Understanding of Mathematics and University Tea­cher Training

The main goal of mathematics edu­cation - the development of thinking -and the main activity of school practice - the solving of problems - are compa­red. The traditional method of problem-solving consisting of four parts (analysis, construction, proof and discussion) is compared with Hadamard's conception of four chronological stages in the pro­cess of creation: preparation, incubation, illumination and verification. The trans­mission of new knowledge by means of lectures leads only to the reproduction of information. The alternative, i.e. con­struction of knowledge in the minds of students, is promising but difficult to put into practice. It offers a way of getting over what Felix Klein and Jaroslav Šedivý have formulated as the problem of two kinds of forgetting (connected with the beginning of university studies and the beginning of teaching practice).