Application of mathematical methods in research. Mathematical Methods and Models in the Social Sciences: Patterns, Specifics and Stages of Application

The solution of practical problems by mathematical methods is consistently carried out by the mathematical formulation of the problem (development of a mathematical model), the choice of a method for conducting a study of the obtained mathematical model, analysis, and the results obtained.

The mathematical formulation of the problem is usually presented in the form of numbers, geometric images, functions, systems of equations, etc.

Mathematical model is a system of mathematical relationships - formulas, functions, equations, systems of equations that describe certain aspects of the object, phenomenon, process under study.

At the stage of choosing the type of mathematical model, by analyzing the data of a search experiment, the following are established: linearity or non-linearity, dynamism or static, stationarity or non-stationarity, as well as the degree of determinism of the object or process under study.

Establishing the general characteristics of the object allows you to choose the mathematical apparatus on the basis of which the mathematical model is built. The choice of mathematical apparatus can be carried out in accordance with the scheme presented on rice. 1.2 .

Rice. 2. Mathematical apparatus for building a mathematical model

As can be seen from this scheme, the choice of the mathematical apparatus is not unambiguous and rigid.

To describe complex objects with a large number of parameters, it is possible to divide the object into elements (subsystems), establish a hierarchy of elements and describe the relationships between them at different levels of the hierarchy.

A special place at the stage of choosing the type of mathematical model is occupied by the description of the transformation of input signals into output characteristics of the object.

If at the previous stage it was established that the object is static, then the construction of a functional model is carried out using algebraic equations. In this case, in addition to the simplest algebraic dependencies, regression models and systems of algebraic equations are used.

If the nature of the change in the indicator under study is known in advance, then the number of possible structures of algebraic models is sharply reduced and preference is given to the structure that expresses the most general pattern or well-known law.

If the nature of the change in the indicator under study is not known in advance, then a search experiment is set up. Preference is given to the mathematical formula that gives the best match with the data of the search experiment.

Search experiment results and a priori information array allow you to establish a scheme for the interaction of an object with the external environment by the ratio of input and output values.

In principle, it is possible to establish four interaction schemes:

one-dimensional-one-dimensional scheme ( rice. 1.3, a ) - the object is affected by only one factor, and its behavior is considered according to one indicator (one output signal);

one-dimensional-multidimensional scheme ( rice. 1.3 b ) - the object is affected by one factor, and its behavior is evaluated by several indicators;

multidimensional-one-dimensional scheme ( rice. 1.3, in ) - the object is affected by several factors, and its behavior is evaluated by one indicator;

multidimensional-multidimensional scheme ( rice. 1.3, g ) - the object is affected by many factors and its behavior is evaluated by a variety of indicators.

mathematical model synthesis

Rice. 3. Schemes of interaction of an object with the external environment

The process of choosing a mathematical model of an object ends with its preliminary control.

In this case, the following types of control are carried out : dimensions; orders; nature of dependencies; extreme situations; boundary conditions; mathematical isolation; physical meaning; model stability.

Dimensional control is reduced to checking the fulfillment of the rule, according to which only quantities of the same dimension can be equated and added.

Order control aimed at simplifying the model. In this case, the orders of the summed quantities are determined and obviously insignificant terms are discarded.

Controlling the nature of dependencies is reduced to checking the direction and rate of change of some quantities when others change. The directions and speed following from the mathematical model must correspond to the physical meaning of the problem.

Control of extreme situations is reduced to checking the visual meaning of the solution when the model parameters approach zero or infinity.

Boundary Condition Control consists in checking the correspondence of the mathematical model to the boundary conditions arising from the meaning of the problem. At the same time, it is checked whether the boundary conditions are really set and taken into account when constructing the desired function and that this function actually satisfies such conditions.

Control of mathematical closure comes down to checking that the mathematical model gives a unique solution.

Physical sense control is reduced to checking the physical content of the intermediate relationships used in the construction of a mathematical model.

Stability control The model consists in checking that the variation of the initial data within the framework of the available data on the real object will not lead to a significant change in the solution.

and stimulation of the "generation" of ideas arising from personal contact. In addition, it requires a significant investment of time.

Lit.: Doctorov B.Z, Expert poll as a method of studying societies, opinions // Sotsiol. research. 1985. Ns 4; Construction of expert systems. M., 1987; Acquisition of knowledge. M., 1990; Golubeva L.N. Technol. relation to knowledge: methodol. aspect. Rybinsk, 1993; Yadov V.A. Sociological strategy. research. Methodology, program, methods. M., 1998; Miles J, Moore C, Practical Knowledge-Based Systems in Conceptual Design. L., 1994.

N.I. Rostegaeva

METHODOLOGY OF SOCIOLOGICAL RESEARCH- 1. A means of implementing general theories. and methodol. principles of sociology. research-I at the empirical level in terms of concret. research situation, each of which is characterized by both typical and unique features. Sociol. ideals and norms of scientific character in M.S.I. adapt in each department. research and to the specifics of the research tasks being solved, to the features of the subject and object being studied, to the organizational and economic capabilities of the research team.

In the sociological program research-I provides a special section containing the rationale for the adequacy of research methods to the subject, object and organizational and economic possibilities of research-I. In part., the rationale for adequacy is given (the shaft of id no-sti - see. validity) sampling procedures, methods for collecting empirical data (technical and instrumental options for methods of survey, observation, analysis of documents, experiment), methods for processing and analyzing the collected empirical data. A necessary element of justification yavl. trial (pilot) research, in which the developed methods are tested in the field conditions and improved in acc. with the results obtained (see trial study).

2. M.S.I. are developed to solve class. similar research tasks requiring standard methods, solutions tested in research experience and validated in specialized methods, studies. They contain normative instructions regarding the development of a method, tools, conditions and rules for its use, criteria for assessing the quality of conc. research situation, boundaries of interpretation. Usually these are author's works, including the names of the subject of research, for which a technique or department has been developed. method, tool: “Methodology for processing and analyzing data on the budget of our time”, “Methodology of a telephone survey *, “Methodology for a focused interview in marketing research”, etc. This genre is often called method, recommendations, since the description and justification of the method, solutions are given for typical research situations, and the sociologist addressing them must creatively take into account the unique features of the research problem being solved.

Lit.: Andreenkov V.G., Sotnikova T.N. Telephone surveys of us. (Method, recommendations for conducting selective mass surveys). M., 1985; Dridze T.M. Informative-target analysis of the content of textual sources // Methods of collecting information in sociol. research. Book. 2. M., 1990. S. 85-102; Method, problems of data analysis on the use of time us. M., 1991; Keselman L. Street poll in sociology. Research: Method, manual. Samara; SPb., 2001.

O.M. Maslova

METHODOLOGY FOR APPLYING MATHEMATICAL METHODS- a set of principles reflecting the correlation of math. formalism and a fragment of reality modeled with its help and allowing the use of mathematics. apparatus as a means of cognition of social. phenomena. It should be distinguished M.p.m.m. from the method of applying Math. methods - descriptions of the sequence of steps, the implementation of which and

QUALITATIVE METHODS

lays down the essence of the application of the method. For example, under the technique of applying the chi-square test to assess the relationship between features (see. Pair connection coefficients of nominal features) refers to the sequence of actions aimed at calculating this criterion, the definition of Table. values, comparison of the sample value of the criterion with the table. value, etc. The methodology for using the same criterion is a set of statements about in what tasks and in what sense this criterion can be used as an indicator of the connection, how it correlates with the cause-and-effect relationships of interest to the researcher, how these relationships can be studied more deeply by using the criterion under consideration in combination with other methods of communication measurement.

The development and observance of the discussed principles are aimed at solving Ch. tasks - ensuring the adequacy of the formalism of the essence of the problem being solved (see. Adequacy of the mathematical method, p./). When using any method, the choice of elements of formalism should be determined by the theory. concepts of the sociologist. Such points should be allocated separately for each method (group of related methods) and for each socio-tsiol. tasks (group of tasks of the same type). But there are also general principles inherent in any methods and tasks. One of the main principles yavl. the requirement to proceed not “from the method”, but “from the task*. The researcher should not “apply factor analysis”, not “use classification methods”, but solve the substantive tasks facing him: study the structure of cause-and-effect relationships, build a typology, etc. Common yavl. pl. principles interpretation of the results of applying the mathematical method, measurement in sociology and data analysis.

Lit.: Tolstova Yu.N. Logic of Math. sociological analysis. data. M., 1991; She is. Sociological analysis. data. M., 2000.

Yu.N. Tolstova

QUALITATIVE METHODS - methods for collecting and analyzing empirical information in qualitative social and (qualitative research techniques), resp. her theory. grounds and methods. principles.

M.k. were formed to solve specific research problems when quantitative methods were insufficient: the study of closed subcultures (age, professional gr.), deviant gr. (criminal communities, drug addicts), as well as social problem gr. (migrants, disabled, unemployed). There are methods, to-rye allow to obtain empirical data in the form of descriptions of everyday life of people, lang., social. meanings of events, actions, phenomena lived by people.

