Methods and techniques for forming mathematical concepts in preschoolers

MAGAZINE Preschooler.RF

Svetlana Gennadievna Antonova Teacher, MBDOU kindergarten No. 9, Pavlovo, Nizhny Novgorod region

Arithmetic is the foundation on which the ability to correctly perceive reality is built, and creates the basis for the development of intelligence and intelligence in relation to practical issues. I. Pestalozzi

Abstract: the article examines one of the current problems of preschool pedagogy - the problem of forming elementary mathematical concepts in preschool children, showing its importance in the formation of cognitive interests and cognitive actions of the child in various types of activities.

Key words: formation of elementary mathematical concepts in preschool children, mathematical development, goals and objectives of FEMP.

At the present stage of development of education, one of the leading principles of preschool education is the principle of developmental education. The development of initial mathematical knowledge and skills stimulates the comprehensive development of children, forms abstract thinking and logic, improves attention, memory and speech, which will allow the child to actively explore and master the world around him. An entertaining journey to the land of geometric shapes and arithmetic problems will be an excellent help in developing such qualities as curiosity, determination and organization.

Formation of elementary mathematical representations (FEMP), as defined by A.V. Beloshistaya, is a purposeful and organized process of transferring and assimilating knowledge, techniques and methods of mental activity of preschool children in the field of mathematics [1, p. 14].

The result of the FEMP process is mathematical development. The mathematical development of preschool children is changes in their cognitive activity that occur as a result of the formation of elementary mathematical concepts and related logical operations [1, p. 16].

I would like to note that this area of ​​knowledge has been developing for more than a century. So, in the ΧVΙΙ - ΧΙΧ centuries. issues of the content and methods of teaching preschool children arithmetic and the formation of ideas about sizes, measures of measurement, time and space were considered in the works of great teachers - L. S. Vygotsky, Ya. A. Komensky, I.G. Pestalozzi, K.D. Ushinsky, L.N. Tolstoy and others, at the beginning of the twentieth century. these issues were dealt with by R.L. Berezina, Z.A. Mikhailova, R.L. Richterman, A.A. Stolyar and others. Since the second half of the 1950s. issues of the formation of mathematical concepts in preschool children were considered by R.L. Berezina, V.V. Davydovs, V.V. Danilov, A.M. Leushin, Z.A. Mikhailova, R.L. Nepomnyashchia, T.V. Taruntaeva and others [4]. Based on the works of T.V. Tarantueva, the initial mathematical training of preschool children included learning to count, the development of quantitative concepts within the first ten, learning to solve and compose elementary arithmetic problems. In addition, it involves performing operations with sets (visually presented), taking measurements using conventional measures, as well as developing children’s eye, developing ideas about time, geometric figures, and developing an understanding of spatial relationships. T.V. Taruntaeva notes that the concept of “formation of mathematical concepts” includes such a volume of material that significantly goes beyond the development of exclusively counting skills and abilities in preschool children, representing, in fact, a full-fledged program of initial mathematical development. Such a program, according to the teacher, should provide children with a deep understanding of quantitative and other relationships, as well as lay the basis for the further development of mathematical thinking in children [7, p. 12].

According to L. S. Vygotsky, “the best age for learning arithmetic ranges approximately between 6 and 8 years” [3, p. 336]. K.D. Ushinsky, in his works, paid great attention to the issue of the development of mental activity. In his opinion: “If reason is a special ability innate to man, then it can work equally well, no matter what it is applied to, and the development of reason is possible equally on any object that only exercises its power” [9, p. 360]. T. D. Richterman believed that it is necessary to pay great attention to the formation in preschool children of the concept of time, as a component of mathematical concepts. He wrote this: “It is vitally important for children already in preschool age to learn how to navigate themselves in time...... The ability to regulate and plan activities in time creates the basis for the development of such personality qualities as organization, composure, focus, accuracy, which are necessary for a child when studying at school and in everyday life" [6, p. 3]. I. G. Pestalozzi also dealt with the problem of memory development: “Memory, which holds series of numbers, chains the mind to their internal relationships” [5, p. 72]. G. A. Uruntaeva dealt with the same problem and the conditions for the successful functioning of memory; she came to the following conclusions: “The effectiveness of arbitrary figurative memory directly depends on the degree of difficulty in remembering and reproducing the features of an object, increasing in the following order: color, shape, size, quantity, spatial arrangement of objects and their parts” [8, p. 6]. In her opinion, mental development is a fundamental direction in the formation of the psyche of preschool children [8].

Today, education uses the accumulated experience of pedagogical science, takes into account modern conditions for the development of a child’s mathematical thinking and new developments of modern pedagogical science in the field of mathematics. Developing the ability to cognize objects and phenomena, highlight properties, identify dependencies and patterns, compare objects by property - all this allows you to form logical structures of thinking, which are the foundation of general mental and mathematical development.

The term “mathematics” as a subject of study is absent in the Federal State Educational Standard for Education. The task of forming mathematical concepts is “penetrating” and has meaningful correspondences in various educational areas, such as, for example, cognitive development.

In accordance with the Federal State Educational Standard for Educational Education, the formation of elementary mathematical concepts is classified in the educational field “Cognitive Development” . This area involves the development of children’s interests, the formation of their consciousness; development of imagination and creative activity, as well as the formation of primary ideas about oneself, about other people, about objects in the surrounding world (shape, color, size, material, sound, rhythm, tempo, quantity, number, part and whole, space and time, movement and rest, causes and effects, etc.) [10c, art. 2. 6]. One of the basic principles of preschool education, in accordance with the Federal State Educational Standard for Preschool Education, is “the formation of the child’s cognitive interests and cognitive actions in various types of activities” [10, art. 1. 4, paragraph 7].

