How to develop in mathematical abilities. V.A

"Not neither one child not capable, mediocre. Important, to this mind, this talent become basis success in teaching, to neither one student not studied below their opportunities" (Sukhomlinsky V.A.)

What is mathematical ability? Or are they nothing more than a qualitative specialization of general mental processes and personality traits, that is, general intellectual abilities developed in relation to mathematical activity? Is mathematical ability a unitary or integral property? In the latter case, we can talk about the structure of mathematical abilities, about the components of this complex education. Psychologists and educators have been looking for answers to these questions since the beginning of the century, but there is still no single view on the problem of mathematical abilities. Let's try to understand these issues by analyzing the work of some of the leading experts who worked on this problem.

Great importance in psychology is attached to the problem of abilities in general and the problem of the abilities of schoolchildren in particular. A number of psychologists' studies are aimed at revealing the structure of schoolchildren's abilities for various types of activity.

In science, in particular in psychology, the discussion continues about the very essence of abilities, their structure, origin and development. Without going into the details of traditional and new approaches to the problem of abilities, we point out some of the main controversial points of the various points of view of psychologists on abilities. However, among them there is no single approach to this problem.

The difference in understanding the essence of abilities is found, first of all, in whether they are considered as socially acquired properties or are recognized as natural. Some authors understand abilities as a complex of individual psychological characteristics of a person that meet the requirements of this activity and are a condition for its successful implementation, which are not reduced to preparedness, to existing knowledge, skills and abilities. Here you should pay attention to several facts. First, abilities are individual characteristics, that is, what distinguishes one person from another. Secondly, these are not just features, but psychological features. And, finally, abilities are not all individual psychological characteristics, but only those that meet the requirements of a certain activity.

With a different approach, most pronounced in K.K. Platonov, any quality of the "dynamic functional structure of the personality" is considered an ability, if it ensures the successful development and performance of activities. However, as noted by V.D. Shadrikov, "with this approach to abilities, the ontological aspect of the problem is transferred to makings, which are understood as the anatomical and physiological characteristics of a person, which form the basis for the development of abilities. The solution of the psychophysiological problem led to a dead end in the context of abilities as such, since abilities, as a psychological category, were not considered as a property of the brain. The sign of success is no more productive, because the success of an activity is determined by the goal, motivation, and many other factors. "According to his theory of abilities, it is possible to productively define abilities as features only in relation to their individual and universal.

Universal (general) for each ability of V.D. Shadrikov names the property on the basis of which a specific mental function is realized. Each property is an essential characteristic of a functional system. It was in order to realize this property that a specific functional system was formed in the process of human evolutionary development, for example, the property to adequately reflect the objective world (perception) or the property to capture external influences (memory) and so on. The property is manifested in the process of activity. Thus, it is now possible to define abilities from the standpoint of the universal as a property of a functional system that implements individual mental functions.

There are two types of properties: those that do not have intensity and therefore cannot change it, and those that have intensity, that is, they can be more or less. The humanities deal mainly with the properties of the first kind, the natural sciences with the properties of the second kind. Mental functions are characterized by properties that have intensity, a measure of severity. This allows you to determine the ability from the standpoint of a single (separate, individual). A single will be represented by a measure of the severity of the property;

Thus, according to the theory presented above, abilities can be defined as properties of functional systems that implement individual mental functions, which have an individual measure of severity, manifested in the success and qualitative originality of the development and implementation of activities. When evaluating an individual measure of the severity of abilities, it is advisable to use the same parameters as when characterizing any activity: productivity, quality and reliability (in terms of the considered mental function).

One of the initiators of studying the mathematical abilities of schoolchildren was the outstanding French mathematician A. Poincaré. He stated the specificity of creative mathematical abilities and singled out their most important component - mathematical intuition. Since that time, the study of this problem began. Subsequently, psychologists identified three types of mathematical abilities - arithmetic, algebraic and geometric. At the same time, the question of the presence of mathematical abilities remained insoluble.

In turn, researchers W. Haeker and T. Ziegen identified four main complex components: spatial, logical, numerical, symbolic, which are the "core" of mathematical abilities. In these components, they distinguished between understanding, memorization, and operation.

Along with the main component of mathematical thinking - the ability for selective thinking, for deductive reasoning in the numerical and symbolic spheres, the ability for abstract thinking, A. Blackwell also highlights the ability to manipulate spatial objects. He also notes the verbal ability and the ability to store data in their exact and strict order and meaning in memory.

A significant part of them is of interest today. In the book, which was originally called "The Psychology of Algebra", E. Thorndike first formulates general mathematical capabilities: the ability to handle symbols, choose and establish relationships, generalize and systematize, select essential elements and data in a certain way, bring ideas and skills into a system. He also highlights special algebraic capabilities: the ability to understand and compose formulas, express quantitative relations as a formula, transform formulas, write equations expressing given quantitative relations, solve equations, perform identical algebraic transformations, graphically express the functional dependence of two quantities, etc.

One of the most significant studies of mathematical abilities since the publication of the works of E. Thorndike belongs to the Swedish psychologist I. Verdelin. He gives a very broad definition of mathematical ability, which reflects the reproductive and productive aspects, understanding and application, but he focuses on the most important of these aspects - the productive one, which he explores in the process of solving problems. The scientist believes that the teaching method can affect the nature of mathematical abilities.

The leading Swiss psychologist J. Piaget attached great importance to mental operations, distinguishing in the ontogenetic development of the intellect the stage of slightly formalized specific operations associated with specific data, and the stage of generalized formalized operations, when operator structures are organized. He correlated the latter with the three fundamental mathematical structures identified by N. Bourbaki: algebraic, order structures, and topological. J. Piaget discovers all types of these structures in the development of arithmetic and geometric operations in the child's mind and in the features of logical operations. Hence the conclusion is drawn about the need for the synthesis of mathematical structures and operator structures of thinking in the process of teaching mathematics.

In psychology, V.A. Krutetsky. In his book "Psychology of mathematical abilities of schoolchildren" he gives the following general scheme of the structure of mathematical abilities of schoolchildren. Firstly, obtaining mathematical information is the ability to formalize the perception of mathematical material, grasping the structure of the problem. Secondly, the processing of mathematical information is the ability for logical thinking in the field of quantitative and spatial relations, numerical and symbolic symbolism, the ability to think in mathematical symbols, the ability to quickly and broadly generalize mathematical objects, relationships and actions, the ability to curtail the process of mathematical reasoning and the system appropriate actions, the ability to think in folded structures. It also requires the flexibility of thought processes in mathematical activity, the desire for clarity, simplicity, economy and rationality of decisions. An essential role is played here by the ability to quickly and freely restructure the direction of the thought process, switch from the direct to the reverse course of thought (the reversibility of the thought process in mathematical reasoning). Thirdly, the storage of mathematical information is mathematical memory (generalized memory for mathematical relations, typical characteristics, reasoning and proof schemes, methods for solving problems and principles for approaching them). And, finally, the general synthetic component is the mathematical orientation of the mind. All the studies cited above suggest that the factor of general mathematical reasoning underlies general mental abilities, and mathematical abilities have a general intellectual basis.

