Mathematical modeling as a condition for designing qualitative education systems | Статья в журнале «Молодой ученый»

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Рубрика: Педагогика

Опубликовано в Молодой учёный №20 (154) май 2017 г.

Дата публикации: 18.05.2017

Статья просмотрена: 7 раз

Библиографическое описание:

Босимова, М. А. Mathematical modeling as a condition for designing qualitative education systems / М. А. Босимова. — Текст : непосредственный // Молодой ученый. — 2017. — № 20 (154). — С. 431-433. — URL: (дата обращения: 01.06.2020).

The purposeful design of interdisciplinary complexes (MDCs) for the formation of professional culture (PC) is to determine the indicators under which achievement of the planned results is achieved. To organize a set of control actions, it is necessary to know what factors determine the state of the system, what are the permissible limits of their variation, so that the system works reliably and steadily.

As you know, any system can be described mathematically using a polynomial of the n-th order, starting with the simplest models. To do this, it is necessary to determine the significant factors xi (input parameters) and unknown coefficients bi for them, that is, the task to make «parametric».

Accounting for the influence of all the factors of the pedagogical process leads to problems of great complexity, so there is a need to identify the main factors or signals that determine the state of the didactic system, as well as the ranges of their change and the possibility of using factor analysis methods for mathematical modeling of the educational process. Hence, we can conclude that the design of the MDC can be reduced to the design of the parameters of the learning process, which we will dwell on in more detail.

Formalizing the task, we want to get the desired quality indicators (the product of the MDC function) as a result of the design. The product of the ideal quality always reacts equally to the consumer's effects (signals). If, in response to the same signal, different reactions are obtained, then we have a quality less than ideal. With regard to education, it can be said that if a prescribed level of training or the formation of personal characteristics of students is achieved, if such a set of psychological and pedagogical standards are observed, then such a didactic system will be of an ideal quality. To minimize differences in obtaining the final results in educational systems in response to uncontrolled factors (noise) and at the same time limiting the possibility of obtaining a guaranteed result, it is necessary to use the ideas of mathematical statistics relating to statistical methods of experimental planning. If you specify a possible number of levels for each factor, a complete search of all possible combinations of factors at all levels form a complete factorial experiment (FPE) — one of the possible experiment plans [1. p. 151].

G. Taguchi, considering the methods of experimental planning, for the first time divided the factors considered into fundamentally the main factors that have a regulating effect on the result, and secondary factors. Since usually the spread of conditions is great, G. Taguchi proposed to characterize the manufactured products with the stability of technical characteristics [2. p. 113–175]. To assess the influence of factors on the result, he used the idea of the signal-to-noise ratio adopted in telecommunications. In general, if we denote the value of the parameter being studied at the input through M, the noise components through x1, x2,..., xn, the value of the parameter at the output through y, then y will be a function of M and noise: y = f (M, x1, x2,..., x n).

Calculation of the stability of parameters in accordance with the methods of G. Taguchi is carried out not by complicated labor-consuming and expensive methods, but by a new method of experimental planning using variance analysis. In the process of experimental design, the values of the parameters are selected in such a way that the signal is as large as possible, and the noise is as small as possible. This idea can be used in pedagogical design.

In pedagogical design, effective management of quality should involve the use of offline methods. With the help of offline methods, you can determine the requirements for the performance characteristics of the learning process, which are set in the form of the best (ideal) values and allowable deviations from them. Any design work begins with the design of the system.

Provision of technological process of the educational process requires a deep knowledge of organizational forms, means and methods, and other characteristics of learning activities. Parameter design is the process of setting the nominal values of the process parameters, which will reduce the sensitivity of the MDC to sources of deviations (to noise). The use of mathematical dependencies of performance characteristics on factors in order to reduce the sensitivity of the project to sources of deviations is the essence of designing parameters.