The most important result, according to A.V. White-haired, pre-school mathematical preparation of a child is not only the accumulation of a certain stock of knowledge and skills, but also the mental development of the child, the formation in him of the necessary cognitive and mental skills, which are basic for the successful assimilation of mathematical and any other general content in the future [1]. This idea meets the requirements of the Federal State Educational Standard for Preschool Education, in which the child must acquire the prerequisites for educational activities. Only after a child has mastered elementary mathematical concepts can it be easier to prepare him to master more complex mathematical problems at subsequent stages of development.

The goals of FEMP for preschool children are: the comprehensive development of children’s personality, their preparation for school, and the cognitive development of children [1, p. 24].

For FEMP in preschool children, the following tasks are set:

• formation in children of a system of elementary mathematical concepts;
• development of the elementary foundations of mathematical thinking;
• development of sensory processes and abilities in children;
• expansion, enrichment of vocabulary and improvement of connected speech[2].

When performing work on FEMP, it is necessary to observe the following principles: consciousness and activity, activity approach, scientific nature and accessibility, strength, consistency and systematicity, connection with life, the principles of individual and differentiated approaches, etc. are also observed [2].

The process of forming elementary mathematical concepts in preschool children includes a number of sections: “Quantity and counting” , “Value” , “Form” , “Orientation in space” , “Orientation in time” . This structure of the program for the formation of elementary mathematical concepts remains constant in all groups.

1. “Quantity and counting” involves the development of children’s ideas about number, counting, set, arithmetic operations, and also involves children solving elementary word problems.
2. “Magnitude” is aimed at developing children’s ideas about quantities, about ways to compare and measure them (thickness, area, length, width, height, volume, mass, time).
3. “Form” reflects the development of children’s ideas about the shape of objects, geometric figures (flat and three-dimensional), their properties, as well as existing relationships.
4. “Time Orientation” contains program material aimed at developing children’s ideas about the time of day and their parts, days of the week, months and seasons, and also involves the development of a “sense of time” .
5. “Orientation in space” assumes that children learn to navigate in relation to their own body, in relation to themselves, another person, objects located around them, and learn to navigate in space and on a plane, incl. on a sheet of paper and in motion[2].

Today, there is a large number of general education programs for preschool education, but each of them adheres to one program material in the formation of elementary mathematical concepts in preschool children. The program of the second younger group is focused on the pre-numerical period - at this stage, children learn to compare objects, become familiar with their shape, learn to distinguish spatial directions, and also navigate time at an elementary level. In the middle group, the program becomes more complicated and filled with new content, counting is introduced using numerals, the size and shape of objects are studied, and they learn to navigate in space and time. In the senior group, work continues on teaching children to count, the acquired skills are consolidated, children learn to compare, measure using a conventional measure (dividing an object into equal parts is introduced, developing the eye). Work is being done to develop ideas about geometric shapes and develop skills of orientation in space and time. In the preparatory group, the material becomes more complex: children are required to master the material studied in previous groups; the acquired knowledge, skills and abilities of children are expanded and deepened. This applies to numeracy skills, incl. ordinal, the quantitative composition of a number from individual units, etc. is introduced, counting backwards, children learn to compose and solve simple arithmetic problems.

To summarize all of the above, I would like to note that FEMP is a purposeful and organized process of transferring and assimilating knowledge, techniques and methods of mental activity of preschool children in the field of mathematics. The result of the process of forming elementary mathematical concepts is the mathematical development of preschool children.

Bibliography

1. Beloshistaya, A.V. Theory and methodology of organizing mathematical development of preschoolers / A.V. White-haired. – Murmansk: MSPU, 2010. – 256 p.
2. Veraksa N.E. Basic general educational program of preschool education “From birth to school” : An exemplary general educational program of preschool education. [Text]/ N.E. Veraksa, T. S. Komarova, M. A. Vasiliev. – M.: Mosaic – Synthesis, 2014. – 368 p.
3. Vygotsky, L.S. Educational psychology / L.S. Vygotsky / Pod. Ed. V.V. Davydova. – M.: Pedagogika-Press, 1996. – 536 p.
4. Gabova, M.A. Means of mathematical development of a child: history and modernity / M.A. Gabova // Kindergarten: theory and practice. – 2011. – No. 3. – P.18-27.
5. Pestalozzi, I. G. Selected pedagogical works: T. 2. / pod. Ed. V. A. Rotenberg, V. M. Clarina. – M.: Pedagogy, 1981. – 416 p.
6. Richterman, T.D. Formation of ideas about time in preschool children: A book for kindergarten teachers [Text] / T.D. Richterman - 2nd ed. – M.: Education, 1991. – 47 p.
7. Taruntaeva, T.V. Development of elementary mathematical concepts in preschoolers / T.V. Taruntaeva. – M.: Education, 2010. 64 p.
8. Uruntaeva, G. A., Afonkina Yu.A. Learning to remember / G. A. Uruntaeva, Yu. A. Afonkina. – Murmansk: Scientific and Methodological Center of the Education System, 1993. – 93 p.
9. Ushinsky, K. D. Selected pedagogical works.: T. 1. / K. D. Ushinsky; edited by A. I. Piskunova (responsible editor), [etc.] – M.: “Pedagogy” , 1974. – 584 pp., T.2 – P. 360.
10. Federal State Educational Standard for Preschool Education//

Methods and techniques for forming mathematical concepts in preschoolers

Svetlana Shurinova

Methods and techniques for forming mathematical concepts in preschoolers

In the process of forming elementary mathematical concepts in preschoolers, the teacher uses a variety of teaching methods : practical, visual, verbal, and playful.

When choosing a method, a number of factors are taken into account: program tasks being solved at this stage, age and individual characteristics of children, availability of necessary didactic tools, etc.