From a different understanding of the essence of abilities, a different approach to the disclosure of their structure follows, which, according to different authors, appears as a set of different qualities, classified on different grounds and in different proportions.

There is no single answer to the question of the genesis and development of abilities, their connection with activity. Along with the assertion that abilities in their generic form exist in a person before activity as a prerequisite for its implementation. Another, contradictory point of view was also expressed: abilities do not exist before the activity of B.M. Thermal. The last provision leads to a dead end, since it is not clear how activity begins to be performed without the ability to do so. In reality, abilities at a certain level of their development exist before activity, and with the beginning of it they manifest themselves and then develop in activity, if it makes ever higher demands on a person.

However, this does not reveal the correlation of skills and abilities. The solution to this problem was proposed by V.D. Shadrikov. He believes that the essence of the ontological differences between abilities and skills is as follows: an ability is described by a functional system, one of its essential elements is a natural component, which is the functional mechanisms of abilities, and skills are described by an isomorphic system, one of its main components are abilities, performing in this system those functions that in the system of abilities implement functional mechanisms. Thus, the functional system of skills, as it were, grows out of the system of abilities. This is a system of the secondary level of integration (if we take the system of abilities as primary).

Speaking about abilities in general, it should be pointed out that abilities are of different levels, educational and creative. Learning abilities are associated with the assimilation of already known ways of performing activities, the acquisition of knowledge, skills and abilities. Creativity is associated with the creation of a new, original product, with finding new ways to perform activities. From this point of view, there are, for example, the ability to assimilate, study mathematics and creative mathematical abilities. But, as J. Hadamard wrote, "between the work of a student solving a problem ... and creative work, the difference is only in level, since both works are of a similar nature" .

Natural prerequisites matter, however, they are not actually abilities, but are inclinations. The inclinations themselves do not mean that a person will develop the corresponding abilities. The development of abilities depends on many social conditions (upbringing, the need for communication, the education system).

Ability types:

1. Natural (natural) abilities.

Are common to humans and animals: perception, memory, the ability to elementary communication. These abilities are directly related to innate inclinations. On the basis of these inclinations, a person, in the presence of elementary life experience, through the mechanisms of learning, develops specific abilities.

2. Specific abilities.

General: determine the success of a person in various activities (thinking abilities, speech, accuracy of manual movements).

Special: determine the success of a person in specific activities, for the implementation of which the inclinations of a special kind and their development are necessary (musical, mathematical, linguistic, technical, artistic abilities).

In addition, abilities are divided into theoretical and practical. Theoretical ones predetermine a person's inclination to abstract-theoretical reflections, and practical ones - to concrete practical actions. Most often, theoretical and practical abilities are not combined with each other. Most people have either one or the other type of ability. Together they are extremely rare.

There is also a division into educational and creative abilities. The former determine the success of training, the assimilation of knowledge, skills, and the latter determine the possibility of discoveries and inventions, the creation of new objects of material and spiritual culture.

3. Creative abilities.

This is, first of all, the ability of a person to find a special look at familiar and everyday things or tasks. This skill is directly dependent on the horizons of a person. The more he knows, the easier it is for him to look at the issue under study from different angles. A creative person is constantly striving to learn more about the world around him, not only in the field of his main activity, but also in related industries. In most cases, a creative person is, first of all, an original thinking person, capable of non-standard solutions.

Ability Development Levels:

1) Inclinations - natural prerequisites for abilities;
2) Abilities - a complex, integral, mental formation, a kind of synthesis of properties and components;
3) Giftedness - a kind of combination of abilities that provides a person with the opportunity to successfully perform any activity;
4) Mastery - excellence in a particular type of activity;
5) Talent - a high level of development of special abilities (this is a certain combination of highly developed abilities, since an isolated ability, even a very highly developed one, cannot be called talent);
6) Genius - the highest level of development of abilities (in the entire history of civilization there were no more than 400 geniuses).

General mental capabilities- these are the abilities that are necessary to perform not one, but many types of activities. General mental abilities include, for example, such qualities of the mind as mental activity, criticality, systematic, focused attention. Man is naturally endowed with general abilities. Any activity is mastered on the basis of general abilities that develop in this activity.

As V.D. Shadrikov, " special capabilities" there are general abilities that have acquired the features of efficiency under the influence of the requirements of the activity. "Special abilities are the abilities that are necessary for the successful mastery of any one specific activity. These abilities also represent the unity of individual private abilities. For example, in the composition mathematical abilities mathematical memory plays an important role; ability to logical thinking in the field of quantitative and spatial relations; fast and wide generalization of mathematical material; easy and free switching from one mental operation to another; striving for clarity, economy, rationality of reasoning, and so on. All particular abilities are united by the core ability of the mathematical orientation of the mind (which is understood as the tendency to isolate spatial and quantitative relationships, functional dependencies during perception), associated with the need for mathematical activity.

A. Poincare came to the conclusion that the most important place in mathematical abilities is the ability to logically build a chain of operations that will lead to the solution of a problem. In addition, it is not enough for a mathematician to have a good memory and attention. According to Poincaré, people capable of mathematics are distinguished by the ability to grasp the order in which the elements necessary for mathematical proof should be located. The presence of this kind of intuition is the basic element of mathematical creativity.

L.A. Wenger refers to mathematical abilities such features of mental activity as the generalization of mathematical objects, relations and actions, that is, the ability to see the general in various specific expressions and tasks; the ability to think in "contracted", large units and "economically", without too much detail; the ability to switch from direct to reverse thought.

In order to understand what other qualities are required to achieve success in mathematics, the 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: what is not, and cannot be, the only pronounced mathematical ability is a cumulative characteristic that reflects the features of various mental processes: perception, thinking, memory, imagination.

The selection of the most important components of mathematical abilities is shown in Figure 1:

Picture 1

Some researchers also single out as an independent component mathematical memory for schemes of reasoning and evidence, methods for solving problems and ways of approaching them. One of them is V.A. Krutetsky. He defines mathematical abilities as follows: “Under the ability to study mathematics, we mean individual psychological characteristics (primarily the characteristics of mental activity) that meet the requirements of educational mathematical activity and determine, on other equal conditions, the success of creative mastery of mathematics as an educational subject, in particular, relatively fast, easy and deep mastery of knowledge, skills and abilities in the field of mathematics".

In our work, we will mainly rely on the research of this particular psychologist, since his research on this problem is still the most global, and his conclusions are the most experimentally substantiated.

