, 1984, p. 374)