home Audience 1 Developed mathematical abilities are classified in psychology. General diagram of the structure of mathematical abilities at school age according to V. A. Krutetsky. What is spatial thinking?

Developed mathematical abilities are classified in psychology. General diagram of the structure of mathematical abilities at school age according to V. A. Krutetsky. What is spatial thinking?

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Analysis of abilities makes it necessary to distinguish between the concepts of abilities, on the one hand, and abilities and skills, on the other. These categories are interconnected and interdependent. S.L. Rubinstein wrote about “a peculiar dialectic between abilities and skills.” On the one hand, in the process of acquiring knowledge, skills and abilities, abilities develop. Their formation and development is impossible outside of this process. On the other hand, abilities allow you to quickly, easily and deeply master relevant knowledge, skills and abilities.

We believe that the real close connection and interdependence of abilities and skills does not “close” the possibility of differentiating these categories. Just as it would be wrong to tear them apart, it would also be wrong to identify them.

How to distinguish abilities from abilities and skills? The definition of the concept of “ability” is based on the characteristics of a person’s individual psychological characteristics. On the other hand, all definitions of skills are based on the concept of activity. A.N. Leontyev speaks of skill as the expedient execution of actions. This is the difference: when they talk about abilities, they mean the psychological characteristics of a person in activity, when they talk about abilities (skills), they mean the psychological characteristics of a person’s activity.

All this gives grounds to differentiate these concepts as follows. Abilities are understood as individual psychological characteristics of a person that are conducive to mastering a certain, for example, mathematical activity, mastering relevant skills and abilities; skills and abilities are understood as specific acts of activity (for example, mathematical), which are carried out by a person at a relatively high level (this concept comes from the analysis of this specific activity).

It must be emphasized that when analyzing both skills, abilities, and abilities, activity is analyzed. Both the presence of abilities and the presence of skills must be judged by the characteristics of a person’s performance of the corresponding (for example, mathematical) activity.

Classification of human abilities.

In the theory of abilities, first of all, natural, or natural and social human abilities that have a socio-historical origin are distinguished.

Natural abilities include such elementary abilities as perception, memory, thinking, and the ability for elementary communications at the level of expression.

Social abilities include general and special higher intellectual abilities.

General abilities include those that determine a person’s success in a wide variety of activities. These, for example, include mental abilities, subtlety and accuracy of manual movements, developed memory, perfect speech and a number of others. Special abilities determine a person’s success in specific types of activities, the implementation of which requires inclinations of a special kind and their development. Such abilities include musical, mathematical, linguistic, technical, literary, artistic and creative, sports and a number of others.

The presence of general abilities in a person does not exclude the development of special ones and vice versa. Often general and special abilities coexist, mutually complementing and enriching each other.

Depending on the activity that a person performs, special abilities can be classified as:

1) Theoretical and practical abilities. These abilities differ in that the former predetermine a person’s inclination to abstract theoretical thinking, and the latter to concrete, practical actions. Such abilities, unlike general and special ones, often do not combine with each other, occurring together only in gifted, multi-talented people.

2) Abilities for communication, interaction with people, as well as subject-activity, or subject-cognitive, abilities. They are most socially conditioned. Examples of abilities of the first type include human speech as a means of communication (speech in its communicative function), the ability of interpersonal perception and evaluation of people, the ability of socio-psychological adaptation to various situations, the ability to come into contact with different people, to win them over, influence them, etc.

3) Educational and creative are different from each other according to R.S. Nemov in that the former determine the success of training and education, a person’s assimilation of knowledge, abilities, skills, the formation of personal qualities, while the latter determine the creation of objects of material and spiritual culture, the production of new ideas, discoveries and inventions, in a word - individual creativity in various areas of human activity. But it seems to us that the difference between the two abilities is not absolute. When studying the mathematical abilities of schoolchildren, we mean not just learning ability.

In our study, we will talk about the educational abilities of schoolchildren, but also about creative educational abilities associated with independent creative mastery of mathematics in school conditions, with independent formulation of simple mathematical problems and finding ways and methods for solving them, inventing proofs, independent deriving formulas. All this is undoubtedly also a manifestation of mathematical creativity. If the criterion of mathematical thinking itself is the presence of a creative principle, then we must not forget that mathematical creativity can be not only objective, but also subjective.

Establishing specific criteria that distinguish a creative thought process from a non-creative one, A. Newell, D. Shaw and G. Simon note the following signs of creative thinking:

1) the product of mental activity has novelty and value in both the subjective and objective sense;

the thought process is also novel in the sense that it requires the transformation of previously accepted ideas or the abandonment of them.

The creative thought process is characterized by strong motivation and persistence, occurring either over a significant period of time or with great intensity.

Abilities and successful performance of activities

The success of any activity is determined not by individual abilities, but only by their successful combination, exactly what is necessary for this activity. There is practically no activity in which success is determined by only one ability. On the other hand, the relative weakness of any one ability does not exclude the possibility of successfully performing the activity with which it is associated, since the missing ability can be compensated by others included in the complex that ensures this activity. For example, poor vision is partially compensated by the special development of hearing and skin sensitivity.

Abilities not only jointly determine the success of an activity, but also interact, influencing each other. The combination of various highly developed abilities is called giftedness, and this characteristic refers to a person who is capable of many different activities.

The versatility and variety of activities in which a person is simultaneously involved acts as one of the most important conditions for the comprehensive and diversified development of his abilities. In this regard, it is necessary to discuss the basic requirements that apply to activities that develop human abilities. R.S. Nemov, in the theory of social learning, identified the following requirements: the creative nature of the activity, the optimal level of difficulty for the performer, proper motivation and ensuring a positive emotional mood during and after the completion of the activity.