So, V.A. Krutetskiy distinguishes nine components mathematical abilities:

1. The ability to formalize mathematical material, to separate form from content, to abstract from specific quantitative relations and spatial forms and to operate with formal structures, structures of relations and connections;
2. The ability to generalize mathematical material, isolate the main thing, digressing from the inessential, see the general in outwardly different;
3. Ability to operate with numerical and symbolic symbols;
4. The ability to "consistent, properly divided logical reasoning", associated with the need for evidence, justification, conclusions;
5. The ability to shorten the process of reasoning, to think in folded structures;
6. The ability to reversibility of the thought process (to the transition from direct to reverse thought);
7. Flexibility of thinking, the ability to switch from one mental operation to another, freedom from the constraining influence of patterns and stencils;
8. Mathematical memory. It can be assumed that its characteristic features also follow from the features of mathematical science, that it is a memory for generalizations, formalized structures, logical schemes;
9. The ability for spatial representations, which is directly related to the presence of such a branch of mathematics as geometry.

In addition to those listed, there are also such components, the presence of which in the structure of mathematical abilities, although useful, is not necessary. The teacher, before classifying a student as capable or incapable of mathematics, must take this into account. The following components are not mandatory in the structure of mathematical talent:

1. The speed of thought processes as a temporal characteristic.
2. The individual pace of work is not critical. The student can think slowly, slowly, but thoroughly and deeply.
3. Ability to fast and accurate calculations (in particular in the mind). In fact, computational abilities are far from always associated with the formation of truly mathematical (creative) abilities.
4. Memory for numbers, numbers, formulas. As academician A.N. Kolmogorov, many outstanding mathematicians did not have any outstanding memory of this kind.

Most psychologists and teachers, speaking of mathematical abilities, rely on this very structure of V.A. Krutetsky. However, in the course of various studies of the mathematical activity of students who show abilities for this school subject, some psychologists have identified other components of mathematical abilities. In particular, we were interested in the results of the research work of Z.P. Gorelchenko. He noted the following features in students capable of mathematics. First, he clarified and expanded the component of the structure of mathematical abilities, called in modern psychological literature "generalization of mathematical concepts" and expressed the idea of the unity of two opposite tendencies of the student's thinking towards generalization and "narrowing" of mathematical concepts. In this component, one can see a reflection of the unity of the inductive and deductive methods of learning new things in mathematics by students. Secondly, the dialectical rudiments in the thinking of students during the assimilation of new mathematical knowledge. This is manifested in the fact that in almost any single mathematical fact, the most capable students tend to see, understand the fact opposite to it, or, at least, consider the limiting case of the phenomenon under study. Thirdly, he noted a special increased attention to emerging new mathematical patterns that are opposite to those previously established.

One of the characteristic signs of increased mathematical abilities of students and their transition to mature mathematical thinking can be considered a relatively early understanding of the need for axioms as initial truths in proofs. An accessible study of the axioms and the axiomatic method greatly contributes to the acceleration of the development of students' deductive thinking. It has also been noted that the aesthetic feeling in mathematical work manifests itself in different ways for different students. In different ways, different students also respond to an attempt to educate and develop in them an aesthetic sense that corresponds to their mathematical thinking. In addition to the indicated components of mathematical abilities that can and should be developed, it is also necessary to take into account the fact that the success of mathematical activity is a derivative of a certain combination of qualities: an active positive attitude towards mathematics, interest in it, the desire to engage in it, turning into a passionate one at a high level of development. passion. You can also highlight a number of characteristic features, such as: diligence, organization, independence, dedication, perseverance, as well as stable intellectual qualities, a sense of satisfaction from hard mental work, the joy of creativity, discovery, and so on.

The presence in the time of the implementation of activities favorable for the performance of mental states, for example, a state of interest, concentration, good "mental" well-being, etc. A certain fund of knowledge, skills and abilities in the relevant field. Certain individual psychological characteristics in the sensory and mental spheres that meet the requirements of this activity.

The students most capable of mathematics are distinguished by a special aesthetic warehouse of mathematical thinking. It allows them to relatively easily understand some theoretical subtleties in mathematics, to capture the flawless logic and beauty of mathematical reasoning, to fix the slightest roughness, inaccuracy in the logical structure of mathematical concepts. An independent steady striving for an original, unconventional, elegant solution of a mathematical problem, for a harmonious unity of the formal and semantic components of the solution of a problem, brilliant guesses, sometimes ahead of logical algorithms, sometimes difficult to translate into the language of symbols, indicate the presence in thinking of a sense of a well-developed mathematical foresight, which is one of the aspects of aesthetic thinking in mathematics. Increased aesthetic emotions during mathematical thinking are primarily inherent in students with highly developed mathematical abilities and, together with the aesthetic warehouse of mathematical thinking, can serve as a significant sign of the presence of mathematical abilities in schoolchildren.

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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 suspend 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 sides 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 unique 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 relationships and dependencies given in the problem “suggest” visual representations.

Representatives of this type are not distinguished by the ability of visual-figurative representation and, therefore, use a more difficult and complex logical-analytical path of solution 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. Schoann 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 is incorrect, when it is believed that there is no regularity in natural mental development. 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 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.

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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Steps

Ask for help.

During the lesson, ask to explain to you the meaning of a particular concept. If the answer still does not shed light on all the dark spots, stay after the lesson and talk to the teacher again. Maybe in a one-on-one conversation, he will explain the material to you in more detail and more than what fit in the assigned time.

Make sure you understand the meaning of all words. Mathematics, if we talk about problems of a higher level, is, as a rule, a set of simple operations. For example, multiplication uses addition, while division requires subtraction. Before you learn any concept, you need to understand what mathematical operations it includes. For each mathematical term (for example, "variable"), do this:
- Learn the textbook definition: "The symbol for a number we don't know is usually a letter, like x or y."
- Practice solving examples on the topic. For example, "4x - 7 = 5," where x is an unknown variable, and 4, 7, and 5 are "constant" (the definition for this concept should also be found in the textbook).
Pay special attention to the study of mathematical rules. Properties, formulas, equations, and methods for solving problems are all the basic tools of mathematical science. Learn to rely on them in the same way that a good carpenter relies on his saw, tape measure, hammer, etc.

