How to develop mathematical abilities in a preschooler? What is spatial thinking? How to use educational games to teach your child the basics of counting

Both parents and teachers know that mathematics is a powerful factor in the intellectual development of a child, the formation of his cognitive and creative abilities. It is also known that the success of teaching mathematics in primary school depends on the effectiveness of a child’s mathematical development in preschool age.

Why do many children find mathematics so difficult not only in elementary school, but even now, during the period of preparation for educational activities? Let's try to answer this question and show why generally accepted approaches to the mathematical preparation of a preschool child often do not bring the desired positive results.

In modern primary school educational programs, important importance is attached to the logical component. The development of a child’s logical thinking implies the formation of logical techniques of mental activity, as well as the ability to understand and trace the cause-and-effect relationships of phenomena and the ability to build simple conclusions based on cause-and-effect relationships. So that the student does not experience difficulties literally from the first lessons and does not have to learn from scratch, already now, in the preschool period, it is necessary to prepare the child accordingly.

Many parents believe that the main thing in preparing for school is to introduce the child to numbers and teach him to write, count, add and subtract (in fact, this usually results in an attempt to memorize the results of addition and subtraction within 10). However, when teaching mathematics using textbooks of modern developmental systems (L. V. Zankov’s system, V. V. Davydov’s system, the “Harmony” system, “School 2100”, etc.), these skills do not help the child in mathematics lessons for very long. The stock of memorized knowledge ends very quickly (in a month or two), and the lack of development of one’s own ability to think productively (that is, to independently perform the above-mentioned mental actions based on mathematical content) very quickly leads to the appearance of “problems with mathematics.”

It is no coincidence that in recent years, many schools working on developmental programs have conducted interviews with children entering the first grade, the main content of which is questions and tasks of a logical, and not just arithmetic, nature. Is this approach to selecting children for education logical? Yes, it is natural, since the mathematics textbooks of these systems are structured in such a way that already in the first lessons the child must use the ability to compare, classify, analyze and generalize the results of his activities.

However, one should not think that developed logical thinking is a natural gift, the presence or absence of which should be accepted. There is a large number of studies confirming that the development of logical thinking can and should be done (even in cases where the child’s natural abilities in this area are very modest). First of all, let's figure out what logical thinking consists of.

Logical techniques of mental actions - comparison, generalization, analysis, synthesis, classification, seriation, analogy, systematization, abstraction - are also called logical techniques of thinking in the literature. When organizing special developmental work on the formation and development of logical thinking techniques, a significant increase in the effectiveness of this process is observed, regardless of the initial level of development of the child.

It is most advisable to develop the logical thinking of a preschooler in line with mathematical development. The process of a child’s assimilation of knowledge in this area is further enhanced by the use of tasks that actively develop fine motor skills, that is
tasks of a logical and constructive nature. In addition, there are various methods of mental action that help enhance the effectiveness of using logical-constructive tasks.

Seriation

Construction of ordered increasing or decreasing series based on a selected characteristic. A classic example of seriation: nesting dolls, pyramids, insert bowls, etc.

Series can be organized by size, by length, by height, by width if the objects are of the same type (dolls, sticks, ribbons, pebbles, etc.), and simply by size (with an indication of what is considered size) if the objects different types (seat toys according to height). Series can be organized by color, for example, by the degree of color intensity (arrange jars of colored water according to the degree of color intensity of the solution).

Analysis

Selecting the properties of an object, or selecting an object from a group, or selecting a group of objects based on a certain characteristic.

For example, the attribute is given: “Find all sour”. First, each object in the set is checked for the presence or absence of this attribute, and then they are isolated and combined into a group based on the “sour” attribute.

Synthesis

Combination of various elements (signs, properties) into a single whole. In psychology, analysis and synthesis are considered as mutually complementary processes (analysis is carried out through synthesis, and synthesis is carried out through analysis).

Tasks to develop the ability to identify the elements of a particular object (features), as well as to combine them into a single whole, can be offered from the very first steps of the child’s mathematical development. Let us give, for example, several such tasks for children two to four years old.

1. A task to select an object from a group based on any criterion: “Take the red ball”; “Take the red one, but not the ball”; "Take the ball, but not the red one."

2. A task to select several objects according to the specified criterion: “Choose all the balls”; “Choose round balls, but not balls.”

3. A task to select one or more objects based on several specified characteristics: “Choose a small blue ball”; "Pick a big red ball." The last type of task involves combining two characteristics of an object into a single whole.