Various are used. modification of the polling method. These are semi-formalized interviews: interviews with open and closed questions, focused interviews, directed (with a guide) and non-formalized interviews: narrative (narrative), which can be biographical or leitmotif (thematic) (see. Biographical method, Classification of interviews). A specific method yavl. analysis of conversations (conversion analysis) recorded on audio and/or videotape, transcribed as texts.

When using the method of non-formalized included observation, the following are observed. requirements: the ability to create and maintain the trust of informants throughout the study, not to violate nature. the course of everyday life; implementation of the rules for maintaining field diaries for their afterbirth, processing and analysis. Method of analysis doc. sources involves, in addition to analyzing personal documents (letters, family archives, memoirs, family histories), referring to photographs, collections (books, postcards, gramophone records, audio and video recordings, etc.), and so on. I see the dock. sources providing valuable information for understanding about-va through people. fate. Formation of an empirical object in qualitative and village

CLASSIFICATION METHODS

is based on a case study strategy (cf. case studies).

Analysis of empirical information in qualitative research and yavl. iterative and is provided by special techniques for coding elements of the content of texts (narratives of free interviews, records of observations, etc.).

The task of coding is to translate the content of texts that describe the studied reality at the level of everyday consciousness and language. everyday life, at the level of scientific. descriptions, interpretations of those meanings, to-rye contained in the analyzed narratives of social. actors.

When encoding, a cut may consist of several. stages, a triangulation procedure is used to reduce possible subjective biases in interpretation: comparison of the results of the analysis of the narrative by different researchers, and / or comparison of data obtained by different methods. Res-vol. formation of the concept (microtheory) of each analyzed case, taking into account each of the previous cases (iterative analysis) to obtain a rich description of typical phenomena that reflect the reality under study.

Lit .: Biographical method in social sciences: history, methodology, practice. M., 1994; Romanov P.V., Yarskaya-Smirnova E.R.“Making the familiar unknown...”: an ethnographic method in social sciences // Sotsiol. magazine. 1998. No. 1/2; Semenova V.V. Qualitative methods: an introduction to humanistic social science. M., 1998; Kovalev E.M., Shteinberg I.E. Qualitative methods in field sociology. research. M., 1999; Strause A., Corbin D. Fundamentals of quality NSSL-I. grounded theory. Procedures and techniques / Per, from English. M., 2001; Isupova O.G. Conversion analysis: presentation of the method // Sots-ya: methodology, methods, math. models. 2002. No. 15; Yadov V.L. Sociological strategy. research. M., 2007.

OM. Maslova

CLASSIFICATION METHODS- a component part of multivariate analysis methods. M.k. allow splitting

nie set of objects on otd. class so that objects assigned to one class are considered similar, close, of the same type, and to different - dissimilar, distant, heterogeneous. In the general case, the desired cl. def. the manifestation in them of certain empirical patterns (definition of combinations of feature values; regression relationships between features; partitioning satisfies a given optimality criterion, etc.). Cl. may or may not overlap. Both the partitioning procedure and its result (set of cells) are called classification. M.k. are used either to compress information, or as a tool typological analysis in order to detect typological syndromes or test the hypothesis of the existence of types in the sense given by the researcher. In the first case, as a rule, a division into a relatively small number of homogeneous groups is required, and the task of determining the natures is not set. bundles of the original objects, as in the second case.

The first algorithms of M.K. arose from a geometric representation: objects are points of a multidimensional space of classification features. The similarity of objects is the proximity of their location in this space; class - Condensation of objects defined. configuration. The variety of problem statements of typological analysis gave rise to the existence of dec. classification procedures, each of which involves a definition. a criterion (set explicitly or implicitly) for the similarity of objects and a classification algorithm.

In a number of M. to. similarity criterion is given as proximity measure between any two objects. In sociology. research classification features often have a nominal level of measurement, so they are converted into binary (dichotomous). It is important to be able to vary the proximity measures, but not in any algorithm it is possible to set the required measure. In some M. to. the proximity measure is already implicit in the algorithm itself.

CLASSIFICATION METHODS

A classification algorithm is a procedure by which objects are divided into classes, i.e. gr., on which a certain regularity is satisfied (partially formalized already by the introduction of the criterion of similarity of objects). The algorithm is implemented under the definitions, restrictions specified in the form of M.c. parameters. (number of cells, threshold of distinguishability of objects and cells, etc.).

Each algorithm is characterized by certain properties. I. Stability Regarding the reordering of objects. Implementation of M.c. presupposes a sparkly ordering of objects with the so-called. the order of arrival at the "input" of the algorithm (some object is called the first, some - the second, etc.). By changing the order and applying the algorithm once more. receive a new result, which may not coincide with the previous one. In case of a match, the algorithm is considered to have the admissibility property with respect to reordered objects. 2. Stability with respect to duplication class. This means that if the objects of the isk-rogo class. add (duplicate) to the original population and repeat the classification procedure, the boundaries of class. will not change. 3. Stability against removal class. This means that if objects of the same class removed from the original population and repeat the classification, then the boundaries of class. will not change. 4. Stability with respect to duplication of objects. This property is similar to the second one, with the only difference that instead of cl. the object is being considered. Among the important ones is the property related to the fact that not every proximity measure (given explicitly) can be used in any algorithm. This applies to those algorithms in which, for example, despite the explicit form of specifying the measure of proximity, the algorithm itself can be implemented only if proximity is understood as Euclidean distance.

The totality of M. to. can be grouped according to grounds. So, depending on the volume of the classified population and on a priori information about the number of cells. It is customary to distinguish three types of M.C.: hierarchical, parallel

A few years ago, when the author of this book worked as a mathematical statistics consultant for a small medical research group, talk of the possibility of cutting a mathematical path through the dense jungle of environmental factors often ended with a rather skeptical shake of the head and the statement that “medicine is after all art". This is partly true, of course, in the sense that intuition and imagination are really necessary for a doctor. At the same time, most patients and potential patients undoubtedly hope for the continuous development and expansion of the scientific aspects of medicine. And science means the application of mathematics.

The question of the areas in which mathematical methods are applicable is essential. In sec. 1.1, we have already noted that the need for a mathematical description arises in any attempt to discuss in precise terms, and that this applies even to such complex areas as art and ethics. In this section, we will consider in more detail the areas of application of mathematics in biology and medicine.

It is well known that one of the approaches to describing the picture of nature is the construction of a hierarchy of levels of organization studied by various sciences; according to the level of abstraction inherent in each of them, these sciences can be arranged in the following sequence: physics, chemistry, biochemistry, physiology, psychology, sociology. We begin with the basic material elements of the real world, i.e., from the subatomic level, and end with unusually versatile manifestations of the spiritual life of human society. In this succession of levels, organization and complexity continually increase. Each level has its own laws, and therefore they can be studied to some extent independently of each other. However, any of them is inextricably linked with patterns operating at lower levels. Thus, the laws of physics and chemistry partly extend to psychology, although the concepts and laws of the latter go beyond the limits of physical and chemical laws.

Problems concerning the organization and operation of hospitals should be attributed to a higher level of abstraction than, say, human physiology and pathology. But although to a certain extent the logical content of this higher level is independent of the lower one, questions of physiology and pathology must inevitably be taken into account in solving any problem concerning the organization of hospital services. We do not intend to delve into these philosophical considerations here or discuss their individual details, but we only want to emphasize that the described sequence of levels corresponds approximately to the order of increasing difficulties in using scientific methods and conducting mathematical research.

As we have noted, applied mathematics has made major and undeniable advances in physics and chemistry, but we will not deal with these issues in this book. In sec. 1.2 it was shown that mathematical descriptions related to biological forms cover a wide range of issues and can be carried out quite accurately. In sec. 1.3 we got acquainted with dynamic models of development and touched upon the problems associated with random fluctuations in the size of populations. The presentation of these questions required a sufficient degree of abstraction, but it was the use of simplifying assumptions that allowed us to get some idea of ​​the laws governing population growth. It was noted that when considering such problems, one inevitably has to deal with the factor of statistical variability, a detailed discussion of which is carried over to Chap. 2.

As we move to higher levels of abstraction, we face not only more complex issues, but also an increasing degree of variability, mostly unpredictable. For example, a complete picture of competition between several species living in a particular environment includes a huge variety of factors. Significant progress has been made in the field of scientific ecological descriptions, mainly in verbal form, but the development of mathematical models is still at the most elementary level. Another example is the field of medical diagnostics. To make a diagnosis, the doctor, together with other specialists, is often forced to take into account a wide variety of facts, based partly on his own personal experience, and partly on materials cited in numerous medical manuals and journals. The total amount of information is increasing with increasing intensity, and there are diseases about which so much has already been written that one person is not able to accurately study, evaluate, explain and use all the information available in making a diagnosis in each particular case.

Of course, a good diagnostician, using his extensive experience and intuition, can select the necessary part of the important data and give a fairly accurate conclusion. However, paradoxical as it may sound, as knowledge accumulates, the situation worsens.