If a child’s activity is creative, non-routine in nature, then it constantly forces him to think and in itself becomes quite an attractive activity as a means of testing and developing abilities. Such activity is always associated with the creation of something new, the discovery of new knowledge, the discovery of new possibilities in oneself. This in itself becomes a strong and effective incentive to engage in it, to make the necessary efforts aimed at overcoming the difficulties that arise. Such activities strengthen positive self-esteem, increase the level of aspirations, generate self-confidence and a sense of satisfaction from the success achieved.

If the activity being performed is in the zone of optimal difficulty, i.e. at the limit of the child’s capabilities, then she leads the development of his abilities, realizing what L.S. Vygotsky called the zone of potential development. Activities not located within this zone lead to the development of abilities to a much lesser extent. If it is too simple, then it only ensures the implementation of existing abilities; if it is overly complex, it becomes impossible to implement and, therefore, also does not lead to the formation of new skills.

Maintaining interest in an activity through stimulating motivation means turning the goal of the corresponding activity into an actual human need. In line with the theory of social learning, the fact was especially emphasized that in order for a person to acquire and consolidate new forms of behavior, learning is necessary, and it does not occur without appropriate reinforcement. The formation and development of abilities is also the result of learning, and the stronger the reinforcement, the faster the development will proceed. As for the necessary emotional mood, it is created by such an alternation of successes and failures in activities that develop a person’s abilities, in which failures (they are not excluded if the activity is in the zone of potential development) are necessarily followed by emotionally supported successes, and their number in general is greater than the number of failures.

Mathematical ability

The study of mathematical abilities was carried out by such outstanding representatives of certain trends in foreign psychology as A. Binet, E. Trondike and G. Reves, and such outstanding mathematicians as A. Poincaré and J. Hadamard. 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 only thing on which all researchers agree is, perhaps, the opinion that it is necessary to distinguish between ordinary, “school” abilities for the assimilation of mathematical knowledge, for its reproduction and independent application, and creative mathematical abilities associated with the independent creation of something original and of social value. product. 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 mathematician are an innate formation, a favorable environment is necessary only for their manifestation and development. Regarding “school” (learning) abilities, foreign psychologists are not so unanimous. Here, perhaps, the dominant theory is the parallel action of two factors - biological potential and environment. The main question in the study of mathematical abilities (both educational and creative) abroad was and remains the question of the essence of this complex psychological education. There are three important problems.

The problem of specificity of mathematical abilities. Do mathematical abilities actually exist as a specific education, different from the category of general intelligence? Or are mathematical abilities a qualitative specialization of general mental processes and personality traits, that is, general intellectual abilities developed in relation to mathematical activity? In other words, is it possible to say that mathematical giftedness is nothing more than general intelligence plus an interest in mathematics and a tendency to do it?

The problem of the structure of mathematical abilities. Is mathematical talent a unitary (single indecomposable) or integral (complex) property? In the latter case, one can raise the question about the structure of mathematical abilities, about the components of this complex mental formation.

The problem of typological differences in mathematical abilities. Are there different types of mathematical talent or, given the same basis, are there differences only in interests and inclinations towards certain branches of mathematics?

For a mathematician, it is not enough to have a good memory and attention. According to Poincaré, people who are capable of mathematics are distinguished by the ability to grasp the order in which the elements necessary for a mathematical proof should be arranged. The presence of intuition of this kind is the main element of mathematical creativity. Some people do not have this subtle sense and do not have strong memory and attention and therefore are not able to understand mathematics. Others have weak intuition, but are gifted with good memory and the ability to pay intense attention and therefore can understand and apply mathematics. Still others have such a special intuition and, even in the absence of excellent memory, can not only understand mathematics, but also make mathematical discoveries. Here we are talking about mathematical creativity, accessible to few. But, as J. Hadamard wrote, “between the work of a student solving a problem in algebra or geometry and creative work, the difference is only in level, in quality, since both works are of a similar nature.” In order to understand what qualities are still required to achieve success in mathematics, researchers analyzed mathematical activity: the process of solving problems, methods of proof, logical reasoning, features of mathematical memory. This analysis led to the creation of various variants of the structures of mathematical abilities, complex in their component composition. At the same time, the opinions of most researchers agreed on one thing - that there is not and cannot be a single clearly expressed mathematical ability - this is a cumulative characteristic that reflects the characteristics of different mental processes: perception, thinking, memory, imagination.

The main position of Russian psychology in this matter is the position on the decisive importance of social factors in the development of abilities, the leading role of a person’s social experience, the conditions of his life and activity. Mental characteristics cannot be innate. This also applies entirely to abilities. Abilities are always the result of development. They are formed and developed in life, in the process of activity, in the process of training and education. Individuals must have prerequisites, internal conditions for the development of abilities. A.N. Leontyev and A.R. Luria also speaks about the necessary internal conditions that make the emergence of abilities possible. Abilities are not contained in inclinations. In ontogenesis they do not appear, but are formed. The inclination is not a potential ability (and the ability is not a developmental inclination), since an anatomical and physiological feature under no circumstances can develop into a mental feature.

Among the most important components of mathematical abilities are the specific ability to generalize mathematical material, the ability to spatial representations, and the ability to abstract thinking. Some researchers also identify mathematical memory for patterns of reasoning and proof, methods for solving problems and principles of approach to them as an independent component of mathematical abilities. Russian psychologist who studied mathematical abilities in schoolchildren, V.A. Krutetsky gives the following definition of mathematical abilities: “By ability to study mathematics we understand individual psychological characteristics (primarily characteristics of mental activity) that meet the requirements of educational mathematical activity and determine, other things being equal, the success of creative mastery of mathematics as an academic subject, in particular the relatively rapid , easy and deep mastery of knowledge, skills and abilities in the field of mathematics.”

