On the meaning and content of school mathematical education (methodological analysis)

Rozin Vadim Markovich Doctor of Philosophy Chief Scientific Associate, Institute of Philosophy of the Russian Academy of Sciences  109240, Russia, Moskovskaya oblast', g. Moscow, ul. Goncharnaya, 12 str.1, kab. 310 rozinvm@gmail.com Other publications by this author

Problematic situation

Currently, education is changing significantly, I even called this process a "quiet revolution" in a joint article with Tatiana Kovaleva ^[15]. Its main features are a turn from the concept of formation to the concepts of "individualization" and "personalization", recognition of different trajectories of student development, setting up conditions for the initiation of independence (in terms of learning and creativity), criticism of the classroom system, changing the functions of the teacher (not only teaches, but also accompanies, helps, initiates, organizes) ^{[13; 14; 15]}. Mathematical education is no exception. Here are the same attitudes, but expressed less clearly, unsystematically. Here is an example.

The teacher of mathematics and physics of the Krasnoyarsk "School of Distance Education" Zheglataya Elena Dmitrievna in the article "Modern approaches in teaching mathematics", in fact, solves two main tasks: on the one hand, she tries to renew the meaning and interest in mathematical education, which, indeed, are practically absent for the bulk of students, on the other ? formulates attitudes, similar to those named. "One of the main goals of the subject "Mathematics" as a component of general secondary education," she writes, "relating to each student, is the development of thinking, primarily the formation of abstract thinking. In the process of studying mathematics in its purest form, logical and algorithmic thinking can be formed, many qualities of thinking - such as strength and flexibility, constructiveness, criticality, etc ... In other words, teaching mathematics is focused on education with the help of mathematics. In accordance with this principle, the main task of teaching mathematics is not the study of the basics of mathematical science as such, but general intellectual development ? the formation of students in the process of studying mathematics of the qualities of thinking necessary for the full functioning of a person in modern society, for the dynamic adaptation of a person to this society.…

A math teacher simply has to be a researcher at least at the level of school math problems, learn to identify key tasks, key methods and key ideas and equip the student with these tasks, methods and ideas…

A math teacher should be very patient, because you can't expect instant results from students. If everything is done (in the sense of reasonable sufficiency), done professionally and honestly, then sooner or later the student will prove himself. We need to wait patiently.

Mathematics is a wonderful science, you need to notice it. The teacher should encourage the students to search for the truth. What does it mean? This means that at every stage of school mathematical education, children need to be taught to observe, compare, notice patterns, formulate hypotheses, learn to prove or reject a hypothesis if a counterexample is found. It is important to teach students to build definitions and their negations independently, to show that in mathematics almost nothing should be memorized ? you should understand, learn to apply and then everything will be remembered by itself…

The teacher should not be a moral teacher, but an adviser, an assistant. One of the most important tips that a good teacher can give to children: mathematics cannot be taught, it can only be learned!..

Here it seems appropriate to formulate one of the principles of teaching schoolchildren, which I call the principle of the “four CO”. A math lesson is about cooperation, empathy, joy, creation…

The merit of mathematics is that it is a very effective tool for self-knowledge of the human mind. And although a person does not always have the opportunity to create something new in a particular field of activity, but being a person, he, nevertheless, cannot but be ready for creative self-expression. Mathematics helps him by awakening creative potencies. This is one of the main purposes of the subject of mathematics" ^[3].

Probably, it would be worth agreeing with these provisions right away, but let's not rush, questions still remain. The thesis that mathematics promotes the development of thinking is very old. For example, back in the 70s of the century before last, our wonderful teacher and methodologist V.A.Latyshev wrote that the purpose of teaching mathematics is "the development of abstract reasoning", and also: "Geometry should be introduced into secondary schools for the development of correct and accurate thinking in students" ^{[7, p. 1324]}. And in the 60s of the last century, E.V. Ilyenkov, G.P. Shchedrovitsky, V.V. Davydov wrote about this (that the school should teach thinking).  But what is currently to be understood by correct thinking, besides who has shown that it is mathematics that contributes to its formation?