It is in this kind of situation, when the mind of one person is not able to cope with the complexities of the tasks before him and describe their solution even in a general verbal form, specialists in the field of the so-called inexact sciences (including, of course, biology and medicine) often argue that mathematical the analysis is imperfect, inappropriate, leads to erroneous conclusions, or is impossible, and is therefore best avoided. This objection contains a rational grain in the sense that modern mathematics may not yet be perfect enough; however, time will pass and we will see that just the opposite is true. In cases where the problem contains a large number of significant interdependent factors, each of which is largely subject to natural variability, only with the help of a properly chosen statistical method can one accurately describe, explain and study in depth the entire set of interrelated measurement results. If the number of factors or important results is so great that the human mind is not able to process them even with the introduction of some statistical simplifications, then data processing can be done on an electronic computer. The use of statistical methods and computer technology is discussed in Chap. 2 and 5, respectively.

The main reason for distrust of mathematical and computational methods seems to be the following. A mathematical model of some biological phenomenon will be acceptable to a biologist only if the verbal information about this phenomenon that he has is complete enough to judge the adequacy of the model. It is clear that obtaining such information is the first and most important step in biological research, and that mathematics plays a secondary role in this step. Naturally, the thought arises that as the questions become more difficult and complex, mathematics becomes less and less important. However, the circumstance is not always taken into account that, having reached a sufficient degree of complexity, mathematics develops further according to its own laws and gives the biologist concepts and a way of thinking that he did not have before. Let's hope that this book at least to some extent illustrates the validity of this statement.

So far, we have had in mind mainly those biological and medical studies that require a higher level of abstraction than physics and chemistry, but are closely related to the latter. Next, we will move on to problems related to animal behavior and human psychology, that is, to the use of applied sciences to achieve some more general goals. This area is rather vaguely referred to as operations research, and is discussed in more detail in Chap. 4. For now, we will only note that we will be talking about the application of scientific methods in solving administrative and organizational problems, especially those that are directly or indirectly related to biology and medicine. Forestry, animal husbandry, general agricultural production, hospital design and medical care are just a few of the issues that fall into this category.

Of course, not all problems of administrative management can be solved on a scientific basis using the methods of operations research. However, the use of these methods, where possible (and they are applicable to many problems of this kind), has great advantages, since it allows you to expand the area of ​​​​exact research and reduce the area of ​​\u200b\u200bvague verbal reasoning. Thanks to this, the intuition and common sense of a person can be directed to solving those issues where it is impossible to use template methods. Even more complex are questions that involve some kind of ethical considerations. But sometimes mathematical analysis can help even in these cases. For example, in medicine there are often complex problems associated with the use of drugs that are still in the testing phase. The doctor is morally obligated to offer his patient the best remedy available, but in fact he cannot make a choice until the trial is over. In these cases, the use of well-designed sequential statistical tests can reduce the time required to obtain final results. Ethical problems are not removed, however, such a mathematical approach somewhat facilitates their solution. Sequential methods are discussed in more detail in Sect. 2.3.

The main point of this section is that mathematical methods are applicable to the widest range of issues - from elementary particle physics to moral problems. It is convenient (though not necessary) to consider a certain hierarchy of levels. As one moves to more abstract levels, mathematical methods become less developed and more difficult to apply.

However, when properly applied, the mathematical approach does not differ significantly from the approach based on simple common sense. Mathematical methods are simply more precise and use clearer formulations and a broader range of concepts, but they should ultimately be compatible with ordinary verbal reasoning, although they probably go beyond them.

• unity of categories of quality and quantity;
• the growing role of abstraction in modern scientific knowledge, the unity of its various forms in the knowledge of objects of the surrounding world;
• the unity of the logical and intimate-psychological spheres in the intellectual activity of the student (11, 11).

In addition to these principles, many researchers (S.I. Arkhangelsky, P.N. Volovik, V.I. Zagvyazinsky, T.V. Ilyasova, I.M. Kantor, V.P. Mizintsev, L.M. Fridman and others (19; 72; 116; 132; 146; 236; 237; 238; 366)) pointed to conditions (sometimes calling them principles) for the application of mathematical methods in systemic pedagogical research:

• the application of these methods should be carried out only at certain stages of a system study, for example, the stage of describing the structure and organization of a system or the stage of describing (mathematical modeling) its behavior;
• the application of mathematical methods is possible only to certain aspects, connections and parameters of pedagogical objects that can be quantified;
• probabilistic-statistical methods can only be applied to the study of statistically stable random events;
• the study of a random event (for example, the effectiveness of the method of education) requires a clear establishment of the unit of the general population for this event;
• the sample must be representative;
• to interpret the results of mathematical methods, it is necessary to use meaningful methods of pedagogy, psychology, and sociology.

The last principle was especially important for the synthesis of meaningful and formal methods in pedagogical systems research. In this regard, A.M. Sohor noted that “mathematical, quantitative methods are not universal in the sense of unconditional and rational solution of any problems. Mathematics, considered in the methodological aspect, is, first of all, a special kind of language, inconceivable without content ... The expediency of using quantitative methods can be determined both by the present level of development of quantitative methods, and by the specific task of the study ”(325, 28).

The cybernetic approach, like mathematical methods, began to be used in pedagogy long before it turned to a systematic approach. But only its interpretation as one of the means of a systematic approach, and not as a universal method for solving pedagogical problems, ensured its wide recognition as a method of pedagogical research. The peak of interest in cybernetic methods in the systematic study of pedagogical objects also falls on the 70s.

Assessing the heuristic potential of cybernetics in various fields of knowledge, philosophers came to the conclusion that the use of cybernetic ideas does not lead to the discovery of new facts and the acquisition of new knowledge, but their significance lies in the ability to foresee the still unknown, generate new ideas, express already existing ideas, identify similarities and analogies between different areas (1, 80).

One of the most famous cybernetic methods is the "black box" method, which focuses on the behavioral characteristics of the system as a whole with a certain abstraction from the internal structure of each functional unit. This method has found its application in pedagogical system studies, as a rule, at the stages of describing the behavior of the system in its relationship with the external environment and other systems, at the stages of studying the issues of managing the pedagogical system.

This, obviously, can explain the wide popularity of the “black box” method in systemic pedagogical research related to the pedagogical management that took shape in the second half of the 80s into a scientific discipline (V.P. Bespalko, V.I. Bondar, M.I. .Kondakov, L.M. Sidon and others (35; 60; 167; 303)).

The fact is that by the time the systematic approach penetrated into domestic pedagogy (the end of the 60s), its section, then called school studies, was a poorly structured and theoretically insufficiently meaningful set of empirical material, completed mainly for reasons of common sense and worldly experience. The conceptual scheme of the systems approach served as a kind of matrix on which the accumulated empirical material was structured. This made it possible to move from descriptiveness to building a theory of pedagogical management based on cybernetics, management theory, system analysis and other system theories that serve as the foundation for the scientific management of any system. At the same time, issues related to the humanitarian nature of the pedagogical system, in the logic of the emerging scheme of a systematic approach based on cybernetic formalization and reduction, faded into the background. That is, at a certain stage in the formation of pedagogical management as a science, the logic of means prevailed over the logic of content.

Even in modern works on pedagogical management, the emphasis is often placed precisely on the cybernetic aspect of the systems approach: “if previous approaches to management were turned “inside the organization”, then the systems approach, first of all, is turned “outside” - on the behavior of the organization in a large system, and then to what determines this behavior” (354, 14).

Another cybernetic method - the method of information decomposition - was widely used in system-pedagogical research at the stage of describing the behavior of the system and managing it. If the pedagogical system is considered as an information system (for which there are certain grounds), then the behavior of the system can be described as the exchange of information within the system and with the external environment. Then the process of managing the system in form can be represented as a process of information processing: collection, processing-interpretation, issuance of a management decision. In order to manage the system, it is necessary to learn how to regulate the flow of information.

Structural management information is a set of controlled parameters of the control object - indicators, which, in turn, are formed from details - logically indivisible elements of the indicator, correlated with a certain property of the object or process displayed by the information.

From a cybernetic point of view, details, as units of information, should be:

1. measurable, that is, quantifiable;
2. are independent, that is, the description parameters should not “overlap”, have connections with each other;
3. are uniquely defined (363, 33).

Understanding that the pedagogical system, due to its humanitarian nature, cannot have requisites that absolutely meet these requirements, many researchers noted the conditional nature of the allocation of requisites and used the decomposition method at the most general level of research, as “abstraction of abstraction” (194, 74) (I. Ya .Lerner, G.N. Prozumentova, V.D. Semenov and others (209; 270; 297)).

However, during the period of the "systemic boom" there were many researchers who considered the discrepancy between pedagogical details to the requirements of measurability, non-intersecting and unambiguousness as shortcomings in pedagogical knowledge and temporary difficulties, and who tried to contribute to the improvement of pedagogical information, bringing it to the level of cybernetic requirements (B.P. Bitinas , I.N. Zolotarev, I.P. Prokopiev, V.S. Khanchin and others (41; 125; 272; 368)).

