Lecture session Topic: Methods of teaching mathematics to junior schoolchildren as an academic subject.

Purpose of the lesson:

1).Didactic:

To achieve students’ understanding of the methods of teaching mathematics to junior schoolchildren as an academic subject.

2). Developmental:

Expand the concepts of methods of teaching mathematics to primary schoolchildren. Develop students' logical thinking.

3). Educating:

Teach students to realize the importance of studying this topic for their future profession.

6.Form of training: frontal.

7. Teaching methods:

Verbal: explanation, conversation, questioning.

Practical: independent work.

Visual: handouts, teaching aids.

Lesson plan:

1. Methods of teaching mathematics to junior schoolchildren as a pedagogical science and as a field of practical activity.
2. Methods of teaching mathematics as an academic subject. Principles of designing a mathematics course in elementary school.
3. Methods of teaching mathematics.

Basic concepts:

Methods of teaching mathematics is the science of mathematics as a scientific subject and the principles of teaching mathematics to students of various age groups; in its research, this science is based on various psychological, pedagogical, mathematical foundations and generalizations of the practical experience of mathematics teachers.

1. Methods of teaching mathematics to junior schoolchildren as a pedagogical science and as a field of practical activity.

Considering the methodology of teaching mathematics to junior schoolchildren as a science, it is necessary, first of all, to determine its place in the system of sciences, outline the range of problems that it is designed to solve, determine its object, subject and features.

In the system of sciences, methodological sciences are considered in the block didactics. As is known, didactics is divided into education theory And theory training. In turn, in the theory of learning, general didactics (general issues: methods, forms, means) and particular didactics (subject-specific) are distinguished. Private didactics are called differently - teaching methods or, as has become common in recent years - educational technologies.

Thus, methodological disciplines belong to the pedagogical cycle, but at the same time, they represent purely subject areas, since the methods of teaching literacy will certainly be very different from the methods of teaching mathematics, although both of them are private didactics.

The methodology of teaching mathematics to primary schoolchildren is a very ancient and very young science. Learning to count and calculate was a necessary part of education in ancient Sumerian and ancient Egyptian schools. Rock paintings from the Paleolithic era tell stories about learning to count. The first textbooks for teaching children mathematics include “Arithmetic” by Magnitsky (1703) and the book by V.A. Lai “Guide to the initial teaching of arithmetic, based on the results of didactic experiments” (1910). In 1935 S.I. Shokhor-Trotsky wrote the first textbook “Methods of teaching mathematics”. But only in 1955, the first book “The Psychology of Teaching Arithmetic” appeared, the author of which was N.A. Menchinskaya turned not so much to the characteristics of the mathematical specifics of the subject, but to the patterns of mastering arithmetic content by a child of primary school age. Thus, the emergence of this science in its modern form was preceded not only by the development of mathematics as a science, but also by the development of two large areas of knowledge: general didactics of learning and the psychology of learning and development.

The teaching technology is based on a methodological system of meaning that includes the following 5 components:

2) learning goals.

3) means

Didactic principles are divided into general and basic.

When considering didactic principles, the main provisions determine the content of the organizational forms and methods of educational work of the school. In accordance with the goals of education and the laws of the learning process.

Didactic principles express what is common to any academic subject and are a guideline for planning the organization and analysis of a practical task.

In the methodological literature there is no single approach to identifying systems of principle:

A. Stolyar identifies the following principles:

1) scientific character

3) visibility

4) activity

5) strength

6) individual approach

Yu.K. Babansky identifies 5 groups of principles:

2) to select the learning task

3) to select the form of training

4) choice of teaching methods

5) analysis of results

The development of modern education is based on the principle of lifelong learning.

The principles of learning are not established once and for all; they deepen and change.

The scientific principle, as a didactic principle, was formulated by N.N. Skatkin in 1950.

Feature of the principle:

Displays, but does not reproduce the accuracy of the scientific system, preserving, as far as possible, the general features of their inherent logic, stages and system of knowledge.

Reliance for subsequent knowledge on previous ones.

A systematic pattern of arrangement of material by year of study in accordance with the age characteristics and age of the students, as well as the further development of the teachers.

Disclosure of internal connections between the concepts of patterns and connections with other sciences.

The redesigned programs emphasized the principles of clarity.

The principle of visibility ensures the transition from living contemplation to real thinking. Visualization makes it more accessible, concrete and interesting, develops observation and thinking, provides a connection between the concrete and the abstract, and promotes the development of abstract thinking.

Excessive use of visualization can lead to undesirable results.

Types of visibility:

natural (models, handouts)

visual clarity (drawings, photos, etc.)

symbolic clarity (schemes, tables, drawings, diagrams)

2.Methods of teaching mathematics as an academic subject. Principles of designing a mathematics course in elementary school.

Methods of teaching mathematics (MTM) is a science whose subject is teaching mathematics, and in a broad sense: teaching mathematics at all levels, from preschool institutions to higher education.

MPM develops on the basis of a certain psychological theory of learning, i.e. MPM is a “technology” for applying psychological and pedagogical theories to primary mathematics teaching. In addition, the MPM should reflect the specifics of the subject of study - mathematics.

The goals of primary mathematics education: general education (mastery of a certain amount of mathematical knowledge by students in accordance with the program), educational (formation of a worldview, the most important moral qualities, readiness for work), developmental (development of logical structures and mathematical style of thinking), practical (formation of the ability to apply mathematical knowledge in specific situations, when solving practical problems).

The relationship between teacher and student occurs in the form of information transfer in two opposite directions: from teacher to student (direct), from teaching to teacher (reverse).

Principles of constructing mathematics in elementary school (L.V. Zankov): 1) teaching at a high level of difficulty; 2) learning at a fast pace; 3) the leading role of theory; 4) awareness of the learning process; 5) purposeful and systematic work.

The learning challenge is key. On the one hand, it reflects the general goals of learning and specifies cognitive motives. On the other hand, it allows you to make the process of performing educational actions meaningful.

Stages of the theory of the gradual formation of mental actions (P.Ya. Galperin): 1) preliminary familiarization with the purpose of the action; 2) drawing up an indicative basis for action; 3) performing an action in material form; 4) speaking the action; 5) automation of action; 6) performing an action mentally.

Techniques for consolidating didactic units (P.M. Erdniev): 1) simultaneous study of similar concepts; 2) simultaneous study of reciprocal actions; 3) transformation of mathematical exercises; 4) preparation of tasks by students; 5) deformed examples.

3.Methods of teaching mathematics.

Question about methods of primary mathematics teaching and their classification has always been the subject of attention from methodologists. In most modern methodological manuals, special chapters are devoted to this problem, which reveal the main features of individual methods and show the conditions for their practical application in the learning process.

Beginning mathematics course consists of several sections, different in content. This includes: problem solving; studying arithmetic operations and developing computational skills; studying measures and developing measurement skills; study of geometric material and development of spatial concepts. Each of these sections, having its own special content, at the same time has its own, private, methodology, its own methods, which are in accordance with the specifics of the content and form of training sessions.

Thus, in the methodology of teaching children to solve problems, the logical analysis of the problem conditions using analysis, synthesis, comparison, abstraction, generalization, etc. comes to the fore as a methodological technique.

But when studying measures and geometric material, another method comes to the fore - laboratory, which is characterized by a combination of mental work and physical work. It combines observations and comparisons with measurements, drawing, cutting, modeling, etc.

The study of arithmetic operations occurs on the basis of the use of methods and techniques that are unique to this section and differ from the methods used in other branches of mathematics.

Therefore, developing mathematics teaching methods, it is necessary to take into account psychological and didactic patterns of a general nature, which are manifested in general methods and principles related to the course as a whole.

The most important task of the school at the present stage of its development is to improve the quality of education. This problem is complex and multifaceted. During today's lesson, our attention will be focused on teaching methods, as one of the most important links in improving the learning process.

Teaching methods are ways of joint activity between teacher and students aimed at solving learning problems.

The teaching method is a system of purposeful actions of the teacher that organizes the cognitive and practical activities of the student, ensuring that he masters the content of education.

Ilyina: “Method is the way in which the teacher directs the teacher’s cognitive activity” (there is no student as an object of activity or educational process)

The teaching method is a way of transferring knowledge and organizing cognitive practical activities of students in which students master knowledge of knowledge, while developing their abilities and forming their scientific worldview.

Currently, intensive attempts are being made to classify teaching methods. It is of great importance for bringing all known methods into a certain system and order, identifying their common features and features.

The most common classification is teaching methods

- by sources of knowledge;

- for didactic purposes;

- according to the level of activity of students;

- by the nature of students’ cognitive activity.

The choice of teaching methods is determined by a number of factors: the objectives of the school at the current stage of development, the academic subject, the content of the material being studied, the age and level of development of students, as well as their level of readiness to master the educational material.

Let's take a closer look at each classification and its inherent purposes.

In the classification of teaching methods for didactic purposes allocate :

Methods of acquiring new knowledge;

Methods of developing skills and abilities;

Methods of consolidating and testing knowledge, abilities, skills.

Often used to introduce students to new knowledge story method.

In mathematics, this method is usually called - method of presenting knowledge.

Along with this method, the most widely used conversation method. During the conversation, the teacher poses questions to the students, the answers to which involve the use of existing knowledge. Based on existing knowledge, observations, and past experience, the teacher gradually leads students to new knowledge.

At the next stage, the stage of formation of skills and abilities, practical teaching methods. These include exercises, practical and laboratory methods, and work with a book.

Contributes to the consolidation of new knowledge, the formation of skills and abilities, and their improvement independent work method. Often, using this method, the teacher organizes the students’ activities in such a way that the students acquire new theoretical knowledge on their own and can apply them in a similar situation.

The following classification of teaching methods by student activity level- one of the early classifications. According to this classification, teaching methods are divided into passive and active, depending on the degree of student involvement in learning activities.

TO passive These include methods in which students only listen and watch (story, explanation, excursion, demonstration, observation).

TO active - methods that organize independent work of students (laboratory method, practical method, work with a book).