The next question is, what should be the modern content of mathematical education? On the one hand, Zheglataya seems to set a fundamentally new content ("observe, compare, notice patterns, formulate hypotheses, teach to prove or reject a hypothesis if a counterexample is found ... independently build definitions and their negations, show that in mathematics almost nothing should be memorized ? you should understand, learn to apply and then everything it will be remembered by itself"), but on the other hand, it tries to save the traditional, disciplinary, in fact, knowledge mathematical content.

"Russian school mathematics," she writes, "has always stood on three pillars: arithmetic (arithmetic calculations), text problems (arithmetic and algebraic), geometry. The rejection of traditional content, the desire to modernize school math programs, and recently direct imitation of not the best Western models has become another reason for the crisis phenomena observed today in our school mathematics education. The second very important traditional feature of Russian mathematical education is the principle of evidence. This principle is very clearly visible in traditional school textbooks on mathematics. Not a single unproven statement, not a single formula without a conclusion. And this is how our mathematical education differs from the American one" [ ].  

The same contradiction can be seen in Western proposals. For example, the Finnish approach ? "through and" lists traditional and new content (and ordinary mathematical knowledge, and meta-knowledge, and mathematical concepts and operations). "C5 Geometry: Students expand their understanding of the concepts of point, segment, straight line and angle and get acquainted with the concepts of line and ray. They explore properties related to lines, angles, and polygons. They strengthen their understanding of the concepts of similarity and congruence. Students are engaged in geometric construction. They learn to use the Pythagorean theorem, the inverse Pythagorean theorem, and trigonometric functions. They learn about the inscribed angle and the central angle and get acquainted with Thales' theorem. Students calculate circles and areas of polygons. Students practice calculating the area, circumference, arc and area of the circle sector. Three-dimensional figures are considered. Students learn to calculate the areas and volumes of a sphere, cylinder and cone. Students strengthen and expand their knowledge of measurement units and unit conversion" ^[3].

One more question. Zheglataya hints that mathematics contributes to the formation of personality, but only hints. But some Western reformers speak about it directly. The position of the rector of the Moscow City Pedagogical University Igor Remorenko is intermediate and problematic. "Even before the advent of the Internet," he explains, "there was such a concept: dry knowledge ceased to be relevant, it is necessary to form some abilities, sometimes they say "competence", although this is not the same thing. And then different countries interpret this maxim differently, offer different solutions according to their specifics. I can give such an example: one student who defended her master's thesis with us said at the exam: "Well, of course, 4K ? no one canceled them!" 4K is a well?known concept in education: there are four core competencies that were once formulated at the Davos Forum, and then they are concretized in different texts ? communication, cooperation, critical thinking and creativity. Over time, this has become the base from which developers of standards and educational programs in different countries are based. Moreover, they are also used in different ways. For example, one of the regions of Canada says: "Well, we take 4K as a basis, but we need two more competencies and we will have 6K (6S)." To these four they add character, character education, they have an educational component, and citizenship — citizenship, it means being a citizen, knowing how society works, and making informed decisions. A certain global structure is taken as a base, on the basis of which education in different countries can be compared with each other, make predictions about the relationship between education and the labor market, and plan research. Over the past 20 years, there have been many such comparative studies of the quality of education in which Russia took an active part ? a study of the quality of reading in elementary school, natural science and mathematical literacy in adolescents, civic skills in high school students, even adults were compared according to their ability to solve problems and certain special approaches in organizing kindergartens. The results of the comparison were one of the goals of the national project "Education"" ^[9].

So the question is: can mathematical education really contribute to the formation of the right personality? At one time, as a graduate student, I asked this question to my supervisor Vasily Vasilyevich Davydov. He thought about it and said, summarizing: "Yes, mental development simultaneously forms the right personality." I remember, and then I didn't really believe it, because there are many good mathematicians and physicists who are at the same time very dubious personalities, from the point of view of morality or citizenship. 

