Eternal truths - Pedagogy. Theoretical pedagogy

Eternal Truths

Each science begins its development with the establishment of axioms. More precisely, in the analysis of the process of the historical development of science, there are found provisions on which all are based and which can be recognized as axiomatic, therefore axioms are sometimes called eternal truths.

Most of us know and will not challenge, for example, such eternal pedagogical truths:

- the school begins with the teacher;

- education is inseparable from the personality of the educator;

- first a teacher, then a science.

These truths have been known for a long time. Thousands of times they are repeated in different variations, and there is not a single teacher who would argue the opposite. Comenius and Rousseau, Pestalozzi and Ushinsky left us with unsurpassed evidence of the existence in pedagogy of some general provisions coming from the depths of centuries. Researchers rely on them as a foundation. Many of the basic provisions of education were known and practically used already when the science of pedagogy was not even in sight. How long have they been born? No one knows. It is only known that from ancient Greeks to our times teachers repeat them without changes.

Once, in more severe times, people could not afford a light-weight attitude to knowledge. Ignorance, ignorance of knowledge meant one thing - perdition. Only the knowledge that has always been confirmed has survived. Together with them, people who rely on the truth survived. Truth experienced centuries, and half-truths, lies and errors were rejected sooner or later.

All sciences, how many there are in the world, have the same passport, without which they consider themselves unthinkable: the pursuit of truth! Each of them, even some pharmacognosy, has as its goal not benefit, not convenience in life, but the truth - wrote A. P. Chekhov. The law of motion to the truth is simple: how many villages do not swindle, and on the road you have to leave. To the desired goal of creating an effective technology of modern education will lead us to a path that is unmistakable at all times - the path of logic, objectivity, calculation. Entering it, it is extremely important to start moving in the right direction. The general audit of our knowledge, without which we can not do without creating modern educational technologies, begins with a comprehension of the beginnings that set the strategy for renewal.

When a breakthrough occurs in a new technology, the question inevitably arises on which theoretical foundation it was possible to obtain previously unavailable results. It is usually assumed that there must be new discoveries and few people know that no less revolutionary transformations can be achieved by rethinking and new applications of already known knowledge. Another angle of view is sometimes preferable to new knowledge. In pedagogy there are so many eternal and great, but forgotten or unclaimed knowledge that there is no sense in discovering new ones. Maybe it's time to take a closer look at what has long been known. Kozma Prutkov was right, urging him to look more often at the rear to avoid noble mistakes.

Science rests on great truths, for which there is no statute of limitations. The cornerstone of continuity of scientific knowledge is the principle that the deep essence of phenomena remains unchanged. Today, with us and around us, the same processes occur that have always occurred. B. Prus says this in his Pharaoh so: "How many dynasties and priests have been replaced in Egypt for three thousand years, how many cities and temples have turned into ruins, into which new layers of land have been layered! Everything has changed, except that twice two is four, that the triangle is half a rectangle, that the moon can close the sun, and boiling water throws the stone into the air. In the transitory world, only wisdom remains unchanged. "

Teaching knowledge comes from the depths of the centuries, first as an experience, later as theoretically purified generalizations, in our days - as irrefutably valid patterns and laws. All the combined knowledge is based on a set of self-evident truths - axioms. Axiom (from the Greek axioma - an indisputable truth that does not require proof) is true judgment, which, when constructing a theory, is taken without evidence as a starting point, and which is used as the basis for proving all other provisions of this theory . Axioms follow from practice. In the history of science, the idea that a man's practical activity was billions of times must lead a person's consciousness to repeating the same logical figures, so that these figures could get the meaning of axioms, is repeatedly stressed in the history of science. This means that axioms can be verified, and then formulated only as a result of a prolonged generalization, as a result of the development of cognition. On the one hand, they summarize what has been achieved, on the other hand they serve as the initial level of further knowledge.

