ABSTRACT
Introduction Mathematics which stresses results over proof is often called applied mathematics, mathematics which stresses proof tends to be called pure mathematics (or just mathematics).
(Hersh, 1997, p. 6)
Not everyone agrees that mathematics splits easily in this way, but it is a useful distinction, provided that we remember that there is an important place for both activities. Moreover, the search for a result, something that works, may stimulate us to look at the principles behind our method and to verify some of our reasoning. Having proved something we may then fi nd that the result has a useful application. The growth of mathematics through history has often relied upon a symbiotic relationship between pure and applied mathematics. At the heart of both pure and applied mathematics, however, are problems:
But anyone who has done mathematics knows what comes first – a problem. Mathematics is a vast network of interconnected problems and solutions. (Hersh, 1997)
Many of the problems which pupils encounter at school are little more than practice exercises, and we have included some of these in the book. They have a place in the learning of mathematics but they are just the tip of an iceberg. For a start they can usually be solved quite quickly, provided the learner can recognise the technique needed, and the problem usually has just the right amount of information, no more and no less. These are quite unlike some of the famous problems of mathematics, some of which have remained unsolved for hundreds of years.