Analytical-synthetic mental activity allows the child to consider the same object from different points of view: as big or small, red or yellow, round or square, etc. However, we are not talking about introducing a large number of objects, quite the contrary, in a way organizing a comprehensive review is the technique of setting various tasks for the same mathematical object.
As an example of organizing activities that develop a child’s ability to analyze and synthesize, we will give several exercises for children five to six years old.

Exercise 1
Material: set of figures - five circles (blue: large and two small, green: large and small), small red square.

Assignment: “Determine which of the figures in this set is extra. (Square.) Explain why. (All the rest are circles.).”

Exercise 2
Material: the same as for Exercise 1, but without the square.
Assignment: “Divide the remaining circles into two groups. Explain why you divided them this way. (By color, by size.).”

Exercise 3
Material: the same and cards with numbers 2 and 3.
Assignment: “What does the number 2 mean on the circles? (Two large circles, two green circles.) The number 3? (Three blue circles, three small circles.).”

Exercise 4
Material: the same didactic set (a set of plastic figures: colored squares, circles and triangles).
Assignment: “Remember what color was the square that we removed? (Red.) Open the box, Didactic set.” Find the red square. What other colors are there squares? Take as many squares as there are circles (see exercises 2, 3). How many squares? (Five.) Can you make one big square out of them? (No.) Add as many squares as needed. How many squares did you add? (Four.) How many are there now? (Nine.)".
The traditional form of tasks for the development of visual analysis are tasks for choosing an “extra” figure (object). Here are a few tasks for children five to six years old.

Exercise 5
Material: drawing of figurines-faces.

Assignment: “One of the figures is different from all the others. Which one? (The fourth one.) How is it different?”

Exercise 6
Material: drawing of human figures.

Task: “Among these figures there is an extra one. Find it. (Fifth figure.) Why is it extra?”
A more complex form of such a task is the task of isolating a figure from a composition formed by superimposing some forms on others. Such tasks can be offered to children five to seven years old.

Exercise 7
Material: drawing of two small triangles forming one large one.

Assignment: “There are three triangles hidden in this picture. Find and show them.”
Note. You need to help the child show the triangles correctly (circle with a small pointer or finger).
As preparatory tasks, it is useful to use tasks that require the child to synthesize compositions from geometric shapes at the material level (from material material).

Exercise 8
Material: 4 identical triangles.

Assignment: “Take two triangles and fold them into one. Now take two other triangles and fold them into another triangle, but of a different shape. How are they different? (One is tall, the other is low; one is narrow, the other is wide.) You can Is it possible to make a rectangle out of these two triangles? (Yes.) A square? (No.) "
Psychologically, the ability to synthesize is formed in a child earlier than the ability to analyze. That is, if a child knows how it was assembled (folded, designed), it is easier for him to analyze and identify its component parts. That is why such serious importance is given in preschool age to activities that actively form synthesis - design.
At first, this is a patterned activity, that is, performing tasks of the “do as I do” type. At first, the child learns to reproduce the object, repeating the entire construction process after the adult; then - repeating the process of construction from memory, and finally moves on to the third stage: independently restores the method of constructing a ready-made object (tasks like “make the same one”). The fourth stage of tasks of this kind is creative: “build a tall house”, “build a garage for this car”, “build a rooster”. The tasks are given without a sample, the child works according to the idea, but must adhere to the given parameters: a garage specifically for this car.
For construction, any mosaics, construction sets, cubes, cut-out pictures are used that are suitable for this age and make the child want to tinker with them. An adult plays the role of an unobtrusive assistant; his goal is to help bring the work to completion, that is, until the intended or required whole object is obtained.

Comparison
- a logical method of mental action that requires identifying similarities and differences between the characteristics of an object (object, phenomenon, group of objects).
Performing a comparison requires the ability to isolate some features of an object (or group of objects) and abstract from others. To highlight various features of an object, you can use the game “Find it using the specified features”: “Which (of these objects) is big yellow? (Ball and bear.) What is big yellow and round? (Ball),” etc.
The child should use the role of the leader as often as the answerer, this will prepare him for the next stage - the ability to answer the question: “What can you tell about him? (The watermelon is large, round, green. The sun is round, yellow, hot.)” . Or: “Who will tell you more about this? (The ribbon is long, blue, shiny, silk.).” Or: “What is this: white, cold, crumbly?” etc.
It is recommended to first teach the child to compare two objects, then groups of objects. It is easier for a small child to first find signs of differences between objects, then signs of their similarity.
Types of comparison tasks:
1. Tasks on dividing a group of objects according to some criteria (large and small, red and blue and

etc.).
2. All games of the “Find the same” type. For a child two to four years old, the set of characteristics by which similarities are sought should be clearly defined. For older children, exercises are offered in which the number and nature of similarities can vary widely.
Let us give examples of tasks for children five to six years old, in which the child is required to compare the same objects according to various criteria.