Take an active part in class work. If you don't know the answer to a question, ask for an explanation. Tell the teacher exactly what you have already understood so that he can pay more attention to the points that caused you difficulty.
- Consider the situation on the example of the above-mentioned problem with a variable. Tell the teacher this: "I understand that if you multiply the unknown variable (x) by 4, subtract 7, you get 5. Where should I start the solution?" Now the teacher will know what exactly is causing you difficulty and how to involve you in solving the task. But if you simply said: “I don’t understand,” the teacher might think that he needs to explain to you first of all what a variable and a constant are.
- Never be afraid to ask questions. Even Einstein asked questions (and then answered them himself)! The solution will not come to you by itself if you do nothing. If you don't want to ask the teacher, then ask a classmate or friend for help.
Seek outside help. If you still need help, and the teacher is unable to explain the material to you in a way that you understand, ask someone to recommend you for more detailed lessons. Find out if there are any special courses or tutoring programs available, or ask your teacher to work with you before or after school.
- Along with different ways of studying the material (audio, visual perception, etc.), there are also different approaches to teaching. If you perceive information best visually, and your teacher, even the best in the world, is guided in the learning process by those who perceive information well by ear, then it will be difficult for you to study with such a teacher. Therefore, it would be useful to get additional help from those who teach in a way that is more convenient for you.
Write down each action in the solution. For example, when solving equations, divide your solution into separate steps and write down everything you did before moving on to the next step.
- A detailed record will help to trace the path of the solution and find errors.
- A step-by-step written solution will show you exactly where you went wrong.
- By writing down each action in a mathematical solution, you will repeat it again and better remember what you already knew.
Try to solve all the tasks that you were given. After a few examples, you will get the hang of it. If tasks are still difficult, then you will understand exactly where you are having difficulty.
Review your teacher-reviewed assignments. Study his notes and corrections and sort out your mistakes. If not everything is clear, ask the teacher to understand together.
- Feel free to ask for help, learn from your mistakes!
- Even if math is hard for you, don't be afraid of it. Worry just makes things worse. Instead, be patient and learn it step by step.
- Don't forget to do your homework! You can even create your own examples and problems to practice.
- Don't sit back for fear of making mistakes. Try to solve something, even if you are not completely sure of the correctness of your decision.
- Ask if you don't understand. Ask the teacher to explain anything you don't understand during or after the lesson. Don't let fear run ahead of the engine. Do not lose faith in yourself and do not pay attention to others.
- When arithmetic is left behind and you study algebra and geometry, know that everything new that you will learn in these sections of mathematics will be based on the material already studied earlier. So make sure you learn each lesson well before moving on.
- It will be much easier for you if you show your teacher your work.
- Always ask your teacher for help if you don't understand something.
- Try to understand everything you do, and not just mindlessly solve similar tasks in the same way. Say, if you are learning to add large numbers, then consider why the number representing tens needs to be added to the sum in the next column. And if you still don't understand, then ask.
- Whether we like it or not, the ability to quickly and correctly count plays an important role in our business and personal lives.
- Enjoy. After all, even if you are not very interested in it yet, nevertheless, mathematics can be truly beautiful in its elegant order.
- Practice math for at least half an hour a day.
Warnings
- Do not try to memorize the analyzed examples by heart. Instead, insist that the teacher explain them to you and make sure you understand what he is saying. Each example has its own solution, and the main thing is to understand why they need to be solved in this way. Also, don't memorize the wrong formulas.

REPORT

ON THE TOPIC:

"Development of mathematical abilities of younger students in teaching mathematics"

Performed:

Sidorova Ekaterina Pavlovna

MOU "Bendery middle

secondary school №15 "

primary school teacher

Bender, 2014

Topic: "The development of mathematical abilities of younger students in teaching mathematics"

Chapter 1: Psychological and pedagogical foundations for the formation of mathematical abilities in younger students

1.1 Definition of the concept of "Mathematical ability"

1.3. Teaching mathematics is the main way to develop the mathematical abilities of younger students

Chapter 2: Methods for identifying the features of the formation of mathematical abilities in the process of solving mathematical problems

2.1.experimental work on the formation of mathematical abilities in a younger student in the process of solving mathematical problems. His results

2.2. Determination of the level of mathematical abilities in children of primary school age

Introduction

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 is no talk of an accurate and rigorous understanding of the content of this concept. At the same time, it should be noted the undying interest in this problem in all currents of psychology, which makes the problem of developing mathematical abilities relevant.

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. As V. A. Krutetsky stated: “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.

The development of effective means for the development of mathematical abilities is important for all levels of the school, but it is especially relevant for the primary education system, where the foundation of school performance is laid, the main stereotypes of educational activity are formed, and attitudes towards educational work are brought up.

Such prominent representatives of certain trends in foreign psychology as A. Binet, E. Trondike and G. Reves made their contribution to the study of mathematical abilities. S. L. Rubinshtein, A. N. Leontiev, A. R. Luria studied the influence of social factors on a child’s abilities. Conducted research on the inclinations underlying the abilities of A.G. Kovaleva, Myasishcheva. The general scheme of the structure of mathematical abilities at school age was proposed by V. A. Krutetsky.

aim work is the development of mathematical abilities of younger students in the process of solving mathematical problems.

Object of study: educational process in the primary grades, aimed at developing the mathematical abilities of students.

Subject of research are the features of the formation of mathematical abilities in younger students.

The research hypothesis is the following: in the process of solving mathematical problems, the development of mathematical abilities in younger students occurs if:

offer younger students to solve heuristic problems;

tasks for the study of symbols of mathematics and geometric images of numbers;

Research objectives:

Reveal the content of the concept of mathematical abilities.

To study the experience of effective psychological activity for the development of mathematical abilities in younger students;

Reveal the content of the concept of mathematical abilities;

Take into account the experience of effective psychological activity in the formation of mathematical abilities in younger students;

Research methods:

Studying the experience of the effective activity of psychological services in the formation of mathematical abilities in younger students in the process of solving mathematical problems.

Monitoring the educational activities of younger students and the process of solving mathematical problems.

pedagogical experiment.

The practical significance of the study lies in the fact that the identified system of classes with children for the development of mathematical abilities, which includes various types of mathematical problems, can be used by psychologists, teachers and parents in working with children of primary school age. The methods proposed in the course work for the development of mathematical abilities in children of primary school age through problem solving, using methods of concretization, abstraction, variation, analogy, posing analytical questions, can be used in the work of a school psychologist.

Chapter I . Psychological and pedagogical foundations for the formation of mathematical abilities in younger students.

The study of cognitive features underlying the acquisition of knowledge is one of the main directions in the search for reserves to increase the effectiveness of school education.

The modern school is faced with the task of providing a general education, ensuring the development of general abilities, and in every possible way supporting the sprouts of special talents. At the same time, it is necessary to take into account that training and education "have a formative effect on the mental capabilities of adolescents not directly, but through internal conditions - age and individual."

According to Teplov, abilities are understood as individual psychological characteristics that determine the ease and speed of acquiring knowledge and skills, which, however, are not limited to these features. As natural prerequisites for the development of abilities, anatomical and physiological features of the brain and nervous system are considered, typological properties of the nervous system, the ratio of 1 and 2 signal systems, individual structural features of analyzers and the specifics of interhemispheric interaction.

One of the most difficult questions in the psychology of abilities is the question of the ratio of innate (natural) and acquired in abilities. The main position in domestic 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. Psychological features cannot be innate. It's all about abilities. They are formed and developed in life, in the process of activity, in the process of training and education.

A.N.Leontiev spoke about the need to distinguish between two kinds of human abilities, natural or natural (basically biological, for example, the ability to quickly form conditional connections) and specifically human abilities (socio-historical origin). "A person is endowed from birth with only one ability - the ability to form specific human abilities." In the following, we will only talk about specifically human abilities.