The teacher’s constant attention to the informed choice of methods and techniques and their rational use in each specific case ensures:

— successful formation of elementary mathematical concepts and their reflection in speech;

- the ability to perceive and highlight relations of equality and inequality (by number, size, shape , sequential dependence (decrease or increase in size, number, highlight quantity, shape , value as a common feature of the analyzed objects, determine connections and dependencies) ;

- orienting children to the use of mastered methods of practical actions (for example, comparison by matching, counting, measuring)

in new conditions and an independent search for practical ways to identify, discover signs, properties, and connections that are significant in a given situation. For example, in a game, identify the sequence order, the pattern of alternation of features, the commonality of properties.

in the formation of elementary mathematical concepts is the practical method . Its essence lies in organizing the practical activities of children, aimed at mastering strictly defined methods of acting with objects or their substitutes (images, graphic drawings, models, etc.)

.

Characteristic features of the practical method in the formation of elementary mathematical concepts :

— performing various practical actions;

- widespread use of didactic material ;

- the emergence of ideas as a result of practical actions with didactic material :

- development of counting skills, measurement and calculations in the most elementary form ;

- widespread use of formed ideas and mastered actions in everyday life, play, work, i.e. in various types of activities.

This method involves the organization of special exercises, which can be offered in the form of a task , organized as actions with demonstration material , or proceed in the form of independent work with handout didactic material .

Exercises can be collective - performed by all children at the same time - and individual - performed by an individual child at the board or teacher’s table. Collective exercises, in addition to assimilation and consolidation of knowledge, can be used for control.

Individuals, performing the same functions, also serve as a model by which children are guided in collective activities.

Game elements are included in exercises in all age groups: and younger ones - in the form of a surprise moment, imitation movements, a fairy-tale character, etc.; in older children they take on the character of search and competition.

As children age, the exercises become more complicated: they consist of a larger number of links, the educational and cognitive content in them is not masked by a practical or gaming task, in many cases, their implementation requires performance actions , the manifestation of ingenuity, and ingenuity. So, in the younger group, the teacher invites the children to take carrots and treat each hare; in the senior class - determine the number of circles on a card posted on the board, find the same number of objects , prove the equality of the circles on the card and the group of objects . If in the first case the exercise consists of a conventionally selected one link, then in the second - of three.

The most effective are complex exercises that make it possible to simultaneously solve program problems from different sections, organically combining them with each other, for example: “quantity and counting”

and
“magnitude”
,
“quantity and counting”
and
“geometric figures”
,
“geometric figures”
,
“magnitude”
and
“quantity and counting”
, etc. Such exercises increase the efficiency of the lesson.

In kindergarten, exercises of the same type (i.e., pursuing the same goal and carried out with the same content) are widely used, thanks to which the necessary methods of action are developed; mastery of counting, measurement, and simple calculations; a circle of elementary mathematical concepts is formed .

The currently existing system of exercises in all age groups is based on the following principle; each previous and subsequent exercise has common elements - material , methods of action, results, etc.

From the point of view of children’s manifestation of activity, independence, and creativity in the process of execution, reproductive (imitative)

and productive exercises.

Reproductive ones are based on simple reproduction of the method of action. At the same time, the actions of children are completely regulated by adults in the form of a sample, explanation, requirement, rule that determines what and how to do. Strict adherence to them gives a positive result, ensures the correct completion of the task, and prevents possible mistakes .

Productive exercises are characterized by the fact that children must fully or partially discover the method of action themselves. This develops independent thinking, requires a creative approach, and develops focus and dedication. The teacher usually says what needs to be done, but does not tell or demonstrate the method of action. When performing exercises, the child resorts to mental and practical tests, shows intelligence, ingenuity, etc. When performing such exercises, the teacher provides help not directly, but indirectly , invites children to think and try again, and approves of the correct actions.

When forming elementary mathematical concepts, the game acts as an independent teaching method . But it can also be classified as a group of practical methods , bearing in mind the special significance of different types of games in mastering various practical actions, such as composing a whole from parts, rows of figures, counting, superposition and application, grouping, generalization, comparison, etc.

Didactic games are the most widely used. Thanks to a learning task put into a game form (game concept, game actions and rules, the child unintentionally learns certain cognitive content. All types of didactic games ( objective , board-printed, verbal)

are an effective means and
method for forming elementary mathematical concepts . Subject and word games are carried out in and outside of mathematics . Desktop - printed, as a rule - in free time from classes. All of them perform the basic functions of education: educational, educational and developmental. There are didactic games for the formation of quantitative ideas , ideas about size , shape , figures, space, time. Thus, it is very promising to present each section of the “
mathematics ” in kindergarten with a system of didactic games that serve to exercise children in applying knowledge.

Game as a method of teaching and forming elementary mathematical concepts involves the use in classes of individual elements of different types of games (plot, movement, etc., game techniques (surprise moment, competition, search, etc.) Currently, a system of so-called educational games.

All didactic games for the formation of elementary mathematical concepts are divided into several groups:

1. Games with numbers and numbers

2. Time travel games

3. Games for orientation in space

4. Games with geometric shapes

5. Logical thinking games

Visual and verbal methods in the formation of “elementary”
mathematical concepts are not independent; they accompany practical and game methods .
Techniques for forming mathematical representations .

In kindergarten, techniques that relate to visual, verbal and practical methods and are used in close unity with each other:

1. Show (demonstration)

method of action in combination with an explanation or example from the teacher.
This is the main teaching method , it is visual, practical and effective in nature, carried out using a variety of didactic means, and makes it possible to develop skills and abilities in children. The following requirements
are imposed on it - clarity, dissection of the demonstration of methods of action;

— consistency of actions with verbal explanations;

- accuracy, brevity and expressiveness of speech accompanying the show:

- activation of perception, thinking and speech of children.

2. Instructions for performing independent exercises. This technique is associated with the teacher’s demonstration of methods of action and follows from it. The instructions reflect what and how to do to get the desired result. In older groups, the instructions are given in full before the task begins; in younger groups, they precede each new action.

3. Explanations, clarifications, instructions. These verbal techniques are used by the teacher when demonstrating a method of action or while children are performing a task in order to prevent mistakes , overcome difficulties, etc. They must be specific, short and figurative.