Exercise 9
Material: images of two apples, a small yellow one and a large red one. The child has a set of shapes: a blue triangle, a red square, a small green circle, a large yellow circle, a red triangle, a yellow square.

Assignment: “Find one that looks like an apple among your figures.” An adult offers to look at each image of an apple in turn. The child selects a similar figure, choosing a basis for comparison: color, shape. “Which figure can be called similar to both apples? (Circles. They are similar in shape to apples.).”

Exercise 10
Material: the same set of cards with numbers from 1 to 9.
Assignment: “Put all the yellow figures to the right. What number fits this group? Why 2? (Two figures.) What other group can be matched to this number? (A blue and red triangle - there are two of them; two red figures, two circles; two square - all options are analyzed.)". The child makes groups, uses a stencil frame to sketch and paint them, then signs the number 2 under each group. “Take all the blue figures. How many are there? (One.) How many colors are there in total? (Four.) Figures? (Six.) ".
The ability to identify the characteristics of an object and, focusing on them, to compare objects is universal, applicable to any class of objects. Once formed and well developed, this skill will then be transferred by the child to any situations requiring its use.
An indicator of the maturity of the comparison technique will be the child’s ability to independently apply it in activities without special instructions from an adult on the signs by which objects need to be compared.
Classification
- division of a set into groups according to some criterion, which is called the basis of classification. Classification can be carried out either according to a given basis, or with the task of searching for the basis itself (this option is more often used with children six to seven years old, as it requires a certain level of formation of the operations of analysis, comparison and generalization).
It should be taken into account that when classifying a set, the resulting subsets should not intersect in pairs and the union of all subsets should form this set. In other words, each object must be included in only one set, and with a correctly defined basis for classification, not a single object will remain outside the groups defined by this basis.
Classification with preschool children can be carried out:
- by name (cups and plates, shells and pebbles, skittles and balls, etc.);
- by size (large balls in one group, small ones in another, long pencils in one box, short pencils in another, etc.);
- by color (this box has red buttons, this one has green buttons);
- in shape (this box contains squares, and this box contains circles; this box contains cubes, this box contains bricks, etc.);
- based on other non-mathematical characteristics: what can and cannot be eaten; who flies, who runs, who swims; who lives in the house and who in the forest; what happens in summer and what happens in winter; what grows in the garden and what in the forest, etc.
All of the examples listed above are classifications based on a given basis: the adult communicates it to the child, and the child carries out the division. In another case, classification is performed on a basis determined by the child independently. Here, the adult sets the number of groups into which many objects (objects) should be divided, and the child independently looks for the corresponding basis. Moreover, such a basis can be determined in more than one way.
For example, tasks for children five to seven years old.

Exercise 11
Material: several circles of the same size, but different colors (two colors).
Assignment: “Divide the circles into two groups. By what criteria can this be done? (By color.).”

Exercise 12
Material: several squares of the same colors are added to the previous set (two colors). The figures are mixed.
Assignment: “Try to divide the figures into two groups again.” There are two options for separation: by shape and by color. An adult helps the child clarify the wording. The child usually says: “These are circles, these are squares.” The adult generalizes: “So, they divided it according to shape.”
In exercise 11, the classification was unambiguously specified by the corresponding set of figures on only one basis, and in exercise 12, the addition of a set of figures was deliberately made in such a way that classification on two different grounds became possible.
Generalization
- this is the presentation in verbal form of the results of the comparison process.
Generalization is formed in preschool age as the identification and fixation of a common feature of two or more objects. A generalization is well understood by a child if it is the result of an activity carried out by him independently, for example, classification: these are all big, these are all small; these are all red, these are all blue; these all fly, these all run, etc.
All of the above examples of comparisons and classifications ended with generalizations. For preschoolers, empirical types of generalization are possible, that is, generalization of the results of their activities. To lead children to this kind of generalization, the adult organizes work on the task accordingly: selects objects of activity, asks questions in a specially designed sequence to lead the child to the desired generalization. When formulating a generalization, you should help the child construct it correctly, use the necessary terms and verbiage.
Here are examples of generalization tasks for children five to seven years old.

