TEACHING STUDENTS TO PROVE MATHEMATICAL STATEMENTS IN THE SELECTIVE COURSE «INTRODUCTION TO CRYPTOLOGY»

Introduction. Students' mastering in the art of proof is one of the most important educational results of mathematics teaching (especially of profile or in depth math teaching). Significant potential in this aspect has both mathematics lessons and elective courses, such as the interdisciplinary elective course «Introduction to Cryptology» (for students of 9th, 10th grades with in depth learning of mathematics). Its theoretical basis are, in particular, the elements of the theory of divisibility and the theory of congruence in the ring of integers, the basis of algorithmization and programming (informatics course for 8th-9th grades). New mathematical facts that we offer to prove in this course: necessary and sufficient condition of the mutual simplicity of two numbers, the property of the multiplicative Euler function, the formula for finding the Euler function's value for an arbitrary natural number and for one that is a power of a prime number, the Euler theorem, Fermat's small theorem. The educational result can be investigated in students' mastering of the ways of thinking in accordance with analytical, synthetic, analytical and synthetic methods of proof, the method by contradiction, the method of complete induction, and the constructive method of the proof of mathematical facts. Purpose is to reveal the peculiarities of the teaching the constructive proofs of mathematical statements in the elective course «Introduction to Cryptology» (for students who are learning mathematics in depth). Methods. Theoretical analysis of psychological and pedagogical literature on the problem were used, comparison, generalization, systematization. Empirical systematization and generalization of advanced pedagogical experience in relation to the problematic issues. Results. It is known that the constructive method of proof is widely used in the proof of geometric facts. However, the counts of sets of rational, integer numbers, and Fermat's small theorem are proved constructively in Algebra textbook for 8th grade with in depth math teaching. We offer to use the constructive method for proving the multiplicative Euler function's property, the theorem on value of the Euler function for a natural number, which is a power of a prime number, theorems on the inverse class of residues modulo. At the same time, in our opinion, it is advisable to use both the teaching as a ready-made proof and an independent investigation of the way of proof by the students. It is advantageous to familiarize students with the method of thinking in the ready-made constructive proof on the example of proving the multiplicative property of Euler's function. It is possible to teach students to construct proofs by analogy, according to the given structure, according to the proposed teacher's idea or method on the example of theorem of the value of the Euler function for a power of a prime number. An independent students' investigation of the proofs must be ensured while carrying out exercises (to prove that a congruence is performed; to prove that the Euler's function is a pair number for natural numbers not less than 3, etc.) Originality. The author's analysis of some points with respect to the teaching of mathematical proofs in the context of the elective course «Introduction to Cryptology» for students who are learning math in depth is provided. Conclusion. In teaching students of the constructive method of proof it makes sense: 1) to carry out the detailed examination of the finished proofs for the purpose of their subsequent reproduction, taking into account the variability of the auxiliary constructions and their symbols; variations of methods of proof are possible; 2) to foresee the students' independent construction of the proof by analogy or on the basis of the teacher's previously prescribed method or the technique of proof; 3) to provide an opportunity of independent search and realization of proofs.