Mathematical competence and mathematical abilities: structural relations and development methodology

Abstract In the presented work, the concept of mathematical abilities is rational as both an essential internal characteristic of mathematical competence and an immanent attribute that prevails in its personal and psychological dimension. In view of the spatial Cartesian implementation the role and place of mathematical abilities in the three-dimensional structure of the internal manifestation of mathematical competence are substantiated. The semantic and system analysis of structural components of the studied phenomenon is made. It is established that complex correlations of four structural components of mathematical abilities (system-forming, coding-formalized, cognitive-generalizing, mnemonic-generalizing) can exist with three dimensions of the external manifestation of mathematical competence (semantic-theoretical, procedural-active, personal-psychological). It is introduced that the development of mathematical abilities is provided by updating the external dimensions of mathematical competence in educational and mathematical activities. The method of developing the mathematical abilities of students is presented according to the results of the implementation of this idea and the principle of developmental continuity. The basis of this technique is the developmental-problem method of teaching mathematics as a four-level problem structure which embodies the methods of mathematical and educational (educational-theoretical) modeling, the method of ascent from the abstract to the concrete, provides a reflection of the process and results of educational and mathematical activities. Due to the problem raised in the work the authors highlight the content and results of a pilot study on the development of mathematical abilities of future mathematics teachers (freshmen and graduates). The effectiveness of the created developmental methods of mathematical education was experimentally tested.