Parents who want to teach their child mathematics are faced with the question - what exactly should be taught to the child. What abilities can and should be developed at preschool age in order to ensure the successful assimilation of the school curriculum.

What abilities are related to mathematical in children under 7 years old

Do not think that mathematical abilities mean only the ability to quickly and accurately count. It's a delusion. Mathematical abilities include a whole range of skills aimed at creativity, logic, and counting.

The speed of counting, the ability to memorize a large array of numbers and data are not true mathematical abilities, since even a slow and thorough child who is thoughtfully engaged can successfully comprehend mathematics.

Mathematical skills include:

  1. Ability to generalize mathematical material.
  2. The ability to see things in common.
  3. The ability to find the main thing in a large amount of different information and exclude the unnecessary.
  4. Use numbers and signs.
  5. Logical thinking.
  6. The child's ability to think in abstract structures. The ability to distract from the problem being solved and see the resulting picture as a whole.
  7. Think both forward and backward.
  8. The ability to think independently without using templates.
  9. Developed mathematical memory. Ability to apply acquired knowledge in different situations.
  10. Spatial thinking - confident use of the concepts of "up", "down", "right" and "left".

How are mathematical abilities formed?

All abilities, including mathematical ones, are not a predetermined skill. They are formed and developed through training and reinforced by practice. Therefore, it is important not only to develop this or that ability, but also to improve it through practical exercises, bringing it to automatism.

Any ability goes through several stages in its development:

  1. Cognition. The child gets acquainted with the subject and learns the necessary material;
  2. Application. Applies new knowledge in independent play;
  3. Consolidation. Returns to classes and repeats previously learned;
  4. Application. Use of fixed material during independent play;
  5. Extension. There is an expansion of knowledge about a subject or ability;
  6. Application. The child supplements independent play with new knowledge;
  7. Adaptation. Knowledge is transferred from the game situation to life.

Any new knowledge must go through the application stage several times. Give the child the opportunity to use the data obtained in an independent game. Children need some time to comprehend and consolidate every minor change in knowledge.

In the event that a child cannot master the acquired skill or knowledge through independent play, there is a high probability that it will not be consolidated. Therefore, after each lesson, let the baby play or get distracted, play with him. During the game, show how to use new knowledge.

How to develop math skills in a child

You need to start mathematical development in the form of a game and use things that will interest the baby. For example, toys and household items that he encounters every day.

From the moment when the child shows interest in a particular object, the parent begins to show the child that the object can not only be examined and touched, but also perform various actions with it. Focusing on some features of an object (color, shape), in an unobtrusive manner, you can show the difference in the number of objects, introduce the first concepts of plurality and spatial position.

After the child learns to separate objects into groups, you can show that they can be counted and sorted. Pay attention to geometric features.

The development of mathematical abilities should go simultaneously with the basics of operations with numbers.

Any new knowledge should be presented with the child's clear interest in learning. In the absence of interest in the subject and its study, the child should not be taught. It is important to strike a balance in a child's education in order to develop a love of mathematics. Almost all the problems associated with the study of the foundations of this discipline have their origin in the initial lack of desire to know.

What to do if the child is not interested

If a child leaves and gets bored with every attempt to teach him the basics of mathematics, then you need to:

  • Change the presentation of the material. Most likely, your explanations are too complex for a child to understand and do not contain game elements. Preschool children cannot perceive information in the classical form of a lesson; they need to be shown and told new material during the game or entertainment. Dry text is not perceived by the child. Apply in teaching or try to involve the child directly in teaching;
  • Show interest in the subject without the participation of the child. Young children are interested in everything that is interesting to their parents. They love to imitate and copy adults. If the child does not show interest in any activity, then try to start playing with the selected items in front of the child. Talk out loud about what you are doing. Show your interest in the process of the game. The child will see your interest and join;
  • If the child still quickly loses interest in the subject, you need to check whether the knowledge and skill that you want to instill in him is too complicated or easy;
  • Keep in mind the duration of classes for different ages. If a child under 4 years of age has lost interest in a subject after 5 minutes, then this is normal. Since at this age it is difficult for him to concentrate on one subject for a long time.
  • Try introducing one element at a time into the lesson. For children 5-7 years old, the duration of classes should not exceed 30 minutes.
  • Do not be upset if the child does not want to study on a particular day. You need to try to involve him in training after a while.

The main thing to remember:

  1. The material must be adapted to the age of the child;
  2. The parent should show interest in the material and the results of the child;
  3. The child must be ready to go.

How to develop mathematical thinking

The order of teaching a child to think mathematically is a series of related activities that are presented in order of increasing complexity of the material.

1. You need to start learning with the concepts of the spatial arrangement of objects

The child must understand where the right is left. What is "above", "below", "before" and "for". The presence of this skill allows you to perceive all subsequent classes easier. Orientation in space is fundamental knowledge not only for the development of mathematical abilities, but also for teaching a child to read and write.

You can offer the child the following game. Take some of his favorite toys and put them in front of him at different distances. Ask him to show which toy is closer, which is further, which is to the left, etc. If you have any difficulty in choosing, tell me the correct answer. Use in this game different variants of words that determine the location of objects relative to the baby.

Use this approach to study and repetition, not only in the classroom, but also in everyday life. For example, ask your child to determine the spatial arrangement of objects on the playground. More often in ordinary life, ask to submit something, orienting the baby in space.

In parallel with spatial thinking, they teach generalization and classification of objects according to their external features and functional affiliation.

2. Learn the concept of multiple items

The child must distinguish between the concepts of many - few, one - many, more - less and equally. Offer toys of different types in different quantities. Offer to count them and say a lot of them or few, which toys are fewer and vice versa, also show the equality of toys.

A good game to reinforce the concept of a set is “What’s in the box”. The child is offered two boxes or boxes containing a different number of items. By moving objects between the boxes, the child is invited to make the number of objects more or less, to equalize. Under the age of 3 years, the number of objects should not be large so that the child can visually assess the difference in objects without counting.

3. It is important to teach a child simple geometric shapes in early childhood.

Teach your child to see them in the world around them. It is good to use applications from mathematical forms for the development of knowledge of geometric shapes. Show the child a drawing of an object with clear contours (house, car). Offer to make an image of an object from the prepared triangles, squares and circles.

Show and explain what the angle of the figures is, invite the child to guess why the “triangle” has such a name. Offer the child to familiarize the figure with a large number of angles.

Consolidate geometric knowledge through drawing the studied material, folding different shapes from other objects (sticks, pebbles, etc.). Plasticine and other materials can be used to create various shapes.

Ask to draw a series of figures of different types, count them together with the child. Ask which figures are many and which are few.

When walking with a child, pay attention to the shape of houses, shops, cars, etc. Show how different shapes can be combined to create new and familiar objects.