An important task in the formalization of the object of investigation is to establish the variety in which the experimenter must make a choice, and also to have methods of influencing the object that would allow one to obtain a distinguishable state. Factors serve for this purpose. A factor is a measured variable that takes at some time a certain value and corresponds to one of the possible ways of influencing the object of research. The number of ways to select the object of the pedagogical system is practically unlimited; hence, the huge number of factors that one has to meet in real pedagogical situations. The projection of any object, including pedagogical systems, as Ashby showed [3. p. 320], boils down to fixing, stabilizing the values of a number of factors. The current practice of optimizing the teaching and upbringing process is based mostly on the study of already designed objects. Despite the widespread prevalence of such an approach in pedagogy, described in detail by Yu. K. Babansky [4], in many cases it should be considered logically unsatisfactory, since optimization is based on possible erroneous solutions arising in the design of didactic systems. Therefore, optimization should be carried out at the design stage, taking into account the controllability or partial controllability of the research object, which allows the experimenter to attach to the factor in question any possible value and maintain this value until the end of the experiment. Like the optimization parameter, the factors must be determined operationally. Each factor has a domain of definition. Requirements for factors — the lack of correlation between any two factors and the compatibility of factors.

After the optimization parameter u has been determined, the goal of the system (to form a specific professional culture) and the set of factors {x i} defining the set of states v of the research object, it is necessary to establish a correspondence between the set of factor values and the values of the optimization parameter:

u = f (x 1, x 2,..., x n),(1)

Where u — is the optimization parameter; Xi are factors, i = 1,2,..., n.

In what follows, we will not make a distinction between the concepts of the statistical mathematical model and the function (1), which is usually called the response function [4.p. 192].

The ultimate goal of the experiment is to determine the set of optimal values of factors and to study the factor space in the neighborhood of this set. In accordance with the intended program, the variation intervals of factors and the number of values that will be tested in the experiment are determined.

The calculation of coefficients is a problem solved by the method of least squares. Due to the optimal organization (orthogonally), the solution is very simple. The method of least squares is used to find the unknown coefficients of a polynomial approximating the original function. If the degree of the polynomial is not specified a priori, the calculations are carried out several times, gradually increasing the degree of the polynomial until the resulting model becomes adequate.

To solve the problem, we compile the X-matrix, the matrix of experimental conditions, consisting of the number of columns corresponding to the number of unknown coefficients, and the number of rows corresponding to the number of experiments. A Y-matrix of the values of the desired function obtained as a result of the experiments is also compiled.

The matrix X is rectangular, with dimensions [n x (k + 1)], n> k. Using the rule of least squares, we obtain a matrix system of normal equations of the form:

(X T x X) B = XT x Y(2)

Where XT — is the transposed matrix of the experimental conditions; B — the matrix-column of the required coefficients of the polynomial mathematical model or regression equations;

(XT x X) — is the matrix of coefficients of normal regression equations.

As is known, (XT x X) is a square non-degenerate matrix of order (k + 1). Hence, for it one can find an inverse matrix (XT x X) – 1. From (2) we can find the matrix-column B

B = (X T x X) — 1 X T x Y(3)

This is the solution of the problem of finding the coefficients of the regression equation.

In the expanded form from formulas (1) — (4) we obtain an expression for the coefficients of regression

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Where — Cij — are elements of the inverse matrix.

If the orthogonality condition is imposed on the experimental matrix X, then the matrix (X T x X) will be diagonal. The elements of the inverse for the diagonal matrix are equal to the reciprocals of the corresponding elements of the direct matrix. It is this circumstance that makes it possible to use the simplest calculation formulas in the planning of experiments and to do the matrix inversion operation practically in the mind. In addition, it makes it possible to evaluate independently of each other all the coefficients of regression.

In pedagogical processes, when starting an experiment, we do not have complete information on the error of experience. Therefore, it is necessary to check the uniformity of the dispersions. This is one of the main requirements of regression analysis. It is recommended to put 3–5 experiments at the zero point, and calculate the variance, assuming that it is valid at all other experimental points [5. p. 155]. In practice, experiments are usually duplicated without resorting to the implementation of the conditions for the zero point.


  1. Беллтан Р. Динамическое программирование. — М.: Изд-во ин. лит-ры, 1960. — 151 с.
  2. Taguchi G. Linear Graphs for Orthogonal Arrays and their Applications to Experimental Design with the Aid of Varions Technigues // Rep. Stat. App. Res., JUSE, 1959, 6, № 4, p. 113–175.
  3. Эшби У. Введение в кибернетику. – М.: Изд-во ин. литер.,1969. – 320 с.
  4. Бабанский Ю. К. Оптимизация учебно-воспитательного процесса: Методические основы. — М.: Просвещение, 1982. — 192 с.
  5. Адлер Ю. П. Введение в планирование эксперимента. М.: Изд-во «Металлургия», 1968. — 155 с.
Основные термины (генерируются автоматически): MDC, FPE, JUSE.

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