The system of scientific knowledge can be constructed without axioms, but at a certain stage of its development, it naturally comes to the formulation of axiomatic propositions and presents a number of its initial principles axiomatically. There are, however, exceptions. There are theories for which axiomatic positions are artificially invented in advance. This applies primarily to various mathematical theories, the subject of study of which goes beyond the realities observed.

For a long time in science, the assertion was dominated by the belief that an axiom is a truth that can not be proved; it is accepted on faith & quot ;, all other provisions are derived from it. However, according to Academician VM Glushkov, this is only a certain convention accepted by everyone, which proceeds from the fact that everything that can not be proved within the framework of one system can be proved in the framework of another. Philosophers warn against the erroneous interpretation of the evidence of axioms. The term axiom was already applied by Aristotle (384-322 BC) as a true beginning, because of clarity and simplicity, not requiring proof. Subsequently, clarity and simplicity are mistakenly interpreted as evidence. "The dissatisfaction with such an axiom definition is," writes N. I. Kondakov, "that the requirement of" evidence "is subjective, since what one seems to be obvious to the other is not obvious."

On the basis of the axioms, all scientific theories have been built, with a few exceptions, and, first of all, mathematical ones. Each scientific theory is built from a finite number of axioms, of which, with the help of certain pre-determined rules, true conclusions can be obtained, formulated in the language of the theory. The scientific theory in this case consists of two parts: 1) the set of necessary true positions - axioms and 2) sets of true positions that are derived logically or otherwise from axioms.

Of course, axiomatic, like any other single method, is limited in its capabilities. "Axiomatics is imperfect, but the best way to form a theory at our disposal, therefore the rejection of axiomatics because of its limitations is analogous to the call to abandon the continuation of the human race because of the imperfection of its representatives," writes the famous Argentine scientist M. Bunge.

What are the pedagogical conclusions that follow from the above philosophical positions?

First. In the course of a long historical development of pedagogical practice and theory, a number of statements (statements) crystallized, the essence of which is comprehended in the depths of centuries and remains unchanged to the present day. Confirming again and again, these provisions have become self-evident truths that do not require new evidence, and must be accepted as axioms.

Second. Modern pedagogical theory is so strong that it can formulate axiomatic positions that open a new stage in the development of the theory.

Third. It is now clear that the construction of a full scientific pedagogical theory is possible only on the condition that some of its provisions will be introduced without evidence, by an axiomatic method. You can not endlessly prove what was self-evident already at the time of Plato and Aristotle.

Fourth. A superstructure develops on the foundation of axiomatic positions - a network of logical assertions that can be deduced by means of theoretical analysis, modeling, experiment, etc.


Fifth. If the system of axioms is chosen correctly, and if we apply the laws of logic correctly to it, then the results will correspond to reality.

In theoretical pedagogy, until recently, the term "axiom" was avoided, preferring others to it - experience, traditions, etc. Some attempts to formulate pedagogical axioms and represent part of the theory in an axiomatic form are noted. In fact, there is not a single fundamental work on the generalization and systematization of knowledge acquired by science, wherever the grounds common to all times and peoples are not confirmed. There is not only the word axiom & quot ;. Nobody denies the main pedagogical statements that have come down to us from the depths of the century, successfully polished in the writings of Plato, Aristotle, Quintilian, Comenius; everyone, analyzing them, can be convinced that they have become the cornerstones of later discoveries and new theories. Is this not a proof of the long-begun and fruitful axiomatization of pedagogy?

Axiomatic method in pedagogy

In order to axiomatize the content of any branch of knowledge, it is necessary to outline his main ideas in a certain orderly way. We will reflect on the method by which axioms are introduced into pedagogy. It is quite obvious that the process of developing axiomatic grounds can not be arbitrary, but must be scientifically justified in accordance with the logical-philosophical and pedagogical requirements. To develop and correctly apply this method is by no means an easy task. At first, one has to resort to borrowing the methods of developed sciences.

