The following, written by Paul Cohen, is an extract from the 1967 Stanford Quad (see http://quad.stanford.edu/):
It is a widely held belief that mathematics is merely a tool to be used by the more applied sciences, and that research in mathematics is impossible. In reality, mathematics is undergoing a great period of expansion and development with perhaps the most spectacular work being done in the more abstract and pure branches of the subject. To mathematicians, mathematics often seems more of an art than a science. On the other hand, a complete divorce from physical science would be unwise, and it is reassuring that pure mathematics does find new and surprising applications.
The misconceptions which the average student brings to the Calculus course often causes him to see it merely as a set of rules for handling special problems. For its discoverers, Newton and Leibniz, however, the essential element of the calculus was a new point of view rather than special problems. When teaching undergraduates, the most challenging problem thus is to give the students a feeling for the power which is in the great mathematical discoveries. For the student specialising in mathematics, the problem is to bring then as quickly as possible to the frontier. This is done more through seminars than formal classes. These informal contacts allow the professor to help the student overcome the diffidence he feels before such a highly developed discipline. The professor must reveal the essentials of mathematics and supply the personal encouragement and direction which will enable the student to make a contribution of his own.