Igor N. Sergeyev

This paper describes the authors’ long-term pedagogical experience in creating and implementing a special program for systematic repetition, expansion and deepening of the full school course of mathematics in general. This program is based on the principle of the didactic spiral and is applicable both in the framework of regular school lessons and in the implementation of additional education. The work highlights three main stages of student preparation: basic, advanced and creative. Together they contain exactly seven clearly distinguished turns of the didactic spiral. After passing each such round, the student (in this case, as a rule, the future entrant) is more or less ready to solve problems in all sections of the school curriculum at once, which is especially important for participation in Olympiads held during the school year. The first, basic, stage covers two turns of the spiral, which can be conditionally called linear and quadratic. The second stage, advanced, already consists of four turns corresponding to the four main methods of a logical nature: the enumeration of cases, equivalent transformations, the introduction of notation and the conclusion of consequences. The third, creative stage is represented by just one turn, at which all the material passed is already passed from a research point of view. It examines four main methodological approaches available within the school curriculum: algebraic, logical, functional and graphical. The choice and implementation of these approaches in solving specific tasks is the essence of this final round and stage.