4. The ability to navigate in space and classify objects allows you to teach how to measure the size of an object

Early learning to measure length with a ruler and using centimeters is not recommended, as this will be difficult material to understand. Try measuring things with your child using sticks, ribbons, and other handy materials. In this training, not the measurement itself is invested, but the principle of its implementation.

Most educators advise teaching your child how to measure with counting sticks. They justify this by the convenience for the child and teaching him to use special material. These sticks will come in handy when learning units of counting. They can also be used as visual material when working with books (put the wand aside according to the number of characters), studying geometric shapes (the child can lay out the desired figure with chopsticks), etc.

5. Quantitative measurements

After learning the basic mathematical concepts, you can move on to quantitative measurements and the study of numbers. The study of numbers and their written designation occurs from an early age according to a certain system.

6. Addition and subtraction

Only after mastering quantitative measurements and numbers should you introduce addition and subtraction. Addition and subtraction are introduced at the age of 5-6 years and are the simplest operations for one action with small numbers.

7. Division

Division at preschool age is introduced only at the level of shares, when the child is asked to divide the object into equal shares. The number of such parts should not exceed four.

Examples of activities with a child to develop mathematical abilities

To solve this problem, you do not need any sophisticated methods, you just need to make some additions to your ordinary life.

  • When walking on the street, invite the child to count any objects or objects (tiles, cars, trees). Point to many objects, ask to find a generalizing sign;
  • Invite the child to solve problems to find the right answer, orienting him. For example, Masha has 3 apples, and Katya has 5, Lena has one apple more than Masha and one less than Katya. The problem can be simplified by asking what number is between 1 and 3;
  • Explain to your child what addition and subtraction are. Do this on apples, toys, or any other object. Let the child feel the objects and show these simple operations by adding or subtracting the object;
  • Ask the child about the difference between the objects;
  • Show what scales are and how they work. Explain that weight can not only be felt by picking up an object, but can also be measured in numbers;
  • Learn to use clocks with arrows;
  • Pay special attention to the spatial arrangement of objects;
  • Forms can be studied not only on cards, but also to look for them in objects around;
  • Show your child that mathematics is in everything that surrounds him, you just have to look closely.

What additional materials will help teach a child math

  • Cards and pictures with a different number of objects, with numbers and mathematical signs, geometric shapes;
  • Magnetic or chalkboard;
  • Watch with an arrow and scales;
  • Sticks for counting;
  • Constructors and puzzles;
  • Checkers and chess;
  • Lotto and dominoes;
  • Books that have an account and allow you to carry out mathematical operations;
  • Methodological aids for the development of logic and other abilities according to the age of the child.

Tips for parents who want to teach their child the basics of mathematics

1. Encourage your child to find answers. Help him find them by reasoning. Do not scold for mistakes and do not laugh at wrong answers. Each attempt of the child to draw a conclusion or solve a problem trains his abilities and allows him to consolidate knowledge;

2. Use the time of joint games to develop the necessary skills. Focus on what has been learned previously, show how new and already fixed material can be used in practice. Create situations in which the child will need to use knowledge in order to achieve a certain result;

3. Do not overload the child with a large amount of new information. Give him time to comprehend the knowledge gained through free play;

4. Combine the development of mathematical abilities with spiritual and physical development. Incorporate counting into PE classes and logic into reading and role-playing. Versatile development of the child - path to the full development of the baby. A physically and spiritually developed child comprehends mathematics much easier;

5. When teaching a child, try to use all channels of information absorption. In addition to the oral story, show it on various objects, let's feel and appreciate the weight and texture. Use a variety of ways to present information. Show how you can use the acquired knowledge in life;

6. Any material should be in the form of a game that will interest the child. Excitement and involvement in the process contributes well to memorization. If the child is not interested in the material, stop. Think about what went wrong and fix it. Each child is individual. Find a way that works for your little one and use it;

7. Important for the successful development of mathematical foundations is the ability to concentrate on the task and memorize the conditions. Ask a question about what the child understood from the given task after each condition. Work to improve concentration;

8. Before inviting the child to decide on their own, show an example of how to reason and decide. Even if the child has repeatedly carried out a certain calculation operation, remind him of the procedure. It is better to show the correct course of action than to allow the child to reinforce the wrong approach;

9. Do not force the child to study if he does not want to. If the kid wants to play, then give him this opportunity. Offer to work out after a while;

10. Try to diversify knowledge in one lesson. It would be better if during the day you pay a little attention to the most diverse areas of mathematical knowledge than if you memorize the same type of material, bringing it to automatism;

11. The task of a parent at preschool age is not to teach counting and make calculations, but to develop abilities. If you don’t teach your child to fold and take away before school, it’s not scary. If a child has mathematical thinking and knows how to draw conclusions, then he will be able to comprehend any complex operations quickly and at school.

What books help develop math skills

The solution to the problem of teaching mathematics to a child under 7 years of age with the help of books begins at an early age. So, for example, the fairy tale "Teremok". In it, the appearance of various characters occurs as they increase in size. In this example, you can teach a child the concepts of big - small. Try to play this fairy tale in the paper theater. Invite the child to arrange the figures of the heroes of the fairy tale in the correct order and tell the story. The tale "Turnip" also teaches the child the concepts of more and less, but its plot develops from the opposite (from large to small).

From a mathematical point of view, it will be useful to study the fairy tale "Three Bears" through the concepts of large, medium and small, the child easily learns to count up to three.

When choosing books to read to your child, pay attention to the following:

  • The presence of an account in the book and the possibility of comparing heroes according to some criteria;
  • Images in the book should be large and interesting. Using them, you can show the child which geometric shapes are used to create different objects (the house is a triangle and a square, the hero's head is a circle, etc.);
  • Any plot should develop linearly and contain certain conclusions at the end. Avoid books with complex plots that do not develop linearly. Teach your child that every action has consequences and how to draw conclusions. This approach will make it easier to understand the principles of logical thinking;
  • Books should be sorted by age.

There are a large number of different publications on sale that allow you to get acquainted with the majority of mathematical operations and terms using the examples of heroes. The main thing is to discuss the material read with the child and ask leading questions that will stimulate the development of mathematical abilities.

Purchase methodical books for the development of mathematical abilities in a child according to his age. Now there are a large number of different materials that contain tasks for the development of the mathematical abilities of the child. Bring such publications into the game. Remind your child about the tasks that he performed earlier on such a publication to solve new problems.

Developing math skills in a child is not an easy task. A child under 7 years of age is looking for new knowledge himself and is happy when they are presented to him in a playful way. Find an activity that suits your child and enjoy learning the basics of math.