In short, the quiet revolution in mathematical education is accompanied by a deep crisis concerning key problems ? the definition of the goals and content of mathematical education, the renewal of its attractiveness (meaning) for students. Let's try to start a discussion of these issues.

Mathematics, from the point of view of modern study

In the book "Mathematics: Origin, Nature, Teaching" published the year before last, I show that in mathematics it is necessary to distinguish three main areas: firstly, mathematical systems formed on the basis of two sources (theoretical reflection of a certain subject area and construction), secondly, the application of mathematical knowledge and objects in physics and in other scientific disciplines, thirdly, mathematics as a field of activity and scientific ethos. For example, if the first geometric knowledge and objects were representations in the language of "ideal objects" of the relations that developed in the Ancient World between the areas of fields in agriculture and their elements, then the subsequent ones were obtained during the description of geometry objects constructed on the basis of the original geometric objects. At the same time, ideal objects were constructed in such a way that the theoretical knowledge assigned to them was consistent and described the objects of this practice. The application of geometric knowledge in physics involved the interpretation of geometric figures as schemes and models, as well as new ways of proving.

If we talk about contradictions in mathematics, which applies not only to epistemology, but also to the field of scientific ethos, then contradictions in mathematics are not a misunderstanding, as some philosophers of mathematics (David Hilbert and others) believe, but the normal state of affairs. "Naturally, the activity of resolving contradictions and substantiation is no less normal, and it has been going on since the very beginning of the existence of mathematics. The analysis of the works of Aristotle and Lakatos shows that the resolution of aporias implies, on the one hand, the restructuring of ideal objects of mathematics, on the other ? the updating of ideas about mathematical proof…The crisis of modern mathematics does not arise at all because many contradictions have been discovered in mathematics. It is caused by both the complication of the field of mathematics and the contradictions of modernity. Conceptualization of mathematics should, in theory, respond to both of these factors, i.e. the awareness of mathematics should have kept pace with changes in the external and internal conditions of its existence and development. This probably didn't happen. Mathematics is probably facing a big reform. Not the last place in it will be occupied by the issues of the new organization of the mathematical community, because it will be necessary to develop principles that allow not only to reconsider views on mathematical proof, but also to build new relationships between mathematicians and their disciplines" ^{[10, pp. 235-236]}. 

Historically, the concepts of Plato, Roger Bacon and Kuzansky contributed to the belief that mathematics is involved in the creation of the world and man, and therefore its study and development is a necessary condition for the formation and even salvation of the individual. But now we have a better understanding of what mathematics is, and therefore we can more correctly characterize its meaning, including for the field of education. Firstly, mathematics is an important historical phenomenon, one of the first types of science ("The Beginnings of Euclid", the works of Archimedes and Apollonius). Secondly, mathematics is a language of mathematical schemes and models used in physics and a number of other scientific disciplines (for example, in sociology and economics). Thirdly, mathematics is a kind of creativity and thinking (mathematical), which can be joined by someone who is fascinated by mathematics.

But it's worth noting right away that the kind of creativity and thinking is one among many others. At present, we cannot, as in the time of Kant, consider natural science and mathematical thinking to be the only and most correct a priori in terms of cognition. There are other types of thinking (and creativity) (humanitarian, social, interdisciplinary, technological, esoteric, etc.) that differ significantly from mathematical. And mathematical language is not universal, for example, in the humanities and social sciences, as well as in philosophy and art, not mathematical models, but schemes are in use ^[11].    

Thus, the importance of mathematics is quite great, but not universal and not in the sense that the assimilation of mathematics contributes to the formation of correct or abstract thinking or a perfect personality (contributes to the formation of mathematical thinking only, and has nothing to do with personality). In education, it is worth abandoning these myths and introducing students to the meanings of mathematics indicated here, of course, not excluding the use of mathematics also for practical purposes (counting, multiplication table, etc.).

