Abstract:

One of the major problems of mathematical education is teaching students the ability to solve mathematical tasks. This explains the relevance of intrinsic understanding of math problem characteristics by a teacher. The key characteristic of a mathematical problem is the level of difficulty, in which it is possible to select the objective (structural and content-related features) and subjective (associated with the experience of a particular individual) components. The account of these components determines the choice of the classification of mathematical tasks used in teaching. From the point of view of subjectivity, the classification of tasks is based, firstly, on the requirements for activities to be carried out by a student in the process of its solution, and, secondly, on his experience. The classification of tasks according to requirements for activities provides reproductive and productive levels of learning. For the most efficient and effective teaching of mathematics it is necessary to develop clear criteria for the selection of tasks for each of these levels and for the evaluation of solved task by students.

68 10-grade students (16-17 years old) from one of the lyceum schools in Kazan (Russia) took part in an empirical study to identify the dependence of the quality of the solution of plane geometry tasks on the level of their difficulty. The students had taken the course of plane geometry, and they were preparing for the unified state exam in mathematics after grade 11. In accordance with the training program and the requirements for the mathematical preparation of students on this stage of learning, the system of tasks on the theme of "Circles and polygons" of reproductive ("student" and the model) and productive (creative and non-typical) levels of learning activities (according to the classification by V.P. Bespalko) was designed. This topic is one of the most difficult in the school course of plane geometry, as for solving such problems there is the necessity for systemic understanding of the construction "circle – polygon". The experiment was conducted for two days with a one week break.

The analysis of the solved problems has shown that 72% of the students have a sufficient level of reproductive activity and 97% – unsatisfactory level of productive activity in solving problems. Obviously, the vast majority of students has no experience in productive activities when solving plane geometry problems. The qualitative analysis of the results has revealed a number of deficiencies in the ability to work with a geometric task: to carry out the analysis of the content correctly, construct a geometrical drawing, find the relationship between elements, etc. The result of this research is to develop criteria for the selection of plane geometry tasks aimed at the development of the productive level of learning activities. Using that by a teacher will enhance the efficiency of teaching mathematics and raise the level of students' mathematical preparation.

Keywords:

Teaching mathematics, school mathematical task, reproductive activity, productive activity.