Such representatives of certain trends in psychology as A. Binet, E. Thorndike and G. Reves, and such outstanding mathematicians as A. Poincaré and J. Hadamard contributed to the study of mathematical abilities. A wide variety of directions also determines a wide variety in approaches to the study of mathematical abilities. All scientists agree that it is necessary to distinguish between ordinary, "school" abilities to master mathematical knowledge, to reproduce it, to apply it independently, and creative mathematical abilities associated with the independent creation of an original and socially valuable product.

A. Rogers notes two aspects of mathematical abilities: reproductive (associated with the function of memory) and productive (associated with the function of thinking). W. Betz defines mathematical abilities as the ability to clearly understand the internal connection of mathematical relations and the ability to think accurately in mathematical concepts.

In the article “Psychologists of Mathematical Thinking”, D. Mordukhai-Boltovsky attached particular importance to the “unconscious thought process”, arguing that “the thinking of a mathematician is deeply embedded in the unconscious sphere, either surfacing to its surface, or plunging into depth. The mathematician is not aware of every step of his thought, like a virtuoso of bow movements. The sudden appearance in the mind of a ready-made solution to a problem that we cannot solve for a long time, we explain by unconscious thinking, which continued to deal with the task, and the result emerges beyond the threshold of consciousness. According to D. Mordukhai-Boltovsky, our mind is able to perform painstaking and complex work in the subconscious, where all the “rough” work is done, and the unconscious work of thought is even less error than the conscious one.

D. Mordukhai-Boltovsky notes the completely specific nature of mathematical talent and mathematical thinking. He argues that the ability to do mathematics is not always inherent even in brilliant people, that there is a significant difference between the mathematical and non-mathematical mind.

There are the following components of mathematical abilities:

  • - “strong memory” (memory, rather than for facts, but for ideas and thoughts);
  • - “wit” as the ability to “embrace in one judgment” concepts from two loosely connected areas of thought, to find in the already known something similar to the given, to look for something similar in the most remote, completely heterogeneous objects;
  • - "speed of thought" (speed of thought is explained by the work that the unconscious mind does to help the conscious mind).

D. Morduchai-Boltovsky distinguishes types of mathematical imagination that underlie different types of mathematicians - "algebraists" and "geometers". Arithmeticians, algebraists, and analysts in general, whose discovery is made in the most abstract form of breakthrough quantitative symbols and their relationships, cannot imagine, since "geometr".

The domestic theory of abilities was created by the joint work of the most prominent psychologists, of which B.M. Teplov, as well as L.S. Vygotsky, A.N. Leontiev, S.L. Rubinstein and B.G. Ananiev. In addition to general theoretical studies of the problem of mathematical abilities, V.A. Krutetsky, with his monograph "The Psychology of Schoolchildren's Mathematical Abilities", laid the foundation for an experimental analysis of the structure of mathematical abilities. Under the ability to study mathematics, he understands individual psychological characteristics (primarily features of mental activity) that meet the requirements of educational mathematical activity and determine, all other things being equal, the success of creative mastery of mathematics as an educational subject, in particular, relatively quick, easy and deep mastery of knowledge and skills. , skills in mathematics.

D.N. Bogoyavlensky and N.A. Menchinskaya, speaking of individual differences in the learning ability of children, introduce the concept of psychological properties that determine success in learning, all other things being equal.

Mathematical abilities are a complex structural mental formation, a kind of synthesis of properties, an integral quality of the mind, covering its various aspects and developing in the process of mathematical activity. This set is a single qualitatively original whole - only for the purposes of analysis, we single out individual components, not considering them as isolated properties. These components are closely connected, influence each other and form in their totality a single system, the manifestation of which is called the “mathematical giftedness syndrome”.

A great contribution to the development of this problem was made by V.A. Krutetsky. The experimental material collected by him allows us to speak about the components that occupy a significant place in the structure of such an integral quality of the mind as mathematical talent. V.A. Krutetsky presented a diagram of the structure of mathematical abilities at school age:

  • · Obtaining mathematical information (the ability to formalize the perception of mathematical material, covering the formal structure of the problem).
  • Processing of mathematical information
  • A) The ability for logical thinking in the field of quantitative and spatial relations, numerical and sign symbolism. The ability to think in mathematical symbols.
  • B) The ability to quickly and broadly generalize mathematical objects, relationships and actions.
  • C) the ability to curtail the process of mathematical reasoning and the system of corresponding actions. The ability to think in folded structures.
  • D) Flexibility of thought processes in mathematical activity.
  • E) Striving for clarity, simplicity, economy and rationality of decisions.
  • E) The ability to quickly and freely restructure the direction of the thought process, switching from direct to reverse thought (reversibility of the thought process in mathematical reasoning).
  • · Storage of mathematical information.

Mathematical memory (generalized memory for mathematical relations, typical characteristics, reasoning schemes, proofs, problem solving methods and principles of approach to them).

· General synthetic component. Mathematical mindset.

Not included in the structure of mathematical giftedness are those components whose presence in this structure is not necessary. They are neutral in relation to mathematical giftedness. However, their presence or absence in the structure (more precisely, the degree of development) determines the types of mathematical mentality. The speed of thought processes as a temporary characteristic, the individual pace of work are not of decisive importance. A mathematician can think slowly, even slowly, but very thoroughly and deeply. Calculating abilities (the ability to quickly and accurately calculate, often in the mind) can also be attributed to the neutral components. It is known that there are people who are able to reproduce complex mathematical calculations in their minds (almost instantaneous squaring and cube of three-digit numbers), but who are not able to solve any complex problems. It is also known that there were and still are phenomenal "counters" that did not give anything to mathematics, and the outstanding mathematician A. Poincret wrote about himself that even addition cannot be done without error.

Memory for figures, formulas and numbers is neutral in relation to mathematical giftedness. As academician A.N. Kolomogorov, many outstanding mathematicians did not have any outstanding memory of this kind.

The ability for spatial representations, the ability to visualize abstract mathematical relationships and dependencies also constitute a neutral component.

It is important to note that the diagram of the structure of mathematical abilities refers to the mathematical abilities of the student. It is impossible to say to what extent it can be considered a general scheme of the structure of mathematical abilities, to what extent it can be attributed to well-established gifted mathematicians.

It is known that in any field of science, giftedness as a qualitative combination of abilities is always diverse and unique in each individual case. But with the qualitative diversity of giftedness, it is always possible to outline some basic typological characteristics of differences in the structure of giftedness, to single out certain types that differ significantly from one another, coming in different ways with equally high achievements in the corresponding field.

Analytic and geometric types are mentioned in the works of A. Poincret, J. Hadamard, D. Mordukhai-Boltovsky, but with these terms they rather associate a logical, intuitive way of creativity in mathematics.

Among domestic researchers, N.A. Menchinskaya. She singled out students with a relative predominance of: a) figurative thinking over abstract c) the harmonious development of both types of thinking.

One cannot think that the analytic type appears only in algebra, and the geometric type in geometry. The analytical warehouse can manifest itself in geometry, and the geometric one - in algebra. V.A. Krutetsky gave a detailed description of each type.