Demonstration is appropriate in all age groups when familiarizing with new actions (application, measurement, but this requires activation of mental activity, excluding direct imitation. In the course of mastering a new action, developing the ability to count , measure, it is advisable to avoid repeated demonstration.

Mastering an action and improving it is carried out under the influence of verbal techniques : explanations, instructions, questions. At the same time, the verbal expression of the method of action is being mastered.

4. One of the main methods of forming elementary mathematical concepts in all age groups is asking questions to children. In pedagogy, the following classification of questions is accepted:

- reproductive-mnemonic (How much? What is it? What is this figure called? What is the difference between a square and a triangle);

- reproductive-cognitive (How many cubes will be on the shelf if I put one more? Which number is greater (smaller)

: nine or seven);

-productive-cognitive (What needs to be done so that there are 9 circles? How to divide the strip into equal parts? How can you determine which flag in a row is red).

Questions activate children’s perception, memory, thinking, and speech, ensuring comprehension and assimilation of the material . When forming elementary mathematical concepts , the most significant series of questions is: from simpler ones, aimed at describing specific features, properties of an object , results of practical actions, i.e., stating, to more complex ones, requiring the establishment of connections, relationships, dependencies, their justification and explanation, use the simplest evidence. Most often, such questions are asked after the teacher demonstrates a sample or the children perform exercises. For example, after the children have divided a paper rectangle into two equal parts, the teacher asks: “What did you do? What are these parts called? Why can each of these two parts be called a half? What shape did the parts turn out to be ? How to prove that the result is squares? What must be done to divide the rectangle into four equal parts?

Questions of different nature cause different types of cognitive activity: from reproductive, reproducing the studied material , to productive, aimed at solving problematic problems.

Basic requirements for questions as a methodological device :

- accuracy, specificity, laconicism:

— logical sequence;

- variety of wording , i.e. the same thing should be asked in different ways.

— the optimal balance between reproductive and productive issues depending on the age of the children and the material ;

- questions should develop the child’s thinking, make him think, highlight what is required, carry out analysis, comparison, juxtaposition, generalization;

-the number of questions should be small, but sufficient to achieve the set didactic goal;

- Prompt and alternative questions should be avoided.

The teacher usually asks a question to the whole group, and the called child answers it. In some cases, choral responses are possible, especially in younger groups. Children need to be given the opportunity to think about their answer.

Older preschoolers should be taught to formulate questions independently. In a specific situation, using didactic material , the teacher invites children to ask about the number of objects , their ordinal place, size, shape , method of measurement, etc. The teacher teaches them to ask questions based on the results of direct comparison: “Kolya compared a square and a rectangle. What can I ask him?”

, following the practical action performed at the board: “Ask Galya what she learned by arranging
the objects into two rows ? Look what I did. What can you ask me?”, based on the action performed by the child sitting next to him: “What can you ask Anya?”
. Children successfully master the ability to ask questions if they are addressed to a specific person - a teacher, a friend.

Children's answers should be:

- short or complete, depending on the nature of the question;

- independent, conscious;

— accurate, clear, loud enough;

- grammatically correct (observance of word order, rules of their agreement, use of special terminology).

When working with preschoolers , an adult often has to resort to the technique of reformulating an answer , giving it the correct sample and asking them to repeat it . For example: “There are four mushrooms on the shelf.”

, says the kid.
“There are four mushrooms on the shelf
,” the teacher clarifies.

5. Control and evaluation. These techniques are interrelated . Control is carried out through monitoring the process of children completing tasks, the results of their actions, and answers. These techniques are combined with instructions, explanations, clarifications, demonstration of methods of action to adults as a model, direct assistance, and include correction of errors.

The teacher corrects errors during individual and collective work with children. Practical and speech errors are subject to correction. The adult explains their reasons, gives an example, or uses the actions and responses of other children as an example. Gradually, the teacher begins to combine control with self- and mutual control. Knowing the typical mistakes that children make when counting, measuring, simple calculations, etc., the teacher carries out preventive work.

The methods and results of actions and the behavior of the children are subject to evaluation. The assessment of an adult who teaches one to be guided by a model begins to be combined with the assessment of comrades and self-esteem. This technique is used during and at the end of an exercise, game, or lesson.

The use of control and assessment has its own specifics depending on the age of the children and the degree to which they have mastered knowledge and methods of action. Control is gradually transferred to the result, the assessment becomes more differentiated and meaningful. These techniques , in addition to teaching, also perform an educational function: they help to cultivate a friendly attitude towards comrades, the desire and ability to help them, etc.

6. During the formation of elementary mathematical concepts in preschoolers, comparison , analysis, synthesis, and generalization act not only as cognitive processes (operations), but also as methodological techniques that determine the path along which the child’s thought moves in the learning process.

Comparison is based on establishing similarities and differences between objects. Children compare objects by quantity , shape , size, spatial location, time intervals by duration, etc. First, they are taught to compare the minimum number of objects . Then the number of objects is gradually increased, and the degree of contrast of the compared features is correspondingly reduced.

Analysis and synthesis as methodological techniques appear in unity. An example of their use is the formation in children of ideas about “many”

and
“one”
, which arise under the influence of observation and practical actions with
objects .
The teacher brings a large number of identical toys into the group at once - as many as there are children. Gives one toy to each child, and then collects them together. Before the children's eyes, a group of objects is split into parts, and the whole is recreated from them.

Based on analysis and synthesis, children are led to a generalization, which usually summarizes the results of all observations and actions. These techniques are aimed at understanding quantitative, spatial and temporal relationships, at highlighting the main, essential. A summary is made at the end of each part and the entire lesson. First, the teacher generalizes, and then the children.

Comparison, analysis, synthesis, generalization are carried out on a visual basis using a variety of didactic means. Observations, practical actions with objects , reflection of their results in speech, questions to children are the external expression of these methodological techniques , which are closely related to each other and are most often used in combination.