Consider the following classification of teaching methods by source of knowledge. This classification is most widely used due to its simplicity.

There are three sources of knowledge: word, visualization, practice. Accordingly, they allocate

- verbal methods(the source of knowledge is the spoken or printed word);

- visual methods(sources of knowledge are observed objects, phenomena, visual aids );

- practical methods(knowledge and skills are formed in the process of performing practical actions).

Let's take a closer look at each of these categories.

Verbal methods occupy a central place in the system of teaching methods.

Verbal methods include story, explanation, conversation, discussion.

The second group according to this classification consists of visual teaching methods.

Visual teaching methods are those methods in which the assimilation of educational material is significantly dependent on the methods used. visual aids.

Practical methods training is based on the practical activities of students. The main purpose of this group of methods is the formation of practical skills.

Practical methods include exercises, practical and laboratory work.

The next classification is teaching methods by the nature of students’ cognitive activity.

The nature of cognitive activity is the level of mental activity of students.

The following methods are distinguished:

Explanatory and illustrative;

Methods of problem presentation;

Partially search (heuristic);

Research.

Explanatory and illustrative method. Its essence lies in the fact that the teacher communicates ready-made information through various means, and students perceive it, realize it and record it in memory.

The teacher conveys information using the spoken word (story, conversation, explanation, lecture), the printed word (textbook, additional manuals), visual aids (tables, diagrams, pictures, films and filmstrips), practical demonstration of methods of activity (showing experience, work on a machine, a method for solving a problem, etc.).

Reproductive method assumes that the teacher communicates and explains knowledge in a ready-made form, and students assimilate it and can reproduce and repeat the method of activity according to the teacher’s instructions. The criterion for assimilation is the correct reproduction (reproduction) of knowledge.

Method of problem presentation is a transition from performing to creative activity. The essence of the problem presentation method is that the teacher poses a problem and solves it himself, thereby showing the train of thought in the process of cognition. At the same time, students follow the logic of presentation, mastering the stages of solving holistic problems. At the same time, they not only perceive, understand and remember ready-made knowledge and conclusions, but also follow the logic of evidence and the movement of the teacher’s thoughts.

A higher level of cognitive activity carries with it partially search (heuristic) method.

The method was called partially search because students independently solve a complex educational problem not from beginning to end, but only partially. The teacher involves students in performing individual search steps. Some of the knowledge is imparted by the teacher, and some of the knowledge is acquired by students on their own, answering questions or solving problematic tasks. Educational activities develop according to the following scheme: teacher - students - teacher - students, etc.

Thus, the essence of the partially search method of teaching comes down to the fact that:

Not all knowledge is offered to students in a ready-made form; some of it needs to be acquired on their own;

The teacher’s activity consists of operational management of the process of solving problematic problems.

One of the modifications of this method is heuristic conversation.

The essence of a heuristic conversation is that the teacher, by asking students certain questions and joint logical reasoning with them, leads them to certain conclusions that constitute the essence of the phenomena, processes, rules under consideration, i.e. Students, through logical reasoning, in the direction of the teacher, make a “discovery.” At the same time, the teacher encourages students to reproduce and use their existing theoretical and practical knowledge, production experience, compare, contrast, and draw conclusions.

The next method in classification according to the nature of students’ cognitive activity is research method training. It provides for the creative assimilation of knowledge by students. Its essence is as follows:

The teacher, together with the students, formulates the problem;

Students resolve it independently;

The teacher provides assistance only when difficulties arise in solving the problem.

Thus, the research method is used not only to generalize knowledge, but mainly so that the student learns to acquire knowledge, investigate an object or phenomenon, draw conclusions and apply the acquired knowledge and skills in life. Its essence comes down to organizing the search and creative activities of students to solve problems that are new to them.

1. Homework:

Prepare for practical training

The new paradigm of education in the Russian Federation is characterized by a personality-oriented approach, the idea of ​​developmental education, the creation of conditions for self-organization and self-development of the individual, the subjectivity of education, the focus on designing the content, forms and methods of teaching and upbringing that ensure the development of each student, his cognitive abilities and personal qualities.

The concept of school mathematical education highlights its main goals - teaching students the techniques and methods of mathematical knowledge, developing in them the qualities of mathematical thinking, corresponding mental abilities and skills. The importance of this line of work is enhanced by the increasing importance and application of mathematics in various fields of science, economics and industry.

The need for mathematical development of younger schoolchildren in educational activities is noted by many leading Russian scientists (V.A. Gusev, G.V. Dorofeev, N.B. Istomina, Yu.M. Kolyagin, L.G. Peterson, etc.). This is due to the fact that during the preschool and primary school period, the child not only intensively develops all mental functions, but also lays the general foundation of the cognitive abilities and intellectual potential of the individual. Numerous facts indicate that if the corresponding intellectual or emotional qualities for one reason or another do not receive proper development in early childhood, then subsequently overcoming such shortcomings turns out to be difficult and sometimes impossible (P.Ya. Galperin, A.V. Zaporozhets , S.N. Karpova).

Thus, the new paradigm of education, on the one hand, presupposes the maximum possible individualization of the educational process, and on the other, requires solving the problem of creating educational technologies that ensure the implementation of the main provisions of the Concept of School Mathematics Education.

In psychology, the term “development” is understood as consistent, progressive significant changes in the psyche and personality of a person, manifesting themselves as certain new formations. The position on the possibility and feasibility of education focused on the development of the child was substantiated back in the 1930s. outstanding Russian psychologist L.S. Vygotsky.

One of the first attempts to practically implement the ideas of L.S. Vygotsky in our country was undertaken by L.V. Zankov, who in the 1950-1960s. developed a fundamentally new system of primary education, which found a large number of followers. In the L.V. system Zankov, for the effective development of students’ cognitive abilities, the following five basic principles are implemented: learning at a high level of difficulty; the leading role of theoretical knowledge; moving forward at a fast pace; conscious participation of schoolchildren in the educational process; systematic work on the development of all students.

Theoretical (rather than traditional empirical) knowledge and thinking, educational activity were placed at the forefront by the authors of another theory of developmental education - D.B. Elkonin and V.V. Davydov. They considered the most important thing to change the student's position in the learning process. Unlike traditional education, where the student is the object of the teacher’s pedagogical influences, in developmental education conditions are created under which he becomes the subject of learning. Today, this theory of educational activity is recognized throughout the world as one of the most promising and consistent in terms of implementing the well-known provisions of L.S. Vygotsky about the developmental and anticipatory nature of learning.

In domestic pedagogy, in addition to these two systems, the concepts of developmental education by Z.I. Kalmykova, E.N. Kabanova-Meller, G.A. Tsukerman, S.A. Smirnova and others. It should also be noted the extremely interesting psychological searches of P.Ya. Galperin and N.F. Talyzina based on the theory they created of the stage-by-stage formation of mental actions. However, as noted by V.A. Tests, in most of the mentioned pedagogical systems, the development of the student is still the responsibility of the teacher, and the role of the former is reduced to following the developmental influence of the latter.

In line with developmental education, many different programs and teaching aids in mathematics have appeared, both for primary grades (textbooks by E.N. Alexandrova, I.I. Arginskaya, N.B. Istomina, L.G. Peterson, etc.), and for secondary school (textbooks by G.V. Dorofeev, A.G. Mordkovich, S.M. Reshetnikov, L.N. Shevrin, etc.). Textbook authors have different understandings of personality development in the process of learning mathematics. Some focus on the development of observation, thinking and practical actions, others - on the formation of certain mental actions, others - on creating conditions that ensure the formation of educational activities and the development of theoretical thinking.

It is clear that the problem of developing mathematical thinking in teaching mathematics at school cannot be solved only by improving the content of education (even with good textbooks), since the implementation of different levels in practice requires the teacher to have a fundamentally new approach to organizing students’ learning activities in the classroom , in home and extracurricular work, allowing him to take into account the typological and individual characteristics of students.

It is known that primary school age is sensitive and most favorable for the development of cognitive mental processes and intelligence. Developing students' thinking is one of the main tasks of primary school. It is on this psychological feature that we concentrated our efforts, relying on the psychological and pedagogical concept of the development of thinking by D.B. Elkonin, position of V.V. Davydov on the transition from empirical to theoretical thinking in the process of specially organized educational activities, based on the works of R. Atakhanov, L.K. Maksimova, A.A. Stolyara, P. - H. van Hiele, related to identifying the levels of development of mathematical thinking and their psychological characteristics.

The idea of ​​L.S. Vygotsky’s idea that learning should be carried out in the zone of proximal development of students, and its effectiveness is determined by which zone (large or small) it prepares, is well known to everyone. At the theoretical (conceptual) level, it is shared almost throughout the world. The problem lies in its practical implementation: how to define (measure) this zone and what should be the teaching technology so that the process of learning the scientific foundations and mastering (“appropriating”) human culture takes place in it, providing the maximum developmental effect?

Thus, psychological and pedagogical science has substantiated the expediency of mathematical development of younger schoolchildren, but the mechanisms for its implementation have not been sufficiently developed. Consideration of the concept of “development” as a result of learning from a methodological point of view shows that it is an integral continuous process, the driving force of which is the resolution of contradictions that arise in the process of change. Psychologists argue that the process of overcoming contradictions creates conditions for development, as a result of which individual knowledge and skills develop into a new holistic formation, into a new ability. Therefore, the problem of constructing a new concept for the mathematical development of younger schoolchildren is determined by contradictions.

The problem of the formation and development of mathematical abilities of younger schoolchildren is relevant at the present time, but at the same time it receives insufficient attention among the problems of pedagogy. Mathematical abilities refer to special abilities that manifest themselves only in a separate type of human activity.

Teachers often try to understand why children studying in the same school, with the same teachers, in the same class, achieve different successes in mastering this discipline. Scientists explain this by the presence or absence of certain abilities.