A significant majority of educators-researchers who used the systems approach considered cybernetic methods as one of the components of systemological tools, "working" at certain stages of system research. At the same time, some researchers have absolutized cybernetic methods, reducing the essence of the system approach to the implementation of the "black box" model. V.P. Bespalko presented the most detailed logic and tool of such systemic research. He characterized the pedagogical system as a closed structure with a function set by the social order - the only factor that determines the quality of the transition from an applicant to a specialist (35, 26). From this point of view, the pedagogical system was portrayed by the author as a “black box” scheme:

social

The management of such a system, first of all, according to the author, involves the diagnostic setting of goals. That is, the initial concepts that designate the goal must be precisely defined, measurable and correlated with a certain scale (35, 45). Then at the output it is possible to correlate goals and results and draw a conclusion about the effectiveness of functioning, depending on the magnitude of the discrepancy between the stated goals and the results obtained.

From the diagnostically set goals, a model of a graduate of the pedagogical system is formed. Noting that science still knows too little about the personality of a person, V.P. Bespalko proposes to derive any standards by expert means that can be used for diagnostic goal setting and refined as science develops (35, 49). As an initial model for the decomposition of the goals of the pedagogical process, the author proposes a model of the logical structure of the personality, in which the personality, in accordance with cybernetic rules, is decomposed into sides, the sides into properties, the properties into diagnosed qualities, and for the diagnosed qualities the formation criteria must be determined (35 , 17). In addition, a vertical decomposition of goals was also proposed - the so-called "tree of goals", according to the goals of the hierarchical levels of the system - its subsystems.

Yu.K. Lazichnaya, applying the decomposition method to the goals of moral education, in order to overcome the vagueness of the concepts and criteria of moral education, proposed to single out 12 ethical concepts, the mastery of which, according to the researcher, will be the result of moral education (202; 203). The concepts themselves were taken by the author from the "Exemplary Educational Program for 8-Year and Secondary School Students" (264), and their content was defined using an ethical dictionary. To increase the diagnostics of setting a goal, five features were identified in each concept. For example, in the concept of “principledness”, the author highlights the following features: firmness of convictions, readiness to act in accordance with one's convictions; readiness to defend them; high demands on oneself and others; complete lack of stubbornness (203; 30).

At each lesson, in the context of studying program topics, it was supposed to reveal one sign of the concept. The implementation of "hidden influence" for each concept consists of a threefold disclosure of all five features of the concept. Thus, the “single complex of influence” for each concept requires fifteen lessons and two lessons for the final (in terms of the content of the concept) presentation, that is, for seventeen lessons, the child, according to Yu.K. Lazichnaya, is guaranteed to form a solid knowledge of the content of one moral categories. According to the author, direct introduction into pedagogical practice of theoretical models pedagogical object (moral education), built exclusively cybernetic methods, quite legitimately (highlighted by me - A.K.).

The examples given are examples of the typical use of cybernetics methods in systemic studies of pedagogical objects, demonstrating both the procedures for applying these methods and the limitations that these methods impose on pedagogical research.

The widespread use of cybernetic methods provided vast empirical material for methodological reflection, both philosophical and pedagogical. The first conditions for the use of these methods in pedagogy were determined at the end of the 60s and were associated with the need for their methodological interpretation “on the basis of dialectical materialism, which is the general methodology of science. On this basis, the methodological expansion of cybernetics and mathematics in the field of pedagogical phenomena should be regarded positively” (126, 5).

Further understanding of the experience of using cybernetic means in various sciences made it possible to constructively assess the possibilities and limits of "cybernetization". The philosopher N.T. Abramova, analyzing the role of cybernetics in the theorization of scientific knowledge, noted: “however, we have to admit that all those theoretical formations that are built in the humanitarian and non-humanitarian areas on the basis of cybernetics turn out to be non-specific for these areas.… Therefore, we can say that in such cases no original conceptual content is created” (1, 85).

The discovery of the analogy of systems of various nature with information systems and the conclusions drawn on the basis of cybernetic isomorphism of systems turn out to be the same in any study: “since the extrapolated scientific statements are known in advance, the class of new models and hypotheses formed on this basis is predetermined in a certain sense” (1, 85).

The main cognitive limitation, according to N.T. Abramova, is imposed by the focus of cybernetic methods on the study of information and management structures, which are similar for most qualitatively different objects, all other aspects of complex self-managing systems cannot be “captured” by these ideas. Meanwhile, it is precisely the specific regularities that constitute the special knowledge about these objects. “These specific laws turn out to be indefinite and unpredictable from the point of view of cybernetics, here there are other laws inherent in each sphere, for the knowledge of which there are specific methods” (1, 85). Therefore, the resulting cybernetic image of an object can be considered only as one of the stages in the process of constructing its theory.

The conclusions of the philosopher of the 70s are confirmed by modern researchers. So, N.M. Komarova, exploring the history of cybernetic pedagogy, notes that it was developed primarily as a means of achieving greater rigor and accuracy in the description and analysis of pedagogical phenomena. “This result was partly achieved, but theoretical work was often associated with the replacement of the categorical apparatus of pedagogy with the apparatus of cybernetics, which had negative consequences for the actual interaction of the relevant theories and theorists” (162, 6).

Analyzing the processes of decreasing interest in cybernetic methods in pedagogy, the author discovers three reasons for this phenomenon:

• disappointment in the heuristic potential of cybernetic pedagogy due to a sharp discrepancy between the hopes placed on its means and the real results of their use;
• contradiction (according to the author, seeming, in our opinion objective - A.K.) between the "technocracy" of the cybernetic approach and the humanitarian paradigm of education, the ideas of humane pedagogy;
• the emergence in the 70s of a synergetic trend in science, which, unlike cybernetics, emphasized the processes of self-development and self-organization, and not rigid and deterministic control (162, 8).

Thus, the main conditions for the productive use of cybernetic methods in system research of pedagogical objects it could be considered:

1. understanding that the resulting "cybernetic image" of a pedagogical object is only one of its "slices", reflecting information and management communications and non-reflective all other, no less, but in terms of the specifics of the object, and more important connections;
2. understanding that obtaining a cybernetic description of a pedagogical object, therefore, is only one of a number of steps in a system-pedagogical research;
3. awareness of the need for a special interpretation of the results of the application of cybernetic methods in a pedagogical context;
4. understanding that the image of a pedagogical object obtained by cybernetic means is only a theoretical, conditional, limited model, from which there is a huge distance from the real object.

That is, the conditions for overcoming cybernetic reduction in a systematic study are the awareness of the objective reduction potential of cybernetic methods and its compensation with the help of meaningful methods of pedagogy and related humanities.

If cybernetic and mathematical methods were first used in pedagogy independently and only with an appeal to the system approach began to “fit” into the systemological toolkit, then the modeling method in its broadest sense entered pedagogy in the first works substantiating the possibility of using the system approach in pedagogical research.

A.T. Kurakin and L.I. Novikova called modeling the main method of systematic research, “in relation to which all other methods act as private ones, are determined by it” (198, 7). This thesis was seriously criticized in the works of M.A. Danilov, V.I. Zagvyazinsky, F.F. Korolev. The main objections were indications of possible damage to the content side of the study in the case of the priority of the formal research method - modeling; poor development of this method (96, 94; 116.34; 170, 369).

However, a careful reading of the work of A.T. Kurakin and L.I. Novikova reveals that their theses and criticism of the methodologists were on different planes. Obviously, because the processes of conceptualization of the domestic methodology of pedagogy have just begun, the methodological formulation of ideas has not always been adequate to the content of these ideas.

So, A.T. Kurakin and L.I. Novikova, describing the essence of the modeling method, noted that the scientist, in accordance with the principles of system research, begins his work with the synthesis of his ideas about the object under study as an integral system, that is with the creation of an abstract model of the object. This is a first-order model based on his prior knowledge of the object, including observation, experiments, abstractions, assumptions, conjectures. To refine the model obtained, the scientist relies on the system private methods. The data obtained with their help and characterizing the object as an integral system, again suggest a subsequent synthesis, as a result of which an abstract model is obtained again, but of a higher order. The scientist goes to the theory as the final final model through a chain of models of the first, second, etc. orders (198, 8) (highlighted by me - A.K.).

In our opinion, this is not about the suppression of form by content. It is obvious that the authors assign the role of a tool for the theoretical synthesis of all ideas about the pedagogical system obtained by meaningful methods to the modeling method.

Listing the varieties of models, A.T. Kurakin and L.I. Novikova first of all name a conceptual model that reflects an object in the form of a certain set of interrelated assumptions, statements, and conclusions. The second they call a figurative model that reproduces the main aspects, elements, connections, relations of an object in the form of descriptions, photo and film models, graphs, diagrams. And only then they name the mathematical and physical model (198, 9). Such a logic of presentation and the characteristics that describe the types of models also indicate, in our opinion, an understanding of the priority of the content of the model over its form.

