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The purpose of this paper is to state what the author believes is a good way to develop the powers of understanding of talented college science students. After making a careful and somewhat novel distinction between knowledge and understanding, the paper closely defines the main problem in terms of the new interpretation of "understanding." A short discussion follows which concludes that what is needed is more and better schooling in the central ideas of mathematics. At this point, the tenor of the paper is changed and a semipopular discussion is given on the fundamental type of mathematics the author has in mind. Finally, a proposal is made which is general enough to be adapted to a wide variety of situations.