7. In the methodology for the formation of elementary mathematical concepts, some special methods of action leading to the formation of concepts and the development of mathematical relations act as methodological techniques . These are techniques of application and application, examining the shape of an object , “weighing”
an object “on the hand”
, introducing counters - equivalents, counting and counting by unit, etc. Children master these
techniques in the process of showing, explaining, performing exercises and then resort to to them for the purpose of verification, proof, in explanations and answers, in games and other activities.
8. Modeling is a visual and practical technique that includes the creation of models and their use in order to form elementary mathematical concepts in children . The technique is extremely promising due to the following factors:

— the use of models and modeling puts the child in an active position and stimulates his cognitive activity;

- a preschooler has some psychological prerequisites for the introduction of individual models and elements of modeling: the development of visual-effective and visual-figurative thinking.

Models can perform different roles: some reproduce external connections, help the child see those that he does not notice on his own, others reproduce the sought-after but hidden connections, the directly not perceived properties of things.

Models are widely used in the formation

· temporary representations : model of parts of the day, week, year, calendar;

· quantitative; numerical ladder, numerical figure, etc., spatial: (models of geometric figures)

etc.

· when forming elementary mathematical concepts, subject - subject-schematic , and graphical models are used.

9. Experimentation is a method of mental education that ensures the child’s independent identification through trial and error of connections and dependencies hidden from direct observation. For example, experimentation in measurement (size, measurement, volume)

.

10. Training - a method of familiarization with social reality (the world of money)

.

LIST OF SOURCES USED

1. Belous, T.K. et al. Organization of work in mathematics in a small kindergarten. / T. K. Belous. // Doshk . Education, 1999, No. 10.

2. Berezina, R.I. Teaching children of the preparatory group to measure. / R.I. Berezina. // Doshk . Education, 1999, No. 10.

3. Veraksa, N. S. Formation of unified temporal-spatial representations . / N. S. Veraksa. // Doshk . Education, 1996, No. 5.

4. Vodopyanov, E. N. Formation of initial geometric concepts in preschoolers . / E. N. Vodopyanov. // Doshk . Education, 2000, No. 3.

5. Raising children through play: A manual for educators of children. garden / Comp. A. K. Bondarenko, A. I. Matusik. — 2nd ed., revised. And additional - M.: Education, 1983.

6. Godinai, G. N., Pilyugina E. G. Education and training of children of primary preschool age . - Moscow Enlightenment, 1988.

7. Let's play. Mathematical games for children 5-6 years old. — Ed. A. A. Stolyar. - M.: Education, 1991).

8. Danilova, V.V. Mathematical training of children in preschool institutions . - M.: Education, 1987.

9. Didactic games and exercises for sensory education of preschoolers : A manual for kindergarten teachers. — Ed. L.A. Wenger. 2nd ed., revised. and additional - M.: Education, 1998.

10. Dyachenko, O. M., Agaeva, E. L. What doesn’t happen in the world? - M.: Education, 1991.

11. Erofeeva, T. I., Pavlova, L. N., Novikova, V. P. Mathematics for preschoolers : Book. For the teacher of children. garden - M.: Education, 1992.

12. Zhitomirsky, V. G., Shevrin, L. N. Geometry for kids. - M.: 1996.

13. Karazanu, V. N. Orientation in space (senior preschool age )

.
/ V. N. Karazan. // Doshk .
Education, 2000, No. 5. 14. Korneeva, G. A., Museibova, T. A. Methodological instructions for studying the course “ Formation of elementary mathematical concepts in preschool children .” - M., 2000.

15. Korneeva, G. A. The role of objective actions in the formation of the concept of number in preschoolers . / G. A. Korneeva. // Question Psychology, 1998, No. 2.

Library of old Soviet mathematics textbooks

The methodology for forming elementary mathematical concepts in preschoolers is constantly developing, improving and enriching with the results of scientific research and advanced pedagogical experience.

Currently, thanks to the efforts of scientists and practitioners, a scientifically based methodological system for the formation of elementary mathematical concepts in preschoolers has been created, is successfully functioning and is being improved. Its main elements - the goal, content, methods, means and forms of organizing work - are closely interconnected and mutually condition each other. The leading and determining one among them is the goal, since it is socially determined and objective in nature. The kindergarten fulfills the social order of society, preparing children to study the basics of science (including mathematics) at school.

Soviet pedagogy and psychology, based on Marxist-Leninist teaching, considers the development of personality as a process of assimilation of the socio-historical experience of mankind. This experience in its generalized form is passed on to the younger generation by adults in the learning process. F. Engels wrote that the individual experience of a child is replaced by the result of the experience of his ancestors. And the assimilation of mathematical axioms by children is nothing more than the assimilation of the heredity accumulated by people.

Training and development are in a dialectical relationship. Based on the current level of development, training should be somewhat ahead of it. This means that in the learning process it is necessary to focus not only on what the child himself is capable of doing, but also on what he can do with the help of adults, under their guidance, i.e., on the future, on the zone of proximal development, in which usually contains new and more complex actions and operations than those that the child already knows. When mastering them, “not only the development process that has already been completed today, not only its cycles that have already been completed, not only the maturation processes that have already been completed, but also those processes that are now in a state of formation, that are only maturing, are only developing”2. What a child could recently do with the help of an adult, after some time, as a result of training, he can do independently. The “zone of proximal development” becomes the “actual” level of development.

Learning leads to development, being its source and paving the way for it. Each of these interrelated processes has its own patterns. It is unlawful to both identify and oppose them to each other.

However, to this day, both in theory and in practice, the opinion that the younger the child is, the less interference there should be in the process of his development, has not completely outlived its usefulness. It is believed that the acquisition of quantitative, spatial, and temporal concepts occurs naturally, spontaneously in children’s everyday life and various activities. There are attempts to strictly define age-related opportunities for acquiring knowledge and to deny the programmatic nature of teaching young children. Thus, the Swiss psychologist J. Piaget considers it a big mistake to think that a child perceives the concept of number and other mathematical concepts directly in learning. In his opinion, these concepts are formed in the child independently and spontaneously.