“A very big and difficult question: does this student have mathematical abilities or not?

First of all, what do we mean by the presence of abilities: creative abilities or the ability to successfully overcome the school mathematics program or the university program?

There is too much variation in the initial data in the source material: some have not learned to learn and believe that if they memorized the rules and solution methods without understanding them, then that is all that is required of them; others, from early childhood, were taught to first understand, and then remember, and to independently search for solutions; third - to use the rules of solution invented for different types of problems, but not to think independently.

The third type is well known to teachers; they know these rules-trained boys and girls who instantly roll off their memorized formulations, but do not have the habit of looking for an independent solution.

I had to meet with schoolchildren of all three indicated types of initial mathematical preparation. Of course, those who were accustomed to understanding and thinking independently stood out sharply against the background of the rest of the dull mass. But then, when, after two or three years of retraining, the others came to the need to understand the material and abandoned the habit of memorizing without understanding, bright personalities appeared among them, capable of contributing something new, offer unexpected solutions, show your true abilities.

My belief is that all normal children have the ability to master mathematics well, at least in school and university mathematics. They just need to be taught to learn. Teach to use the gift that nature has endowed man with - the ability to think. Some schoolchildren literally changed radically when gaps in knowledge and skills were eliminated in their initial mathematical education. Therefore, I strongly condemn those who label a student as incapable of mathematics too early. Let me give myself as an example: up until the sixth grade, I had a hard time with mathematics, and I felt a constant fear of tasks.

I remember telling my parents: “how good it would be to study if there were no mathematics.” In 1925 the family moved to Saratov. It turned out that at the Saratov school they took more mathematics, and I had to catch up with the class. I independently studied the necessary sections and turned to the previous material, in which I also had gaps.

Then I came across a collection of competitive problems offered upon admission to the St. Petersburg Institute of Transport. I solved a significant number of problems on my own. Six months later, I was known as the best student in the class in mathematics. The whole point is that when I worked on the textbook independently, I brought the matter to a point of understanding and only then moved on, first reinforcing the material I had covered by solving problems on my own. Then at the university I also took the position of a mathematical leader, although it was only about the educational process, and not about my own creativity. It took many years for me to come up with problems for research and begin to influence the creative interests of others.

As a university student, I adhered to the following rule: I listened carefully to lectures, on the same day I looked through the short notes I had made and expanded on the information received by reading the relevant passages in the textbook. He immediately reinforced what he had learned with several independently solved problems. This method of repetition helped me avoid exam fever. It was enough for me to refresh my memory of what I had previously learned.

I never allowed myself to move on without understanding the previous one. Perhaps it makes sense to say that immediately after the lectures, after reflection, I briefly wrote down the contents of the lecture, paying attention to the clarity of the formulation of definitions and theorems. I also placed additional information gleaned from books after recording the contents of the lecture. My notes were a success during the course; they were taken, rewritten, and asked for during the holidays to retake them. As a result, I was not able to save a single such notebook; they all went to different hands.

I believe that writing notes has brought me two benefits. Firstly, from the very beginning I thoroughly studied everything new that was presented to us and, secondly, I learned to briefly state the basic things that I needed to know and be able to apply. This habit of concise and clear formulations remained with me for the rest of my life.

If we talk about the ability to comprehend the course of school and university mathematics, then I am convinced that in most cases the lack of ability is attributed to those who do not want to study or have serious gaps in the previous parts of the course and do not consider it necessary to restore the unknown in a timely manner. Many years of experience in communicating with students, schoolchildren and their parents have convinced me that, as a rule, failures in mastering a mathematics course are associated not with a lack of mathematical abilities, but with a lack of solid knowledge of fundamental concepts, with laziness of mind, which interferes with systematic work on the material, and with the desire to reduce all knowledge to memorization without understanding. We must remember that only in overcoming difficulties on our own is the key to knowledge and confidence in our geniuses and knowledge.

In the overwhelming majority of cases, when they talk about a student’s lack of mathematical abilities to complete a compulsory course, we are talking about something else - either inability or unwillingness to learn.

The conclusion about the lack of abilities is usually pedagogically unfounded and harmful. Such a conclusion can have a depressing effect on the student’s psyche. This is the first thing. And secondly, it seems to give an indulgence to someone who is lazy or has not learned to learn.

The ability to learn does not come by itself, but requires systematic education, constant attention from teachers and serious efforts from students. The purpose of school education is not to overload students' memory with information that does not turn into a tool of labor, but to make the mind inquisitive, agile, capable of analyzing new situations, and finding approaches to solving emerging problems. Anyone who relies only on memory, on cramming, disconnects thought and reason from the work of cognition. Memory must play the role of an active assistant to the mind, and the unusual role of the only means of cognition should not be imposed on it. Basic information and ideas should be stored in memory, which, as needed, are turned into active methods.

In the same way, it is impossible to teach someone to speak a foreign language if only by providing the memory with words and rules. This is not enough. It is also necessary to accustom a person to actively use the acquired knowledge. And for this you need to speak, that is, force knowledge not to lie as a dead weight in the depths of memory, but to actively act. For mathematics, exercises for solving problems and drawing logical conclusions are as necessary as speaking a foreign language when learning it.”

Gnedenko B.V., Mathematics and life, M., “Komkniga”, 2006, pp. 118-121.

Mathematical abilities in children are classified as innate talents. Children take their first steps towards learning mathematics in preschool age. Mathematical thinking is closely related to creativity and the level of development of mental abilities. But not all children easily master exact science. Why is this happening? Is it possible to develop mathematical abilities in a child?

