ACCOMPANYING TRIANGLE OF MATHEMATICAL COMPETENCE

Considering the modern concept of development of the educational system in Ukraine most relevant studies is those that are studying the dual nature of competence, being rethought the duality of its manifestations from a scientific perspective. The problem field of research consists of structure and qualities of personality, which reflect both external and internal manifestations of mathematical competence. In the presented paper the dual nature of mathematical competence is represented as socially accepted and individual-psychological dimensions of personality. Its purpose is to determine the structure and substantive characteristics of the internal manifestation of mathematical competence justify fractality of the structures of its external and internal manifestations and build a Cartesian interpretation of the phenomenon under the study. For this, methods of structural-systemic and fractal analysis, abstraction and theoretical modeling, ranking and generalization are applied. It is proved that the external and internal manifestations of mathematical competence have three basic dimensions. The three-dimensional structure of the external manifestation of mathematical competence seems to be substantive-theoretical, process-active, and referential communicative measurements. The same structure of its internal manifestation is represented by value-motivational, reflective-evaluative, and personal-psychological measurements with an appropriate number of ranked indicators. This allowed us to interpret mathematical competence as one of the varieties of a fractal, it is a structure that consists of a similar substructure. It was found that the accompanying trihedral of mathematical competence dynamically determines the three-dimensional structure of its internal manifestation and, at the same time, establishes a connection with the three-dimensional structure of the external manifestation. The ranking of indicators on each dimension made it possible to state that the internal manifestation of mathematical competence most reveals the values of mathematical activity, its self-esteem, and mathematical abilities. But the fundamental indicators of the internal manifestations of mathematical competence are the needs of mathematical activity, its introspection and memory for mathematical material. It is proved that the ratio of the number of substantial characteristics of both external and internal dimensions of mathematical competence reflects the feature of the Egyptian triangle, the sides of which form the simplest Pythagorean triple – 3, 4, 5. Keywords: mathematical competence, duality of manifestations, fractality, accompanying trihedral.