Abilities are formed and developed in the process of learning, mastering relevant activities, therefore it is necessary to form, develop, educate and improve the abilities of children. In the period from 3-4 years to 8-9 years, rapid development of intelligence occurs. Therefore, during primary school age the opportunities for developing abilities are the highest. The development of the mathematical abilities of a junior schoolchild is understood as the purposeful, didactically and methodically organized formation and development of a set of interrelated properties and qualities of the child’s mathematical thinking style and his abilities for mathematical knowledge of reality.

The first place among academic subjects that pose particular difficulties in learning is given to mathematics, as one of the abstract sciences. For children of primary school age, it is extremely difficult to perceive this science. An explanation for this can be found in the works of L.S. Vygotsky. He argued that in order “to understand the meaning of a word, you need to create a semantic field around it. To construct a semantic field, a projection of meaning into a real situation must be carried out.” It follows from this that mathematics is complex, because it is an abstract science, for example, it is impossible to transfer a number series to reality, because it does not exist in nature.

From the above it follows that it is necessary to develop the child’s abilities, and this problem must be approached individually.

The problem of mathematical abilities was considered by the following authors: Krutetsky V.A. “Psychology of mathematical abilities”, Leites N.S. “Age giftedness and individual differences”, Leontyev A.N. "Chapter on Abilities" by Zach Z.A. “Development of intellectual abilities in children” and others.

Today, the problem of developing the mathematical abilities of younger schoolchildren is one of the least developed problems, both methodological and scientific. This determines the relevance of this work.

The purpose of this work: systematization of scientific points of view on this problem and identification of direct and indirect factors influencing the development of mathematical abilities.

When writing this work, the following questions were set: tasks:

1. Studying psychological and pedagogical literature in order to clarify the essence of the concept of ability in the broad sense of the word, and the concept of mathematical ability in the narrow sense.

2. Analysis of psychological and pedagogical literature, periodical materials devoted to the problem of studying mathematical abilities in historical development and at the present stage.

ChapterI. The essence of the concept of ability.

1.1 General concept of abilities.

The problem of abilities is one of the most complex and least developed in psychology. When considering it, first of all, it should be taken into account that the real subject of psychological research is human activity and behavior. There is no doubt that the source of the concept of abilities is the indisputable fact that people differ in the quantity and quality of productivity of their activities. The variety of human activities and the quantitative and qualitative differences in productivity make it possible to distinguish between types and degrees of abilities. A person who does something well and quickly is said to be capable of this task. Judgment about abilities is always comparative in nature, that is, it is based on a comparison of productivity, the skill of one person with the skill of others. The criterion of ability is the level (result) of activity that some people manage to achieve and others do not. The history of social and individual development teaches that any skillful skill is achieved as a result of more or less intense work, various, sometimes gigantic, “superhuman” efforts. On the other hand, some achieve high mastery of activity, skill and skill with less effort and faster, others do not go beyond average achievements, others find themselves below this level, even if they try hard, study and have favorable external conditions. It is the representatives of the first group that are called capable.

Human abilities, their different types and degrees, are among the most important and complex problems of psychology. However, the scientific development of the issue of abilities is still insufficient. Therefore, in psychology there is no single definition of abilities.

V.G. Belinsky understood abilities as the potential natural forces of the individual, or its capabilities.

According to B.M. Teplov, abilities are individual psychological characteristics that distinguish one person from another.

S.L. Rubinstein understands ability as suitability for a particular activity.

The psychological dictionary defines ability as quality, opportunity, skill, experience, skill, talent. Abilities allow you to perform certain actions at a given time.

Ability is an individual's readiness to perform an action; suitability is the existing potential to perform any activity or the ability to achieve a certain level of development of ability.

Based on the above, we can give a general definition of abilities:

Ability is an expression of the correspondence between the requirements of activity and the complex of neuropsychological properties of a person, ensuring high qualitative and quantitative productivity and growth of his activity, which is manifested in a high and rapidly growing (compared to the average person) ability to master this activity and master it.

1.2 The problem of developing the concept of mathematical abilities abroad and in Russia.

A wide variety of directions also determined a wide variety in the approach to the study of mathematical abilities, in methodological tools and theoretical generalizations.

The study of mathematical abilities should begin with defining the subject of research. The only thing that all researchers agree on is the opinion that it is necessary to distinguish between ordinary, “school” abilities for assimilation of mathematical knowledge, for their reproduction and independent application, and creative mathematical abilities associated with the independent creation of an original and socially valuable product.

Back in 1918, Rogers' work noted two sides of mathematical abilities, reproductive (related to the memory function) and productive (related to the thinking function). In accordance with this, the author built a well-known system of mathematical tests.

The famous psychologist Revesh, in his book “Talent and Genius,” published in 1952, considers two main forms of mathematical abilities - applicative (as the ability to quickly discover mathematical relationships without preliminary tests and apply the corresponding knowledge in similar cases) and productive (as the ability to discover relationships, not directly arising from existing knowledge).

Foreign researchers show great unity of views on the issue of innate or acquired mathematical abilities. If here we distinguish between two different aspects of these abilities - “school” and creative abilities, then in relation to the latter there is complete unity - the creative abilities of a scientist - mathematics are an innate education, a favorable environment is necessary only for their manifestation and development. This is, for example, the point of view of mathematicians who were interested in questions of mathematical creativity - Poincaré and Hadamard. Betz also wrote about the innateness of mathematical talent, emphasizing that we are talking about the ability to independently discover mathematical truths, “for probably everyone can understand someone else’s thought.” The thesis about the innate and hereditary nature of mathematical talent was vigorously promoted by Revesh.

Regarding “school” (learning) abilities, foreign psychologists do not speak so unanimously. Here, perhaps, the dominant theory is the parallel action of two factors - biological potential and environment. Until recently, even in relation to school mathematical abilities, the ideas of innateness dominated.

Back in 1909-1910. Stone and independently Curtis, studying achievements in arithmetic and abilities in this subject, came to the conclusion that it is hardly possible to talk about mathematical abilities as a single whole, even in relation to arithmetic. Stone pointed out that children who are skilled at calculations often lag behind in arithmetic reasoning. Curtis also showed that it is possible to combine a child's success in one branch of arithmetic and his failure in another. From this they both concluded that each operation required its own special and relatively independent ability. Some time later, Davis conducted a similar study and came to the same conclusions.

One of the significant studies of mathematical abilities must be recognized as the study of the Swedish psychologist Ingvar Werdelin in his book “Mathematical Abilities”. The author’s main intention was to, based on the multifactor theory of intelligence, analyze the structure of schoolchildren’s mathematical abilities and identify the relative role of each factor in this structure. Werdelin takes as a starting point the following definition of mathematical abilities: “Mathematical ability is the ability to understand the essence of mathematical (and similar) systems, symbols, methods and proofs, to memorize, retain them in memory and reproduce, combine them with other systems, symbols, methods and proofs, use them in solving mathematical (and similar) problems.” The author examines the question of the comparative value and objectivity of measuring mathematical abilities using teachers' grades and special tests and notes that school grades are unreliable, subjective and far from a real measurement of abilities.

The famous American psychologist Thorndike made a great contribution to the study of mathematical abilities. In his work “The Psychology of Algebra” he gives a lot of all kinds of algebraic tests to determine and measure abilities.

Mitchell, in his book on the nature of mathematical thinking, lists several processes that, in his opinion, characterize mathematical thinking, in particular:

1. classification;

2. ability to understand and use symbols;

3. deduction;

4. manipulation of ideas and concepts in an abstract form, without reference to the concrete.

Brown and Johnson, in the article “Ways to Identify and Educate Students with Potential in the Sciences,” indicate that practicing teachers have identified those features that characterize students with potential in mathematics, namely:

1. extraordinary memory;

2. intellectual curiosity;

3. ability for abstract thinking;

4. ability to apply knowledge in a new situation;

5. the ability to quickly “see” the answer when solving problems.

Concluding a review of the works of foreign psychologists, it should be noted that they do not give a more or less clear and distinct idea of ​​the structure of mathematical abilities. In addition, we must also keep in mind that in some works the data were obtained using a less objective introspective method, while others are characterized by a purely quantitative approach, ignoring the qualitative features of thinking. Summarizing the results of all the studies mentioned above, we will obtain the most general characteristics of mathematical thinking, such as the ability for abstraction, the ability for logical reasoning, good memory, the ability for spatial representations, etc.

In Russian pedagogy and psychology, only a few works are devoted to the psychology of abilities in general and the psychology of mathematical abilities in particular. It is necessary to mention the original article by D. Mordukhai-Boltovsky “Psychology of Mathematical Thinking”. The author wrote the article from an idealistic position, attaching, for example, special importance to the “unconscious thought process,” arguing that “the thinking of a mathematician ... is deeply embedded in the unconscious sphere.” The mathematician is not aware of every step of his thought “the sudden appearance in the consciousness of a ready-made solution to a problem that we could not solve for a long time,” the author writes, “we explain by unconscious thinking, which ... continued to engage in the task, ... and the result floats beyond the threshold of consciousness.” .

The author notes the specific nature of mathematical talent and mathematical thinking. He argues that the ability for mathematics is not always inherent even in brilliant people, that there is a difference between a mathematical and a non-mathematical mind.

Of great interest is Mordecai-Boltovsky’s attempt to isolate the components of mathematical abilities. He refers to such components, in particular:

1. “strong memory”, it was stipulated that this meant “mathematical memory”, memory for “a subject of the type with which mathematics deals”;

2. “wit,” which is understood as the ability to “embrace in one judgment” concepts from two poorly connected areas of thought, to find similarities with the given in what is already known;

3. speed of thought (speed of thought is explained by the work that unconscious thinking does in favor of conscious thinking).

D. Mordecai-Boltovsky also expresses his thoughts on the types of mathematical imagination that underlie different types of mathematicians - “geometers” and “algebraists”. “Arithmeticians, algebraists and analysts in general, whose discovery is made in the most abstract form of discontinuous quantitative symbols and their relationships, cannot express it like a geometer.” He also expressed valuable thoughts about the peculiarities of the memory of “geometers” and “algebraists.”

