The purpose of the educational system was always concentrated upon the formation in the oncoming generation those behavioral models and values which will allow it to become successful out of the school walls. The scientists from different countries consider that the expansion of the possibilities of the social choice consists in the formation of the key competences of the citizens as the complex of the individual characteristics which are necessary and sufficient for the effective realization of the professional work in the predetermined conditions and on the predetermined level of the quality.
The idea of the competency building approach has become the answer for the new social request not only in the vocational training, but also in the training to any subject at the comprehensive school, especially in mathematics. According to the educational standard, the formation of the key competences in mathematical training is understood as the formation of the readiness of the pupils to use the acquired knowledge, the abilities and the ways of an activity in a real life for the solution of the practical problems.
The important place in mathematical training is occupied by such problems as: both the purpose, and the means of study. Besides, the mathematical problems possess the huge potential in the sense of the methodical reserves. The activity of the solution of a mathematical problem is adequate to the activity of any kind, including a professional one. The ability to search and to choose the resources to plan the solution, to reach the result of the activity, to estimate and to correct is always and repeatedly necessary in the course of the solution of a mathematical problem. The work with a mathematical problem represents didactically adapted social experience of the solutions of the problems (informative, world outlook, moral, political and others).
Thus, the mathematical problems are the effective means of the formation of the key competences. And the ability to solve a problem is an indicator of the development of the pupil, his social maturity, an activity of a vital position, an independence of the person.
In the designated sense, the geometrical problems are especially valuable. Their maintenance and the methods used for the solution allow the complex apply of the major knowledge and skills in a non-standard situation. The combination of the constructions, the calculations, the proofs, the researches aggregated in one problem is the best way to improve the formation and the development of key competences.
The work is submitted to the scientific international conference «Current problems of education, October, 2-9, 2010, Greece, came to the editorial office on 17.08.2010.