The article is about the role of communication process both in everyday life and in professional pedagogical activity. The article deals with the disclosure of communication functions in various pedagogical situations related to the teacher’s activities as a source of knowledge, as a researcher of students and as an organizer of collective activity. The importance of taking into account the individual psychological characteristics of the pupils, especially adolescents is noted. The article compares the different classifications of accentuations and considers the results of the experiment to identify them in the tenth graders, the authors provides recommendations on the organization of classes in the study groups and the preferred type of tasks for all types of accentuations described by A.E. Lichko.

The article discusses the technology of full learning, based on a model developed by American psychologists J. Carroll and V.S. Bloom. The features of its use in the construction of the educational process in institutions of higher education are described. The experience of its use in conducting classes in the disciplines of the mathematical cycle with students in undergraduate economic specialties is described. To implement this technology, the training material is divided into separate modules with an introductory theoretical part, practical tasks and final verification work. Practical tasks are individual variants of typical tasks, which are compiled in such a way that some of them can be performed during classroom work, for example, working at the blackboard, and some at homework. The effectiveness of this technology is confirmed by comparing the results of students' examinations in the control and experimental groups.

In the course of mathematics of secondary school it is not provided the consideration of a separate topic “Theory of Diophantine equations”, but such equations are increasingly appearing in the texts of additional entrance tests for technical, physical, mathematical and economic specialties of prestigious universities and in the last task of tests of the profi le level of the exam in mathematics. This material is very scattered in various publications, so for its systematization the authors developed an elective course “The solution of Diophantine equations”. Within the framework of this course, the classification of the main methods of solving equations in integers by elementary means and their adaptation for eleventh grades was carried out. Each of the considered methods of solution is provided with a detailed illustration of the features of its application on examples and a selection of tasks for solution without help of teacher.