Abstract:

The main objective of the mathematical theory of training is as follows: on the basis of the pupil’s characteristics, his initial knowledge level and distribution of the studied material, it is necessary to define quantity of the pupil’s knowledge during training and after its ending. The aim of the work is the research of various mathematical models of training with using of the computer and creation of imitating (computer) models of the didactic system. Methodological basis of the research are works by N. Winer, K. Shannon, V. M. Glushkov, D. A. Pospelov (cybernetics, the theory of information), R. Atkinson, L. P. Leontyev, F. S. Roberts, L. B. Itel’son (mathematical modelling of training), B. Skinner, S. I. Arkhangelsky, V. P. Bespalko, E. I. Mashbits, I. V. Robert (cybernetic approach in pedagogy, the automated training systems).

We use information–cybernetic approach in the analysis of the didactic system “teacher–pupil” on the basis of which three models are offered. The first model is based on the assumption that information given by the teacher consists of the blocks uniting 4–6 elements of a learning material (ELM). The pupil understands the new block of the given information only when he manages to comprehend each entering ELM before arrival of the next information block. For determination of the average speed of assimilation the method of statistical tests is used. The computer program simulates transferring of 2000 blocks and counts the number of the blocks comprehended by the pupil in the given interval of time at various speed of the information reporting by the teacher.

The second model considers that different ELM–s have unequal complexity and are remembered with various durability (strength). That knowledge which is included in the pupil’s educational activity and which is demanded by him, is remembered much better and forgotten more slowly than the knowledge which the pupil doesn't use. At increase in number of the pupil’s addressing to this ELM: 1) the time of his using it decreases, tending to some limit; 2) the forgetting coefficient decreases, tending to zero. This model allows to simulate the training at multiple addressing to a set of ELM–s during one or several lessons.

The third model is two-component model of training which considers that:
1) the state of the pupil in each timepoint is defined by the quantity of the weak knowledge and strong knowledge (abilities, skills);
2) weak knowledge is forgotten quicker than strong knowledge.
3) while training the quantity of the pupil’s weak knowledge increases, and a piece of weak knowledge turn into stronger (more durable) knowledge;
4) after the end of training forgetting takes place: strong knowledge gradually turns into less strong, and the quantity of the weak knowledge decreases according to the exponential law.

This model also takes into account:
1) the distribution of educational information during the whole time of training at school;
2) the pupil's learning and forgetting coefficients in various years of training;
3) shares of educational information of the previous levels (forms) which are used by the pupil when studying new material, and also using them during vacation and after the training completion.

The mathematical equations modeling training are presented in the article and the graphs of dependence of the pupil’s knowledge different types on the time which are turning out as the result of their solution on the computer are analysed.

Keywords:

Computer modeling, didactics, education, mathematical methods, pedagogy, pupil, simulations, teacher, training.