The theory of abilities was created over a long period of time by the joint work of the most prominent psychologists of that time: B.M. Teplov, L.S. Vygotsky, A.N. Leontyev, S.L. Rubinstein, B.G. Anafiev and others.

In addition to general theoretical studies of the problem of abilities, B.M. Teplov, with his monograph “Psychology of Musical Abilities,” laid the foundation for an experimental analysis of the structure of abilities for specific types of activities. The significance of this work goes beyond the narrow question of the essence and structure of musical abilities; it found a solution to the basic, fundamental questions of research into the problem of abilities for specific types of activities.

This work was followed by studies of abilities similar in idea: to visual activity - V.I. Kireenko and E.I. Ignatov, literary abilities - A.G. Kovalev, pedagogical abilities - N.V. Kuzmina and F.N. Gonobolin, design and technical abilities - P.M. Jacobson, N.D. Levitov, V.N. Kolbanovsky and mathematical abilities - V.A. Krutetsky.

A number of experimental studies of thinking were carried out under the leadership of A.N. Leontyev. Some issues of creative thinking were clarified, in particular, how a person comes to the idea of ​​solving a problem, the method of solving which does not directly follow from its conditions. An interesting pattern was established: the effectiveness of exercises leading to the correct solution varies depending on at what stage of solving the main problem auxiliary exercises are presented, i.e. the role of guiding exercises was shown.

A series of studies by L.N. is directly related to the problem of abilities. Landes. In one of the first works in this series - “On some shortcomings of studying students’ thinking” - he raises the question of the need to reveal the psychological nature, the internal mechanism of the “ability to think.” To cultivate abilities, according to L.N. Landa means “to teach the technique of thinking”, to form the skills of analytical and synthetic activity. In his other work - “Some Data on the Development of Mental Abilities” - L. N. Landa discovered significant individual differences in schoolchildren’s mastery of a new method of reasoning when solving geometric proof problems - differences in the number of exercises required to master this method, differences in the pace of work, differences in the formation of the ability to differentiate the use of operations depending on the nature of the task conditions and differences in the assimilation of operations.

Of great importance for the theory of mental abilities in general and mathematical abilities in particular are the studies of D.B. Elkonin and V.V. Davydova, L.V. Zankova, A.V. Skripchenko.

It is usually believed that the thinking of children 7-10 years old is figurative in nature and has a low ability for distraction and abstraction. Experiential learning conducted under the guidance of D.B. Elkonin and V.V. Davydov, showed that already in the first grade, with a special teaching method, it is possible to give students in alphabetic symbolism, i.e. in general form, a system of knowledge about the relationships of quantities, dependencies between them, to introduce them to the field of formal sign operations. A.V. Skripchenko showed that, under appropriate conditions, third- and fourth-grade students can develop the ability to solve arithmetic problems by composing an equation with one unknown.

1.3 Mathematical ability and personality

First of all, it should be noted that what characterizes capable mathematicians and is necessary for successful work in the field of mathematics is the “unity of inclinations and abilities in vocation”, expressed in a selective positive attitude towards mathematics, the presence of deep and effective interests in the relevant field, the desire and need to engage in it, passionate passion for the business.

Without a penchant for mathematics, there can be no genuine aptitude for it. If a student does not feel any inclination towards mathematics, then even good abilities are unlikely to ensure a completely successful mastery of mathematics. The role played here by inclination and interest boils down to the fact that a person interested in mathematics is intensively engaged in it, and, consequently, vigorously exercises and develops his abilities.

Numerous studies and characteristics of gifted children in the field of mathematics indicate that abilities develop only if there are inclinations or even a unique need for mathematical activity. The problem is that often students are capable of mathematics, but have little interest in it, and therefore do not have much success in mastering this subject. But if the teacher can arouse their interest in mathematics and the desire to do it, then such a student can achieve great success.

At school, such cases often occur: a student capable of mathematics has little interest in it, and does not show much success in mastering this subject. But if the teacher is able to awaken his interest in mathematics and the inclination to engage in it, then such a student, “captured” by mathematics, can quickly achieve great success.

From this follows the first rule of teaching mathematics: the ability to get students interested in science and encourage them to independently develop their abilities. The emotions experienced by a person are also an important factor in the development of abilities in any activity, not excluding mathematical activity. The joy of creativity, the feeling of satisfaction from intense mental work, mobilize his strength and force him to overcome difficulties. All children with an aptitude for mathematics are distinguished by a deep emotional attitude towards mathematical activity and experience real joy caused by each new achievement. Awakening the creative spirit in a student and teaching him to love mathematics is the second rule of a mathematics teacher.

Many teachers point out that the ability to quickly and deeply generalize can manifest itself in one subject without characterizing the student’s educational activity in other subjects. An example is that a child who is able to generalize and systematize material in literature does not show similar abilities in the field of mathematics.

Unfortunately, teachers sometimes forget that mental abilities, which are general in nature, in some cases act as specific abilities. Many teachers tend to use objective assessment, i.e. if a student is weak in reading, then in principle he cannot achieve heights in the field of mathematics. This opinion is typical for primary school teachers who teach a range of subjects. This leads to an incorrect assessment of the child's abilities, which in turn leads to a lag in mathematics.

1.4 Development of mathematical abilities in younger schoolchildren.

The problem of ability is a problem of individual differences. With the best organization of teaching methods, the student will progress more successfully and faster in one area than in another.

Naturally, success in learning is determined not only by the student’s abilities. In this sense, the content and methods of teaching, as well as the student’s attitude to the subject, are of key importance. Therefore, success and failure in learning do not always provide grounds for making judgments about the nature of the student’s abilities.

The presence of weak abilities in students does not relieve the teacher from the need, as far as possible, to develop the abilities of these students in this area. At the same time, there is an equally important task - to fully develop his abilities in the area in which he demonstrates them.

It is necessary to educate the capable and select the capable, while not forgetting about all schoolchildren, and to raise the overall level of their training in every possible way. In this regard, various collective and individual methods of work are needed in their work in order to intensify the activities of students.

The learning process should be comprehensive, both in terms of organizing the learning process itself, and in terms of developing in students a deep interest in mathematics, problem-solving skills, understanding the system of mathematical knowledge, solving with students a special system of non-standard problems that should be offered not only lessons, but also on tests. Thus, a special organization of the presentation of educational material and a well-thought-out system of tasks help to increase the role of meaningful motives for studying mathematics. The number of result-oriented students is decreasing.

In the lesson, not just problem solving, but the unusual way of solving problems used by students should be encouraged in every possible way; in this regard, special importance is placed not only on the result in solving the problem, but on the beauty and rationality of the method.

Teachers successfully use the “problem formulation” technique to determine the direction of motivation. Each task is assessed according to a system of the following indicators: the nature of the task, its correctness and relation to the source text. The same method is sometimes used in a different version: after solving the problem, students were asked to create any problems that were somehow related to the original problem.

To create psycho-pedagogical conditions for increasing the efficiency of organizing the learning process system, the principle of organizing the learning process in the form of substantive communication using cooperative forms of student work is used. This is group problem solving and collective discussion of grading, pair and team forms of work.

Chapter II. The development of mathematical abilities in primary schoolchildren as a methodological problem.

2.1 General characteristics of capable and talented children

The problem of developing children's mathematical abilities is one of the least developed methodological problems of teaching mathematics in primary school today.

The extreme heterogeneity of views on the very concept of mathematical abilities determines the absence of any conceptually sound methods, which in turn creates difficulties in the work of teachers. Perhaps this is why there is a widespread opinion not only among parents, but also among teachers: mathematical abilities are either given or not given. And there’s nothing you can do about it.

Of course, abilities for one or another type of activity are determined by individual differences in the human psyche, which are based on genetic combinations of biological (neurophysiological) components. However, today there is no evidence that certain properties of nerve tissue directly affect the manifestation or absence of certain abilities.

Moreover, targeted compensation for unfavorable natural inclinations can lead to the formation of a personality with pronounced abilities, of which there are many examples in history. Mathematical abilities belong to the group of so-called special abilities (as well as musical, visual, etc.). For their manifestation and further development, the assimilation of a certain stock of knowledge and the presence of certain skills are required, including the ability to apply existing knowledge in mental activity.

Mathematics is one of those subjects where the individual mental characteristics (attention, perception, memory, thinking, imagination) of a child are crucial for its mastery. Behind important characteristics of behavior, behind the success (or failure) of educational activities, those natural dynamic features mentioned above are often hidden. They often give rise to differences in knowledge—its depth, strength, and generality. Based on these qualities of knowledge, which relate (along with value orientations, beliefs, and skills) to the content side of a person’s mental life, children’s giftedness is usually judged.

Individuality and talent are interrelated concepts. Researchers dealing with the problem of mathematical abilities, the problem of the formation and development of mathematical thinking, despite all the differences in opinions, note, first of all, the specific features of the psyche of a mathematically capable child (as well as a professional mathematician), in particular, flexibility of thinking, i.e. unconventionality, originality, the ability to vary ways of solving a cognitive problem, ease of transition from one solution path to another, the ability to go beyond the usual way of activity and find new ways to solve a problem under changed conditions. It is obvious that these features of thinking directly depend on the special organization of memory (free and connected associations), imagination and perception.

Researchers identify such a concept as depth of thinking, i.e. the ability to penetrate into the essence of each fact and phenomenon being studied, the ability to see their relationships with other facts and phenomena, to identify specific, hidden features in the material being studied, as well as purposeful thinking, combined with breadth, i.e. the ability to form generalized methods of action, the ability to cover the whole problem without missing out on details. Psychological analysis of these categories shows that they should be based on a specially formed or natural inclination towards a structural approach to the problem and extremely high stability, concentration and a large amount of attention.

Thus, the individual typological characteristics of the personality of each student separately, by which we mean temperament, character, inclinations, and somatic organization of the personality as a whole, etc., have a significant (and maybe even decisive!) influence on the formation and the development of the child’s mathematical thinking style, which, of course, is a necessary condition for preserving the child’s natural potential (inclinations) in mathematics and its further development into pronounced mathematical abilities.