It is wrong to think that children's minds are limited and cannot understand mathematics. Like any other natural gift, mathematical abilities will open only as a result of correct, systematic development. This means that in teaching children it is not only possible, but it is very important from early preschool age to pay attention to the development of these inclinations.

It is all the more important to do this because a new generation of children will look for their calling in a world ruled by digital technologies. Any profession is related to mathematics, even the most humanitarian or creative ones. Thanks to mathematics, a child learns holistic and quick thinking, analysis, and makes informed conclusions.

How to develop the mathematical abilities of a child under 7 years old? The results depend not only on the age at which you started training, but also on the methods chosen. Diagnosis of the mathematical abilities of children aged 5, 6 and 7 years will help determine the course and load in teaching preschoolers. It will allow you to assess the presence and level of development of children’s mathematical thinking and basic knowledge in mathematics.

Diagnosis of mathematical abilities in a child according to A. V. Beloshistaya

If a child quickly learns numbers and learns to count, this does not mean that a mathematician is growing up in the family. Mental arithmetic is the simplest topic in exact science. Mathematical abilities are judged by mental qualities such as:

analysis and logic;
ability to read diagrams and formulas;
understanding of abstract concepts;
the ability to accurately perceive the shapes of objects in space.

Doctor of Sciences V. A. Beloshistaya has been working on the issue of diagnosing and developing mathematical abilities in preschool children (younger - 5 and 6 years old, older - 6 and 7 years old). Her method of assessing children's mathematical talents has several courses:

Diagnostics for children 5-6 years old. It is carried out in two stages in order to assess the ability of synthesis and analysis. Individual testing. Based on its results, one can judge whether the child understands the difference between figures and shapes of objects, whether he can divide things into groups according to an independently chosen criterion, and whether he has the skills of generalization and comparison.
Diagnostics for figurative analysis in preschoolers 5 and 6 years old.
Testing older preschoolers (5-7 years old) to determine the level of development of analysis and synthesis skills. In the task, children need to identify specific figures in complex images from many intersecting figures.
Diagnostics of basic mathematical concepts: counting, comparison, knowledge of the concepts of “more” and “less”, “wider” and “narrower”, etc.

For a more complete picture of the development of mathematical abilities in preschoolers in dynamics, the first two types of diagnostics are carried out at the beginning of the school year, and the second two - in May (at the end of the year).

The material at hand for tests should be bright, easy to use, and understandable for the child. Different tasks are used for each age.

Method of Kolesnikova E.V. to diagnose a child’s mathematical abilities

The well-known teacher and scientist E.V. Kolesnikova in Russia has more than a dozen books and manuals on the preparation of primary and secondary preschoolers. One of the main courses of her work is diagnosing mathematical abilities in children 6-7 years old. Kolesnikova’s method has been approved by the Federal State Educational Standard, as one that meets the standards of pedagogical diagnostics in Russia. However, the method is successfully used to assess the level of mathematical abilities of preschoolers in different countries.

The purpose of the methodology: assessing the child’s level of readiness for school, searching for gaps in the study of basic mathematical knowledge to correct learning deficiencies at the stage of preparation for school. The advantage of the method is the accurate and most complete diagnosis of the child’s knowledge.

Tips for parents on developing their child’s mathematical abilities

Albert Einstein called play the highest form of exploration. When choosing methods for developing children, it is useful for parents to use play activities.

Developing science abilities in children in this way helps:

better understand the world around us;
assess your capabilities;
become sociable;
train thinking;
gain basic understanding of mathematics as a science;
become more confident and independent.

The following games are used in training:

Counting sticks. With their help, children learn to distinguish the shapes of objects, compare, develop attention, memory, intelligence and perseverance.
Puzzles. They perfectly develop logic and analytical thinking, teach how to synthesize information, summarize and classify data. That is, mathematical riddles comprehensively develop mathematical intelligence, and also cultivate perseverance and strong-willed qualities that help solve assigned problems despite difficulties.
Puzzles. They train spatial thinking, develop memory and logic, observation and ingenuity. In solving them, the child learns to calculate his steps and masters counting (simple, ordinal).

Developing math skills through play activities is beneficial for several reasons:

it is easier for a child to perceive knowledge;
a positive attitude towards the subject is formed, and therefore internal interest;
the game provides an opportunity to apply a creative approach to solving problems (develops creative potential);
the game is interesting, which means the child sees meaning in learning (motivation).

Is it possible to develop the mathematical abilities of preschoolers with the help of fairy tales?

You cannot force anything into a child’s memory - through cramming and many repetitions. If the knowledge is associated with a very real emotion, it will probably settle in the child’s memory for a long time. Therefore, the task of parents is to delight, surprise and delight their little students during the lessons. How to do it? It is unlikely that I will reveal a secret if I say that a fairy tale is ideal for this matter - the first guide in getting to know the peculiarities of the surrounding world, relationships between people.

For children, a fairy tale plot is no less real than the events of real life. Fairy tales develop imagination, speech, flexibility of thinking, create a special vision of the world, and teach good qualities (honesty, kindness, loyalty). Developing mathematical abilities through fairy tales is easy if you show a little imagination:

It’s fun to learn simple counting with the tale of a little goat who could count to ten, “The Wolf and the Seven Little Goats.”
Ordinal counting will help you master “Teremok” and even “Turnip”.
In “Three Bears” the child gets acquainted with the concepts of “big”, “small” and “medium”, and learns to count to three.

Activities with fairy tales can be endlessly changed and complicated. For example, invite your child to compare animals with geometric shapes. Searching for similarities between fairy tale characters and figures develops the ability to think abstractly.