The content of modern education

Now there is a difficult question about the nature of the content of mathematical education. In the traditional paradigm (Komensky, Pestalozzi, Froebel, Disterveg, Ushinsky, etc.), this issue was solved unambiguously ? only mathematical knowledge and disciplines, this is what should be taught at school. However, Latyshev already doubted this. "It is known," he wrote, "that all methods of thinking are reduced to a very limited number of basic ones and that the number of different abilities is small. Finally, different methods of thinking are found in the same subject, which means that the occupation of one of them should prepare for the others. Are we not convinced that general education should make a student capable of any mental work?" ^{[7, p. 1322]}. If Latyshev only guessed that it was necessary to teach not knowledge, but methods of thinking, then philosophers and methodologists of the 60-70s, for example, Ilyenkov and Shchedrovitsky directly reduced the content of education to ways of thinking. In my PhD thesis on pedagogy, I proposed the genesis of geometry, focused specifically on the analysis of methods of geometric thinking ^[10].

Between these two points of view there is another one ? not knowledge, but meta-knowledge, meta-subject contents, for example, mathematical relations, simplified structures of ala Gilbert or Bourbaki, and so on. A group of Soviet mathematicians (V. Ashkinuse, V.Boltyansky, N.Vilenkin, V.Levin, A.Semushin, I.Yaglom) proposed to replace some outdated knowledge in the geometry subject with other, more advanced, meta-subject ^{[1; 2; 8]}. However, it turned out that students do not understand and do not assimilate the content offered to them.

 Currently, teachers seem to be ready to abandon the interpretation of the content of mathematical education as mathematical knowledge and disciplines. However, what instead is unclear. It is unlikely that teachers of mathematics can be completely satisfied, for example, with the "didactics of big ideas" (fundamental concepts and concepts, technological packages, everyday use, big challenges), all this is, of course, heuristic, but why exactly these ideas, and will their implementation lead to the mastery of mathematics that is needed today? In short, teachers found themselves at a crossroads: it is no longer possible in the old way, and the new content is questionable and eclectic.

I will express my point of view. The content of mathematical education (and other scientific disciplines) is the meanings obtained during the reconstruction of the formation and development of these disciplines. Meanings as a response to the problems of misunderstanding of the relevant disciplinary texts and provisions with the aim of clarifying situations, activities and thinking, which probably led to the creation of these texts and provisions. I will give one example. When learning geometry begins, as a rule, students do not understand for a long time what geometric shapes and their inherent relations of equality, similarity, parallelism are. By myself, I remember that for about half a year in the fifth or sixth grade I memorized all these definitions and proofs purely formally, without understanding; then something happened and I began, not to understand, but stopped misunderstanding.  

But after all, it is possible to offer students, but only after the misunderstanding is fixed, not geometric knowledge and proofs, but such a reconstruction. In the geometry textbook there is such a theorem: "The diagonal of a rectangle divides it into two equal right triangles" (originally, in another formulation, this is Theorem 41 of "Euclid's Beginnings"). We draw attention to the legend according to which geometry originated in ancient Egypt and Sumer from the needs of agriculture. We analyze a typical practical problem that the ancient scribes solved ? the division of a rectangular field (and there were most of them) by a diagonal dividing line into two equal parts, and explain that the measurement of the areas required to determine the amount of tax showed that the original area was twice the area of each triangular field obtained from the division. Later we can discuss how the scribes came to the concept of "area" and how they defined it (calculated).