Experienced subject teachers know that mathematical abilities are a “piecemeal commodity,” and if such a child is not dealt with individually (individually, and not as part of a club or elective), then the abilities may not develop further.

That is why we often see how a first-grader with outstanding abilities “levels off” by the third grade, and in the fifth grade completely ceases to differ from other children. What is this? Research by psychologists shows that there may be different types of age-related mental development:

. “Early rise” (in preschool or primary school age) is due to the presence of bright natural abilities and inclinations of the corresponding type. In the future, consolidation and enrichment of mental qualities may occur, which will serve as a start for the development of outstanding mental abilities.

Moreover, the facts show that almost all scientists who distinguished themselves before the age of 20 were mathematicians.

But “alignment” with peers can also occur. We believe that this “leveling off” is largely due to the lack of a competent and methodologically active individual approach to the child in the early period.

“Slow and extended rise”, i.e. gradual accumulation of intelligence. The absence of early achievements in this case does not mean that the prerequisites for great or outstanding abilities will not emerge in the future. Such a possible “rise” is the age of 16-17 years, when the factor of “intellectual explosion” is the social reorientation of the individual, directing his activity in this direction. However, such a “rise” can also occur in more mature years.

For a primary school teacher, the most pressing problem is “early rise”, which occurs at the age of 6-9 years. It is no secret that one such brightly capable child in the class, who also has a strong type of nervous system, is capable, literally, of preventing any of the children from opening their mouths in class. And as a result, instead of maximally stimulating and developing the little “prodigy,” the teacher is forced to teach him to remain silent (!) and “keep his brilliant thoughts to himself until asked.” After all, there are 25 other children in the class! Such “slowing down,” if it occurs systematically, can lead to the fact that after 3-4 years the child “evens out” with his peers. And since mathematical abilities belong to the group of “early abilities,” then perhaps it is precisely the mathematically capable children that we lose in the process of this “slowing down” and “leveling off.”

Psychological research has shown that although the development of educational abilities and creative talent in typologically different children proceeds differently, children with opposite characteristics of the nervous system can achieve (achieve) an equally high degree of development of these abilities. In this regard, it may be more useful for the teacher to focus not on the typological characteristics of the nervous system of children, but on some general characteristics of capable and talented children, which are noted by most researchers of this problem.

Different authors identify a different “set” of general characteristics of capable children within the framework of the types of activities in which these abilities were studied (mathematics, music, painting, etc.). We believe that it is more convenient for a teacher to rely on some purely procedural characteristics of the activity of capable children, which, as shown by a comparison of a number of special psychological and pedagogical studies on this topic, turn out to be the same for children with different types of abilities and giftedness. Researchers note that most capable children have:

Increased propensity for mental action and a positive emotional response to any new mental challenge. These children don't know what boredom is - they always have something to do. Some psychologists generally interpret this trait as an age-related factor in giftedness.

The constant need to renew and complicate mental workload, which entails a constant increase in the level of achievement. If this child is not burdened, then he finds his own activity and can master chess, a musical instrument, radio, etc., study encyclopedias and reference books, read specialized literature, etc.

The desire to independently choose things to do and plan your activities. This child has his own opinion about everything, stubbornly defends the unlimited initiative of his activities, has high (almost always adequate) self-esteem and is very persistent in self-affirmation in his chosen field.

Perfect self-regulation. This child is capable of fully mobilizing forces to achieve a goal; able to repeatedly renew mental efforts in an effort to achieve a goal; has, as it were, an “initial” attitude towards overcoming any difficulties, and failures only force him to strive to overcome them with enviable tenacity.

Increased performance. Long-term intellectual stress does not tire this child; on the contrary, he feels good precisely in the situation of having a problem that requires a solution. Purely instinctively, he knows how to use all the reserves of his psyche and his brain, mobilizing and switching them at the right moment.

It is clearly seen that these general procedural characteristics of the activity of capable children, recognized by psychologists as statistically significant, are not uniquely inherent in any one type of human nervous system. Therefore, pedagogically and methodologically, the general tactics and strategy of an individual approach to a capable child should obviously be built on such psychological and didactic principles that ensure that the above-mentioned procedural characteristics of the activities of these children are taken into account.

From a pedagogical point of view, a capable child most of all needs an instructive style of relationship with a teacher, which requires greater information content and validity of the requirements put forward on the part of the teacher. The instructive style, as opposed to the imperative style that dominates in elementary school, involves appealing to the student’s personality, taking into account his individual characteristics and focusing on them. This style of relationship contributes to the development of independence, initiative and creative potential, which is noted by many teacher-researchers. It is equally obvious that, from a didactic point of view, capable children need, at a minimum, to ensure an optimal pace of progress in content and an optimal volume of learning load. Moreover, what is optimal for yourself, for your abilities, i.e. higher than for ordinary children. If we take into account the need for constant complication of mental workload, the persistent craving for self-regulation of their activities and the increased performance of these children, we can say with sufficient confidence that at school these children are by no means “prosperous” students, since their educational activities are constantly not carried out in zone of proximal development (!), and far behind this zone! Thus, in relation to these students, we (wittingly or unwittingly) constantly violate our proclaimed credo, the basic principle of developmental education, which requires teaching the child taking into account his zone of proximal development.

Working with capable children in primary school today is no less a “sick” problem than working with unsuccessful ones.

Its lesser “popularity” in special pedagogical and methodological publications is explained by its lesser “conspicuousness,” since a poor student is an eternal source of trouble for a teacher, and only the teacher (and not always), but Petya’s parents (if they deal with this issue specifically). At the same time, the constant “underload” of a capable child (and the norm for everyone is an underload for a capable child) will contribute to insufficient stimulation of the development of abilities, not only to the “non-use” of the potential of such a child (see points above), but also to the possible extinction of these abilities as unclaimed in educational activities (leading during this period of the child’s life).

There is also a more serious and unpleasant consequence of this: it is too easy for such a child to learn at the initial stage, as a result, he does not sufficiently develop the ability to overcome difficulties, does not develop immunity to failure, which largely explains the massive “collapse” in the performance of such children when transition from primary to secondary level.

In order for a public school teacher to successfully cope with working with a capable child in mathematics, it is not enough to identify the pedagogical and methodological aspects of the problem. As thirty years of practice in implementing a developmental education system have shown, in order for this problem to be solved in the conditions of teaching in a mass primary school, a specific and fundamentally new methodological solution is needed, fully presented to the teacher.

Unfortunately, today there are practically no special teaching aids for primary school teachers intended for working with capable and gifted children in mathematics lessons. We cannot cite a single such manual or methodological development, except for various collections such as the “Mathematical Box”. To work with capable and gifted children, you do not need entertaining tasks; this is too poor food for their minds! We need a special system and special “parallel” teaching aids to existing ones. The lack of methodological support for individual work with a capable child in mathematics leads to the fact that primary school teachers do not do this work at all (club or extracurricular work, where a group of children solves entertaining tasks with the teacher, which, as a rule, are not systematically selected, cannot be considered individual). One can understand the problems of a young teacher who does not have enough time or knowledge to select and systematize appropriate materials. But even an experienced teacher is not always ready to solve such a problem. Another (and, perhaps, the main!) limiting factor here is the presence of a single textbook for the entire class. Working according to a single textbook for all children, according to a single calendar plan, simply does not allow the teacher to implement the requirement of individualizing the pace of learning of a capable child, and the same content volume of the textbook for all children does not allow implementing the requirement of individualizing the volume of the educational load (not to mention the requirement of self-regulation and independent activity planning).

We believe that the creation of special teaching materials in mathematics for working with capable children is the only possible way to implement the principle of individualization of education for these children in the context of teaching a whole class.

2.2 Methodology for long-term assignments

The methodology for using the system of long-term assignments was considered by E.S. Rabunsky when organizing work with high school students in the process of teaching German at school.

A number of pedagogical studies have considered the possibility of creating systems of such tasks in various subjects for high school students, both to master new material and to eliminate knowledge gaps. In the course of research, it was noted that the vast majority of students prefer to perform both types of work in the form of “long-term tasks” or “delayed work.” This type of organization of educational activities, traditionally recommended mainly for labor-intensive creative work (essays, abstracts, etc.), turned out to be the most preferable for the majority of schoolchildren surveyed. It turned out that such “deferred work” satisfies the student more than individual lessons and assignments, since the main criterion for student satisfaction at any age is success at work. The absence of a sharp time limit (as happens in a lesson) and the possibility of freely returning to the content of the work many times allows you to cope with it much more successfully. Thus, tasks designed for long-term preparation can also be considered as a means of cultivating a positive attitude towards the subject.

For many years, it was believed that everything said applies only to older students, but does not correspond to the characteristics of the educational activities of primary school students. Analysis of the procedural characteristics of the activities of capable children of primary school age and the work experience of Beloshista A.V. and teachers who took part in the experimental testing of this methodology, showed the high efficiency of the proposed system when working with capable children. Initially, to develop a system of tasks (hereinafter we will call them sheets in connection with the form of their graphic design, convenient for working with a child), topics related to the formation of computational skills were selected, which are traditionally considered by teachers and methodologists as topics that require constant guidance at the stage acquaintance and constant monitoring at the consolidation stage.

During the experimental work, a large number of printed sheets were developed, combined into blocks covering an entire topic. Each block contains 12-20 sheets. The worksheet is a large system of tasks (up to fifty tasks), methodically and graphically organized in such a way that as they are completed, the student can independently approach the understanding of the essence and method of performing a new computational technique, and then consolidate the new way of activity. A worksheet (or a system of sheets, i.e. a thematic block) is a “long-term task”, the deadlines for which are individualized in accordance with the desires and capabilities of the student working on this system. Such a sheet can be offered in class or instead of homework in the form of a task with a “delayed deadline” for completion, which the teacher either sets individually or allows the student (this path is more productive) to set a deadline for himself (this is a way to form self-discipline, since independent planning of activities in connection with independently determined goals and deadlines is the basis of human self-education).