At the same time, criticism of the enthusiasm for the modeling method should, obviously, be recognized as fair, although not entirely addressed to A.T. Kurakin and L.I. Novikova. The fact is that many researchers understood the ratio of formal and substantive methods as a synonym for the ratio of theoretical and empirical methods. We found such an understanding in A.T. Kurakin and L.I. Novikova in describing the essence of the modeling method. We find the same thesis in T.V. Ilyasova: “the essence of pedagogical system research is to ensure theoretical synthesis(rather than a simple generalization of the material of a number of sciences) at the highest, pedagogical level. ... This is ensured through the establishment of the relationship of methods of cognition: in a systematic study, the usual division into empirical and theoretical methods is abandoned. The leading method is modeling as their dialectical unity” (132, 23) (highlighted by the author – A.K.).

In this execution, this thesis is also invulnerable to criticism of modeling as a method that is detrimental to the content side of the study. However, the very understanding of the opposition content-formal methods as a synonym for the opposition empirical-theoretical is dangerous, since then the assignment of all content methods to the group of empirical, that is, methods of a lower level of cognition, and all formal methods to the group of theoretical ones is legitimized. As a result, theoretical science becomes the ideal for the development of a scientific discipline, which is understood as a science that has an arsenal of formalization tools and contains formalized knowledge, in full accordance with the classical scientist standards and norms of cognition. In this logic, the fear of methodologists is completely justified, seeing in the expanding use of formal methods (cybernetic, mathematical, modeling) a threat to the scientization of pedagogical science.

In the 70s, many works were devoted to the problems of modeling in domestic pedagogy, a significant part of the authors of which connected the modeling method with certain stages of the system research program (S.I. Arkhangelsky, T.V. Ilyasova, E.G. Kostyashkin, V.V. .Kraevsky, V.P.Mizintsev, V.I.Mikheev, O.Yu.Ovakimyan, L.G.Turbovich, A.Yu.Uvarov, A.A.Chentsov and others). Summarizing from this point of view the accumulated experience of using and understanding the modeling method, it can be found that, depending on the objectives of the study and understanding of the essence modeling method it could be used for

• parametric description of the system - modeling the initial level of the system, based on empirical observations and available knowledge about the object;
• morphological description of the system - modeling of the elemental composition, relationships, properties, features of the object;
• functional description of the system - modeling the dependencies between the parameters of the system or its parts based on its characterization as part of the metasystem;
• study of the behavior of the system - modeling of the operating modes of the system, the processes of its development, system management;
• constructive and technological description of the system - modeling the desired image of the system as a technological prescription for practice.

Modeling was thus applied at all levels of systematic research of the pedagogical object - from ontological to praxeological. It should be noted that there is a certain dependence of the degree of formalization of the model on the level of its use. The least formalized are, as a rule, models of the ontological and praxeological levels, or systems of models that reflect the description of the pedagogical object from the ontological to the praxeological level, or models that seek to synthesize various ideas about the object into a complete picture (works by V.V. Zagvyazinsky, A.T. .Kurakin, L.I. Novikova, E.G. Kostyashkin, V.V. Kraevsky, R.S. Shaduri and others (116; 197; 182; 198; 374)). At the same time, the degree of formalization at the epistemological and methodological levels of system research is higher, especially in cases where the models reflect one aspect of the system under study (works by V.P. Bespalko, V.P. Mizintsev, Yu.O. Ovakimyan, A.Yu. Uvarova, A.A. Chentsova and others (33; 34; 35; 36; 236; 237; 238; 248; 353; 370)). The modeling method, especially in situations of constructing formalized models of pedagogical systems, was used in close connection with the methods of mathematics and cybernetics, however, unlike these methods, it was potentially more focused on reflecting an integral object, on synthesizing knowledge about it, than on fixing some kind of his side.

Methodological reflection of modeling experience in pedagogy allowed researchers to form certain methodological norms for the application of the modeling method in pedagogical system research.

1. First of all, the main norm is the recognition that the model is the result of schematization, however, the degree of this schematization depends on the general intention and goals of the analysis, on the expected completeness and accuracy of the solution (248, 4).
2. An expediently constructed model should clearly reflect the most significant features of the phenomenon; secondary details are not reproduced by the model (182, 43; 248, 4).
3. When modeling, the situation is deliberately simplified for research purposes; there are no models without scientifically defined simplifications. At the same time, too far-reaching simplifications can interfere with the mastery of the object, and the rejection of simplifications can make it difficult to understand (248, 5; 370, 7; 374, 7).
4. Simplification and schematization make it possible to use the methods of mathematics and cybernetics in modeling (240, 16; 263, 10-12; 353, 8).
5. Determination of the measure of acceptable simplification; retention in the mind of the researcher of the fact that simplification is only a special method of theoretical research, and its result - a model - is only a partial image of a complex real pedagogical object; "revival" of the model in practice is possible only with the use of meaningful methods of pedagogy and related humanities (132, 24; 182, 43; 197, 225).
6. The methods of modeling, mathematics and cybernetics in pedagogical system research are of a particular, auxiliary nature, since only certain aspects of the pedagogical object are known by their means, some of the problems of interest to pedagogy are solved with greater rigor, but cannot replace their own methods of pedagogy (96, 94; 116 , 34; 170, 369; 182, 43).

Thus, building a special methodology at each stage of a systematic study turned out to be more or less connected with the use of formalization methods, the fundamental basis of which, by definition, was reduction. To a large extent, this was facilitated by the epistemological nature of the systemic approach itself, based, as an "engineering" style of thinking, on reduction.

The combination of substantive and formal methods in the methodology of systematic research, according to methodologists, should have compensated for the reduction essence of the systematic approach with the maximum use of its heuristic potential. But the "measure" of this combination, its cognitive forms, each researcher determined himself. On this occasion, in the period under review, there was the only methodological prescription that urgently required to take into account the specific nature of pedagogical objects (M.A. Danilov, V.I. Zagvyazinsky, F.F. Korolev, V.V. Kraevsky, etc.).

The implementation of this prescription and, in general, the scientistic or humanitarian orientation of the systemic research of the pedagogical object was determined by the “individual scientific picture of the world” (V.I. Drygin) of researchers, the pedagogical paradigm, within the limits of scientific norms and cognitive standards of which the methodology of a specific systemic research was formed. Those scientists who managed to carry out methodological activities within the framework of the humanitarian paradigm, without compromising the specifics of the subject, correlating research manipulations (means) with the nature and internal laws of the humanitarian system, the values ​​of humanitarian culture (content) were able, with the help of a new cognitive means, to obtain new knowledge, more fully revealing the complex nature of the integrity of pedagogical systems (Y.K. Babansky, M.A. Danilov, V.I. Zagvyazinsky, F.F. Korolev, V.V. Kraevsky, A.T. Kurakin, L.I. Novikova, A .M.Sidorkin and others).

Those researchers whose cognitive standards and values ​​lay within the limits of the natural-science paradigm built a systematic research methodology in which the logic of means subordinated the logic of content to itself. The products of the implementation of such a research methodology were schematized formal representations of pedagogical phenomena and processes and the corresponding formalized prescriptions for pedagogical practice (V.P. Bespalko, L.V. Bespalko, G.N. Zubenko, Yu.K. Lazichnaya, A.Yu. Uvarov, A.A. Chentsov and others).

Comparison of these two lists reveals another dependency. The first group includes methodologists or theorists who have devoted a significant part of their work to the development of methodological issues. In the second - theorists and researchers of applied issues of pedagogy. The period we are considering in the history of Russian pedagogy (the end of the 60s - 80s) chronologically coincides with the stage of the beginning of the conceptualization of the national methodology of pedagogy, according to the periodization of S.I. Koltash (160, 7-8). Since the process of forming the main provisions of the national methodology of pedagogy continued, a certain gap between the theory and methodology of pedagogy persisted, although gradually became smaller. Far from all the provisions of the national methodology of pedagogy were accepted and assimilated by scientists who developed theoretical and applied problems of pedagogy.

The consequences of the conceptual design of the methodology of pedagogy was the growth of its status as an independent discipline, the growth of the methodological culture of teacher-researchers. The methodological activity of researchers, that is, the activity of forming a research methodology, began to turn into methodological activity associated with a deep preliminary development of the methodological foundations of the research (and not just with the selection of appropriate quotations from the classics of Marxism-Leninism, which, with minimal dexterity, could justify everything anything) and building on this basis the methodology of a particular study.

Thus, the relationship between the breadth and degree of development of the methodological foundations of the study, on the one hand, and the humanitarian orientation of the systematic research methodology, a measure in combination of formal and meaningful research methods, on the other hand, is obvious.

At the same time, the development of a systematic research methodology based on the search for ways and forms of combining meaningful and formal methods, and, in fact, humanitarian and natural science disciplinary standards, can also be regarded as a response of pedagogy to postmodern trends in culture associated with the destruction of the "Berlin wall between two kinds of unified knowledge” (290, 55). Already in the 70s, the pedagogical discourse reflected the tendencies of the oncoming movement: the "scientists" tried to assimilate the ideas of the integrity and complexity of the human personality, its self-development, the value orientation of pedagogical science and practice, etc., and the "humanists", accepting the idea of ​​the expediency of the pedagogical process , learn to set goals more clearly, look for stricter criteria for pedagogical activity and means of fixing, etc. Thus, it is necessary to recognize a certain role of a systematic approach in the emergence in pedagogy of that general cultural trend, which, according to V.V. Savchuk, consists in the complete confiscation of all “tools” - both natural science and humanitarian ones - for general scientific use (290, 55).