Analytic type. Thinking of this type is characterized by the predominance of a very well-developed verbal-logical component over a weak visual-figurative one. They easily operate with abstract schemes. They have no need for visual supports, for the use of subject or schematic visualization in solving problems, even those when the mathematical relationships and dependencies given in the problem “suggest” visual representations.

Representatives of this type do not differ in the ability of visual-figurative representation and, therefore, use a more difficult and complex logical-analytical solution path where reliance on an image gives a much simpler solution. They very successfully solve problems expressed in an abstract form, while problems expressed in a concrete-visual form try to translate them into an abstract plan as far as possible. Operations associated with the analysis of concepts are easier to carry out than operations associated with the analyzer of a geometric diagram or drawing.

  • - Geometric type. The thinking of representatives of this type is characterized by a very well-developed visual-figurative component. In this regard, we can talk about the predominance of a well-developed verbal-logical component. These students feel the need for a visual interpretation of the expression of abstract material and demonstrate great selectivity in this regard. But if they fail to create visual supports, to use objective or schematic visualization in solving problems, then they hardly operate with abstract schemes. They stubbornly try to operate with visual schemes, images, ideas, even where the problem is easily solved by reasoning, and the use of visual supports is unnecessary or difficult.
  • - Harmonic type. This type is characterized by a balance of well-developed verbal-logical and visual-figurative components, with the former playing the leading role. Spatial representations in representatives of this type are well developed. They are selective in the visual interpretation of abstract relationships and dependencies, but visual images and schemes are subject to their verbal-logical analysis. Using visual images, these students are clearly aware that the content of the generalization is not limited to particular cases. Representatives of this type successfully implement a figurative-geometric approach to solving many problems.

The established types have a general meaning. Their presence is confirmed by many studies.

In foreign psychology, ideas about the age-related features of the mathematical development of a schoolchild, based on the studies of J. Piaget, are still widespread. Piaget believed that a child only by the age of 12 becomes capable of abstract thinking. Analyzing the stages of development of a teenager’s mathematical reasoning, L. Schoanne came to the conclusion that in a visual-specific plan, a student thinks up to 12-13 years old, and thinking in terms of formal algebra, associated with mastering operations, symbols, develops by the age of 17.

A study of domestic psychologists gives different results. P.P. Blonsky wrote about the intensive development of a teenager, generalizing and abstracting thinking, the ability to prove and understand evidence. Research by I.V. Dubrovina give grounds to say that, in relation to the age of younger schoolchildren, we cannot assert any formed structure of mathematical abilities proper, of course, excluding cases of special giftedness. Therefore, the concept of "mathematical ability" is conditional when applied to younger schoolchildren - children aged 7 - 10 years; when studying the components of mathematical abilities at this age, we can only talk about the elementary forms of such components. But the individual components of mathematical abilities are formed already in the primary grades.

Experimental training, which was carried out in a number of schools of the Institute of Psychology (D.B. Elkonin, V.V. Davydov), shows that with a special teaching method, younger students acquire a greater ability for distraction and reasoning than is commonly thought. However, although the age characteristics of the student to a greater extent depend on the conditions in which learning is carried out, it would be wrong to assume that they are entirely created by learning. Therefore, the extreme point of view on this question, when it is believed that there is no regularity in natural mental development, is wrong. A more effective system of teaching can “become” the whole process, but up to certain limits, the sequence of development can change somewhat, but cannot give the line of development a completely different character. There can be no arbitrariness here. For example, the ability to generalize complex mathematical relations and methods cannot be formed earlier than the ability to generalize simple mathematical relations. Thus, age features are a somewhat arbitrary concept. Therefore, all studies are focused on a general trend, on the general direction of development of the main components of the structure of mathematical abilities under the influence of learning.

In foreign psychology, there are works where an attempt is made to identify individual qualitative features of the mathematical thinking of boys and girls. V. Stern speaks of his disagreement with the point of view according to which differences in the mental sphere of men and women are the result of unequal education. In his opinion, the reasons lie in various internal inclinations. Therefore, women are less prone to abstract thinking and less capable in this regard.

In their studies, Ch. Spearman and E. Thorndike came to the conclusion that “there is no big difference in terms of abilities,” but at the same time they note a greater tendency for girls to detail, remember details.

Relevant research in Russian psychology was carried out under the guidance of I.V. Dubrovina and S.I. Shapiro. They did not find any qualitative specific features in the mathematical thinking of boys and girls. The teachers they interviewed did not point out these differences either.

Of course, in fact, boys are more likely to show mathematical ability. Boys are more likely to win Mathematical Olympiads than girls. But this actual difference must be attributed to the difference in traditions, in the education of boys and girls, due to the widespread view of male and female professions. This leads to the fact that mathematics is often outside the focus of the interests of girls.

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SUMMARY ON THE DISCIPLINE

Psychological and pedagogical foundations for teaching mathematics

"Mathematical Ability"

DONE: female student

correspondence department Dudrova L.V.

CHECKED: Gumenskaya O.M.

Saratov 2013

Introduction

1. Mathematical ability

4. Age features of mathematical abilities0

Conclusion

Bibliography

Introduction

Abilities - a set of mental qualities with a complex structure. For example, in the structure of mathematical abilities there are: the ability to generalize mathematically, the ability to stop the process of mathematical reasoning and actions, flexibility in solving mathematical problems, etc.

The structure of literary abilities is characterized by the presence of highly developed aesthetic feelings, vivid images of memory, a sense of the beauty of language, fantasy and the need for self-expression.

The structure of abilities in music, pedagogy, and medicine also has a rather specific character. Among the personality traits that form the structure of certain abilities, there are those that occupy a leading position, and there is also an auxiliary one. For example, in the structure of a teacher’s abilities, the leading ones will be: tact, the ability to selectively observe, love for pupils, which does not exclude exactingness, the need to teach, the ability to organize the educational process, etc. Auxiliary: artistry, the ability to concisely and clearly express one’s thoughts, etc.

It is clear that both the leading and auxiliary elements of the teacher's abilities form a single component of successful education and upbringing.

1. Mathematical ability

Such outstanding representatives of certain trends in psychology as A. Binet, E. Thorndike 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 determines a wide variety in approaches to the study of mathematical abilities. Of course, the study of mathematical abilities should begin with a definition. Attempts of this kind have been made repeatedly, but there is still no established, satisfying definition of mathematical abilities. The only thing that all researchers agree on is, perhaps, the opinion that one should distinguish between ordinary, “school” abilities for mastering mathematical knowledge, for their reproduction and independent application, and creative mathematical abilities associated with the independent creation of an original and of social value. product.

