Innovative approach of mathematical modeling of learning processes as an optimal control object

system "Teacher-student."

Learning processes in secondary schools as an object of social management consists of pupils (receiver of information) and teachers (source of information) of the subject area. The functional structure of the educational process in school education consists of several socio-psychological parts (Fig. 2). The exchange of information and the assessment of their interaction between individual parts is one of the main goals of this work.
The use of a systematic approach to the theory of cybernetics allows us to obtain a control structure with a hierarchy of development in space and time. Since the object under study is an open control system, the practical application of the developed models in non-deterministic objects is relevant both from the theoretical and applied side.

Pic.2. The functional structure of the learning process in the teacher-student didactic system.

Analysis of the situation is reduced to the study of the learning process. At this stage, along with the system under study, the situation of educational processes outside it is analyzed, i.e. in an environment in which the system under consideration is or should be included. It is necessary to get a general idea of ​​how to solve a problem (problem), since they determine the choice of the most realistic goals. The result of the analysis of the learning process (situation) is a list of problems and alternatives to solve them, which are investigated at the stage of the system being developed.

Based on the results of the analysis of the learning process, the goals of the tasks are formulated. At the stage of analysis of the system, its elements are mentally divided, with the properties of each being singled out, connections with each other and with the external environment as an information exchange vertically and horizontally as a hierarchical multiply connected system. The purpose of the analysis is to separate the essential properties and relations in the system from the inessential, to move on to a deeper study of it. For this, the analyzed properties and relations are considered in the relationship, i.e. analysis is associated with the synthesis and provides the didactic unity of these methods of knowledge.
Identification - an accurate description of the structure and parameters of the system model. For identification, modern mathematical-statistical methods are used, such as the method of principal components, the method of heuristic self-organization, regression analysis, etc.
After the mathematical model of the system has been built, the state of the system is checked using simulation. And also with the help of the obtained results of imitation, the adequacy of the model of the system under study is checked. If the required level of reality is not achieved, then the solution of the problems of analysis and synthesis is repeated. The computational experiment continues until a satisfactory quantitative result is obtained. With obtaining the required results of analysis and synthesis, practical application of the developed models is carried out (Fig. 3).
The system under study is a dynamic system, i.e. in it, all changes occur in time, and models that reflect these changes are dynamic models of the system.
In mathematical modeling of the learning process, its specific implementation is described as a correspondence between the elements of the set of possible “values” and the elements of an ordered set of “moments of time”, i.e. as a display.
With the help of these concepts a mathematical model of the system is built [5].