The teacher determines tactics for working with worksheets for the student individually. At first, they can be offered to the student as homework (instead of a regular assignment), individually agreeing on the timing of its completion (2-4 days). As you master this system, you can move on to the preliminary or parallel method of work, i.e. give the student a sheet before learning the topic (on the eve of the lesson) or during the lesson itself for independent mastery of the material. Attentive and friendly observation of the student in the process of activity, “contractual style” of relationships (let the child decide for himself when he wants to receive this sheet), perhaps even exemption from other lessons on this or the next day to concentrate on the task, advisory assistance (on one question can always be answered immediately when passing a child in class) - all this will help the teacher to fully individualize the learning process of a capable child without spending a lot of time.

Children should not be forced to copy assignments from the sheet. The student works with a pencil on a sheet of paper, writing down answers or completing actions. This organization of learning evokes positive emotions in the child - he likes to work on a printed basis. Freed from the need for tedious copying, the child works with greater productivity. Practice shows that although the worksheets contain up to fifty tasks (the usual homework norm is 6-10 examples), the student enjoys working with them. Many children ask for a new sheet every day! In other words, they exceed the work quota for the lesson and homework several times over, while experiencing positive emotions and working at their own discretion.

During the experiment, such sheets were developed on the topics: “Oral and written calculation techniques”, “Numbering”, “Quantities”, “Fractions”, “Equations”.

Methodological principles for constructing the proposed system:

1. The principle of compliance with the mathematics program for primary grades. The content of the sheets is tied to a stable (standard) mathematics program for primary grades. Thus, we believe it is possible to implement the concept of individualizing mathematics teaching for a capable child in accordance with the procedural features of his educational activities when working with any textbook that corresponds to the standard program.

2. Methodically, the principle of dosage is implemented in each sheet, i.e. in one sheet only one technique or one concept is introduced, or one connection, but essential for a given concept, is revealed. This, on the one hand, helps the child clearly understand the purpose of the work, and on the other hand, helps the teacher to easily monitor the quality of mastery of this technique or concept.

3. Structurally, the sheet represents a detailed methodological solution to the problem of introducing or introducing and consolidating one or another technique, concept, connections of this concept with other concepts. The tasks are selected and grouped (i.e., the order in which they are placed on the sheet matters) in such a way that the child can “move” along the sheet independently, starting from the simplest methods of action already familiar to him, and gradually master a new method, which in the first steps fully revealed in smaller actions that are the basis of this technique. As you move through the sheet, these small actions are gradually assembled into larger blocks. This allows the student to master the technique as a whole, which is the logical conclusion of the entire methodological “construction”. This structure of the sheet allows you to fully implement the principle of a gradual increase in the level of complexity at all stages.

4. This structure of the worksheet also makes it possible to implement the principle of accessibility, and to a much deeper extent than can be done today when working only with a textbook, since the systematic use of sheets allows you to learn the material at an individual pace that is convenient for the student, which the child can regulate independently.

5. The system of sheets (thematic block) allows you to implement the principle of perspective, i.e. gradual inclusion of the student in the activities of planning the educational process. Tasks designed for long-term (delayed) preparation require long-term planning. The ability to organize your work, planning it for a certain period of time, is the most important educational skill.

6. The system of worksheets on the topic also makes it possible to implement the principle of individualization of testing and assessing students’ knowledge, not on the basis of differentiating the level of difficulty of tasks, but on the basis of the unity of requirements for the level of knowledge, skills and abilities. Individualized deadlines and methods for completing tasks make it possible to present all children with tasks of the same level of complexity, corresponding to the program requirements for the norm. This does not mean that talented children should not be held to higher standards. Worksheets at a certain stage allow such children to use material that is more intellectually rich, which in a propaedeutic sense will introduce them to the following mathematical concepts of a higher level of complexity.

Conclusion

An analysis of psychological and pedagogical literature on the problem of the formation and development of mathematical abilities shows: without exception, all researchers (both domestic and foreign) connect it not with the content side of the subject, but with the procedural side of mental activity.

Thus, many teachers believe that the development of a child’s mathematical abilities is only possible if there are significant natural abilities for this, i.e. Most often in teaching practice it is believed that abilities need to be developed only in those children who already have them. But experimental research by Beloshistaya A.V. showed that work on the development of mathematical abilities is necessary for every child, regardless of his natural talent. It’s just that the results of this work will be expressed in different degrees of development of these abilities: for some children this will be a significant advance in the level of development of mathematical abilities, for others it will be a correction of natural deficiencies in their development.

The great difficulty for a teacher when organizing work on the development of mathematical abilities is that today there is no specific and fundamentally new methodological solution that can be presented to the teacher in full. The lack of methodological support for individual work with capable children leads to the fact that primary school teachers do not do this work at all.

With my work, I wanted to draw attention to this problem and emphasize that the individual characteristics of each gifted child are not only his characteristics, but, perhaps, the source of his giftedness. And the individualization of such a child’s education is not only a way of his development, but also the basis for his preservation in the status of “capable, gifted.”

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The new paradigm of education in the Russian Federation is characterized by a personality-oriented approach, the idea of ​​developmental education, the creation of conditions for self-organization and self-development of the individual, the subjectivity of education, the focus on designing the content, forms and methods of teaching and upbringing that ensure the development of each student, his cognitive abilities and personal qualities.

The concept of school mathematical education highlights its main goals - teaching students the techniques and methods of mathematical knowledge, developing in them the qualities of mathematical thinking, corresponding mental abilities and skills. The importance of this line of work is enhanced by the increasing importance and application of mathematics in various fields of science, economics and industry.

The need for mathematical development of younger schoolchildren in educational activities is noted by many leading Russian scientists (V.A. Gusev, G.V. Dorofeev, N.B. Istomina, Yu.M. Kolyagin, L.G. Peterson, etc.). This is due to the fact that during the preschool and primary school period, the child not only intensively develops all mental functions, but also lays the general foundation of the cognitive abilities and intellectual potential of the individual. Numerous facts indicate that if the corresponding intellectual or emotional qualities for one reason or another do not receive proper development in early childhood, then subsequently overcoming such shortcomings turns out to be difficult and sometimes impossible (P.Ya. Galperin, A.V. Zaporozhets , S.N. Karpova).

Thus, the new paradigm of education, on the one hand, presupposes the maximum possible individualization of the educational process, and on the other, requires solving the problem of creating educational technologies that ensure the implementation of the main provisions of the Concept of School Mathematics Education.

In psychology, the term “development” is understood as consistent, progressive significant changes in the psyche and personality of a person, manifesting themselves as certain new formations. The position on the possibility and feasibility of education focused on the development of the child was substantiated back in the 1930s. outstanding Russian psychologist L.S. Vygotsky.

One of the first attempts to practically implement the ideas of L.S. Vygotsky in our country was undertaken by L.V. Zankov, who in the 1950-1960s. developed a fundamentally new system of primary education, which found a large number of followers. In the L.V. system Zankov, for the effective development of students’ cognitive abilities, the following five basic principles are implemented: learning at a high level of difficulty; the leading role of theoretical knowledge; moving forward at a fast pace; conscious participation of schoolchildren in the educational process; systematic work on the development of all students.

Theoretical (rather than traditional empirical) knowledge and thinking, educational activity were placed at the forefront by the authors of another theory of developmental education - D.B. Elkonin and V.V. Davydov. They considered the most important thing to change the student's position in the learning process. Unlike traditional education, where the student is the object of the teacher’s pedagogical influences, in developmental education conditions are created under which he becomes the subject of learning. Today, this theory of educational activity is recognized throughout the world as one of the most promising and consistent in terms of implementing the well-known provisions of L.S. Vygotsky about the developmental and anticipatory nature of learning.

In domestic pedagogy, in addition to these two systems, the concepts of developmental education by Z.I. Kalmykova, E.N. Kabanova-Meller, G.A. Tsukerman, S.A. Smirnova and others. It should also be noted the extremely interesting psychological searches of P.Ya. Galperin and N.F. Talyzina based on the theory they created of the stage-by-stage formation of mental actions. However, as noted by V.A. Tests, in most of the mentioned pedagogical systems, the development of the student is still the responsibility of the teacher, and the role of the former is reduced to following the developmental influence of the latter.

In line with developmental education, many different programs and teaching aids in mathematics have appeared, both for primary grades (textbooks by E.N. Alexandrova, I.I. Arginskaya, N.B. Istomina, L.G. Peterson, etc.), and for secondary school (textbooks by G.V. Dorofeev, A.G. Mordkovich, S.M. Reshetnikov, L.N. Shevrin, etc.). Textbook authors have different understandings of personality development in the process of learning mathematics. Some focus on the development of observation, thinking and practical actions, others - on the formation of certain mental actions, others - on creating conditions that ensure the formation of educational activities and the development of theoretical thinking.

It is clear that the problem of developing mathematical thinking in teaching mathematics at school cannot be solved only by improving the content of education (even with good textbooks), since the implementation of different levels in practice requires the teacher to have a fundamentally new approach to organizing students’ learning activities in the classroom , in home and extracurricular work, allowing him to take into account the typological and individual characteristics of students.

It is known that primary school age is sensitive and most favorable for the development of cognitive mental processes and intelligence. Developing students' thinking is one of the main tasks of primary school. It is on this psychological feature that we concentrated our efforts, relying on the psychological and pedagogical concept of the development of thinking by D.B. Elkonin, position of V.V. Davydov on the transition from empirical to theoretical thinking in the process of specially organized educational activities, based on the works of R. Atakhanov, L.K. Maksimova, A.A. Stolyara, P. - H. van Hiele, related to identifying the levels of development of mathematical thinking and their psychological characteristics.

The idea of ​​L.S. Vygotsky’s idea that learning should be carried out in the zone of proximal development of students, and its effectiveness is determined by which zone (large or small) it prepares, is well known to everyone. At the theoretical (conceptual) level, it is shared almost throughout the world. The problem lies in its practical implementation: how to define (measure) this zone and what should be the teaching technology so that the process of learning the scientific foundations and mastering (“appropriating”) human culture takes place in it, providing the maximum developmental effect?