Exercise 13
Material: set of six figures of different shapes.

Assignment: “One of these figures is extra. Find it. (Figure 4.).” Children of this age are unfamiliar with the concept of a bulge, but they usually always point to this shape. They can explain it like this: “Her corner went inward.” This explanation is quite suitable. “How are all the other figures similar? (They have 4 corners, these are quadrilaterals.).”
When selecting material for a task, an adult must ensure that the child does not end up with a set that focuses the child on unimportant features of objects, which will encourage incorrect generalizations. It should be remembered that when making empirical generalizations, the child relies on external visible signs of objects, which does not always help to correctly reveal their essence and define the concept.
For example, in exercise 14, figure 4, in general, is also a quadrilateral, but non-convex. A child will become acquainted with figures of this kind only in the ninth grade of high school, where the definition of the concept “convex flat figure” is formulated in a geometry textbook. In this case, the first part of the task was focused on the operation of comparing and identifying a figure that differs in external shape from other figures in a given group. But the generalization is made based on a group of figures with characteristic features, frequently occurring quadrangles. If a child becomes interested in figure 4, an adult can note that it is also a quadrangle, but of an unusual shape. Forming in children the ability to independently make generalizations is extremely important from a general developmental point of view.
Next, we give an example of several interrelated exercises (tasks) of a logical and constructive nature on the formation of an idea of a triangle for five-year-old children. For modeling constructive activities, children use counting sticks, a stencil frame with slots in the shape of geometric shapes, paper, and colored pencils. The adult also uses sticks and figures.

Exercise 14
The purpose of the exercise is to prepare the child for subsequent modeling activities through simple constructive actions, to update counting skills, and to organize attention.

Assignment: “Take from the box as many sticks as I have (two). Place them in front of you the same way (vertically side by side). How many sticks? (Two.) What color sticks do you have (the sticks in the box are of two colors: red and green)? Make them different colors. What color are your sticks? (One is red, one is green.) How many are there together?

Exercise 15
The purpose of the exercise is to organize constructive activities according to the model. Counting exercises, development of imagination, speech activity.
Material: counting sticks of two colors.
Assignment: “Take another stick and put it on top. How many sticks are there? Let’s count. (Three.) What does the figure look like? (Like a gate, the letter “P.”) What words start with “P”?”

Exercise 16
The purpose of the exercise is to develop observation, imagination and speech activity. Formation of the ability to evaluate the quantitative characteristics of a changing structure (without changing the number of elements).
Material: counting sticks of two colors.
Note: the first task of the exercise is also preparatory to the correct perception of the meaning of arithmetic operations.
Assignment: “Move the top stick like this (the adult moves the stick down so that it is in the middle of the vertical sticks). Has the number of sticks changed? Why hasn’t it changed? (The stick has been rearranged, but not removed or added.) What does the figure look like now? ( With the letter "N".) Name the words starting with "N".

Exercise 17
The purpose of the exercise is to develop design skills, imagination, memory and attention.
Material: counting sticks of two colors.
Assignment: “What else can be put together from three sticks? (The child puts together figures and letters. Names them, comes up with words.).”

Exercise 18
The purpose of the exercise is to form an image of a triangle, a primary examination of the triangle model.
Material: counting sticks of two colors, a triangle drawn by an adult.

Task: “Make a figure out of sticks.” If the child does not fold the triangle himself, an adult helps him. “How many sticks were needed for this figure? (Three.) What kind of figure is this? (Triangle.) Why is it called that? (Three angles.).” If the child cannot name the figure, the adult suggests its name and asks the child to explain how he understands it. Next, the adult asks to trace the figure with a finger, count the corners (vertices), touching them with a finger.

Exercise 19
The purpose of the exercise is to consolidate the image of the triangle on the kinesthetic (tactile sensations) and visual level. Recognition of triangles among other figures (volume and stability of perception). Outlining and shading triangles (development of small muscles of the hand).
Note: the task is problematic because the frame used has several triangles and figures similar to them with sharp corners (rhombus, trapezoid). Material: stencil frame with figures of different shapes.
Assignment: “Find a triangle on the frame. Circle it. Color in the triangle along the frame.” The shading is done inside the frame, the brush moves freely, the pencil “knocks” on the frame.

