Abstract:

Improvement of teaching technique requires assessment of didactic characteristics of various learning material elements (LME). The problem of the didactic complexity (DC) estimation of various issues of the school mathematics, which can be considered as one of indicators of the pupil’s intellectual development, represents a considerable interest. The purpose of the present study is to develop a method for estimating the complexity of various issues of school mathematics. This will answer the question: How does the complexity of the educational material increases during the time of schooling?

This problem lays at the intersection of the following scientific areas: the psychology of the educational process (B.M. Velichkovsky, V.M. Krol), the optimization of textbooks (Ya.A. Mikk, V.P. Bespalko), the complexity measurement of the solution of the educational tasks (G.A. Ball, A.V. Gidlevsky), the formation of the cognitive operations in senior classes (N.N. Pospelov, I.N. Pospelov), the application of the content-analysis method to assess the complexity of education texts (R.V. Mayer, A.M. Sokhor), folding and unfolding of knowledge and operations (S.I. Shapiro), the modeling in pedagogy (G.V.Glass, J.C.Stanley, M.V. Yadrovskaya).

The complexity of the particular LME is proportional to the time (or number of words) which required for explanation this material to a pupil. It depends on: 1) the volume of the LME, i.e. the minimum quantity of words which necessary to say in order to explain this LME; 2) the level of abstraction, the number of mathematical symbols, formulas and objects shown in the pictures; 3) the degree of the information folding, which is characterized by a share of new concepts expressed through simple concepts.

To assess the DC of some LMEs we used:
1) the method of decomposing operations on elementary actions;
2) the method of paired comparisons;
3) the content analysis of the paragraphs of the textbook.

It is taken into account:
1) the paragraph volume;
2) the quantity of mathematical symbols in the formulas;
3) the quantity of new concepts that are not included in a given level of knowledge;
4) the information volumes of the new concepts definitions.

To measure the information volume V of this LME:
1) we replace the text of the given LME with an equivalent text of minimum length containing the same information;
2) we replace drawings and formulas with their verbal description of the minimum length;
3) we count and summarize up the quantities of words in the texts.

To determine the corrected information volume of the LME, it is necessary:
1) to set the level of knowledge (or the system of concepts) relatively to which the DC complexity is determined;
2) to write down definitions of the new concepts used in this LME, which are not known to the pupil, and after that to count the number of their uses;
3) to multiply the quantities of words in the definitions of new concepts by the numbers of their uses and summarize up all these products with information volume V of this LME.

As a result of the didactic complexity assessment of the 27 LMEs, it is established that while schooling the mathematics issues complexity increases 150 – 200 times. It turned out that the LME "Quadratic root and its transformations" is in 3 – 4 times more complicated than the LME "Multiplication of two-digit natural numbers", and the DC of LME "Properties of a definite integral" is 3.5-5 times larger than the DC of LME "Quadratic function".

Keywords:

Didactics, qualimetry, mathematical methods, paired comparisons, pedagogical expertise, complexity, theory of learning, textbook.