Thus, psychological and pedagogical science has substantiated the expediency of mathematical development of younger schoolchildren, but the mechanisms for its implementation have not been sufficiently developed. Consideration of the concept of “development” as a result of learning from a methodological point of view shows that it is an integral continuous process, the driving force of which is the resolution of contradictions that arise in the process of change. Psychologists argue that the process of overcoming contradictions creates conditions for development, as a result of which individual knowledge and skills develop into a new holistic formation, into a new ability. Therefore, the problem of constructing a new concept of mathematical development for younger schoolchildren is determined by contradictions:

between the need for a high level of mathematical development for modern man and the inadequacy of the integral system of the process of teaching mathematics in primary school to this task;

between the discrete nature of the education system and the need to create a holistic picture of the world in the child’s mind;

between the basic postulate of the theory of developmental education, which posits the essence of a child’s personality as a “self-developing system” emerging in the educational process, amenable to controlled processes of formation and development, through the use of developmental education technologies, and the absence of such technologies in primary school mathematics education;

between the need for mathematics teachers to use an activity-based approach to teaching and their practical unpreparedness for such teaching, for thoughtful joint activity between teacher and student in the “zone of proximal development.”

Summarizing the above, it can be argued that the problem of mathematical development of younger schoolchildren is undoubtedly relevant and requires, for its solution, the expansion of general approaches, going beyond the framework of “pure didactics”, taking into account modern achievements not only in the field of psychology and physiology, creating a general concept of the formation and development of students' mathematical thinking on a broader theoretical basis than is currently accepted.

The purpose of our research was to build, on the basis of the dominant individual-typological characteristics of thinking, a concept of mathematical development, which would ensure the continuity of mathematical education at the preschool, primary school level and in grades V-VI of the basic school, its continuity and improving the quality of mathematical training of a child of primary school age , as well as in the development and testing of its applied aspect in the form of educational technology (methods, tools, forms).

We formulate the main provisions of the concept of mathematical development of a child of primary school age as follows.

1. The starting point is the concept of educational and mathematical activity, which should be characterized by a set of interconnected main components and qualities of the child’s mathematical thinking and his abilities for mathematical knowledge of reality. In the process of all educational and mathematical activities at school, such mental actions as analysis, planning, and reflection should be formed, which ensure mastery of generalized methods of solving mathematical problems.

Let's consider the purpose of studying the course “Methods of teaching mathematics in primary school” in the process of preparing a future primary school teacher.

Lecture discussion with students

2. Methods of teaching mathematics to junior schoolchildren as a pedagogical science and as a field of practical activity

Considering the methodology of teaching mathematics to primary schoolchildren as a science, it is necessary, first of all, to determine its place in the system of sciences, outline the range of problems that it is designed to solve, and determine its object, subject and features.

In the system of sciences, methodological sciences are considered in the block didactics. As is known, didactics is divided into theory education Andtheory training. In turn, in the theory of learning, general didactics (general issues: methods, forms, means) and particular didactics (subject-specific) are distinguished. Private didactics are called differently - teaching methods or, as has become common in recent years - educational technologies.

Thus, methodological disciplines belong to the pedagogical cycle, but at the same time, they represent purely subject areas, since the methods of teaching literacy will certainly be very different from the methods of teaching mathematics, although both of them are private didactics.

The methodology of teaching mathematics to primary schoolchildren is a very ancient and very young science. Learning to count and calculate was a necessary part of education in ancient Sumerian and ancient Egyptian schools. Rock paintings from the Paleolithic era tell stories about learning to count. The first textbooks for teaching children mathematics include Magnitsky’s “Arithmetic” (1703) and the book by V.A. Laya “Guide to the initial teaching of arithmetic, based on the results of didactic experiments” (1910)... In 1935, SI. Shokhor-Trotsky wrote the first textbook “Methods of Teaching Mathematics”. But only in 1955, the first book “The Psychology of Teaching Arithmetic” appeared, the author of which was N.A. Menchinskaya turned not so much to the characteristics of the mathematical specifics of the subject, but to the patterns of mastering arithmetic content by a child of primary school age. Thus, the emergence of this science in its modern form was preceded not only by the development of mathematics as a science, but also by the development of two large areas of knowledge: general didactics of learning and the psychology of learning and development. Recently, the psychophysiology of child brain development has begun to play an important role in the development of teaching methods. At the intersection of these areas, answers to three “eternal” questions in the methodology of teaching subject content are being born today:

Why teach? What is the purpose of teaching math to a young child? Is this necessary? And if necessary, then why?

What to teach? What content should be taught? What should be the list of mathematical concepts to be taught to your child? Are there any criteria for selecting this content, a hierarchy of its construction (sequence) and how are they justified?

How to teach? What ways of organizing a child’s activities (methods, techniques, means, forms of teaching) should be selected and applied so that the child can usefully assimilate the selected content? What is meant by “benefit”: the amount of knowledge and skills of the child or something else? How to take into account the psychological characteristics of age and individual differences of children when organizing training, but at the same time “fit” into the allotted time (curriculum, program, daily routine), and also take into account the actual filling of the class in connection with the collective system adopted in our country training (classroom-lesson system)?

These questions actually determine the range of problems of any methodological science. The methodology of teaching mathematics to junior schoolchildren as a science, on the one hand, is addressed to specific content, selection and ordering of it in accordance with the set learning goals, on the other hand, to the pedagogical methodological activity of the teacher and the educational (cognitive) activity of the child in the lesson, to the process of mastering the selected material. content managed by the teacher.

Object of study of this science - the process of mathematical development and the process of forming mathematical knowledge and ideas of a child of primary school age, in which the following components can be distinguished: the purpose of teaching (Why teach?), content (What to teach?) and the activity of the teacher and the activity of the child (How to teach?) . These components form methodological systemmu, in which a change in one of the components will cause a change in the other. The modifications of this system that resulted from a change in the purpose of primary education due to a change in the educational paradigm in the last decade were discussed above. Later we will consider the modifications of this system that entail psychological, pedagogical and physiological research of the last half century, the theoretical results of which gradually penetrate into methodological science. It can also be noted that an important factor in changing approaches to constructing a methodological system is changing the views of mathematicians on defining a system of basic postulates for constructing a school mathematics course. For example, in 1950-1970. The prevailing belief was that the set-theoretic approach should be the basis for constructing a school mathematics course, which was reflected in the methodological concepts of school mathematics textbooks, and therefore required an appropriate focus of initial mathematical training. In recent decades, mathematicians have increasingly talked about the need to develop functional and spatial thinking in schoolchildren, which is reflected in the content of textbooks published in the 90s. In accordance with this, the requirements for a child’s initial mathematical preparation are gradually changing.

Thus, the process of development of methodological sciences is closely connected with the process of development of other pedagogical, psychological and natural sciences.

Let's consider the relationship between the methods of teaching mathematics in elementary school and other sciences.

1. The method of mathematical development of a child uses OSnew ideas, theoretical principles and research resultsknowledge of other sciences.

For example, philosophical and pedagogical ideas play a fundamental and guiding role in the process of developing a methodological theory. In addition, borrowing ideas from other sciences can serve as the basis for the development of specific methodological technologies. Thus, the ideas of psychology and the results of its experimental research are widely used by the methodology to substantiate the content of training and the sequence of its study, to develop methodological techniques and systems of exercises that organize children’s assimilation of various mathematical knowledge, concepts and ways of acting with them. Physiological ideas about conditioned reflex activity, two signaling systems, feedback and age-related stages of maturation of the subcortical zones of the brain help to understand the mechanisms of acquisition of skills, abilities and habits in the learning process. Of particular importance for the development of methods of teaching mathematics in recent decades are the results of psychological and pedagogical research and theoretical research in the field of constructing the theory of developmental learning (L.S. Vygotsky, J. Piaget, L.V. Zankov, V.V. Davydov, D. B. Elkonin, P.Ya. Galperin, N.N. Poddyakov, L.A. Wenger, etc. This theory is based on the position of L.S. Vygotsky that learning is built not only on completed cycles of child development, but primarily on those mental functions that have not yet matured (“zones of proximal development”). Such training contributes to the effective development of the child.

2. The methodology creatively borrows research methods, withchanged in other sciences.

In fact, any method of theoretical or empirical research can find application in methodology, since in the conditions of integration of sciences, research methods very quickly become general scientific. Thus, the method of literature analysis familiar to students (composing bibliographies, taking notes, summarizing, drawing up theses, plans, writing out quotations, etc.) is universal and is used in any science. The method of analyzing programs and textbooks is commonly used in all didactic and methodological sciences. From pedagogy and psychology, the methodology borrows the method of observation, questioning, and conversation; from mathematics - methods of statistical analysis, etc.

3. The methodology uses specific research resultspsychology, physiology of higher nervous activity, mathematicski and other sciences.

For example, the specific results of J. Piaget’s research into the process of young children’s perception of the conservation of quantity gave rise to a whole series of specific mathematical tasks in various programs for primary schoolchildren: using specially designed exercises, the child is taught to understand that changing the shape of an object does not entail a change in its quantity (for example, when pouring water from a wide jar into a narrow bottle, its visually perceived level increases, but this does not mean that there is more water in the bottle than there was in the jar).

4. The technique is involved in complex development studieschild in the process of his education and upbringing.

For example, in 1980-2002. A number of scientific studies have appeared on the process of personal development of a child of primary school age in the course of teaching him mathematics.