But for now we ask: maybe the equality of geometric shapes is due to the equality of the areas of the fields? Analyzing the answers and guesses of the students, we draw attention to the fact that the equality of geometric figures is confirmed in the proof by the procedure of superimposing them on each other. We ask what it is, why it is impossible to verify equality by eye or otherwise. We tell you that the Pythagoreans in the ancient culture of the 6th-5th century BC also did not understand the papyri of the Egyptians and the clay books of the Sumerians, where measurements of the area of fields were given. At the same time, we saw that in one text the comparison of numbers showed that the areas of triangular fields obtained from dividing a rectangular field were the same. We ask how the Pythagoreans could decipher the calculations of the Egyptians and Sumerians, if they considered numbers and drawings sacred objects that were created by the gods. Gradually we come to the conclusion that the Pythagoreans built a new class of sacred objects ? geometric figures, considering that, on the one hand, they are fundamentally different from fields (because the latter are not sacred objects), on the other hand, they can be compared for equality and inequality precisely as sacred objects. To understand what this means, the Pythagoreans came up with a procedure for superimposing some figures on others.

The last link of the reconstruction is a story about the formation of reasoning in ancient culture around the same period (i.e., a new way of obtaining knowledge, some based on others by inference). However, as a result of reasoning, it was possible to obtain ordinary knowledge corresponding to observations and experience, and paradoxes. Analyzing the paradoxes, we bring students to understand why and how Socrates and Plato, and a little later Aristotle, proposed to build definitions, rules of reasoning and categories that allow reasoning without contradictions. All this made up the ancient logic, which allowed the introduction of dialectics and thinking. For students, the dry residue that is proposed to be thought over and discussed is the assumption that the proofs of geometric theorems were formed under the influence of Plato's dialectic and Aristotle's logic, and within the framework of the proofs, geometric figures finally turn into ideal objects of geometry.           

I have disassembled only one reconstruction, but an important one. In reality, for the course of geometry, solving the problems of misunderstanding the basic features of geometry as mathematics (see above its three main characteristics), they probably need to be made several dozen. Each such reconstruction will make it possible to determine the main mathematical contents and establish genetic links between them. For example, the above reconstruction allows us to assume that in a propaedeutic geometry course it is worth introducing children to the concepts of bodies of regular geometric shape (recall the Froebel system), bring them to an understanding of the concept of "area", teach them to compare areas with each other, measure and compare individual elements of regular bodies. In the course itself, you must first talk about the views of the Pythagoreans, then about the invention of reasoning and paradoxes, and then about ways to resolve them. And only then discuss what geometric relations and proofs are.

Moreover, it is advisable to do all this in the form of setting problems and questions, initiating discussion, inducing students to solve problems independently and subsequent reflection. In other words, in addition to reconstruction and genetic connections, the structure of the content of education is determined by another factor ? modern ideas about the laws of development and evolution of individuals. And the latter, as noted, involves initiating independence and tracking different trajectories of personality development. Therefore, it is necessary to proceed from the fact that different students will answer questions differently and solve the problems posed to them differently. Ultimately, they may come to a different understanding of geometry (mathematics). These points need to be identified so that tutors can work individually with their wards (help them, discuss situations and problems that have arisen, initiate, if necessary, the next steps).

Another factor determining the features of the modern content of education is the ability to transfer the bulk of mathematical knowledge and proofs to the Internet, leaving only some for learning, which you can rely on to introduce the necessary understanding of mathematics. This is, if we are talking about a comprehensive school, it is another matter if students will specialize in mathematics (in special classes or at universities).  "Concrete mathematical knowledge," Zheglataya notes, "lying outside, relatively speaking, the arithmetic of natural numbers and the primary foundations of geometry, are not a "matter of first necessity" for the vast majority of people and therefore cannot form the target basis for teaching mathematics as a subject of general education" ^[3].

  At the same time, this transfer means the need to move to a new content of education, to a sharp increase in the role and importance of reflexivity. After all, in fact, the reconstruction of the meaning of mathematical contents proposed by the author is nothing more than a reflection of the situations, activities and thinking that led to the formation of mathematics. Here, however, an important question arises, where can a school teacher find such a reflexive content of mathematics? Of course, I can refer to my research in mathematics, but, firstly, they may not be the best in terms of methodology, and secondly, I was able to analyze only the formation of Euclidean geometry and not completely mathematical logic ^[10]. The reflection of mathematics, in the sense I have indicated, has yet to come, is still waiting for its Newton. A school math teacher can try to swim on his own, and I think it will always justify itself, reflection is a useful thing, but still, I think he will not be able to notice a specialist methodologist with mathematical training. 