Back in 1918, in the work of A. Rogers, two aspects of mathematical abilities were noted, reproductive (associated with the function of memory) and productive (associated with the function of thinking). W. Betz defines mat. abilities as the ability to clearly understand the inner connection of mathematical relations and the ability to think accurately in mathematical concepts. Of the works of Russian authors, it is necessary to mention the original article by D. Mordukhai-Boltovsky "Psychology of Mathematical Thinking", published in 1918. The author, a specialist mathematician, wrote from an idealistic position, giving, for example, special significance to the “unconscious thought process”, arguing that “the thinking of a mathematician is deeply embedded in the unconscious sphere, now surfacing to its surface, now plunging into depth. The mathematician is not aware of every step of his thought, like a virtuoso of the movement of the bow.

Of great interest is Mordukhai-Boltovsky's attempt to isolate the components of mathematical abilities. He refers to such components in particular: “strong memory”, memory for “objects of the type that mathematics deals with”, memory rather than for facts, but for ideas and thoughts, “wit”, which means the ability to “embrace in one judgment" concepts from two loosely connected areas of thought, to find similarities with the given in what is already known, to look for similarities in the most separated, seemingly completely heterogeneous objects.

The Soviet theory of abilities was created by the joint work of the most prominent Russian psychologists, of which B.M. Teplov, as well as L.S. Vygotsky, A.N. Leontiev, S.L. Rubinstein and B.G. Ananiev.

In addition to general theoretical studies of the problem of mathematical abilities, V.A. Krutetsky, with his monograph "The Psychology of Schoolchildren's Mathematical Abilities", laid the foundation for an experimental analysis of the structure of mathematical abilities. Under the ability to study mathematics, he understands individual psychological characteristics (primarily the characteristics of mental activity) that meet the requirements of educational mathematical activity and determine, all other things being equal, the success of the creative mastery of mathematics as an educational subject, in particular, relatively quick, easy and deep mastery of knowledge and skills. , skills in mathematics. D.N. Bogoyavlensky and N.A. Menchinskaya, speaking of individual differences in the learning ability of children, introduces the concept of psychological properties that determine success in learning, all other things being equal. They do not use the term "ability", but in essence the corresponding concept is close to the definition given above.

Mathematical abilities are a complex structural mental formation, a kind of synthesis of properties, an integral quality of the mind, covering its various aspects and developing in the process of mathematical activity. This set is a single qualitatively original whole - only for the purposes of analysis, we single out individual components, by no means considering them as isolated properties. These components are closely connected, influence each other and form in their totality a single system, the manifestations of which we conventionally call the “mathematical giftedness syndrome”.

2. Structure of mathematical abilities

A great contribution to the development of this problem was made by V.A. Krutetsky. The experimental material collected by him allows us to speak about the components that occupy a significant place in the structure of such an integral quality of the mind as mathematical talent.

General scheme of the structure of mathematical abilities at school age

1. Obtaining mathematical information

A) The ability to formalize the perception of mathematical material, covering the formal structure of the problem.

2. Processing of mathematical information.

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

B) The ability to quickly and broadly generalize mathematical objects, relationships and actions.

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

D) Flexibility of thought processes in mathematical activity.

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

E) The ability to quickly and freely restructure the direction of the thought process, switching from direct to reverse thought (reversibility of the thought process in mathematical reasoning.

3. Storage of mathematical information.

A) Mathematical memory (generalized memory for mathematical relations, typical characteristics, reasoning and proof schemes, problem solving methods and principles of approach to them)

4. General synthetic component.

A) Mathematical orientation of the mind.

Not included in the structure of mathematical giftedness are those components whose presence in this structure 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 development) determines the types of mathematical mentality.

1. The speed of thought processes as a temporal characteristic. The individual pace of work is not critical. A mathematician can think slowly, even slowly, but very thoroughly and deeply.

2. Computational abilities (the ability to quickly and accurately calculate, often in the mind). It is known that there are people who are able to perform complex mathematical calculations in their minds (almost instantaneous squaring and cube of three-digit numbers), but who are not able to solve any complex problems. It is also known that there were and still are phenomenal "counters" that did not give anything to mathematics, and the outstanding mathematician A. Poincaré wrote about himself that even addition cannot be done without error.

3. Memory for numbers, formulas, numbers. As academician A.N. Kolmogorov, many outstanding mathematicians did not have any outstanding memory of this kind.

4. Ability for spatial representations.

5. Ability to visualize abstract mathematical relationships and dependencies

It should be emphasized that the scheme of the structure of mathematical abilities refers to the mathematical abilities of the student. It cannot be said to what extent it can be considered a general scheme of the structure of mathematical abilities, to what extent it can be attributed to well-established gifted mathematicians.

3. Types of mathematical mindsets

It is well known that in any field of science, giftedness as a qualitative combination of abilities is always diverse and unique in each individual case. But with the qualitative diversity of giftedness, it is always possible to outline some basic typological differences in the structure of giftedness, to single out certain types that differ significantly from one another, and come to equally high achievements in the corresponding field in different ways. Analytic and geometric types are mentioned in the works of A. Poincaré, J. Hadamard, D. Mordukhai-Boltovsky, but with these terms they rather associate a logical, intuitive way of creativity in mathematics.

Among domestic researchers, N.A. Menchinskaya. She singled out students with a relative predominance of: a) figurative thinking over abstract; b) abstract over figurative c) harmonious development of both types of thinking.

One cannot think that the analytic type appears only in algebra, and the geometric type in geometry. The analytical warehouse can manifest itself in geometry, and the geometric one - in algebra. V.A. Krutetsky gave a detailed description of each type.

Analytical type

The thinking of representatives of this type is characterized by a clear predominance of a very well-developed verbal-logical component over a weak visual-figurative one. They easily operate with abstract schemes. They have no need for visual supports, for the use of objective or schematic visualization in solving problems, even those when the mathematical relations and dependencies given in the problem “suggest” visual representations.

Representatives of this type do not differ in the ability of visual-figurative representation and, therefore, use a more difficult and complex logical-analytical solution path where reliance on an image gives a much simpler solution. They very successfully solve problems expressed in an abstract form, while problems expressed in a concrete-visual form try to translate them into an abstract plan as far as possible. Operations associated with the analysis of concepts are carried out by them easier than operations associated with the analysis of a geometric diagram or drawing.

Geometric type

The thinking of representatives of this type is characterized by a very well-developed visual-figurative component. In this regard, we can conditionally speak of predominance over a well-developed verbal-logical component. These students feel the need for a visual interpretation of the expression of abstract material and demonstrate great selectivity in this regard. But if they fail to create visual supports, use objective or schematic visualization in solving problems, then they hardly operate with abstract schemes. They stubbornly try to operate with visual schemes, images, ideas, even where the problem is easily solved by reasoning, and the use of visual supports is unnecessary or difficult.

harmonic type

This type is characterized by a relative balance of well-developed verbal-logical and visual-figurative components, with the former playing the leading role. Spatial representations in representatives of this type are well developed. They are selective in the visual interpretation of abstract relationships and dependencies, but visual images and schemes are subject to their verbal-logical analysis. Using visual images, these students are clearly aware that the content of the generalization is not limited to particular cases. They also successfully implement a figurative-geometric approach to solving many problems.