According to J. Piaget, his students and followers, mastery of mathematical concepts occurs on the basis of logical operations of classification and seriation, which the child discovers himself and which is almost impossible to learn. They appear quite late, at 11-12 years old, i.e. already at school age. This point of view does not solve the problem of mathematical development and learning of children in preschool age.

A productive approach to solving this problem developed in Soviet pedagogy and psychology based on data from numerous studies. It is as follows: in the conditions of rationally structured training, taking into account the age capabilities of preschoolers, it is possible to form in them a complete understanding of individual mathematical concepts. In this case, learning is considered as an indispensable condition for development, which in turn becomes a controlled process associated with the active formation of elementary mathematical concepts and logical operations. This approach does not ignore spontaneous experience and its impact on the child’s development, but the leading role is given to targeted learning.

The mathematical development of preschoolers should be understood as shifts and changes in the cognitive activity of the individual that occur as a result of the formation of elementary mathematical concepts and related logical operations.

The formation of elementary mathematical concepts is a purposeful and organized process of transferring and assimilating knowledge, techniques and methods of mental activity provided for by program requirements. Its main goal is not only preparation for successful mastery of mathematics at school, but also the comprehensive development of children.

§ 2. Methodology for the formation of elementary mathematical representations and other sciences

The method of forming elementary mathematical concepts in children in kindergarten is associated with many sciences, and above all with science, the subject of study of which is different aspects of the personality and activity of a preschool child, the process of his upbringing and education.

It has the closest connection with preschool pedagogy, the science of communist education of children. Technique? the formation of elementary mathematical concepts is based on the tasks of teaching and mental education of the younger generation developed by preschool pedagogy and didactics: principles, conditions, ways, content, means, methods, forms of organization, etc. This connection is mutual in nature: research and development of problems of formation elementary mathematical concepts in children in its own way” improves pedagogical theory, enriching it with new factual material.

Multilateral contacts exist between private methods that study specific patterns of the process of raising and teaching young children: methods of forming elementary mathematical concepts, speech development, theory and methods of physical education, etc.

Preparing children to master mathematics at school cannot be carried out successfully without connection with the methods of primary teaching of mathematics and those aspects of mathematics itself that are the theoretical basis for teaching preschoolers and primary schoolchildren. Relying on these sciences allows, firstly, to determine the volume and content of knowledge that should be mastered by children in kindergarten and serve as the foundation of mathematical education; secondly, to use teaching methods and means that fully meet the age characteristics of preschoolers and the requirements of the principle of continuity.

The reform of general education and vocational schools has set the task of improving the quality of teaching in all general education subjects, including mathematics. It is well known that when mastering mathematical knowledge, many students experience serious difficulties, the cause of which, as a rule, is insufficient mathematical preparation in preschool age.

Improving the content and methods of teaching mathematics at school requires a new attitude towards preparing children in the period immediately preceding schooling. At present, significant changes have already been made to the program for the development of mathematical concepts in preschoolers (increasing the volume of mental calculation, counting groups of objects, learning to measure individual quantities, expanding geometric knowledge, etc.); More effective methods and means of teaching have been found and tested (modeling, problematic tasks and situations, developmental and educational games, etc.). The connection with the methods of teaching mathematics in primary school allows us to correctly determine the main ways to further improve the methods of forming elementary mathematical concepts in preschoolers.

Education should be structured taking into account the patterns of development of cognitive activity and the child’s personality, which is the subject of study of psychological sciences. Perception, representation, thinking, speech not only function, but also intensively develop during the learning process.

Psychological characteristics and patterns of a child’s perception of many objects, number, space, time serve as the basis for developing methods for the formation of elementary mathematical concepts. Psychology determines the age-related capabilities of children in mastering knowledge and skills that do not appear as something frozen and change depending on the type of training. Modern psychological research shows that the abilities of preschool children in mastering mathematical concepts are great and have not yet been fully revealed or fully studied.

The rational construction of the learning process is associated with the creation of optimal conditions based on the anatomical and physiological characteristics of young children. The patterns of physiological processes in preschoolers serve as the basis for determining the place and duration of classes on the formation of elementary mathematical concepts for each age group of kindergarten, determine their very structure, the combination and alternation of various methods and means of teaching, different types of activities (inclusion of physical education, dosing educational and cognitive tasks, etc.).

The method of forming elementary mathematical concepts is a relatively young scientific pedagogical discipline, but it has ancient origins. A historical excursion shows how the concepts of initial teaching of mathematics gradually changed depending on the demands of life and the level of development of mathematical science itself, makes it possible to critically evaluate the rich heritage, avoid many mistakes, take into account the positive experience of the past, as well as the results of the latest research. In Marxist-Leninist theory, it finds a solid methodological basis, which provides a comprehensive and in-depth examination of the phenomenon in its development, compliance with the principle of objectivity, specificity, unity of theory and practice.

Connection with various sciences creates a theoretical basis for the methods of forming mathematical concepts in children in kindergarten.

§ 3. Study of problems in the formation of elementary mathematical concepts in preschool children

For a long time, the concepts of the initial teaching of number and counting to young children were built either on the basis of speculative theoretical constructions or through empirical experience. Outstanding thinkers of the past (Ya. A. Komenskin, I. G. Pestalozzi, K. D. Ushnsky, L. N. Tolstoy), prominent figures in the field of preschool education abroad (F. Frebel, M. Montessori) and in our country (E.I. Tikheyeva, F.N. Blekher) successfully combined direct work with children with theoretical understanding of its results.

The formation of methods for the formation of mathematical concepts in preschoolers is associated with the use of experimental research methods, which have begun to be introduced recently.

Scientific research in this area is being carried out at the Institute of Preschool Education of the Academy of Sciences of the USSR and in a number of other scientific and educational institutions of the country. Educators, methodologists, and teachers also take part in this work.