The method of system-pedagogical research, connected with the determination of an expedient combination of formal and substantive methods, one way or another, lined up for the implementation of all stages of the system research program. But, both in philosophy and in pedagogy, there was such an understanding of the essence of the systematic approach, which reduced it to the analysis of one of the aspects of the object, its essential, in the opinion of the researcher, defining aspect (314, 55). The implementation of such an understanding of the essence of the systemic approach involved the development of a methodology that would make it possible to carry out in detail the individual stages of the systemological program.

Then a systematic study of the pedagogical object was built with an emphasis on the study of its essential side, and often in general was reduced only to the study of this side (goal, part, structure, function, etc.). Accordingly, from the general program of the systemic approach, then either only one stage was implemented, involving the study of this aspect of the pedagogical system, or all stages of the systemic study were associated with the consideration of only a certain aspect of the system. Since the intention to study one side of the object is associated with determining the angle of view of the object, such a deliberate orientation of the study acquired the status of an approach. This approach was secondary to the system approach, so its name, as a rule, was formed by specifying

• on the aspect of system research (system-structural (32; 252; 371), system-morphological (99) or element-by-element (37), system-functional (345), system-historical (243), etc.);
• the leading method used in the study (system-cybernetic (98), system-target (203), program-target (401));
• on the approach with which the system approach program is integrated (system-activity (295), system-thought-activity (386), system-personality (300), system-optimization (107)).

The methodology of such a study consisted of certain stages of the system program and methods that make it possible to investigate the intended aspect of the pedagogical object.

A special orientation towards the study of one side of the object, even in its systemic connections, necessarily implies a deliberate refusal to consider other sides of the object. That is, this type of system research is initially partial. A number of educators-researchers were aware of the partial "cut" of the integral system obtained by applying this approach, justifying the need for its application by the requirements of the epistemological situation and the objectives of the study, and keeping in mind the possibility and necessity of other analytical planes (99; 130; 345, etc. .).

But quite often, especially in applied and particular-theoretical works, there was an identification of a certain aspect of system research with the system approach as a whole (37; 203; 348, 401, etc.). In this case, the systemic approach turned into its cognitive opposite - elementarism, reflecting the processes of degradation of systemic research to the historically preceding style of thinking - fragmentalism (according to M.S. Kagan's periodization).

The main reason for the fragmentation of research that claims to be systemic is, in our opinion, the insufficient methodological training of researchers, both in terms of the methodology of pedagogy and in questions of the methodology of the systematic approach. It is characteristic that in most of the works that we have classified as “quasi-systemic”, among the sources of ideas about the methodology of the systems approach, there are rarely combinations of a capacious list of solid general systemological works and works on the methodology of the systems approach in pedagogy. As a rule, there are indications of individual works, and not leading systemologists, but their interpreters.

Acquaintance with the systematic approach from "secondary" sources, from "retelling" provided a reduction in ideas about the systematic approach. Since the systems approach was originally based on cognitive reduction, its "simplified" version gave researchers the tool of double reduction. If, at the same time, to implement the chosen aspect of systemic research, a methodology was built based on formalization methods, then the result of the knowledge of the pedagogical object was a "reduction in the cube" - an abstraction that is extremely far from the real object. This situation clearly illustrates one of the paradoxes of rational cognition, the roots of which, according to E.L. Chertkova, “are in the predominance of technical and instrumental tasks over the problems of holistic knowledge of the truth” (quoted from 247, 185).

A significant number of teachers-systemologists, fearing "losing the whole in one of the aspects" (352, 261), turned to the authentic meaning of the systems approach - considering the object holistically, from all sides, combining all the different ideas about the object into a holistic picture. Let's conventionally designate the so understood systemic approach as "holistic" in contrast to its previous version - "partial" systemic approach. Such an understanding of the essence of a systematic approach required the formation of an adequate methodology for systematic research, a methodology associated with obtaining a holistic view of the pedagogical system.

As part of the development of this methodology for a “holistic” systematic approach, two directions have emerged that differ in their tasks:

• methodology of theoretical systemic research focused on obtaining holistic knowledge about a holistic object;
• a method of praxeological systematic research focused on theorization and methodologisation of pedagogical practice with the aim of a holistic transformation of a real holistic pedagogical object.

In theoretical studies, of all methodological searches, two striking trends can be distinguished: the development of a system-holistic approach and complex studies.

The prototype of the system-holistic approach was born in Russian pedagogy in the 1960s in connection with the development of the problem of a holistic pedagogical process (188). As an approach, these ideas were formalized mainly in the works of the Volgograd problem group of the methodology sector of pedagogy of the Research Institute of OP APS of the USSR in the late 70s - 80s during the development of the theory of a holistic educational process (V.S. Ilyin, A.M. .Saranov, N.K. Sergeev, V.V. Serikov and others). The system-holistic approach forces the researcher to focus on integrity as the main quality of the object under study, on the nature of this integrity, its manifestations - the integral properties of the system, development, levels, mechanisms of its formation (130; 217; 242; 295). Thus, the entire program of system research is oriented, each of its stages: composition, structure, functions, etc. are of interest to the researcher precisely from the point of view of their contribution to the formation of the integrity of a given object.

For example, when studying personality as the goal of a holistic educational process, scientists are not engaged in “dividing” it into numerous components and presenting it as a model - the sum of individual qualities and properties, but are trying to consider it as a whole from the point of view of core, integrative properties that are system-forming in her organization. According to researchers, the identification of these properties and the focus of the pedagogical process on their development will provide, according to the laws of system formation, the education of a comprehensively developed and harmonious personality (130, 41). It is this understanding of the structure of personality that today is the leading one in humanitarian disciplinary norms. That is, if we consider the current state of pedagogical science to be higher compared to the previous period, then we can highly appreciate the heuristics of the system-holistic approach.

Another direction in the development of a methodology for a “holistic” systematic approach can be considered the organization of complex studies - studies in which the same pedagogical object is studied from different points of view (88). Moreover, this definition also corresponds to research conducted within the same science and integrating data from different sciences on its basis (30; 83; 151); and studies whose topics are in a relationship of continuity (327, 26); and studies in which the object is examined by specialists from different fields with specific methods (88; 170; 197). In the pedagogical literature of the period under review, all the semantic shades of this concept are found, however, most often, in connection with a systematic approach, it was customary to call complex studies specially organized studies of pedagogical objects conducted by representatives of various scientific disciplines within the framework of specific cognitive norms.

Already in the first works on a systematic approach in pedagogy, which emphasized its “holistic” essence, its interdisciplinary, complex nature, the idea was put forward that only the organization of integrated research would allow realizing a genuine systematic approach to pedagogical processes and phenomena (F.F. Korolev , A.T. Kurakin, L.I. Novikova).

The President of the Academy of Pedagogical Sciences of the USSR V.N. Stoletov in 1976 stated that “all the problems of pedagogical science are complex problems. Any subtle analysis of a pedagogical problem must be completed with a synthesis. Therefore, the problems of pedagogical science can be solved only through complex research” (328, 9).

Especially attention to complex research increased in the first half of the 80s due to the combination of intrascientific trends (return to the authentic essence of the system approach, ideas about the polystructural and polysystem nature of pedagogical systems, the desire to synthesize the results of private studies, understanding the level of complexity of pedagogical objects, etc. .) and external socio-cultural conditions (growth of integrative tendencies in science, party guidelines requiring the efforts of “big science to be concentrated on discoveries capable of making truly revolutionary changes in production” (61, 57), etc.).

If we are talking about the formation of a methodology for complex research based on a systematic approach, then two aspects can be considered - a methodology for organizing a comprehensive research based on a systematic approach, and a methodology for individual disciplinary research within the framework of a comprehensive one, which can also be based on a systematic approach.

In the works on the methodology of complex research during the period of increased interest in them, it was mainly about the organization of these studies. In such a situation, the instrumental-organizational function of the systems approach was actualized, since it was on its basis that the logical, methodological and organizational problems of a comprehensive study were solved in each specific case:

• formulation of a complex problem and formation of the goals of complex research work;
• determination of a set of scientific disciplines, representatives of which should be included in a comprehensive research work;
• questions of organization of scientific teams;
• planning of complex scientific research works and their management;
• specific methods of activity of the scientific team carrying out complex research work;
• registration of the results of the latter (88, 125; 230; 316).

Thus, the opportunities to obtain holistic knowledge about a particular pedagogical object using a systematic approach were associated mainly with the implementation of a system-holistic approach in individual research and the organization of complex systemic research of pedagogical objects.