The established types seem to have a general meaning. Their presence is confirmed by many studies.

4. Age features of mathematical abilities

mathematical ability mind

In foreign psychology, ideas about the age-related features of the mathematical development of a schoolchild, based on the early studies of J. Piaget, are still widespread. Piaget believed that a child only by the age of 12 becomes capable of abstract thinking. Analyzing the stages of development of a teenager's mathematical reasoning, L. Schoanne came to the conclusion that in terms of visual-specific, a student thinks up to 12-13 years old, and thinking in terms of formal algebra, associated with mastering operations, symbols, develops only by 17 years.

A study of domestic psychologists gives different results. More P.P. Blonsky wrote about the intensive development in a teenager (11-14 years old) of generalizing and abstracting thinking, the ability to prove and understand evidence. A legitimate question arises: to what extent can we talk about mathematical abilities in relation to younger students? Research led by I.V. Dubrovina, gives grounds to answer this question in the following way. Of course, excluding cases of special giftedness, we cannot speak of any formed structure of mathematical abilities proper in relation to this age. Therefore, the concept of "mathematical abilities" is conditional when applied to younger schoolchildren - children of 7-10 years old, when studying the components of mathematical abilities at this age, we can usually talk only about the elementary forms of such components. But individual components of mathematical abilities are already formed in the primary grades.

Experimental training, which was carried out in a number of schools by employees of the Institute of Psychology (D.B. Elkonin, V.V. Davydov), shows that with a special teaching method, younger students acquire a greater ability for distraction and reasoning than is commonly thought. However, although the age characteristics of the student to a greater extent depend on the conditions in which learning is carried out, it would be wrong to say that they are entirely created by learning. Therefore, the extreme point of view on this question, when it is believed that there is no regularity in natural mental development, is wrong. A more effective system of teaching can “become” the whole process, but up to certain limits, the sequence of development can change somewhat, but cannot give the line of development a completely different character.

Thus, the age features that are mentioned are a somewhat arbitrary concept. Therefore, all studies are focused on a general trend, on the general direction of development of the main components of the structure of mathematical abilities under the influence of learning.

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 be no question of an accurate and rigorous understanding of the content of this concept.

The books reviewed in this paper confirm this conclusion. At the same time, it should be noted the 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 foundation - the theory of mathematical abilities.

So, as V.A. Krutetsky: "The task of the 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 activity. The development of this problem is of both theoretical and practical interest."

Bibliography

1. Gabdreeva G.Sh. The main aspects of the problem of anxiety in psychology // Tonus. 2000 №5

2. Gurevich K.M. Fundamentals of career guidance M., 72.

3. Dubrovina I.V. Individual differences in the ability to generalize mathematical and non-mathematical material in primary school age. // Issues of psychology., 1966 No. 5

4. Izyumova I.S. Individual-typological features of schoolchildren with literary and mathematical abilities.// Psychol. magazine 1993 No. 1. T.14

5. Izyumova I.S. On the problem of the nature of abilities: the makings of mnemonic abilities in schoolchildren of mathematical and literary classes. // Psych. magazine

6. Eleseev O.P. Workshop on the psychology of personality. SPb., 2001

7. Kovalev A.G. Myasishchev V.N. Psychological characteristics of a person. T.2 "Abilities" Leningrad State University.: 1960

8. Kolesnikov V.N. Emotionality, its structure and diagnostics. Petrozavodsk. 1997.

9. Kochubey B.I. Novikov E.A. Emotional stability of schoolchildren. M. 1988

10. Krutetsky V.A. Psychology of mathematical abilities. M. 1968

11. Levitov V.G. mental state of anxiety, anxiety.//Questions of psychology 1963. No. 1

12. Leitis N.S. Age giftedness and individual differences. M. 1997

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Mathematical abilities have a direct impact on the mental development of a preschooler. A child has to look at the world around him with a “mathematical eye” to a much greater extent than an adult. The reason is that in a short period of time, the child's brain needs to deal with shapes and sizes, geometric shapes and spatial orientation, understand their characteristics and relationships.

What abilities in preschool age are related to mathematical

Many parents think that it is too early to develop the mathematical abilities of children at preschool age. And they mean by this concept some special abilities that allow children to operate with large numbers, or a passion for formulas and algorithms.

In the first case, abilities are confused with natural giftedness, and in the other case, a pleasing result may have nothing to do with mathematics. Perhaps the child liked the rhythm of counting or remembered the images of numbers in an arithmetic example.

To dispel this misconception, it is important to clarify what abilities are called mathematical.

Mathematical abilities are the features of the flow of the thought process with the severity of analysis and synthesis, rapid abstraction and generalization in relation to mathematical material.

It relies on the same mental operations. They develop in all children with varying efficiency. It is possible and necessary to stimulate their development. This does not mean at all that the child will awaken mathematical talent, and he will grow up to be a real mathematician. But, if you develop the ability to analyze, highlight signs, generalize, build a logical chain of thoughts, then this will contribute to the development of the preschooler's mathematical abilities and more general intellectual ones.

Elementary mathematical representations of preschoolers

So, abilities for mathematics go far beyond arithmetic and develop on the basis of mental operations. But, just as the word is the basis of speech, so in mathematics there are elementary ideas, without which it is pointless to talk about development.

Toddlers need to be taught to count, to introduce quantitative relationships, to expand their knowledge of geometric shapes. By the end of preschool age, the child should have basic mathematical representations:

  1. Know all the numbers from 0 to 9 and recognize them in any form of writing.
  2. Count from 1 to 10, both forward and backward (starting with any number).
  3. Have an idea about simple ordinal numbers and be able to operate with them.
  4. Perform addition and subtraction operations within 10.
  5. Be able to equalize the number of items in two sets (There are 5 apples in one basket, 7 pears in the other. What needs to be done to make the fruits in the baskets equally?).
  6. Know the basic geometric shapes and name the features that distinguish them.
  7. Operate with quantitative ratios "more-less", "further-closer".
  8. Operate with simple qualitative ratios: the largest, smallest, lowest, etc.
  9. Understand complex relationships: “more than the smallest, but less than others”, “ahead and above others”, etc.
  10. Be able to identify an extra object that is not suitable for a group of others.
  11. Arrange simple rows in ascending and descending order (The cubes show dots in the amount of 3, 5, 7, 8. Arrange the cubes so that the number of dots on each subsequent one decreases).
  12. Find the corresponding place of an object with a numerical sign (On the example of the previous task: cubes with points 3, 5 and 8 are placed. Where to put a cube with 7 points?).

