The Mathematical Background of Teachers in Training

Abstract

There are many ideas, fundamental to the whole of mathematics, that are difficult to teach explicitly and are rarely examined. They include the concept of mathematical proof, the idea of a converse, necessary and sufficient conditions, and those elements of elegance and rigour without which mathematics can lose its beauty and even its claim to be called mathematics. It is these ideas I shall refer to as mathematical background; it must be emphasised that this background is common to both traditional and “modern” mathematics, and that its cultivation in their pupils would seem a desirable objective for all teachers, both conventional and progressive. Most of these ideas were clearly expressed in the Mathematical Association’s booklet, “Suggestions for Sixth-Form Work in Pure Mathematics”, which seems admirable in everything but its title, since most of the work should be begun much earlier than the sixth-form. (If it is left until then, there is justification for one reviewer’s remark that it is “an attempt to paper over the cracks of sixth-form mathematics”.)