Social experience, social influence, and upbringing play a decisive and decisive role.

The fundamental solution to this issue in Russian psychology is as follows: abilities cannot be innate, only the makings of abilities can be innate - some anatomical and physiological features of the brain and nervous system with which a person is born.

Natural data are one of the most important conditions for the complex process of formation and development of abilities. As S. L. Rubinshtein noted, abilities are not predetermined, but cannot simply be implanted from outside. Individuals must have prerequisites, internal conditions for the development of abilities.

But the recognition of the real significance of innate inclinations in no case means the recognition of the fatal conditionality of the development of abilities by innate characteristics. Abilities are not contained in makings. In ontogeny, they do not appear, but are formed.

A somewhat different understanding of inclinations is given in the works of A.G. Kovalev and V.N. Myasishchev. They understand the inclinations as psychophysiological properties, primarily those that are found 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 at the first try of activity. However, the basic position of the ability in the proper sense of the word is preserved, they are formed, in activity, they are lifetime education.

When talking about the makings of abilities, they usually first of all mean the typological properties of the nervous system. As you know, typological properties are the natural basis of individual differences between people. On this basis, the most complex systems of various temporary connections arise - the speed of their formation, their strength, and the ease of differentiation. They determine the power of concentrated attention, mental performance.

A number of studies have shown that, along with general typological properties that characterize the nervous system as a whole, there are particular typological properties that characterize the work of individual areas of the cortex, revealed in relation to different analyzers and different brain systems. Unlike general typological properties that determine temperament, particular typological properties are of the greatest importance in the study of special abilities.

A.G. Kovalev and V.N. Myasishchev tend to attach somewhat more importance than other psychologists to the natural side, the natural prerequisites for development. A.N.Leontiev and his followers tend to emphasize the role of education in the formation of abilities.

Such outstanding representatives of certain trends in psychology as A. Binet, E. Thorndike and G. Reves, and such outstanding mathematicians as A. Poincare 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. 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, A. Rogers noted two sides of mathematical abilities, reproductive (associated with the function of memory) and productive (associated with the function of thinking), in the work of A. Rogers. 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.

Of the works of domestic authors, it is necessary to mention the originalarticle by D. Mordukhay-Boltovsky "Psychology of mathematical thinking", published in 1918we discussed the need to use sources until the end of the last century!

year. 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. The sudden appearance in consciousness of a ready-made solution to a problem that we cannot solve for a long time, - the author writes, - we explain by unconscious thinking, which continued to deal with the task, and the result emerges beyond the threshold of consciousness. According to Mordukhai-Boltovsky, our mind is capable of performing 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.

The author 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. Of great interest is Mordukhai-Boltovsky's attempt to isolate the components of mathematical abilities. He refers to these 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 is understood 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 separated seemingly completely heterogeneous objects.

* "speed of thought" (speed of thought is explained by the work that unconscious thinking does to help the conscious). Unconscious thinking, according to the author, proceeds much faster than conscious.

D. Mordukhai-Boltovsky also expresses his views on the types of mathematical imagination that underlie different types of mathematicians - "geometers" and "algebraists". Arithmeticians, algebraists, and analysts in general, whose discovery is made in the most abstract form of breakthrough quantitative symbols and their interrelationships, cannot imagine like a "geometer".

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.

In addition to general theoretical studies of the problem of mathematical abilities, V.A. Krutetsky, with his monograph "Psychology of mathematical abilities of schoolchildren", 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, 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, skills , skills in mathematics. D.N. Bogoyavlensky and N.A. Menchinskaya, speaking about individual differences in the learning ability of children, introduces the concept of psychological properties that determine, ceteris paribus, success in learning. They do not use the term "ability", but in essence the corresponding concept is close to the definition given above.

The study of mathematical abilities also includes the solution of one of the most important problems - the search for natural prerequisites, or inclinations, of this type of ability. The inclinations include the innate anatomical and physiological characteristics of the individual, which are considered as favorable conditions for the development of abilities. For a long time, inclinations were considered as a factor fatally predetermining the level and direction of development of abilities. Classics of Russian psychology B.M. Teplov and S.L. Rubinshtein scientifically proved the illegitimacy 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 mandatory development of a particular type of ability. It can only be a favorable condition for this development. The typological properties that make up the inclinations and are an important part of them reflect such individual features of the functioning of the body as the limit of working capacity, the speed characteristics of the nervous response, the ability to restructure the reaction in response to changes in external influences.

The general scheme 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 scheme for the structure of mathematical abilities at school age:

Obtaining mathematical information.

The ability to formalize the perception of mathematical material, grasping the formal structure of the problem.

Processing of mathematical information.

The ability for logical thinking in the field of quantitative and spatial relations, numerical and sign symbolism.

The ability to think in mathematical symbols.

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

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

Flexibility of thought processes in mathematical activity.

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

The ability to quickly and freely restructure the direction of the thought process, switch 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 and proof schemes, methods for solving problems and principles for approaching them).

General synthetic component.

Mathematical mindset.

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

Not included in the structure of mathematical talent are 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.

1.2. Conditions for the formation of mathematical abilities of younger students in the process of teaching mathematics.

Since the purpose of our work is not just a list of recommendations necessary for the successful acquisition of mathematical knowledge by children, but the development of recommendations for classes whose purpose is to develop mathematical abilities, we will dwell in more detail on the conditions for the formation of mathematical abilities proper. As already noted, abilities are formed and developed only in activity. However, in order for an activity to have a positive effect on abilities, it must satisfy certain conditions.

First, the activity should evoke strong and stable positive emotions and pleasure in the child. The child should experience a feeling of joyful satisfaction from the activity, then he has a desire to engage in it on his own initiative, without coercion. A lively interest, a desire to do the job as best as possible, and not a formal, indifferent, indifferent attitude towards it, are necessary conditions for the activity to positively influence the development of abilities. If the child assumes that he cannot cope with the task, he seeks to bypass it, a negative attitude is formed to the task and to the subject in general. To avoid this, the teacher must create a “success situation” for the child, must notice and approve of any achievements of the student, and increase his self-esteem. This is especially true for mathematics, since this subject is not easy for most children.

Since abilities can bear fruit only when they are combined with a deep interest and a steady inclination to relevant activities, the teacher must actively develop the interests of children, striving to ensure that these interests are not superficial, but are serious, deep, stable and efficient.

Secondly, the child's activity should be as creative as possible. The creativity of children in mathematics can be manifested in an unusual, non-standard solution to a problem, in the disclosure of methods and techniques of calculations by children. To do this, the teacher must pose feasible problems for the children and ensure that the children solve them on their own with the help of leading questions.

Thirdly, it is important to organize the child's activity in such a way that he pursues goals that are always slightly superior to his current capabilities, the level of activity he has already achieved. Here we can talk about focusing on the “zone of proximal development” of the student. But in order to comply with this condition, an individual approach to each student is necessary.

