The course of higher mathematics at technical universities is reduced annually due to the reduction of the time allotted for the development of its program. The use of more effective teaching methods can only partially compensate for these losses. Especially depressing is the loss of the opportunity to bring proofs of even the simplest statements and theorems in the course of mathematics. If, as R. Kronecker said, definitions are the soul of mathematics, then its beauty lies in the impeccable application of logic, and the driving forces and its very life are in the ideas of proofs. Due to the introduction of distance learning, the possibility of investigating the effectiveness of newly proposed teaching methods by statistical methods has also been lost. Indirectly, this leads to ignoring the psychological, cognitive features of the contingent, to refusing to take into account the features of memory, intuition, associativity of thinking. Studying mathematics without proving its claims can cause a student of a technical university to be disgusted with it. The question of applying logical rules of inference is not simple, since modern mathematical logic differs significantly from the logic of the times of Euler, especially Aristotle. The evidence may be different, more or less acceptable for this contingen. All modern mathematics consists of fragments that include an algebraic structure and predicate logical calculus, which allows one true statement on a fragment to build others that are also true. The categorical apparatus used (predicative part of the algebraic structure) it can be modified, which can make it easier for students to perceive. The meaning of the phrase “simplicity of perception” is subjective, has not been studied in cognitive psychology, but this does not become less important in practical teaching. In the course of higher algebra at the Technical University, one of the most difficult questions is the concept of the determinant of a square matrix, the definition of which can be descriptive or recursive. The first is more natural and allows you to make some simple statements about the determinant, the second is convenient for calculating the determinant, but it makes it difficult to prove its properties.t. The author managed to find an analogue of the Kronecker symbol for determining the sign oof the product of matrix elements, invariant with respect to the permutation of elements in the product, the sum of which is the determinant. This led to an obvious simplification of the proof of almost all the properties of the determinant and made their understanding quite accessible to a student of a technical university. The author points out a number of problematic fragments of mathematics taught at technical universities, so that the work will continue, and its effectiveness can be checked by statistical methods. At the same time, the approach has no obvious applicability, if only because the issue of easy assimilation of logically related material has not been considered in cognitive psychology, and the search for an optimal system of definitions does not have a clear logical orientation. The proposed approach is not trivial, its effectiveness can be verified by statistical methods, which is not yet possible due to the remote learning system. The proposed approach can be developed by applying it to other similar problematic fragments. Perhaps, in the process of finding solutions to similar problems, there will be a general method for finding the optimal system of definitions that simplifies learning, understanding and assimilation of similar material.

Keywords: formal-logical, theory of determinants, methods of teaching mathematics.