| dc.description.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. | en_EN |
