Strategie rozwiązywania zadań matematycznych jako problem dydaktyki matematyki (fragment badań)

Abstract

One of the main tasks of teaching mathematics is developing an ability to apply the acquired mathematical knowledge and skills for solving problems, as well as developing the student's creative attitude to problems on each level of education. The acquisition of patterns of reasoning by the students in standard cases obviously does not exhaust the task. Elaboration of adequate methods in this domain requires knowledge concerning the specific process of mathematical thinking in the course of solving problems, which does not resolve itself into formal transformations and automatic procedure following some patterns, knowledge of efficient heuristic measures, means of control, etc. In the present article four fragments from an introductory investigation are presented, carried out by the Section of Didactics of Mathematics at the Higher Pedagogical School in Kraków. The aim of the investigation was analysis of the process above mentioned. In each of the presented examples the process of solving one and the same problem by persons of an investigated group was observed and analysed. Hipothetical reconstruction of this process was only based either on a complete set of recordings, sketch solutions and final written version of the solution, or was completed by an interview with those persons who had solved the problem. Mathematical education of the investigated persons was beyond that of an average student which eliminated ordinary inaptitude and lack of knowledge necessary for the solution of the problem. (They were either secondary school pupils participating in the final stage of the mathematical olympic competition, i.e. those interested in mathematics and whose mathematical knowledge and talent were beyond the school curriculum, or students of the mathematical faculty, or lecturers in such faculty). The problems were traditional. For each of them there was a standard way leading to the solution; but standard did not mean the simplest. The appearance of other, more efficient and more rational ways in those circumstances, or, inversely, a failure in missing the conventional way threw light on the investigated problem. The first fragment concerned a trigonometry problem, solved by 10 persons among them 6 students of mathematics, two lecturers and two professors of mathematics, the second and the third ones, related to school analysis and algebra, were solved by 67 participants of the final stage of the nathematical olympic competition. The fourth one dealt with by the 3rd year student of mathematics, considered the role of drawing for the solution of a geometrical problem. Although all the problems were of the traditional type and had standard solutions, the analysis showed a great wealth of ways followed by the solvers, various inductive and recurrent steps, various ways of applying results of the inductive stage for the deductive stage, importance of a right choice of an adequate model, some dangerous tendencies of following analogies, failures resulting from a wrong choice of the criterion for the classification of cases, etc. The fourth fragment of the investigation showed the role of restructuring a drawing and its connection with reasoning where physiognomical perception of a drawing plays only a certain role but not the most essential one. The article comprises some provisional conclusions from the performed soundings, first of all pointing out the necessity of investigation in the difficult domain of the specific character, of mathematical thinking and heuristic problems peculiar to this domain, without which elaboration of an adequate pedagogy is impossible.