It is convenient to develop mathematical abilities with the help of fairy tales, since parents can do this at any time outside of class (at home, on a walk, on a trip). A fairy tale can also become part of the curriculum in kindergarten or school. Based on a plot well known to children, teachers create riddles and labyrinths, using them as the basis for numerical problems and counting rhymes for exercising their fingers. But the most important thing is that children like such activities.

How does mental arithmetic Soroban develop thinking?

Abilities are individually expressed opportunities for the successful implementation of a particular activity. They include both individual knowledge, skills, and readiness to learn new ways and techniques of activity. Different criteria are used to classify abilities. Thus, sensorimotor, perceptual, mnemonic, imaginative, mental, and communicative abilities can be distinguished. Another criterion may be one or another subject area, according to which abilities can be qualified as scientific (mathematical, linguistic, humanitarian); creative (musical, literary, artistic); engineering.

Let us briefly formulate several provisions of the general theory of abilities:

1. Abilities are always there ability for a certain type of activity, they exist only in the corresponding specific human activity. Therefore, they can only be identified on the basis of an analysis of specific activities. Accordingly, mathematical abilities exist only in mathematical activity and must be revealed in it.

2. Abilities are a dynamic concept. They not only appear and exist in activity, they are created in activity, and develop in activity. Accordingly, mathematical abilities exist only in dynamics, in development; they are formed and developed in mathematical activity.

3. During certain periods of human development, the most favorable conditions arise for the formation and development of certain types of abilities, and some of these conditions are temporary, transitory. Such age periods when the conditions for the development of certain abilities are most optimal are called sensitive (L. S. Vygotsky, A. N. Leontiev). Obviously, there are optimal periods for the development of mathematical abilities.

4. The success of an activity depends on a set of abilities. Equally, the success of mathematical activity depends not on a single ability, but on a complex of abilities.

5. High achievements in the same activity can be due to different combinations of abilities. Therefore, in principle, we can talk about different types of abilities, including mathematical ones.

6. Compensation of some abilities by others is possible within a wide range, as a result of which the relative weakness of any one ability is compensated by another ability, which ultimately does not exclude the possibility of successfully performing the corresponding activity. A.G. Kovalev and V.N. Myasishchev understand compensation more broadly - they talk about the possibility of compensating for a missing ability with skill, characterological qualities (patience, perseverance). Apparently, compensation of both types can also occur in the area of mathematical abilities.

7. Complex and not fully resolved in psychology is the question of the relationship between general and special talent. B. M. Teplov was inclined to deny the very concept of general talent, unrelated to specific activity. The concepts of “ability” and “giftedness” according to B. M. Teplov make sense only in relation to specific historically developing forms of social and labor activity. It is necessary, in his opinion, to talk about something else, about more general and more special aspects of giftedness. S. L. Rubinstein rightly noted that general and special giftedness should not be opposed to each other - the presence of special abilities leaves a certain imprint on general giftedness, and the presence of general giftedness affects the nature of special abilities. B. G. Ananyev pointed out that one should distinguish between general development and special development and, accordingly, general and special abilities. Each of these concepts is legitimate, both corresponding categories are interconnected. B. G. Ananyev emphasizes the role of general development in the formation of special abilities.

Study of mathematical abilities in foreign psychology.

Such outstanding representatives of certain trends in psychology as A. Binet, E. Trondike and G. Reves, and such outstanding mathematicians as A. Poincaré and J. Hadamard, also contributed to the study of mathematical abilities.

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 only thing on which all researchers agree is, perhaps, the opinion that it is necessary to distinguish between ordinary, “school” abilities for the assimilation of mathematical knowledge, for its reproduction and independent application, and creative mathematical abilities associated with the independent creation of something original and of social value. product.

Foreign researchers show great unity of views on the issue of innate or acquired mathematical abilities. If we distinguish here 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 mathematician are an innate formation, a favorable environment is necessary only for their manifestation and development. Regarding “school” (learning) abilities, foreign psychologists are not so unanimous. Here, perhaps, the dominant theory is the parallel action of two factors - biological potential and environment.

The main question in the study of mathematical abilities (both educational and creative) abroad has been and remains the question of the essence of this complex psychological formation. In this regard, three important problems can be identified.

1. The problem of specificity of mathematical abilities. Do mathematical abilities actually exist as a specific education, different from the category of general intelligence? Or are mathematical abilities a qualitative specialization of general mental processes and personality traits, that is, general intellectual abilities developed in relation to mathematical activity? In other words, is it possible to say that mathematical giftedness is nothing more than general intelligence plus an interest in mathematics and a tendency to do it?

2. The problem of the structure of mathematical abilities. Is mathematical talent a unitary (single indecomposable) or integral (complex) property? In the latter case, one can raise the question about the structure of mathematical abilities, about the components of this complex mental formation.

3. The problem of typological differences in mathematical abilities. Are there different types of mathematical talent or, given the same basis, are there differences only in interests and inclinations towards certain branches of mathematics?

Study of the problem of abilities in domestic psychology.

So, social experience, social influence, and education play a decisive and determining role. Well, what is the role of innate abilities?

Of course, it is difficult to determine in each specific case the relative role of the congenital and acquired, since both are fused and indistinguishable. But the fundamental solution to this issue in Russian psychology is this: abilities cannot be innate, only the inclinations of abilities can be innate - some anatomical and physiological features of the brain and nervous system with which a person is born.

But what is the role of these innate biological factors in the development of abilities?

As S. L. Rubinstein noted, abilities are not predetermined, but they cannot simply be implanted from the outside. Individuals must have prerequisites, internal conditions for the development of abilities. A. N. Leontiev, A. R. Luria also talk about the necessary internal conditions that make the emergence of abilities possible.