Pedagogical science, the main specific feature of which is the focus on improving practice (329, 9), traditionally solves the problem of introducing the results of theoretical research into pedagogical reality in order to increase the efficiency of pedagogical processes and develop pedagogical systems. Therefore, without exception, all pedagogical research has a praxeological part containing recommendations for evidence-based change in practice.

Between the theoretical conclusions of systematic studies of pedagogical objects, carried out, due to the cognitive properties of the systematic approach, its program and methodology, at a sufficiently high level of abstraction, and the possibility of their practical implementation, there is a certain barrier, overcoming which should be specially instrumented.

The question of using the results of systematic research in pedagogical practice with the aim of its systemic transformation is essentially the question of ontologizing ideal schemes of pedagogical phenomena and processes built on the basis of a systematic approach. The problem of ontologization of ideal schemes, theoretical constructions is one of the most important problems of the philosophy of knowledge, in which principles are developed that normalize the connection of the theorist to the improvement of practice: the principles of conventionality, tolerance, apoliticality, anti-activism, humanism (129, 155). the theoretician who turns to practice lies in the fact that “any scientific activity has a substantive, one-sided character and therefore the best, developed subject knowledge remains inconsistent with the practice of activity. They involve the synthesis of all subject knowledge before their application in practice” (13, 181), according to O.S. Anisimov.

As V.V. Ilyin notes, “a reflexive position regarding the non-union of the real-historical and socio-theoretical series and mode allows us to warn against the characteristic ontologization of abstract-theoretical schemes in the spirit of naive realism” (129, 154).

During the period under review, not all researchers were able to take such a reflexive position, which resulted in projections of abstract structures generated as a result of a systematic study of a particular pedagogical object onto this very object. Examples can be "levels of assimilation" by V.P. Bespalko, which became a tool for monitoring the activities of a teacher; "personality models" of V.P. Bespalko, which became the basis for the design of pedagogical systems and their management; "types of education" by I.S. Maryenko, which are the basis for planning and organizing educational work "according to directions"; “decomposition of moral categories” by Yu.K. Lazichnaya as the basis of the coded analysis of the lesson and much more. Moreover, it is not always possible to accuse the researcher of “naive realism”, of the form his constructions acquired in practice, since often the practical “fine-tuning” of his recommendations was carried out by methodologists and teachers, interpreting them to the best of their own understanding. At the same time, the need for "finishing" was apparently caused by the lack of praxeological interpretation of the research results in the author's performance.

Interest in the results of theoretical research and their intensive “implementation” into practice “without translation” is also the result of a combination of many scientific and sociocultural factors: the party’s course to increase the “scientific” nature of practice, to strengthen the practical significance of theoretical research; the “applied” nature of pedagogy, which still has very little experience of realizing itself as a theoretical science and, accordingly, does not reflect deeply on the problem of ontologization of theoretical constructions; the general high authority of the systematic approach, which reduces the barrier of criticism in the perception of the results of a systematic study; insufficient methodological culture of researchers and practitioners; finally, a purely psychological factor - "operating with constructions gives the impression of manipulating natural relations, but this is an illusion" (129, 155), the illusion of conquering the real world by the power of science.

Nevertheless, in the period under review, it is already possible to find studies related to the reflection of the relationship between the theory and practice of system research, understanding the problem of ontologization of abstract schemes generated as a result of applying a systematic approach to pedagogical objects.

So, B.S. Gershunsky bred the concepts of "integrity", "systematic" and "complexity" according to the levels of methodological knowledge. He believed that "integrity" is a philosophical category, denoting the general orientation of research thinking; "systemic" - a general scientific category, implying the construction of a special research program, instrumental to a "holistic" view of the object; and “complexity” is a practical category, denoting the way of working with the studied object as with a system (81, 19-20).

In this logic, in domestic pedagogy, directions of practice-oriented systemic research arose, specifically building the theory of systemic practice: in the late 70s - an integrated approach (M.V. Kabatchenko, O.G. Panchenko, M.M. Potashnik, E. F. Sulimov, G. N. Filonov and others (254; 260; 343; 364)), in the early 80s - optimization approaches (Yu.K. Babansky, M. M. Potashnik and others (25; 249; 259; 265)), in the first half of the 80s - the concept of educational systems (A.T. Kurakin, L.I. Novikova, V.A. Karakovsky and others (73; 148; 199; 245; 246 ; 312)), at the end of the 80s (as a theoretical understanding of the practice of the late 70s - 80s) - the concept of social and pedagogical (V.D. Semenov (297; 298)) and educational complexes (B. Z. Vulfov, M. M. Plotkin, V. I. Shirinsky (76; 381)).

The first attempt at a special thinking of the ways of ontologizing the products of systematic research in pedagogical practice was the design of an integrated approach. The development of an integrated approach began in 1977 by the problem group of the Research Institute of General Problems of Education of the Academy of Pedagogical Education of the USSR under the leadership of G.N. Filonov and I.S. Maryenko. One of the theoretical issues that scientists solved in the course of developing an integrated approach was the question of its relationship with the system approach. It was determined that “complexity is the practical implementation of a systematic approach in research work” (H.J. Liimets (329, 11)); “It is legitimate to consider a systematic approach as a strategy of education, and a comprehensive approach as its tactics” (M.V. Kabatchenko, M.M. Potashnik (260, 17)); “The systemic approach makes it possible to make a cut of the internal dynamic state of the object of education, to deeply and comprehensively trace the qualitative changes in its elements, to fix the nature of the interaction of subjects, to reveal the causes and mechanism of emerging contradictions…; an integrated approach, being the principle of communist education, determines the content and direction of the activity of the subject of the educational process, allows you to realize the goal of comprehensive and harmonious development for all members of society ”(G.N. Filonov (364, 189)).

Having answered the question about the relationship between the integrated approach and the system approach, the question of the relationship with the system approach of other concepts - optimization, educational systems, socio-pedagogical and educational complexes, and others could no longer be addressed. According to the principle of isomorphism, the relationship was the same: a systematic approach is a tool for understanding pedagogical systems, and these concepts are tools for the practical application of a systematic approach in pedagogical reality.

^ Common features of practice-oriented research methodology there was an understanding of the incongruence of real pedagogical objects and their theoretical images; the requirement to rely on specific socio-pedagogical conditions for the functioning of the system, and not only on the general parameters of the education system when designing a specific pedagogical object; a combination of science, practical experience and common sense, rational and irrational in the design of pedagogical systems and the implementation of projects: focus on the evolutionary path of changing the real pedagogical system, on the “cultivation”, and not the “implementation” of innovations; the idea of ​​integrating the educational efforts of the subjects of the pedagogical process, etc.

Thus, a generalizing analysis of the activities of teachers-researchers in building a methodology for systematic research of a particular pedagogical object allows us to assert that the methodological activity in the course of the system-pedagogical search was aimed at clarifying and interpreting the general program of pedagogical system research.

A generalization of the methodological experience of teachers of the period under review in this area makes it possible to distinguish four types of methodology for systemic pedagogical research: a methodology that repeats, but does not specify the “steps” of the program; methodology associated with the interpretation of "other-scientific" methods and the synthesis of formal and meaningful methods for the implementation of program stages; a methodology that allows you to implement in detail the individual stages of the systemological program; a technique associated with attempts to synthesize all ideas about the system into a complete picture.

The orientation of methodological activity and the choice of methodology for systemic pedagogical research, as we found out, were associated with different ideas about the essence of the systemic approach and its cognitive program; with the scientific and cognitive paradigm, within which the research orientations and scientific standards of the scientist were formed; with the level of methodological culture of the researcher, with the objectives of a particular study.

The historical reconstruction of the genesis of the systematic approach in Russian pedagogy in the late 60s - 80s of the twentieth century makes it possible to make sure that the general scientific systematic approach was not mechanically transferred to pedagogical soil and was further developed in pedagogical research of the period under review both at the level of the methodological program, and at the level of methodology for implementing this program.