Thus, examining the structure of abilities in general and mathematical abilities in particular, as well as the age and individual characterological characteristics of children of primary school age, we can draw the following conclusions:

Psychological science has not yet developed a unified view of the problem of abilities, their structure, origin and development.

If by mathematical abilities we mean all the individual psychological characteristics of a person that contribute to the successful mastery of mathematical activity, then it is necessary to isolate the following groups of abilities: the most general abilities (conditions) necessary for the successful implementation of any activity:

 diligence;

 persistence;

 performance;

in addition, well-developed voluntary memory and voluntary attention, interest and inclination to engage in this activity;

general elements of mathematical abilities, those general features of mental activity that are necessary for a very wide range of activities;

specific elements of mathematical abilities - features of mental activity that are characteristic only of mathematics, specific specifically for mathematical activity, unlike all others.

Mathematical ability is a complex, integrated education, the main components of which are:

Ability to formalize mathematical material;

Ability to generalize mathematical material;

Ability for logical reasoning;

The ability to reversibility of the thought process;

Flexibility of thinking;

Mathematical memory;

The desire to save mental strength.

The components of mathematical abilities at primary school age are presented only in their "embryonic" state. However, in the process of schooling, their development is noticeable, while the younger school age is the most fruitful for this development.

There are also natural prerequisites for the development of mathematical abilities, which include:

High level of general intelligence;

The predominance of verbal intelligence over non-verbal;

High degree of development of verbal-logical functions;

Strong type of nervous system;

Some personality traits, such as reasonableness, prudence, perseverance, independence, self-sufficiency.

When developing classes for the development of mathematical abilities, one should take into account not only the age and individually typological characteristics of children, but also observe certain conditions so that this development is as possible as possible:

The activity should evoke strong and stable positive emotions in the child;

Activities should be as creative as possible;

Activities should be focused on the student's "zone of proximal development".

1.3 Teaching mathematics is the main way to develop the mathematical abilities of younger students

One of the most important theoretical and practical problems of modern pedagogy is the improvement of the process of teaching younger students. The history of the development of foreign and Russian pedagogy and psychology is inextricably linked with the study of various aspects of learning difficulties. According to many authors (N. P. Vaizman, G. F. Kumarina, S. G. Shevchenko and others), the number of children who are already in the primary grades are not able to master the program in the allotted time and in the required volume fluctuates from 20% to 30% of the total number of students. Being mentally intact, not having classical forms of developmental anomalies, such children experience difficulties in social and school adaptation, showing failure in learning.

Difficulties that arise in younger students in the learning process can be grouped into three groups: biogenic, sociogenic and psychogenic, which leads to a weakening of the cognitive abilities (attention, perception, memory, thinking, imagination, speech) of the child and significantly reduces the effectiveness of learning. In addition to the general prerequisites for difficulties in learning, there are specific ones - difficulties in mastering mathematical material.

A number of studies by modern authors (N. B. Istomina, N. P. Lokalova, A. R. Luria, G. F. Kumarina, N. A. Menchinskaya, L. S. Tsvetkova, etc.) are devoted to the problem of teaching an elementary course in mathematics. . As a result of the analysis of the named literary sources and in the course of our own research, the following main difficulties were identified for younger students in teaching mathematics:

Lack of stable counting skills.

Ignorance of the relationship between adjacent numbers.

Inability to move from a concrete plane to an abstract one.

Instability of graphic forms, i.e. lack of formation of the concept of "working line", mirror writing of numbers.

Inability to solve arithmetic problems.

“Intellectual passivity”.

Based on the analysis of the psychological and psychophysical causes underlying these difficulties, the following groups can be distinguished:

Group 1 - difficulties associated with the insufficiency of abstraction operations, which manifests itself when moving from a concrete to an abstract action plan. In this regard, difficulties arise in the assimilation of the number series and its properties, the meaning of the counting action.

Group 2 - difficulties associated with insufficient development of fine motor skills, lack of formation of visual-motor coordination. These reasons underlie such difficulties for students as mastering the writing of numbers, their mirror image.

Group 3 - difficulties associated with insufficient development of associative links and spatial orientation. These reasons underlie such difficulties for students as difficulties in translating from one form (verbal) to another (digital), in determining geometric lines and figures, difficulties in counting, and in performing counting operations with the transition through a dozen.

Group 4 - difficulties associated with insufficient development of mental activity and individual psychological characteristics of the personality of students. In this regard, younger students experience difficulties in the formation of rules based on the analysis of several examples, difficulties in the process of developing the ability to reason when solving problems. These difficulties are based on the insufficiency of such a mental operation as generalization.

Group 5 - difficulties associated with the unformed cognitive attitude to reality, which is characterized by "intellectual passivity". Children perceive an educational task only when it is translated into a practical plan. If it is necessary to solve intellectual problems, they have a desire to use various workarounds (memorization without memorization, guessing, the desire to act according to a model, use hints).

Of no small importance in teaching students is the motivation for future activities. For a younger student, the primary task in organizing motivation is to overcome the fear of difficult, abstract, incomprehensible mathematical information, to awaken confidence in the possibility of its assimilation and interest in learning.

The teacher needs in each case to professionally approach the construction and implementation of the educational process, focusing on the personal growth of the child, taking into account the individual characteristics of his mental activity, creating positive prospects for the development of the student's personality, organizing a student-oriented educational environment that allows in practice to identify and realize creative potential child. Based on theoretical knowledge, the teacher must be able to anticipate the child's difficulties in learning and eliminate them; plan corrective and developmental work, create problem situations to activate the dynamics of the development of cognitive processes; organize productive independent work, create a favorable emotional and psychological background for the learning process. The peculiarity of methodological knowledge and skills lies in the fact that they are closely related to psychological, pedagogical and mathematical knowledge.

The dependence of some mathematical knowledge and skills on others, their consistency and consistency show that gaps at one level or another delay the further study of mathematics and are the cause of school difficulties. A decisive role in preventing school difficulties is played by the diagnosis of mathematical knowledge and skills of students. When organizing and conducting which it is necessary to comply with certain conditions: formulate questions clearly and specifically; provide time to think about the answer; treat student responses positively.

Consider a typical situation that often occurs in practice. The student was given a task: “Insert the missing number so that the inequality is true 5> ? ". The student completed the task incorrectly: 5 > 9. What should the teacher do? Turn to another student or try to figure out the reasons for the mistake?

The choice of the teacher's actions in this case may be due to a number of psychological and pedagogical reasons: the individual characteristics of the student, the level of his mathematical training, the purpose for which the task was offered, etc. Suppose the second path was chosen, i.e. decided to identify the causes of the error.

First of all, it is necessary to invite the student to read the completed record.