Abilities are not contained in inclinations. In ontogenesis they do not appear, but are formed. The inclination is not a potential ability (and the ability is not a developmental inclination), since an anatomical and physiological feature under no circumstances can develop into a mental feature.

A slightly different understanding of inclinations is given in the works of A.G. Kovalev and V.N. Myasishchev. By inclinations they understand psychophysiological properties, primarily those that are detected in the earliest phase of mastering a particular activity (for example, good color discrimination, visual memory). In other words, inclinations are a primary natural ability, not yet developed, but making itself felt during the first attempts at activity.

However, even with this understanding of inclinations, the basic position remains the same: abilities in the proper sense of the word are formed in activity and are lifetime education.

Naturally, all of the above can be attributed to the question of mathematical abilities, as a type of general ability.

Mathematical abilities and their natural prerequisites (works of B. M. Teplov).

Although mathematical abilities were not the subject of special consideration in the works of B. M. Teplov, answers to many questions related to their study can be found in his works devoted to the problems of abilities. Among them, a special place is occupied by two monographic works - “The Psychology of Musical Abilities” and “The Mind of a Commander”, which have become classic examples of the psychological study of abilities and have incorporated universal principles of approach to this problem, which can and should be used when studying any types of abilities.

In both works, B. M. Teplov not only gives a brilliant psychological analysis of specific types of activity, but also, using examples of outstanding representatives of musical and military art, reveals the necessary components that make up bright talents in these areas. B. M. Teplov paid special attention to the issue of the relationship between general and special abilities, proving that success in any type of activity, including music and military affairs, depends not only on special components (for example, in music - hearing, sense of rhythm ), but also on the general characteristics of attention, memory, and intelligence. At the same time, general mental abilities are inextricably linked with special abilities and significantly influence the level of development of the latter.

The role of general abilities is most clearly demonstrated in the work “The Mind of a Commander.” Let us dwell on the consideration of the main provisions of this work, since they can be used in the study of other types of abilities associated with mental activity, including mathematical abilities. Having conducted an in-depth study of the commander’s activities, B. M. Teplov showed the place of intellectual functions in it. They provide analysis of complex military situations, identifying individual significant details that can affect the outcome of upcoming battles. It is the ability to analyze that provides the first necessary stage in making the right decision, in drawing up a battle plan. Following the analytical work comes the stage of synthesis, which allows us to combine the variety of details into a single whole. According to B. M. Teplov, the activity of a commander requires a balance of the processes of analysis and synthesis, with a mandatory high level of their development.

Memory occupies an important place in the intellectual activity of a commander. She is very selective, that is, she retains first of all the necessary, essential details. As a classic example of such memory, B. M. Teplov cites statements about the memory of Napoleon, who remembered literally everything that was directly related to his military activities, from unit numbers to the faces of soldiers. At the same time, Napoleon was unable to memorize meaningless material, but had the important feature of instantly assimilating what was subject to classification, a certain logical law.

B. M. Teplov comes to the conclusion that “the ability to find and highlight the essential and constant systematization of material are the most important conditions that ensure the unity of analysis and synthesis, the balance between these aspects of mental activity that distinguish the work of the mind of a good commander” (B. M. Teplov 1985, p.249). Along with an outstanding mind, a commander must have certain personal qualities. This is, first of all, courage, determination, energy, that is, what, in relation to military leadership, is usually designated by the concept of “will.” An equally important personal quality is stress resistance. The emotionality of a talented commander is manifested in a combination of the emotion of combat excitement and the ability to gather and concentrate.

B. M. Teplov assigned a special place in the intellectual activity of the commander to the presence of such a quality as intuition. He analyzed this quality of the commander's mind, comparing it with the intuition of a scientist. There is a lot in common between them. The main difference, according to B. M. Teplov, is the need for the commander to make an urgent decision, on which the success of the operation may depend, while the scientist is not limited by time frames. But in both cases, “insight” must be preceded by hard work, on the basis of which the only correct solution to the problem can be made.

Confirmation of the provisions analyzed and generalized by B. M. Teplov from a psychological point of view can be found in the works of many outstanding scientists, including mathematicians. Thus, in the psychological study “Mathematical Creativity,” Henri Poincaré describes in detail the situation in which he managed to make one of his discoveries. This was preceded by a long preparatory work, a large proportion of which, according to the scientist, was the process of the unconscious. The stage of “insight” was necessarily followed by the second stage - careful conscious work to put the evidence in order and verify it. A. Poincaré came to the conclusion that the most important place in mathematical abilities is occupied by the ability to logically build a chain of operations that will lead to solving a problem. It would seem that this should be accessible to any person capable of logical thinking. However, not everyone is able to operate mathematical symbols with the same ease as when solving logical problems.

Here we are talking about mathematical creativity, accessible to few. But, as J. Hadamard wrote, “between the work of a student solving a problem in algebra or geometry and creative work, the difference is only in level, in quality, since both works are of a similar nature” (J. Hadamard, p. 98). In order to understand what qualities are still required to achieve success in mathematics, researchers analyzed mathematical activity: the process of solving problems, methods of proof, logical reasoning, features of mathematical memory. This analysis led to the creation of various variants of the structures of mathematical abilities, complex in their component composition. At the same time, the opinions of most researchers agreed on one thing - that there is not and cannot be a single clearly expressed mathematical ability - this is a cumulative characteristic that reflects the characteristics of different mental processes: perception, thinking, memory, imagination.