In recent years, research into the problems of teaching six-year-olds has been widely developed (APN of the USSR, Research Institute of Pedagogical Sciences of Ukraine, Georgia, the Baltic and other republics, Mogilev Pedagogical Institute, etc.). These studies have a direct impact on the theory and practice of forming elementary mathematical concepts in preschoolers.

In modern conditions, in connection with the transition to schooling for children from the age of six, the development of methods for improving the preparation of preschoolers for mastering school mathematics is of particular importance.

Research in the field of the formation of elementary mathematical concepts in children is directly related to practice and provides scientific ways to solve its most important problems. The developed content and methodological techniques, didactic means and forms of organizing work are used in the practice of forming elementary mathematical concepts in children in kindergarten. The publication of the main results of the study makes them available to a wide circle of preschool workers. The recommendations of scientists are taken into account when revising the program for the development of elementary mathematical concepts in kindergarten. Periodically, changes are made to it, new requirements and tasks are introduced, taking into account the results of scientific research. The conclusions and recommendations of scientists contribute to the improvement of the work of kindergartens and the development of mathematical concepts in children, and serve as the basis for subsequent scientific research.

Student, educational and research work (tests, coursework, graduation, diploma), in which the knowledge, skills and abilities necessary for a future specialist are acquired, must meet the requirements of relevance, novelty, theoretical and practical significance, objectivity and reliability, as well as any other scientific works devoted to the problems of mathematical development of preschool children.

Mathematical development and its importance in the development of preschool children

Marina Zverkova

Mathematical development and its importance in the development of preschool children

One of the main goals of preschool education is the child’s mathematical development . It is not just about teaching how to count, measure and solve arithmetic problems. It also implies the development of the ability to see , discover properties, relationships, dependencies in the surrounding world, and be able to convey them with the help of signs and symbols.

Even in early childhood, babies encounter objects that differ in shape, color and quantity. At this age, the child’s basic elementary concepts and abilities begin to form. The first toys resemble geometric shapes: cubes, construction sets, pyramids. The counting begins with mom’s questions: “Tell me, how old are you?”

.
Parents children to name the shapes of toys, their size and quantity.
Through gaming activities, the ability to distinguish between different properties and features of objects is formed. The baby develops his first concept of mathematics , although he does not yet know or realize it. The consciousness of a child in early childhood is chaotic. Parents teach children to compare , group objects, and call them by their proper names.

Through visual and objective actions, they help the child remember what he hears based on objective images. By the age , a child can already group objects according to their external characteristics, color, and shape. So, for example, a child can put green toys away from red ones, select pencils from a pile of other objects and put them together, can put pyramid rings according to size, in order.

When engaging with objects through play activities, the child compares them. This is where the first acquaintance with mathematics .

By the age of four, children can easily count to five, and a little older to ten, but they may make mistakes in counting.

By the age , children already begin to understand when numbers increase and when they decrease. That is why it is important to start systematic classes from kindergarten in order to increase the child’s mental perception.

In today's modern society, one of the requirements for preschool education is that children acquire mathematical knowledge and basic concepts in kindergarten.

In the course of their development, preschoolers receive their first basic understanding of mathematics . The available methods and means of forming elementary mathematical concepts are developed specifically for age categories, taking into account the gradual development of skills and abilities in preschoolers in this direction.

Mathematics is an independent educational subject and is designed to develop intellectual abilities depending on the natural potential of preschool children . Its role in the development of elementary concepts in preschoolers is very great . During this type of activity, the child develops and develops cognitive and personal abilities.

During the learning process, through the means of mathematical classes, the child receives his first ideas about mathematical concepts .

Mathematics is one of the few disciplines that covers different aspects of children's . In the process of forming elementary mathematical concepts and learning, all cognitive processes actively develop in preschoolers This becomes effective if, when organizing classes, the frequency and sequence of development of cognitive processes in a child is taken into account, depending on the psychophysical development of each child .

If a child has not reached the age at which he is able to understand mathematical processes , then classes will not play any role for his consciousness. A child’s capabilities are determined by his psychology. , innovative methods and means are increasingly included in preschool

The abilities of each child depend on his individual psychological characteristics. Mathematical abilities cannot be innate, since only anatomical and physiological characteristics of a person are innate. Mathematical abilities are a special type of ability; they depend on the integral quality of the mind and develop in the process of mathematical activity .

A person’s abilities can manifest themselves in various areas, and here, like everything else, mathematical abilities are revealed during the activities of a preschooler . Preschool age the most favorable period for the development of abilities .

Analysis of scientific research (A. M. Leushina, N. I. Nepomnyashchaya, A. A. Stolyar, etc., pedagogical experience convinces that rationally organized teaching of mathematics to preschoolers ensures the overall mental development of children . (Rationally organized is timely, that is appropriate to the age and interests of children .) At the same time, pedagogical guidance from an adult is important . Children acquire basic knowledge about the set, number, size and shape of objects, learn to navigate time and space. They master counting and measurements of linear and volumetric objects with the help of conventional and generally accepted measures, they establish quantitative relationships between quantities, wholes and parts.

In recent years, the concept of pre-mathematical preparation . Preparing the child and his cognitive world for a mathematical way of thinking . Various ways of forming the cognitive sphere make it possible to prepare a child for studying the subject of mathematics . When organizing classes, there is an impact on visual and logical thinking, memory, creative imagination, perception, and voluntary attention of a preschooler .

Children in preschool age observe and imitate adults, they watch every action and listen carefully to what the teacher says and this is an important property. Children must be taught to act independently, to show and talk about their actions. Preschoolers should be encouraged to repeat after the teacher about the properties and qualities of objects. Games with children should contain mathematical activities .

mathematics classes in kindergarten, the simplest types of practical and mental activity of children . By types of activities - in this case, methods of examination, counting, measurement - are understood objective sequential actions that a child must perform to acquire knowledge: element-by-element comparison of two sets, imposing measures, etc. By mastering these actions, the child learns the purpose and methods of activity, as well as rules ensuring the formation of knowledge.