It may seem to someone that the reflection of mathematics also means that in its teaching it is necessary to adhere to historical forms (for example, representatives of the "genetic system" of teaching mathematics thought so ^[16]). No, historical forms differ significantly from modern ones, therefore, for the purposes of education, they must be transformed, replaced by modern ones.  

Another important consideration concerns the variability of the modern content of education. If modern education proceeds from the recognition of different personality types and trajectories of their development, as well as the principle of "cultural conformity" (with all innovations and independence, a young person should remain in culture and contribute to its evolution), then it is clear that the teacher should strive to provide the student with content in all its cultural diversity, which again, probably, it involves strengthening the reflexive principle. For example, it is not easy to introduce the fifth postulate of the "Principles of Euclid", but to immerse in the situation of attempts to prove it, and to offer material leading to the geometry of Lobachevsky, and then Riemann. And so it is everywhere ? the diversity and variability of the meanings of mathematics.   

At the end of a difficult question for the author: what should a textbook be, with a presentation of the new content of education? For example, what sequence of mathematical contents can we talk about here? One thing is clear ? this is not a traditional presentation, as for example, even in good textbooks like Kiselyov. Maybe, in general, the time has come to search and write textbooks of a fundamentally new type, and the author, however, not in mathematics, but in the humanities (cultural studies), tried to write a similar textbook ^[12]. Talking about the idea of this textbook, which outlines the main paradigms of cultural studies, I write the following: "And here's how L. puts the question today.Ionin, who wrote an excellent textbook on the sociology of culture. “Just as culture as a whole is a diverse, multi-layered phenomenon, so a textbook on the sociology of culture cannot but be a kind of introduction to interdisciplinary research.” Explaining further what should be understood by this, Ionin writes: “In this work, with all the desire, it will not be possible to comprehensively highlight the development of each of the cultural sciences, which, as already mentioned, are also whimsically intertwined with each other. Therefore, in the historical review, we will focus rather not on the development of disciplines, but on the change of global paradigms of the vision of culture. A paradigm shift is something more than an alternation of theories and concepts put forward by one or another author. A paradigm shift is a change in attitudes to the object of research, involving a change in research methods, research goals, the angle of view on the subject, and often even a change in the subject of research itself” ^{[5, p. 7, 24-25]}. To what has been said, we add the following.

         A modern textbook on cultural studies should probably solve two main tasks: to help the teacher and the student to enter, immerse themselves in the reality of cultural work (so to speak, to feel this reality) and equip him with the means to orient and operate in this reality. Therefore, it does not make sense to present cultural theories or ideas by themselves, as some kind of information. But it is necessary to indicate the main approaches and methods created and used in cultural studies, to characterize their purpose and boundaries, to give a kind of guide for orientation in cultural studies as a heterogeneous complex discipline.

         Based on this understanding, I tried to build the material. The textbook begins not with a traditional explanation of the subject and the concept of culture, but with the presentation of three available samples of cultural research. I think it will help to feel what it is even before any understanding (reflection) of the subject of cultural studies. Then the first reflection and review of the subject of cultural studies will be offered. Based on it, it will be possible to take the next step: to characterize the main problems and alternatives that have arisen so far in cultural studies. In turn, such an analysis will allow discussing the methodological foundations of cultural studies and how culture is understood in different areas of cultural studies. But in order for the reader to be able to rely on empirical material, an analysis of the formation of the culture of the ancient world, as well as the main stages of formation in the culture of personality, will be presented beforehand. The final section of the textbook includes a small reference material and appendices devoted to the cultural analysis of art" ^{[12, p. 5-6]}.

 Culturology, of course, differs significantly from mathematics, but I am sure that modern education is united in its principles of construction and opposition to the traditional paradigm of education. However, the author's beliefs are not the ultimate truth, but an invitation to discussion.