Among the most important components of mathematical abilities are the specific ability to generalize mathematical material, the ability to spatial representations, and the ability to abstract thinking. Some researchers also identify mathematical memory for patterns of reasoning and proof, methods for solving problems and principles of approach to them as an independent component of mathematical abilities. The Soviet psychologist, who studied mathematical abilities in schoolchildren, V. A. Krutetsky gives the following definition of mathematical abilities: “By abilities to study mathematics, we understand individual psychological characteristics (primarily characteristics of mental activity) that meet the requirements of educational mathematical activity and determine, other things being equal, conditions for the success of creative mastery of mathematics as an academic subject, in particular the relatively quick, easy and deep mastery of knowledge, skills and abilities in the field of mathematics" (Krutetsky V.A., 1968).

The study of mathematical abilities also includes the solution of one of the most important problems - the search for the natural prerequisites, or inclinations, of this type of ability. Inclinations include the innate anatomical and physiological characteristics of an individual, which are considered as favorable conditions for the development of abilities. For a long time, inclinations were considered as a factor that fatally predetermined the level and direction of development of abilities. The classics of Russian psychology B. M. Teplov and S. L. Rubinstein scientifically proved the illegality of such an understanding of inclinations and showed that the source of the development of abilities is the close interaction of external and internal conditions. The severity of one or another physiological quality in no way indicates the obligatory development of a particular type of ability. It can only be a favorable condition for this development. The typological properties that are part of the inclinations and are an important component of them reflect such individual characteristics of the functioning of the body as the limit of performance, the speed characteristics of the nervous reaction, the ability to rearrange the reaction in response to changes in external influences.

The properties of the nervous system, which are closely related to the properties of temperament, in turn, influence the manifestation of the characterological characteristics of the individual (V.S. Merlin, 1986). B. G. Ananyev, developing ideas about the general natural basis for the development of character and abilities, pointed to the formation in the process of activity of connections between abilities and character, leading to new mental formations, denoted by the terms “talent” and “vocation” (Ananyev B. G., 1980). Thus, temperament, abilities and character form, as it were, a chain of interconnected substructures in the structure of personality and individuality, having a single natural basis (E. A. Golubeva 1993).

General diagram of the structure of mathematical abilities at school age according to V. A. Krutetsky.

The material collected by V. A. Krutetsky allowed him to build a general diagram of the structure of mathematical abilities at school age.

1. Obtaining mathematical information.

1) The ability to formally perceive mathematical material, to grasp the formal structure of a problem.

2. Processing of mathematical information.

1) The ability for logical thinking in the field of quantitative and spatial relations, numerical and symbolic symbolism. Ability to think in mathematical symbols.

2) The ability to quickly and widely generalize mathematical objects, relationships and actions.

3) The ability to curtail the process of mathematical reasoning and the system of corresponding actions. The ability to think in collapsed structures.

4) Flexibility of thought processes in mathematical activity.

5) Striving for clarity, simplicity, economy and rationality of decisions.

6) The ability to quickly and freely rearrange the direction of the thought process, switch from direct to reverse train of thought (reversibility of the thought process in mathematical reasoning).

3. Storage of mathematical information.

1) Mathematical memory (generalized memory for mathematical relations, typical characteristics, patterns of reasoning and proof, methods for solving problems and principles of approach to them).

4. General synthetic component.

1) Mathematical orientation of the mind.

The selected components are closely related, influence each other and form in their totality a single system, an integral structure, a unique syndrome of mathematical giftedness, a mathematical mindset.

The structure of mathematical giftedness does not include those components whose presence in this system is not necessary (although useful). In this sense, they are neutral in relation to mathematical giftedness. However, their presence or absence in the structure (more precisely, the degree of their development) determines the type of mathematical mindset. The following components are not mandatory in the structure of mathematical giftedness:

1. Speed of thought processes as a temporary characteristic.

2. Computational abilities (the ability to make quick and accurate calculations, often in the mind).

3. Memory for numbers, numbers, formulas.

4. Ability for spatial representations.

5. Ability to visualize abstract mathematical relationships and dependencies.

Conclusion.

The problem of mathematical abilities in psychology represents a vast field of action for the researcher. Due to the contradictions between various currents in psychology, as well as within the currents themselves, there can still be no talk of an accurate and strict understanding of the content of this concept.

The books reviewed in this work confirm this conclusion. At the same time, it should be noted that there is undying interest in this problem in all currents of psychology, which confirms the following conclusion.

The practical value of research on this topic is obvious: mathematics education plays a leading role in most educational systems, and it, in turn, will become more effective after the scientific substantiation of its basis - the theory of mathematical abilities.

So, as V. A. Krutetsky argued: “The task of comprehensive and harmonious development of a person’s personality makes it absolutely necessary to deeply scientifically develop the problem of people’s ability to perform certain types of activities. The development of this problem is of both theoretical and practical interest.”

Bibliography:

Hadamard J. Study of the psychology of the invention process in the field of mathematics. M., 1970.
Ananyev B.G. Selected works: In 2 volumes. M., 1980.
Golubeva E.A., Guseva E.P., Pasynkova A.V., Maksimova N.E., Maksimenko V.I. Bioelectric correlates of memory and academic performance in older schoolchildren. Questions of psychology, 1974, No. 5.
Golubeva E.A. Abilities and personality. M., 1993.
Kadyrov B.R. Level of activation and some dynamic characteristics of mental activity.
dis. Ph.D. psychol. Sci. M., 1990.
Krutetsky V.A. Psychology of mathematical abilities of schoolchildren. M., 1968.
Merlin V.S. Essay on an integral study of individuality. M., 1986.
Pechenkov V.V. The problem of the relationship between general and specifically human types of v.n.d. and their psychological manifestations. In the book "Abilities and Inclinations", M., 1989.
Poincare A. Mathematical creativity. M., 1909.
Rubinshtein S.L. Fundamentals of general psychology: In 2 vols. M., 1989.
Teplov B.M. Selected works: In 2 volumes. M., 1985.