The axiomatic method is a method of constructing a section of science or a science in general, in which a subset of all the true assertions of a section (or science) from among these statements, is placed in the basis of the section as initial assumptions (axioms), from which then other logical assertions of this section or scientific theory are derived logically. The most important qualities of the axiomatic method are consistency, independence, and in some cases the completeness of the system of axioms created on the basis of this method.

The axiomatic method presents three main requirements to the order of construction of axiomatic theories for the creation of the axiomatic foundations of the scientific theory.

First, the axiomatic method requires a clear definition of the main concepts that will be used in the axiom system, and then theorems. More precisely, it is rather not even about concepts, but about terms used in the system of axiomatic constructions. These terms must be unique, precise, specific, denoted by all known and understandable language. They are introduced without special evidence and definitions.

Secondly, the use of the axiomatic method requires the proper construction of axioms from primary statements. The set of terms must be combined by logical relationships, and this union can not be uncomfortable, and that more arbitrary. The link is fixed uniquely, so that its interpretation can only be unambiguous.

Third, the application of the method requires the correct deduction from the axiom system of further effects. The latter appear by the introduction of more and more new and more complex objects on the basis of primary concepts and terms, using explicit definitions.

The above requirements of the axiomatic method are formulated to construct rigorous formal-mathematical theories and not all the way can be taken to construct a system of pedagogical axioms, but in the main, if pedagogy pretends to be a scientific theory, it should not deviate from them.

The importance of an axiomatic method for constructing a scientific theory can not be overestimated. It facilitates the organization and systematization of scientific knowledge, allows you to quickly identify the internal, logical connection between the individual sections of the theory, clearly isolates the initial positions and provisions derived from axioms, accustoms to the accuracy and rigor of reasoning. The undeniable advantage of the axiomatic method is that it is a most valuable tool for scientific research, finding new regularities, allows us to identify in the theory subjected to axiomatization those basic, guiding ideas that are often obscured by minor details and which, without this method, would not be easy to distinguish.

Is pedagogy ready to introduce an axiomatic method?

The first step in its application is the streamlining of supporting concepts and terms. Are they singled out? Yes, it is indisputable. The centuries-old practice and relatively young theory have quite firmly mastered the basic concepts and terms. They are relatively few. The following are common: education, training, education, development, formation, teaching, teaching, educational (educational, pedagogical) process, pupil, teacher, school, class, subject, intellectual development, spiritual development, physical development, activity, interest, exercise, system, method, form and some others. One can not yet state that all teachers understand the concepts denoted by terms unambiguously. Many of these concepts in themselves designate an extremely wide class of objects and, naturally, with a nominal enumeration of the latter, the original meaning of the concept may be lost or not in accordance with the context. For example, the term school defines all types and types of existing and existing schools. On the one hand, this is the power of a comprehensive character of concepts, on the other hand, the inconvenience that is enclosed in them creates great difficulties for the axiomatization of pedagogical theory. The way to overcome the contradiction is the use of supporting terms and the concepts they designate at a high level of abstraction, where terms can be used without reference to the meaning. The peculiarities of pedagogical terminology are such that it is very difficult to "cleanly" apply the requirements of the axiomatic method to the construction of a system of pedagogical axioms.

Concepts (terms) are assembled into axiomatic positions by means of links (bundles). The role of tangles in formal-mathematical axiomatic theories is played by signs, symbols. The most common signs are: & Igrave; - the sign of the inclusion of one set in another, the signs for the compounding of utterances (& Ugrave ;, & Uacute ;, - the conjunction, disjunction, implication, negation, the generality of the quantifier), signs for designating the logical connection between utterances, etc. All common notations can be used to formalize pedagogical theory. First, it is necessary to agree on the symbolic designation of primary terms and concepts. Then axioms pedagogy will be written in the form of formal logical expressions. For each section of pedagogy, special notation for concretization of applied concepts and expressions.