Summarizing the question of the connection between the methods of mathematical development and the formation of mathematical concepts in preschoolers, we can note the following:

It is impossible to derive a system of methodological knowledge and methodological technologies from any one science;

Data from other sciences are necessary for the development of methodological theory and practical guidelines;

The technique, like any science, will develop if it is replenished with more and more new facts;

The same facts or data can be interpreted and used in different (and even opposite) ways, depending on what goals are realized in the educational process and what system of theoretical principles (methodology) is adopted in the concept;

The methodology does not simply borrow and use data from other sciences, but processes them in order to develop ways to optimally organize the learning process;

The methodology is determined by the corresponding concept of the child’s mathematical development; Thus, concept - This is not something abstract, far from life and real educational practice, but a theoretical basis that determines the construction of the totality of all components of the methodological system: goals, content, methods, forms and means of teaching.

Let us consider the relationship between modern scientific and “everyday” ideas about teaching mathematics to primary schoolchildren.

The basis of any science is the experience of people. For example, physics relies on the knowledge we acquire in everyday life about the movement and fall of bodies, about light, sound, heat and much more. Mathematics also proceeds from ideas about the shapes of objects in the surrounding world, their location in space, quantitative characteristics and relationships between parts of real sets and individual objects. The first harmonious mathematical theory - Euclid's geometry (IV century BC) was born from practical land surveying.

The situation is completely different with the methodology. Each of us has a store of life experience in teaching someone something. However, it is possible to engage in the mathematical development of a child only with special methodological knowledge. With what differ special (scientific) methodological knowledgeand skills from life Thayan ideas that to teach mathematics to a primary school student, it is enough to have some understanding of counting, calculations and solving simple arithmetic problems?

1. Everyday methodological knowledge and skills are specific; they are dedicated to specific people and specific tasks. For example, a mother, knowing the peculiarities of her child’s perception, through repeated repetitions teaches the child to name numerals in the correct order and recognize specific geometric figures. If the mother is persistent enough, the child learns to name numerals fluently, recognizes a fairly large number of geometric shapes, recognizes and even writes numbers, etc. Many people believe that this is exactly what the child should be taught before going to school. Does this training guarantee the development of a child's mathematical abilities? Or at least this child’s continued success in math? Experience shows that it does not guarantee. Will this mother be able to teach the same to another child who is different from her child? Unknown. Will this mother be able to help her child learn other math material? Most likely not. Most often, you can observe a picture when the mother herself knows, for example, how to add or subtract numbers, solve this or that problem, but cannot even explain to her child so that he learns the solution method. Thus, everyday methodological knowledge is characterized by specificity, limitation of the task, situations and persons to which it applies,

Scientific methodological knowledge (knowledge of educational technology) tends to to generality. They use scientific concepts and generalized psychological and pedagogical principles. Scientific methodological knowledge (educational technologies), consisting of clearly defined concepts, reflects their most significant relationships, which makes it possible to formulate methodological patterns. For example, an experienced, highly professional teacher can often determine by the nature of a child’s mistake which methodological patterns in the formation of a given concept were violated when teaching this child.

2. Everyday methodological knowledge is of an intuitive natureter. This is due to the method of obtaining them: they are acquired through practical trials and “adjustments”. A sensitive, attentive mother follows this path, experimenting and vigilantly noticing the slightest positive results (which is not difficult to do after spending a lot of time with the child. Often the subject “mathematics” itself leaves specific imprints on the perception of parents. You can often hear: “I myself struggled with mathematics at school , he has the same problems. It’s hereditary for us." Or vice versa: “I didn’t have any problems with mathematics at school, I don’t understand who he was born like that!” no, and nothing can be done about it. The idea that mathematical abilities (as well as musical, visual, sports and others) can be developed and improved by most people is perceived with skepticism. This position is very convenient for justifying doing nothing, but from a general methodological point of view. scientific knowledge about the nature, character and genesis of a child’s mathematical development, it is, of course, inadequate.

We can say that, in contrast to intuitive methodological knowledge, scientific methodological knowledge rational And conscious. A professional methodologist will never blame heredity, “planidas”, lack of materials, poor quality of teaching aids and insufficient attention of parents to the child’s educational problems. He has a fairly large arsenal of effective methodological techniques; you just need to select from it those that are most suitable for a given child.

Scientific methodological knowledge can be transferred to anotherto a person. The accumulation and transfer of scientific methodological knowledge is possible due to the fact that this knowledge is crystallized in concepts, patterns, methodological theories and recorded in scientific literature, educational and methodological manuals that future teachers read, which allows them to come even to their first practice in their lives with sufficient a large amount of generalized methodological knowledge.

Everyday knowledge about teaching methods and techniques is gainedusually through observation and reflection. In scientific activity, these methods are supplemented methodical experiment. The essence of the experimental method is that the teacher does not wait for a combination of circumstances as a result of which the phenomenon of interest to him arises, but causes the phenomenon himself, creating the appropriate conditions. Then he purposefully varies these conditions in order to identify the patterns to which this phenomenon obeys. This is how any new methodological concept or methodological pattern is born. We can say that when creating a new methodological concept, each lesson becomes a methodological experiment.

5. Scientific methodological knowledge is much broader, more diverse,than worldly things; it possesses unique factual material, inaccessible in its volume to any bearer of everyday methodological knowledge. This material is accumulated and comprehended in separate sections of the methodology, for example: methods of teaching problem solving, methods of forming the concept of a natural number, methods of forming ideas about fractions, methods of forming ideas about quantities, etc., as well as in certain branches of methodological science, for example : teaching mathematics in groups for correction of mental retardation, teaching mathematics in compensation groups (visually impaired, hearing impaired, etc.), teaching mathematics to children with mental retardation, teaching schoolchildren capable of mathematics, etc.

The development of special branches of methods for teaching mathematics to young children is in itself the most effective method of general didactics for teaching mathematics. L.S. Vygotsky began working with mentally retarded children - and as a result, the theory of “zones of proximal development” was formed, which formed the basis of the theory of developmental education for all children, including teaching mathematics.

One should not think, however, that everyday methodological knowledge is an unnecessary or harmful thing. The “golden mean” is to see small facts as reflections of general principles, and how to move from general principles to real life problems is not written in any book. Only constant attention to these transitions and constant practice in them can form in the teacher what is called “methodological intuition.” Experience shows that the more everyday methodological knowledge a teacher has, the greater the likelihood of forming this intuition, especially if this rich everyday methodological experience is constantly accompanied by scientific analysis and comprehension.

The methodology for teaching mathematics to primary schoolchildren is applied field of knowledge(applied Science). As a science, it was created to improve the practical activities of teachers working with children of primary school age. It was already noted above that the methodology of mathematical development as a science is actually taking its first steps, although the methodology of teaching mathematics has a thousand-year history. Today there is not a single primary (and preschool) education program that does without mathematics. But until recently, it was only about teaching young children the elements of arithmetic, algebra and geometry. And only in the last twenty years of the 20th century. began to talk about a new methodological direction - theory and practice mathematical development child.

This direction became possible in connection with the emergence of the theory of developmental education for young children. This direction in traditional methods of teaching mathematics is still debatable. Not all teachers today support the need to implement developmental education in progress teaching mathematics, the purpose of which is not so much the formation in the child of a certain list of knowledge, abilities and skills of a subject nature, but rather the development of higher mental functions, his abilities and the disclosure of the child’s internal potential.

For a progressively thinking teacher, it is obvious that practicallywhat results from the development of this methodological direction should become incommensurably more significant than the results of simply teaching methods of teaching primary mathematical knowledge and skills to children of primary school age, in addition, they should be qualitatively different. After all, to know something means to master this “something”, to learn it manage.

Learning to manage the process of mathematical development (i.e., the development of a mathematical style of thinking) is, of course, a grandiose task that cannot be solved overnight. The methodology has already accumulated a lot of facts showing that the teacher’s new knowledge about the essence and meaning of the learning process makes it significantly different: it changes his attitude both to the child and to the content of teaching, and to the methodology. By learning the essence of the process of mathematical development, the teacher changes his attitude to the educational process (changes himself!), to the interaction of the subjects of this process, to its meaning and goals. It can be said that methodology is science,constructing teacher as a subject of educational interaction. In real practical activities today, this is reflected in modifications in the forms of work with children: teachers are paying more and more attention to individual work, since the effectiveness of the learning process is obviously determined by the individual differences of children. Teachers are paying more and more attention to productive methods of working with children: search and partial search, children's experimentation, heuristic conversation, organizing problem situations in lessons. Further development of this direction may lead to significant substantive modifications in mathematics education programs for primary schoolchildren, since many psychologists and mathematicians in recent decades have expressed doubts about the correctness of the traditional content of primary school mathematics programs primarily with arithmetic material.

There is no doubt about the fact that child learning process in mathematics is constructive for its development personalities . The process of teaching any subject content leaves its mark on the development of the child’s cognitive sphere. However, the specificity of mathematics as an academic subject is such that its study can significantly influence the overall personal development of the child. 200 years ago this idea was expressed by M.V. Lomonosov: “Mathematics is good because it puts the mind in order.” The formation of systematic thought processes is only one side of the development of a mathematical style of thinking. Deepening the knowledge of psychologists and methodologists about the various aspects and properties of human mathematical thinking shows that many of its most important components actually coincide with the components of such a category as general human intellectual abilities - these are logic, breadth and flexibility of thinking, spatial mobility, laconicism and consistency, etc. And such character traits as determination, perseverance in achieving a goal, the ability to organize oneself, “intellectual endurance”, which are formed through active mathematics, are already personal characteristics of a person.

Today, there are a number of psychological studies showing that a systematic and specially organized system of mathematics classes actively influences the formation and development of an internal action plan, reduces the child’s level of anxiety, developing a sense of confidence and mastery of the situation; increases the level of development of creativity (creative activity) and the general level of mental development of the child. All of these studies support the idea that math content is powerful means of development intelligence and a means of personal development of the child.

Thus, theoretical research in the field of methods of mathematical development of a child of primary school age, refracted through a set of methodological techniques and the theory of developmental education, is implemented when teaching specific mathematical content in the practical activities of the teacher in the classroom.

Lecture 3.Traditional and alternative systems of teaching mathematics to primary schoolchildren

Brief overview of training systems.

Features of the acquisition of mathematical knowledge, skills and abilities by students with severe speech impairments.