Mathematics ability is one of the talents given by nature, which manifests itself from an early age and is directly related to the development of creative potential and the desire to understand the world around the child. But why is learning maths so difficult for some children, and can these abilities be improved?

The opinion that only gifted children can master mathematics is wrong. Mathematical abilities, like other talents, are the result of a child’s harmonious development, and must begin from a very early age.

In the modern computer world with its digital technologies, the ability to “make friends” with numbers is extremely necessary. Many professions are based on mathematics, which develops thinking and is one of the most important factors influencing the intellectual growth of children. This exact science, whose role in the upbringing and education of a child is undeniable, develops logic, teaches one to think consistently, determine the similarities, connections and differences of objects and phenomena, makes the child’s mind fast, attentive and flexible.

For mathematics classes for children five to seven years old to be effective, a serious approach is needed, and the first step is to diagnose their knowledge and skills - to assess at what level the child’s logical thinking and basic mathematical concepts are.

Diagnostics of mathematical abilities of children 5-7 years old using the method of Beloshistaya A.V.

If a child with a mathematical mind has mastered mental calculation at an early age, this is not yet a basis for one hundred percent confidence in his future as a mathematical genius. Mental arithmetic skills are only a small element of an exact science and are far from the most complex. A child’s ability for mathematics is evidenced by a special way of thinking, which is characterized by logic and abstract thinking, understanding of diagrams, tables and formulas, the ability to analyze, and the ability to see figures in space (volume).

To determine whether children from primary preschool (4-5 years old) to primary school age have these abilities, there is an effective diagnostic system created by Doctor of Pedagogical Sciences Anna Vitalievna Beloshista. It is based on the creation by a teacher or parent of certain situations in which the child must apply this or that skill.

Diagnostic stages:

Testing a 5-6 year old child for analysis and synthesis skills. At this stage, you can evaluate how the child can compare objects of different shapes, separate them and generalize them according to certain characteristics.
Testing figurative analysis skills in children aged 5-6 years.
Testing the ability to analyze and synthesize information, the results of which reveal the ability of a preschooler (first grader) to determine the shapes of various figures and notice them in complex pictures with figures superimposed on each other.
Testing to determine the child’s understanding of the basic concepts of mathematics - we are talking about the concepts of “more” and “less”, ordinal counting, the shape of the simplest geometric figures.

The first two stages of such diagnostics are carried out at the beginning of the school year, the rest - at the end, which makes it possible to assess the dynamics of the child’s mathematical development.

The material used for testing should be understandable and interesting for children - age-appropriate, bright and with pictures.

Diagnosis of a child’s mathematical abilities using the method of Kolesnikova E.V.

Elena Vladimirovna has created many educational and methodological aids for the development of mathematical abilities in preschoolers. Her method of testing children 6 and 7 years old has become widespread among teachers and parents in different countries and meets the requirements of the Federal State Educational Standard (FSES) (Russia).

Thanks to Kolesnikova’s method, it is possible to determine as accurately as possible the level of key indicators of the development of children’s mathematical skills, find out their readiness for school, and identify weaknesses in order to fill gaps in a timely manner. This diagnosis helps to find ways to improve the child’s mathematical abilities.

Developing a child’s mathematical abilities: tips for parents

It is better to introduce a child to any science, even something as serious as mathematics, in a playful way - this will be the best teaching method that parents should choose. Listen to the words of famous scientist Albert Einstein: “Play is the highest form of exploration.” After all, with the help of the game you can get amazing results:

– knowledge of yourself and the world around you;

– formation of a mathematical knowledge base;

– development of thinking:

– personality formation;

– development of communication skills.

You can use various games:

Counting sticks. Thanks to them, the baby remembers the shapes of objects, develops his attention, memory, ingenuity, and develops comparison skills and perseverance.
Puzzles that develop logic and ingenuity, attention and memory. Logic puzzles help children learn better spatial awareness, thoughtful planning, simple and backward counting, and ordinal counting.
Mathematical riddles are a great way to develop the basic aspects of thinking: logic, analysis and synthesis, comparison and generalization. While searching for a solution, children learn to draw their own conclusions, cope with difficulties and defend their point of view.

The development of mathematical abilities through play creates learning excitement, adds vivid emotions, and helps the child fall in love with the subject of study that interests him. It is also worth noting that gaming activities also contribute to the development of creative abilities.

The role of fairy tales in the development of mathematical abilities of preschool children

Children's memory has its own characteristics: it records vivid emotional moments, that is, the child remembers information that is associated with surprise, joy, and admiration. And learning “from under pressure” is an extremely ineffective way. In the search for effective teaching methods, adults should remember such a simple and ordinary element as a fairy tale. A fairy tale is one of the first means of introducing a child to the world around him.

For children, fairy tales and reality are closely connected, magical characters are real and alive. Thanks to fairy tales, a child’s speech, imagination and ingenuity develop; they give the concept of goodness, honesty, broaden horizons, and also provide an opportunity to develop mathematical skills.

For example, in the fairy tale “The Three Bears,” the child unobtrusively gets acquainted with counting to three, the concepts of “small,” “medium,” and “large.” “Turnip”, “Teremok”, “The Little Goat Who Could Count to 10”, “The Wolf and the Seven Little Kids” - in these tales you can learn simple and ordinal counting.

When discussing fairy-tale characters, you can invite your child to compare them in width and height, to “hide” them in geometric shapes that are suitable in size or shape, which contributes to the development of abstract thinking.

You can use fairy tales not only at home, but also in school. Children really love lessons based on the plots of their favorite fairy tales, using riddles, labyrinths, and fingering. Such classes will become a real adventure in which the kids will take personal part, which means the material will be learned better. The main thing is to involve children in the game process and arouse their interest.