If a student reads it as “five less than nine”, then the mistake is that the mathematical symbol has not been mastered. To eliminate the error, it is necessary to take into account the peculiarities of the perception of the younger student. Since it has a visual-figurative character, it is necessary to use the method of comparing the sign with a specific image, for example, with a beak, which is open to a larger number and closed to a smaller one.

If the student reads the entry as “five is greater than nine”, then the error is that some of the mathematical concepts have not been mastered: the ratio “more”, “less”; establishing a one-to-one correspondence; quantitative number; natural series of numbers; check. Given the visual-figurative nature of the child's thinking, it is necessary to organize work on these concepts using practical tasks.

The teacher invites one student to lay out 5 triangles on the desk, and the other - 9 and think about how they can be arranged in order to find out who has more or less triangles.

Based on his life experience, the child can independently suggest a course of action or find it with the help of a teacher, i.e. establish a one-to-one correspondence between data elements of subject sets (triangles):

If the student has successfully completed tasks for comparing numbers, then it is necessary to establish how conscious his actions are. Here the teacher will need knowledge of such mathematical concepts as “counting” and “natural series of numbers”, since they are the basis of the rationale: “The number that is called earlier in counting is always less than any number following it.”

The practical activity of a teacher requires a whole range of knowledge in psychology, pedagogy and mathematics. On the one hand, knowledge must be synthesized and united around a specific practical problem that has a multilateral holistic character. On the other hand, they must be translated into the language of practical actions, practical situations, that is, they must become a means of solving real practical problems.

When teaching mathematics to younger students, the teacher must be able to create problem situations for the development of cognitive processes; organize productive independent work, create a favorable emotional and psychological background for the learning process.

In psychological and pedagogical research devoted to the problems of teaching mathematics, the difficulties experienced by elementary school students in mastering the ability to solve arithmetic problems are noted. However, the solution of arithmetic problems is of great importance for the development of cognitive activity of students, because. contributes to the development of logical thinking.

G.M. Kapustina notes that children with learning difficulties at different stages of working on a task experience difficulties: when reading a condition, in analyzing an objective-effective situation, in establishing relationships between quantities, in formulating an answer. They often act impulsively, thoughtlessly, they cannot cover the variety of dependencies that make up the mathematical content of the problem. However, the solution of arithmetic problems is of great importance for the development of cognitive activity of students, because. contributes to the development of their verbal-logical thinking and arbitrariness of activity. In the process of solving arithmetic problems, children learn to plan and control their activities, master the techniques of self-control, they develop perseverance, will, and develop an interest in mathematics.

In her research, M. N. Perova proposed the following classification of mistakes that students make when solving problems:

1. Introducing an extra question and action.

2. Exclusion of the desired question and action.

3. Inconsistency of questions with actions: correctly posed questions and the wrong choice of actions, or, conversely, the right choice of actions and the wrong wording of questions.

4. Random selection of numbers and actions.

5. Errors in the names of quantities when performing actions: a) the names are not written; b) the names are written erroneously, outside the objective understanding of the content of the task; c) names are written only for individual components.

6. Mistakes in calculations.

7. Incorrect wording of the answer to the problem (the formulated answer does not correspond to the question of the problem, it is stylistically constructed incorrectly, etc.).

When solving problems, younger students develop arbitrary attention, observation, logical thinking, speech, quick wits. Problem solving contributes to the development of such processes of cognitive activity as analysis, synthesis, comparison, generalization. Solving arithmetic problems helps to reveal the main meaning of arithmetic operations, to concretize them, to connect them with a certain life situation. Tasks contribute to the assimilation of mathematical concepts, relationships, patterns. In this case, as a rule, they serve to concretize these concepts and relationships, since each plot task reflects a certain life situation.

Chapter II . A technique for identifying the features of the formation of mathematical abilities in the process of solving mathematical problems.

2.1. Experimental work on the formation of mathematical abilities in a younger student in the process of solving mathematical problems.

For the purpose of practical substantiation of the conclusions obtained during the theoretical study of the problem: what are the most effective forms and methods aimed at developing the mathematical abilities of schoolchildren in the process of solving mathematical problems, a study was conducted. Two classes took part in the experiment: experimental 2 (4) "B", control - 2 (4) "C" UVK "School-gymnasium" No. 1 p.g.t. Soviet.

Stages of experimental activity

I - Preparatory. Purpose: determination of the level of mathematical abilities based on the results of observations.

II - The ascertaining stage of the experiment. Purpose: determination of the level of formation of mathematical abilities.

III - Formative experiment. Purpose: creation of the necessary conditions for the development of mathematical abilities.

IV - Control experiment. Purpose: to determine the effectiveness of forms and methods that contribute to the development of mathematical abilities.

At the preparatory stage, students of the control - 2 "B" and experimental 2 "C" classes were observed. Observations were carried out both in the process of studying new material and in solving problems. For observations, those signs of mathematical abilities that are most clearly manifested in younger students were identified:

1) relatively fast and successful mastery of mathematical knowledge, skills and abilities;

2) the ability to consistently correct logical reasoning;

3) resourcefulness and ingenuity in the study of mathematics;

4) flexibility of thinking;

5) the ability to operate with numerical and symbolic symbols;

6) reduced fatigue during mathematics;

7) the ability to shorten the process of reasoning, to think in collapsed structures;

8) the ability to switch from direct to reverse course of thought;

9) the development of figurative-geometric thinking and spatial representations.

In November 2011, we filled out a table of mathematical abilities of schoolchildren, in which we rated each of the listed qualities in points (0-low level, 1-average level, 2-high level).

At the second stage, diagnostics of the development of mathematical abilities was carried out in the experimental and control classes.

For this, the "Problem Solving" test was used:

1. Compose compound problems from these simple problems. Solve one compound problem in different ways, underline the rational one.

The cow of the cat Matroskin on Monday gave 12 liters of milk. Milk was poured into three-liter jars. How many cans did the cat Matroskin get?

Kolya bought 3 pens for 20 rubles each. How much money did he pay?

Kolya bought 5 pencils at a price of 20 rubles. How much do pencils cost?

Matroskin's cow gave 15 liters of milk on Tuesday. This milk was poured into three-liter jars. How many cans did the cat Matroskin get?

2. Read the problem. Read the questions and expressions. Match each question with the correct expression.

a + 18

class 18 boys and a girls.

How many students are in the class?

18 - a

How many more boys than girls?

a - 18

How many fewer girls than boys?

3. Solve the problem.

In his letter to his parents, Uncle Fyodor wrote that his house, the house of the postman Pechkin and the well were on the same side of the street. From the house of Uncle Fyodor to the house of the postman Pechkin 90 meters, and from the well to the house of Uncle Fyodor 20 meters. What is the distance from the well to the house of the postman Pechkin?

With the help of the test, the same components of the structure of mathematical abilities were checked as during observation.

Purpose: to establish the level of mathematical abilities.

Equipment: student card (sheet).

The test tests skills and mathematical abilities:

The skills required to solve the problem.

