Keyword: «principle of phased learning»

Currently, there are many problems in the teaching of mathematics associated with a decrease in students' motivation, many of them do not understand the studied material properly. One of the main reasons for this situation is the reliance in teaching not on the natural features of the perception of mathematical knowledge by children of different ages, but most often on a purely logical sequence of the material. Soviet teaching methods, for ideological reasons, limited direction of attention toward the natural characteristics of students. This led to the fact that a high theoretical level of teaching mathematics at school was combined with the complication of educational programs, absolutization of theoretical thinking in comparison with figurative thinking, and neglect of visual learning tools. The purpose of the article is the development of new approaches to the methodology of teaching mathematics, based on the principle of nature conformity - one of the most famous pedagogical principles. This principle considers the attitude towards a person during training as a part of nature, provides for the reliance on his/her own talents and inclinations given to him/her from birth. For a subject such as mathematics, which is studied from the 1st to the 11th grade, and then also in higher school, the principle of nature-conformity is implemented primarily through the phasing, multi-stage gain of new mathematical knowledge by students. The necessity of preliminary stages, steps in the study of basic mathematical concepts is shown. The phasing of knowledge is examined on the example of the formation of the most important algebraic concept of a group and such a structure as scalar. These two concepts permeate the entire course of school mathematics and some university courses. A new approach to the study of the principle and method of mathematical induction is also considered on the basis of a minimum condition that is clearer for students to understand. Examples of applying this form of induction to solving tasks are given.