This mathematical "baggage" is to be accumulated by the child before entering school. The listed representations are elementary. It is impossible to study mathematics without them.

Among the basic skills, there are quite simple ones that are already available at 3-4 years old, but there are also those (9-12 points) that use the simplest analysis, comparison, generalization. They have to be formed in the process of playing lessons at the senior preschool age.

The list of elementary representations can be used to identify the mathematical abilities of preschoolers. Having offered the child to complete the task corresponding to each item, they determine which skills have already been formed and which ones need to be worked on.

We develop the mathematical abilities of the child in the game

Completing tasks with a mathematical bias is especially useful for children, as it develops. The value lies not only in the accumulation of mathematical concepts and skills, but also in the fact that the general mental development of the preschooler takes place.

In practical psychology, there are three categories of gaming activities aimed at developing individual components of mathematical abilities.

  1. Exercises to determine the properties of objects, identify objects according to a designated feature (analytical and synthetic abilities).
  2. Games for comparing various properties, identifying essential features, abstracting from secondary ones, generalization.
  3. Games for the development of logical conclusions based on mental operations.

The development of mathematical abilities in preschool children should be carried out exclusively in a playful way.

Exercises for the development of analysis and synthesis

1.Get in order! A game to sort objects by size. Prepare 10 one-color strips of cardboard of the same width and different lengths and arrange them randomly in front of the preschooler.

Instruction: "Arrange the "athletes" in height from the shortest to the tallest." If the child is at a loss with the choice of the strip, invite the "athletes" to measure their height.

After completing the task, invite the child to turn away and swap some of the strips. The preschooler will have to return the "hooligans" to their places.

2.Make a square. Prepare two sets of triangles. 1st - one large triangle and two small ones; 2nd - 4 identical small ones. Invite the child to first fold a square of three parts, then of four.

Picture 1.

If a preschooler spends less time compiling the second square, then understanding has come. Capable children complete each of these tasks in less than 20 seconds.

Abstraction and generalization exercises

1.The fourth is redundant. You will need a set of cards that show four items. On each card, three objects should be interconnected by a significant feature.

Instructions: “Find what is odd in the picture. What does not suit everyone else and why?

Figure 2.

Such exercises should start with simple groups of objects and gradually complicate them. For example, a card with the image of a table, a chair, a kettle and a sofa can be used in classes with 4-year-old children, and sets with geometric shapes can be offered to older preschoolers.

2.Build a fence. It is necessary to prepare at least 20 strips of equal length and width or counting sticks in two colors. For example: blue - C, and red - K.

Instruction: “Let's build a beautiful fence where colors alternate. The first will be a blue stick, followed by a red one, then ... (we continue to lay out the sticks in the sequence SKSSKKSK). And now you continue to build a fence so that there is the same pattern.

In case of difficulty, pay the child's attention to the rhythm of the alternation of colors. The exercise can be performed several times with a different rhythm of the pattern.

Logical and mathematical games

1.We're going, we're going, we're going. It is necessary to select 10-12 rectangular pictures depicting objects well known to the child. A child plays with an adult.

Instruction: “Now we will make a train of wagons, which will be firmly interconnected by an important feature. There will be a cup in my trailer (puts the first picture), and in order for your trailer to join, you can select a picture with a picture of a spoon. The cup and spoon are connected because they are dishes. I will complete our train with a picture of a scoop, since the scoop and spoon have a similar shape, etc.”

The train is ready to go if all the pictures have found their place. You can mix pictures and start the game again, finding new relationships.

2. Tasks for finding a suitable “patch” for a rug are of great interest to preschoolers of different ages. To play the game, you need to make several pictures that show a rug with a cut out circle or rectangle. Separately, it is necessary to depict options for “patches” with a characteristic pattern, among which the child will have to find a suitable one for the rug.

You need to start completing tasks with the color shades of the rug. Then offer cards with simple patterns of rugs, and as the skills of logical choice develop, complicate the tasks on the model of the Raven test.

Figure 3

“Repairing” the rug simultaneously develops a number of important aspects: visual-figurative representations, mental operations, the ability to recreate the whole.

Recommendations for parents on the development of mathematical abilities of the child

Oftentimes, liberal arts parents tend to ignore the development of math skills in their children, and this is a misguided approach. At preschool age, these abilities are used by the child to learn about the world around them.

A preschooler needs to be stimulated by a mathematical approach in order to understand the patterns, the cause-and-effect and logical way of real life.

From early childhood, the child should be surrounded by educational toys that require elementary analysis and the search for regular connections. These are various pyramids, mosaics, insert toys, sets of cubes and other geometric bodies, LEGO constructors.

Upon reaching the age of three, it is necessary to supplement the cognitive activity of the child with games that stimulate the formation of mathematical abilities. In this case, several important points should be taken into account:

  • Educational games should be short. Preschoolers with the right inclinations show curiosity about such games, therefore, they should last as long as there is interest. Other children need to be skillfully lured to complete the task.
  • Games of an analytical and logical nature should be carried out using visual material - pictures, toys, geometric shapes.
  • It is easy to prepare stimulus material for the game yourself, focusing on the examples in this article.

Scientists substantiated that the use of geometric material is most effective in the development of mathematical abilities. The perception of figures is based on sensory abilities that are formed in the child earlier than others, allowing the baby to capture the connections and relationships between objects or their details.

Developing logical and mathematical games and exercises contribute to the formation of independent thinking of a preschooler, his ability to highlight the main thing in a significant amount of information. And these are the qualities that are necessary for successful learning.

To explain where the ability to perform mathematical operations developed in a person, experts suggested two hypotheses. One of them was that the aptitude for mathematics is a side effect of the emergence of language and speech. Another suggested that the reason was the ability to use an intuitive understanding of space and time, which has a much more ancient evolutionary origin.

In order to answer the question which of the hypotheses is correct, psychologists put experiment involving 15 professional mathematicians and 15 ordinary people with the same level of education. Each group was presented with complex mathematical and non-mathematical statements that had to be assessed as true, false or meaningless. During the experiment, the brains of the participants were scanned using functional tomography.

The results of the study showed that statements that dealt with calculus, algebra, geometry and topology, activated areas in the parietal, inferotemporal and prefrontal cortex in mathematicians, but not in the control group. These zones differed from those that were excited in all participants in the experiment with ordinary statements. "Mathematical" sections were activated in ordinary people only if the subjects were asked to do simple arithmetic operations.

Scientists explain the result by saying that high-level mathematical thinking involves a neural network that is responsible for the perception of numbers, space and time and is different from the network associated with language. According to experts, based on the study, it is possible to predict whether a child will develop mathematical abilities if assessed spatial thinking skills.

Thus, to become a mathematician you need to develop spatial thinking.