Abilities manifested in mathematical activity.

The ability to distinguish the task from other texts.

Ability to formalize mathematical material.

Ability to write down the solution of the problem, to make calculations.

The ability to operate with numerical and symbolic symbols.

The ability to write the solution of a problem with an expression. Ability to solve problems in different ways.

Flexibility of thinking, the ability to shorten the process of reasoning.

Ability to perform the construction of geometric figures.

The development of figurative-geometric thinking and spatial representations.

At this stage, mathematical abilities have been studied and the following levels have been determined:

Low level: Mathematical ability manifests itself in a general, inherent need.

Intermediate level: abilities appear in similar conditions (according to the model).

High level: creative manifestation of mathematical abilities in new, unexpected situations.

Qualitative analysis of the test showed the main reasons for the difficulty in performing the test. Among them: a) the lack of specific knowledge in solving problems (they cannot determine how many actions the problem is solved, they cannot write down the solution of the problem by the expression (in 2 "B" (experimental) class 4 people - 15%, in 2 "C" class - 3 people - 12%) b) insufficient formation of computational skills (in the 2nd "B" class 7 people - 27%, in the 2nd "C" class 8 people - 31%. The development of mathematical abilities of students is ensured, first of all, by the development of mathematical thinking style.To determine the differences in the development of the ability to reason in children, a group lesson was conducted on the material of the diagnostic task "different-same" according to the method of A. Z. Zak. The following levels of reasoning ability were revealed:

high level - solved tasks No. 1-10 (contain 3-5 characters)

Intermediate level - Problems 1-8 solved (contain 3-4 characters)

low level - solved tasks #1 - 4 (contain 3 characters)

The following methods of work were used in the experiment: explanatory-illustrative, reproductive, heuristic, problem presentation, research method. In real scientific creativity, the formulation of the problem goes through the problem situation. We strived to ensure that the student independently learned to see the problem, formulate it, explore the possibilities and ways to solve it. The research method is characterized by the highest level of cognitive independence of students. At the lessons, we organized independent work of students, giving them problematic cognitive tasks and assignments of a practical nature.

2.2. Determination of the level of mathematical abilities in children of primary school age.

Thus, our study allows us to assert that the work on the development of mathematical abilities in the process of solving word problems is an important and necessary matter. The search for new ways to develop mathematical abilities is one of the urgent tasks of modern psychology and pedagogy.

Our research has a certain practical significance.

In the course of experimental work, based on the results of observations and analysis of the data obtained, it can be concluded that the speed and success of the development of mathematical abilities does not depend on the speed and quality of assimilation of program knowledge, skills and abilities. We managed to achieve the main goal of this study - to determine the most effective forms and methods that contribute to the development of students' mathematical abilities in the process of solving word problems.

As the analysis of research activity shows, the development of children's mathematical abilities develops more intensively, since:

a) appropriate methodological support has been created (tables, instructional cards and worksheets for students with different levels of mathematical abilities, a software package, a series of tasks and exercises for the development of certain components of mathematical abilities;

b) the program of the optional course "Non-standard and entertaining tasks" was created, which provides for the implementation of the development of students' mathematical abilities;

c) diagnostic material has been developed that allows timely determination of the level of development of mathematical abilities and correction of the organization of educational activities;

d) a system for the development of mathematical abilities has been developed (according to the plan of the formative experiment).

The need to use a set of exercises for the development of mathematical abilities is determined on the basis of the identified contradictions:

Between the need to use tasks of different levels of complexity in mathematics lessons and their absence in teaching;

Between the need to develop mathematical abilities in children and the real conditions for their development;

Between the high requirements for the tasks of forming the creative personality of students and the weak development of the mathematical abilities of schoolchildren;

Between the recognition of the priority of introducing a system of forms and methods of work for the development of mathematical abilities and an insufficient level of development of ways to implement this approach.

The basis for the study is the choice, study, implementation of the most effective forms, methods of work in the development of mathematical abilities.

Conclusion

Summing up, it should be noted that the topic we are considering is relevant for the modern school. To prevent and eliminate difficulties in teaching mathematics to younger students, the teacher must: know the psychological and pedagogical characteristics of the younger student; be able to organize and carry out preventive and diagnostic work; create problem situations and create a favorable emotional and psychological background for the process of teaching mathematics to younger students.

In connection with the problem of the formation and development of abilities, it should be pointed out that a number of psychologists' studies are aimed at revealing the structure of preschoolers' abilities for various types of activities. At the same time, abilities are understood as a complex of individual - psychological characteristics of a person that meet the requirements of this activity and are a condition for successful implementation. Thus, abilities are a complex, integral, mental formation, a kind of synthesis of properties, or as they are called components.

The general law of the formation of abilities is that they are formed in the process of mastering and performing those types of activities for which they are necessary.

Abilities are not something once and for all predetermined, they are formed and developed in the process of learning, in the process of exercising, mastering the corresponding activity, therefore it is necessary to form, develop, educate, improve the abilities of children and it is impossible to foresee exactly how far this development can go.

Speaking about mathematical abilities as features of mental activity, one should, first of all, point out several misconceptions that are common among teachers.

First, many believe that mathematical ability lies primarily in the ability to quickly and accurately calculate (in particular in the mind). In fact, computational abilities are far from always associated with the formation of truly mathematical (creative) abilities. Secondly, many people think that preschoolers capable of mathematics have a good memory for formulas, numbers, and numbers. However, as Academician A. N. Kolmogorov points out, success in mathematics is least of all based on the ability to quickly and firmly memorize a large number of facts, figures, formulas. Finally, it is believed that one of the indicators of mathematical abilities is the speed of thought processes. A particularly fast pace of work is not in itself related to mathematical ability. A child can work slowly and unhurriedly, but at the same time thoughtfully, creatively, successfully advancing in the assimilation of mathematics.

Krutetsky V.A. in the book "Psychology of mathematical abilities of preschoolers" distinguishes nine abilities (components of mathematical abilities):

1) The ability to formalize mathematical material, to separate form from content, to abstract from specific quantitative relations and spatial forms and to operate with formal structures, structures of relations and connections;

2) The ability to generalize mathematical material, to isolate the main thing, abstracting from the insignificant, to see the general in the outwardly different;

3) The ability to operate with numerical and symbolic symbols;

4) The ability to "consistent, correctly divided logical reasoning", associated with the need for evidence, justification, conclusions;

5) The ability to reduce the process of reasoning, to think in collapsed structures;

6) The ability to reversibility of the thought process (to the transition from direct to reverse thought);

7) Flexibility of thinking, the ability to switch from one mental operation to another, freedom from the constraining influence of patterns and stencils;

8) Mathematical memory. It can be assumed that its characteristic features also follow from the features of mathematical science, that it is a memory for generalizations, formalized structures, logical schemes;

9) The ability for spatial representations, which is directly related to the presence of such a branch of mathematics as geometry.

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