As a rule, educational tasks in the classroom are solved in combination with educational ones. Thus, the teacher teaches children to be organized, independent, listen carefully , and do work efficiently and on time. This disciplines children and helps them develop focus, organization, and responsibility. Thus, teaching children mathematics from an early age ensures their comprehensive development .

Naturally, the basis of cognition is sensory development , acquired through experience and observation. In the process of sensory cognition, ideas are formed - images of objects, their properties, relationships. Thus, operating with various sets (objects, toys, pictures, geometric shapes), children learn to establish equality and inequality of sets, to name a quantity with words: “more”

,
“less”
,
“equally”
.
Comparing concrete sets prepares children for later mastering the concept of number. It is operations with sets that are the basis to which children turn not only in kindergarten, but also throughout the subsequent years of school. The concept of set forms in children the basis for understanding abstract numbers and the laws of the natural series of numbers.
Although the concepts of a natural number, as well as a geometric figure, magnitude, part and whole are abstract, they nevertheless reflect the connections and relationships of objects in the surrounding reality. So, in the second junior group of kindergarten (fourth year of life)

the focus is on building knowledge about the set.
The concept of set is one of the basic and most general; it runs through all of mathematics . The concept of set is so broad that it is not defined even at the modern level of development of science , but is introduced as initial and explained with specific examples. In the middle group, in the process of studying the basic properties of a set, the concept of number is formed, and in the senior group, the first ideas about the natural series of numbers are formed. At preschool age, understanding of the basic properties of a set is limited. However, awareness of its individual properties (equality and inequality, independence of the power of a set from its qualitative characteristics) is possible already in early preschool age .
In older groups, it is worth teaching children about sets , dividing the set into groups and explaining to them the difference between a smaller and a larger group, as well as the equality of parts. Visually teach preschoolers the sequence of counting to ten and counting backwards. Teach children to count by touch and by ear within ten. Learn to compare the number of objects in different groups, add and remove objects to a given number.

At the same time, preschoolers are taught to compare objects by size (size)

and the results of the comparison
are denoted by the corresponding words-concepts ( “more - less”
,
“narrow - wide”
, etc., build rows of objects according to their size in
increasing or decreasing (large, small, even smaller, smallest)
. However, for In order for a child to master these concepts, it is necessary to form concrete ideas in him, teach him to compare objects with each other, first directly - by superimposing, and then indirectly - by measuring.

The central task of children's mathematical development in kindergarten is learning to count. The main methods for this are overlaying and applying, mastery of which anticipates learning to count with the help of numeral words.

Children in preschool age are able to divide objects and name their parts, for example, dividing an apple into slices or a pie. Preschoolers should understand that a whole apple is larger than a slice or half an apple. Senior students must master and understand that the number 7 is more than six, but less than eight. By the end of the learning period, preschoolers should be able to perform simple mathematical operations .

In the mathematical preparation of children , the development of elementary mathematical concepts, an important role is played by teaching measurement, as the initial way of knowing the quantitative characteristics of the environment. This makes it possible for preschoolers , first of all, to use not generally accepted, but conventional measures when measuring granular, liquid substances and extents. At the same time, children develop their eye sense , which is very important for their sensory development .

mathematics program in kindergarten provides for the development of children's eye in determining the size of objects. To do this, they are trained to estimate the size (size of objects)

in general or according to individual parameters, comparing with the size of known objects.
Attention is paid to developing the ability to check the correctness of the assessment in one’s practical activities, using additions, reductions, etc. Each practical action adds new content to children’s . It has been proven that the formation of elementary mathematical knowledge occurs simultaneously with the development of practical skills.
Practical actions, while fulfilling a certain role in the mathematical development of children , do not themselves remain unchanged. Thus, the activity associated with the account is changed. At first it is based on a practical element-by-element comparison of two specific sets, and later the number as an indicator of the power of the set and the natural series of numbers, which subsequently replaces one of the specific sets, acquires special significance .

First, children take objects with their hands, rearrange them, and then count the objects without touching them, or perceive them only by touch.

Based on practical actions children form such mental operations as analysis, synthesis, comparison, and generalization. The teacher should focus in assessing the results of his work, first of all, on these indicators, on how children can compare, analyze, generalize, and draw conclusions.

In the process of systematic teaching of mathematics, children master special terminology - the names of numbers, geometric shapes (circle, square, triangle, rhombus, etc., elements of shapes (side, vertex, base)

etc. However, it is not recommended to use words-terms such as
“natural rad”
,
“totality”
,
“structure”
,
“elements of a set”
, etc. when working with children. Children should also master measured quantities: meter, centimeter, kilogram, gram, etc. Moreover, the work is not limited to just classes. They learn to find and compare objects in everyday life, on the street and in nature. For example: three birch trees under the window.

One should keep in mind the use of the entire didactic space in an educational situation.

Mathematics is not necessarily a boring activity, as it might seem at first glance. To teach, teachers play with children, come up with various counting rhymes, proverbs, sayings, and riddles. The child masters the first numerical concepts and forms.

There are also didactic forms and means of education that use visual aids, illustrations, and games.

To achieve results, they use various materials : counting sticks, natural materials , and teach how to count and recognize money.

Shcherbakova E.I., among the tasks for the formation of elementary mathematical knowledge and the subsequent mathematical :

-acquisition of knowledge about set, number, size, shape, space and time as the foundations of mathematical development ;

-formation of a broad initial orientation in quantitative, spatial and temporal relations of the surrounding reality;

- formation of skills and abilities in counting, calculations, measurement, modeling, general educational skills;

-mastery of mathematical terminology ;

-development of cognitive interests and abilities, logical thinking, general intellectual development of the child .

These problems are most often solved by the teacher simultaneously in each mathematics , as well as in the process of organizing various types of independent children's activities. Numerous psychological and pedagogical studies and advanced pedagogical experience in preschool institutions show that only properly organized children's activities and systematic training ensure the timely mathematical .