The systemological searches of teachers of the period under review were not theoretically comprehended and generalized, which makes it difficult to use their results in modern systemic pedagogical research. A generalizing analysis of a large array of materials, reflecting the systemological experience of domestic teachers of the period under study, allows us to present this experience in the form of a methodology model of the pedagogical system approach (see Table 2) .

table 2

^ Methodology Model of the Pedagogical Systems Approach

Epistemological level of system-pedagogical research	The act of cognitive activity	Cognitive procedures and methods
Ontological	Isolation of a pedagogical object, its presentation as an organic whole, holistically interacting with the environment	Definition of the concept of "pedagogical system", its qualitative specifics, allocation of a class of pedagogical systems among other systems, classification of pedagogical systems
Epistemological	Occupation of a special research position, which allows to single out the pedagogical system in the pedagogical object as a subject of knowledge	Defining the principles of studying and describing pedagogical systems, understanding the polysystemic, polystructural nature of pedagogical systems, clarifying the categories of the systemic approach in relation to the field of pedagogy, understanding the relativity and limitations of the systemic description of the pedagogical object, the limits of applicability of the systemic approach to pedagogical reality
Methodological	a) the formation of a methodological program as an invariant algorithm - procedures for a systematic study of a pedagogical object	Procedures related to obtaining a static "cut" of the system, its "anatomy": 1.1. representation of the system as part of a metasystem (the choice of a metasystem is associated with the definition of a system-forming factor); description of the composition of the system, that is, the allocation of relatively autonomous material elements, necessary and sufficient for the formation of a system, not having, but potentially capable of forming an integrative systemic quality an indication of the angle of view from which the system will be considered (associated with the definition of a system-forming relationship) Description from this angle subsystem system structure: allocation of subsystems as the smallest components of the system, preserving the integrative systemic quality
		2. Procedures related to obtaining a dynamic description of the system, its "physiology": functional, genetic, managerial, predictive representation of the system (these procedures can only be implemented if special methods are used, that is, building a systematic research methodology - see paragraph b)
	b) building a systematic research methodology as a set of special methods, techniques and tools that allow you to explore the pedagogical object as a system: Technique that repeats the program, without the use of special methods	Reformulation of known knowledge in systemic terms, empirical description of the pedagogical object as an integral set of elements, empirical systematization of existing knowledge about the object
	Methodology associated with the interpretation of foreign scientific methods and the synthesis of meaningful and formal methods	Using the methods of mathematics, cybernetics, modeling, etc. as private, auxiliary, subject to pedagogical interpretation of the results of their application using meaningful methods
	A technique that allows to carry out in detail the individual stages of a systemological program	System-structural, system-morphological, system-functional, system-historical, system-cybernetic, program-target, system-target, system-activity, system-thought-activity, system-personality, system-optimization, etc.
	A technique related to obtaining holistic knowledge about a holistic object	System-holistic approach, complex research, etc.
Praxeological	Methodology focused on building a "methodology of practice" with the aim of holistic transformation of a real holistic pedagogical object	Procedures for "translating" ideal system schemes and models, determining the conditions for their practical application, which allow avoiding naive realism (an integrated approach, an optimization approach, the concepts of educational systems, socio-pedagogical and educational-methodical complexes, etc.)

The methodology model of the pedagogical system approach was developed by us on the basis of an analysis of the systemological experience of teachers of the period under review and correlates both with the model of the general scientific system approach and with the model of modern methodology of pedagogy. The study of modern searches in the field of developing a methodology for a systematic approach will clarify and develop this model, however, within the framework of the objectives of this study, it holistically reflects the results of the efforts of domestic teachers of the late 60s - 80s of the twentieth century to form the logic, methodological program and methodology of systemic pedagogical research

The use of mathematical methods in the field of management is the most important direction in improving management systems. Mathematical methods speed up economic analysis, contribute to a more complete account of the influence of factors on performance, improve the accuracy of calculations. The application of mathematical methods requires:

• a systematic approach to the study of a given object, taking into account the relationships and relationships with other objects (enterprises, firms);
• development of mathematical models that reflect the quantitative indicators of the systemic activity of the employees of the organization, the processes occurring in complex systems, which are enterprises;
• improvement of the information support system for enterprise management using electronic computers.

Solving problems of economic analysis by mathematical methods is possible if they are formulated mathematically, i.e. real economic relationships and dependencies are expressed using mathematical analysis. This necessitates the development of mathematical models.

In management practice, various methods are used to solve economic problems. For example, in network planning and management, various mathematical methods are used, and many authors put different content into the meaning of the term "operations research".

The methods of elementary mathematics are used in traditional economic calculations when substantiating the need for resources, developing a plan, projects, etc.

Classical methods of mathematical analysis are used independently (differentiation and integration) and within the framework of other methods (mathematical statistics, mathematical programming).

Statistical Methods- the main means of studying mass recurring phenomena. They are used when it is possible to represent changes in the analyzed indicators as a random process. If the relationship between the analyzed characteristics is not deterministic, but stochastic, then statistical and probabilistic methods become practically the only research tool. In economic analysis, the methods of multiple and paired correlation analysis are best known.

To study simultaneous statistical aggregates, the law of distribution, the variational series, and the sampling method are used. For multidimensional statistical aggregates, correlations, regressions, dispersion, covariance, spectral, component, factorial types of analysis are used.

Economic Methods are based on the synthesis of three areas of knowledge: economics, mathematics and statistics. The basis of econometrics is an economic model, i.e. a schematic representation of an economic phenomenon or processes, a reflection of their characteristic features with the help of scientific abstraction. The most common method of economic analysis is "input-output". The method represents matrix (balance) models built according to a chess scheme and clearly illustrating the relationship between costs and production results.

Methods of mathematical programming- the main means of solving problems of optimization of production and economic activities. In fact, the methods are means of planned calculations, and they make it possible to assess the intensity of planned targets, the scarcity of results, to determine the limiting types of raw materials, groups of equipment.

Under operations research refers to the development of methods of purposeful actions (operations), the quantitative evaluation of solutions and the choice of the best of them. The goal of operations research is the combination of structural interrelated elements of the system that provides the best economic indicator to the greatest extent.

Game theory as a section of operations research, it is a theory of mathematical models for making optimal decisions under conditions of uncertainty or conflict of several parties with different interests.

Queuing Theory on the basis of probability theory, explores mathematical methods for quantifying queuing processes. A feature of all tasks related to queuing is the random nature of the phenomena under study. The number of requests for service and the time intervals between their receipts are random in nature, but in the aggregate they obey statistical patterns, the quantitative study of which is the subject of queuing theory.

Economic cybernetics analyzes economic phenomena and processes as complex systems from the point of view of the laws of control and the movement of information in them. Methods of modeling and system analysis are most developed in this area.

The application of mathematical methods in economic analysis is based on the methodology of economic and mathematical modeling of economic processes and scientifically substantiated classification of methods and tasks of analysis. All economic and mathematical methods (tasks) are divided into two groups: optimization solutions according to a given criterion and non-optimization ones (solutions without an optimality criterion).

On the basis of obtaining an exact solution, all mathematical methods are divided into exact (according to the criterion or without it, a unique solution is obtained) and approximate (based on stochastic information).

Optimal exact methods include methods of the theory of optimal processes, some methods of mathematical programming and methods of operations research, optimization approximations - part of the methods of mathematical programming, operations research, economic cybernetics, heuristic.

TO non-optimization accurate belong to the methods of elementary mathematics and the classical methods of mathematical analysis, economic methods, to non-optimization approximate- method of statistical tests and other methods of mathematical statistics.

Particularly often used are mathematical models of queues and inventory management. For example, the theory of queues is based on the one developed by scientists A.N. Kolmogorov and A.L. Khanchin queuing theory.

Theory of queuing. This theory allows you to study systems designed to serve a massive flow of requirements of a random nature. Random can be both the moments of the appearance of requirements and the time spent on their maintenance. The purpose of the methods of the theory is to find a reasonable organization of service that ensures its given quality, to determine the optimal (from the point of view of the accepted criterion) standards of duty service, the need for which arises unplanned, irregularly.

Using the method of mathematical modeling, it is possible to determine, for example, the optimal number of automatically operating machines that can be serviced by one worker or a team of workers, etc.

A typical example of objects of the theory of queuing can serve as automatic telephone exchanges - automatic telephone exchanges. The PBX randomly receives “requirements” - calls from subscribers, and “service” consists in connecting subscribers to other subscribers, maintaining communication during a conversation, etc. The problems of the theory, formulated mathematically, are usually reduced to the study of a special type of random processes.

Based on the given probabilistic characteristics of the incoming call flow and service duration, and taking into account the scheme of the service system, the theory determines the corresponding characteristics of the quality of service (failure probability, average waiting time for the start of service, etc.).

Mathematical models of numerous problems of technical and economic content are also problems of linear programming. Linear programming is a discipline devoted to the theory and methods for solving problems of extrema of linear functions on sets defined by systems of linear equalities and inequalities.

The task of planning the work of the enterprise. For the production of homogeneous products, it is necessary to spend various production factors - raw materials, labor, machine park, fuel, transport, etc. Usually there are several proven technological methods of production, and in these methods the costs of production factors per unit of time for the release of products are different.

The number of consumed production factors and the number of manufactured products depends on how long the enterprise will work according to one or another technological method.

The task is to rationally distribute the time of the enterprise's work according to various technological methods, i.e. the one at which the maximum number of products will be produced for a given limited cost of each production factor.

On the basis of the method of mathematical modeling in operational research, many important tasks are also solved that require specific methods of solution. These tasks include:

• product reliability;
• equipment replacement;
• resource allocation;
• pricing;
• resource allocation;
• as well as scheduling theory (the so-called scheduling theory).

The issue of resource allocation is one of the main ones in the process of production management. To address this issue, operational research uses the construction of a linear statistical model.

Pricing problem. For the enterprise, the issue of pricing for products plays an important role. How the pricing is carried out at the enterprise depends on its profit. In addition, in the current conditions of a market economy, price has become an essential factor in the competitive struggle.

Theory of network planning. Network planning and management is a planning system for managing the development of large economic complexes, design and technological preparation for the production of new types.