Keyword: «exploratory learning»

At present, the task of intellectual development of students comes to the forefront in teaching mathematics at school, and the development of their mathematical thinking is becoming increasingly relevant. The aim of the article is to examine those elements in the procedural-cognitive side of the content of teaching mathematics, which act as mental means and methods of mathematical knowledge. The methodological basis of the study is the systemic, meta-subject and activity-based approaches. As a result of learning mathematics, the mind of a person forms structures corresponding to the identified elements of the content of teaching mathematics, which are called mathematical thinking schemes in the article. These include logical, algorithmic, combinatorial and figurative-geometric cognitive structures. The article provides their characteristics and describes the practical experience of their formation in schoolchildren. All these structures are universal, i.e. they are used regardless of the specific mathematical material, and they are of great importance not only for learning, but also for mathematical creativity. All the considered schemes of mathematical thinking have one common characteristic: their formation can be carried out only over a long period of time, using the sensitive possibilities of their development in each age period. A theoretical analysis of the relationship between the processes of differentiation and integration of such structures is made. It is shown that the process of integration should prevail quantitatively in the formation of such structures, gradual, small-step accumulation of mathematical knowledge. However, qualitative leaps and breakthroughs in the creation of knowledge can occur through differentiation, using the deductive method. The recommendations given in the article to teachers on the formation of various types of mathematical thinking schemes in schoolchildren at different levels of education are of practical significance. The greatest attention is paid to figurative-geometric thinking schemes and their role in teaching mathematics. It is noted that the use of computer mathematics systems in teaching, which also help in conducting computer experiments and in research-based learning, is a great help to the teacher in forming such patterns of thinking in the conditions of a digital society.