Formalization of axioms

Well-organized science strives to make its theory compact, observable, readable. This is achieved by coagulation information, presentation of part of it in symbolic, symbolic form. The need for the use of abbreviated symbolic signs is approaching and pedagogy. The sign with its sensual clarity facilitates logical operations, makes the process of thinking more productive. The formalization of the pedagogical theory is a necessary stage in the development of the knowledge system, which indicates its qualitative improvement.

The need for a reduced symbolic representation of a part of the pedagogical theory is strongly encouraged by attempts to electronically simulate pedagogical phenomena aimed at understanding the underlying laws of the educational process and finding more effective ways of managing it. Neither today, nor in the near future computers can not yet understand all the accounts of human thought, expressed in words. There is an urgent need to formalize statements in such a way that they carry as much information as possible, remaining compact and understandable not only for people, but also for machines. In this regard, the development of the rules of "coagulation" pedagogical knowledge and symbols for their recording is an important practical task. Axiomatic positions written down in a collapsed form are entered into the memory of the computer and constitute a bank (or database) of information. When deducing new conclusions, designing teaching and educational systems, the machine relies on a database, checking whether these new conclusions are inconsistent with the available scientific provisions.

Symbolism (from Greek symbolon - symbol) is a system of specially created characters (symbols) for designations of objects, thoughts, feelings, ideas. Symbolics allows in a reduced form to record various complex and long statements. It visually reveals the structure of the relationship. In addition, symbolism is a powerful means of international pedagogical cooperation.

Below is a system of symbols for the formalization of pedagogical utterances and a reduced record of interrelations (Table 8). Some of the notation is borrowed from the international language of mathematical and cybernetic pedagogy. An experienced test confirmed the expediency of a formalized stock of a part of pedagogical statements. The system finds application in the student's practice of processing large amounts of information, drawing up reference schemes (summaries).

Table 8

A system of symbols for the formalization of pedagogical statements and a reduced record of relationships



Value, example record

& Ugrave;

Means and & quot ;. For example, A & Ugrave; In - reads A and In or A takes place and B occurs.

& Uacute;

Means or & quot ;. For example, A & Uacute; In - read A or In


Means implies & quot ;. For example, A ® In - read A entails In .


Means not & quot ;. For example, -A - reads "not A or" it is not true that A

Means denying the whole statement. For example, the line above the A is read "not A & quot ;, and above the - reads is not a foundation


Means if and only if & quot ;. For example, a ~ A reads A appears if and only if and

Means for all & quot ;. For example, x - reads for all x


Means exists such & quot ;. For example, $ x - reads there is such a x

Means is the basis of & quot ;. For example, A In read A is the ® base In . Dash on top in the expression A In means denying the entire expression and reading A is not the foundation of In

Means be the main factor & quot ;. For example, a A read is the main factor for A . The dash on top in this expression gives the entire expression a negative.

® ®

Means accompanied & quot ;. For example, A ® In - reads the A is accompanied by a B A dash from above denies the whole statement. For example, A In - reads A is not accompanied by a phenomenon In

Means there is a reason & quot ;. For example, x y - reads x is the cause of y . The dash above denies the whole expression: x is not the cause of in

& Igrave;

Means contained & quot ;. For example, a & Igrave; A - read a is contained in A or and is included in A . The dash above denies the entire expression. For example, a` & Igrave; A reads a is not included in A


Means be single & quot ;. For example, A ®or In - reads A is with In a single entity or A is organically linked to In


Means achieved by & quot ;. For example, A *** In - read A is achieved through In or the A path is through In

The system of conditional abbreviations for writing some pedagogical concepts can be, for example, the following: В - education; P - development; Rd - spiritual development; РF - physical development; U - the pupil; Uch is a teacher; O - learning; Ш - school; Un - exercise; M - method; And - interest; F - life, practice; Z - knowledge; Zс - knowledge systematized (education); P is a habit, understanding; C - Ability; D - activity; F - form; St - system; Msh - thinking, etc.

Symbolic means and symbols are developed in accordance with the capabilities of electronic computers, therefore the designations may be different, it is important to observe the logic of the links, which must always remain unchanged.

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