What is spatial thinking

To solve a huge number of tasks that our civilization puts before us, a special kind of mental activity is needed - spatial thinking. The term spatial imagination refers to the human ability to clearly represent three-dimensional objects in detail and color.

With the help of spatial thinking, one can manipulate spatial structures - real or imaginary, analyze spatial properties and relationships, transform the original structures and create new ones. It has long been known in the psychology of perception that initially only a few percent of the population possesses the rudiments of spatial thinking.

Spatial thinking is a specific type of mental activity that takes place in solving problems that require orientation in practical and theoretical space (both visible and imaginary). In its most developed forms, this is thinking by patterns in which spatial properties and relationships are fixed.

How to develop spatial thinking

Exercises for the development of spatial thinking are very useful at any age. At first, many people find it difficult to complete them, but over time they gain the ability to solve more and more difficult tasks. Such exercises ensure the normal functioning of the brain, avoid many diseases caused by an insufficient level of work of the neurons of the cerebral cortex.

Children with developed spatial thinking often excel not only in geometry, drawing, chemistry and physics, but also in literature! Spatial thinking allows you to create whole dynamic pictures in your head, a kind of movie based on a read passage of text. This ability greatly facilitates the analysis of fiction and makes the reading process much more interesting. And, of course, spatial thinking is indispensable in the lessons of drawing and labor.

With developed spatial thinking, it becomes much it is easier to read drawings and maps, locate and present a route to the goal. This is simply necessary for orienteering enthusiasts, and for everyone else it will greatly help in everyday life in the city.

Spatial thinking develops from early childhood, when the child begins to make his first movements. Its formation goes through several stages and ends approximately in adolescence. However, in the course of life, its additional development and transformation is possible. You can check the level of development of spatial thinking with the help of a small interactive test.

There are three types of such operation:

  1. Changing the spatial position of the image. A person can mentally move an object without any change in its appearance. For example, movement according to the map, mental rearrangement of objects in the room, redrawing, etc.
  2. Change the structure of the image. A person can mentally change an object in any way, but at the same time it remains motionless. For example, mentally adding one shape to another and combining them, imagining how an object will look if you add a detail to it, etc.
  3. Simultaneous change in both the position and structure of the image. A person is able to simultaneously imagine changes in the appearance and spatial position of an object. For example, a mental rotation of a three-dimensional figure with different sides, an idea of ​​​​how such a figure will look like from one side or the other, etc.

The third type is the most advanced and provides more options. However, to achieve it, you must first master the first two types of operation well. The exercises and tips presented below will be aimed at developing overall spatial thinking and all three types of actions.

3D puzzles and origami

Folding three-dimensional puzzles and paper figures allows you to form images of various objects in your head. After all, before starting work, you should present the finished figure in order to determine the quality and procedure. Folding can take place in several stages:

  • Repetition of actions for someone
  • Work in accordance with the instructions
  • Folding the figure with partial support on the instruction
  • Independent work without relying on the material (may not be carried out immediately, but after several repetitions of the previous steps)

It is important that the student clearly trace each action and remember it. Instead of puzzles, you can also use a regular constructor.

They are divided into two types:

  1. Using visual aids. To do this, you need to have several blanks of various three-dimensional geometric shapes: a cone, a cylinder, a cube, a pyramid, etc. Task: to study the figures; find out how they look from different angles; put figures on top of each other and see what happens, etc.
  2. Without the use of visual material. If the student is well acquainted with various three-dimensional geometric shapes and has a good idea of ​​how they look, then the tasks are transferred to the mental plan. Task: describe what this or that figure looks like; name each side of it; imagine what will happen when one figure is superimposed on another; say what action needs to be performed with a figure in order to turn it into another (for example, how to turn a parallelepiped into a cube), etc.

Redrawing (copying)

Tasks of this type are in order of increasing complexity:

  1. A simple redrawing of the figure. The student has a model/sample of a figure in front of him, which he needs to transfer to paper without changes (the dimensions and appearance must match). Each side of the figure is drawn separately.
  2. Copying with addition. Task: redraw the figure without changes and add to it: 5 cm in length, an additional face, another figure, etc.
  3. Scalable redrawing. Task: copy the figure with a change in its size, i.e. draw 2 times more than the layout, 5 times less than the sample, subtracting 3 cm on each side, etc.
  4. Copy from view. Task: imagine a three-dimensional figure and draw it from different sides.

Representation

Segments and lines will act as presentation objects. Tasks can be very diverse, for example:

  • Imagine three differently directed segments, mentally connect them and draw the resulting figure.
  • Imagine that a triangle is superimposed on two segments. What happened?
  • Imagine two lines coming together. Where will they intersect?

Drawing up drawings and diagrams

They can be based on visual material or based on represented objects. You can draw up drawings, diagrams and plans for any subject. For example, a plan of a room showing the location of each thing in it, a schematic representation of a flower, a drawing of a building, etc.

Game "Guess by touch"

The child closes his eyes and receives some object that he can feel. The object must be of such dimensions that the student has the opportunity to study it in its entirety. A certain amount of time is allotted for this, depending on the age of the student and the volume of the subject (15-90 seconds). After this time, the child must say what exactly it was and why he decided so.

Also in the game you can use different types of fabric, fruits similar in shape (apples, nectarines, oranges, peaches), non-standard geometric shapes and more.

The game "Fly in a cage"

This game requires at least three people. Two are directly involved in the game, and the third monitors its progress and checks the final answer.

Rules: two participants represent a grid of 9 by 9 squares (you can not use a graphic image!). There is a fly in the upper right corner. Taking turns making moves, the players move the fly around the squares. You can use movement symbols (right, left, up, down) and the number of cells. For example, a fly moves up three squares. The third participant has a graphic diagram of the grid and indicates each move (each movement of the fly). He then says "Stop" and the other players have to say where they think the fly is at the moment. The winner is the one who correctly named the square where the fly stopped (checked according to the scheme compiled by the third participant).

The game can be made more difficult by adding a number of cells to the grid or a parameter such as depth (making the grid three-dimensional).

Graphic tasks-simulators

They are performed by eye without the use of any auxiliary items (rulers, pens, compasses, etc.).

1. To what mark should a person move so that a falling tree does not hit him?

2. Which (which) of the figures can (can) pass between object A and object B?

Picture from the book of Postalovsky I.Z. "Training of imaginative thinking"

3. Imagine that the ovals in the picture are cars. Which of them will be at the intersection first, if the speed of movement of cars is equal?

Picture from the book of Postalovsky I.Z. "Training of imaginative thinking"

4. Restore the part of the figure that the ruler covered.

Picture from the book of Postalovsky I.Z. "Training of imaginative thinking"

5. Determine where the ball will fall.

Picture from the book of Postalovsky I.Z. "Training of imaginative thinking"