Exercise 20
The purpose of the exercise is to consolidate the visual image of a triangle. Recognition of the desired triangles among other triangles (perceptual accuracy). Development of imagination and attention. Development of fine motor skills.
Assignment: “Look at this drawing: here is a mother cat, a father cat and a kitten. What shapes are they made of? (Circles and triangles.) What triangle is needed for a kitten? For a mother cat? For a father cat? Draw your cat ". Then the child completes the drawings of the remaining cats, focusing on the sample, but independently. The adult draws attention to the fact that the father cat is the tallest. “Place the frame correctly so that the daddy cat turns out to be the tallest.”

Note: this exercise not only helps the child accumulate reserves of images of geometric figures, but also develops spatial thinking, since the figures on the stencil frame are located in different positions, and to find the one you need, you need to recognize it in a different position, and then rotate the frame to find it drawing in the position required by the drawing.
It is obvious that the child’s constructive activity in the process of performing these exercises develops not only the child’s mathematical abilities and logical thinking, but also his attention, imagination, trains motor skills, eye, spatial concepts, accuracy, etc.
Each of the above exercises is aimed at developing logical thinking techniques. For example, exercise 15 teaches the child to compare; exercise 16 - compare and generalize, as well as analyze; Exercise 17 teaches analysis and comparison; exercise 18 - synthesis; exercise 19 - analysis, synthesis and generalization; exercise 20 - actual classification by attribute; exercise 21 teaches comparison, synthesis and elementary seriation.
The logical development of a child also presupposes the formation of the ability to understand and trace the cause-and-effect relationships of phenomena and the ability to build simple conclusions based on cause-and-effect relationships. It is easy to see that when completing all the above examples of tasks and task systems, the child practices these skills, since they are also based on mental actions: analysis, synthesis, generalization, etc.
Thus, two years before school it is possible to have a significant impact on the development of a preschooler’s mathematical abilities. Even if your child does not become an indispensable winner of mathematical Olympiads, he will not have problems with mathematics in elementary school, and if he does not have them in elementary school, then there is every reason to expect that he will not have them in the future.

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

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

The results of the study showed that statements that related to calculus, algebra, geometry and topology activated areas in the parietal, inferotemporal and prefrontal cortices of the brain in mathematicians, but not in the control group. These zones were different from those that were excited in all participants in the experiment during ordinary statements. “Mathematical” areas were activated in ordinary people only if the subjects were asked to perform simple arithmetic operations.

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

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

What is spatial thinking?

To solve a huge number of problems that our civilization poses to us, we need a special type of mental activity - spatial thinking. The term spatial imagination refers to the human ability to clearly imagine three-dimensional objects in detail and color.

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

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

How to develop spatial thinking

Exercises to develop spatial thinking are very useful at any age. At first, many people have difficulty performing them, but over time they gain the ability to solve increasingly complex problems. Such exercises ensure normal functioning of the brain and help avoid many diseases caused by insufficient functioning of neurons in the cerebral cortex.

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

With developed spatial thinking it becomes much more it is easier to read drawings and maps, determine locations and visualize the route to the goal. This is a must-have for orienteering enthusiasts, and will greatly help everyone else in everyday life in the city.

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

There are three types of such operations:

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

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

3D puzzles and origami

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

Repeating actions after someone
Work according to instructions
Folding a figure with partial support according to the instructions
Independent work without relying on the material (can be carried out not immediately, but after several repetitions of the previous stages)

It is important that the student clearly traces each action and remembers it. Instead of puzzles, you can also use a regular construction set.

Divided into two types:

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

Redrawing (copying)

Tasks of this type proceed in increasing complexity:

Simple re-drawing of a figure. The student faces a mock-up/sample of a figure, which he needs to transfer onto paper without changes (the dimensions and appearance must match). Each side of the figure is drawn separately.
Copying with addition. Task: redraw the figure without changes and add to it: 5 cm in length, an additional edge, another figure, etc.
Scalable redrawing. Task: copy a shape changing its size, i.e. draw 2 times larger than the model, 5 times smaller than the sample, decreasing each side by 3 cm, etc.
Copy from view. Task: imagine a three-dimensional figure and draw it from different sides.

Representation

The representation objects will be segments and lines. Tasks can be very diverse, for example:

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

Drawing up drawings and diagrams

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

Game "Guess by touch"

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

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

Game "Fly in a Cage"

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

Rules: two participants present a grid of 9 by 9 squares (graphics cannot be used!). There is a fly in the upper right corner. Taking turns making moves, players move the fly across the squares. You can use movement symbols (right, left, up, down) and the number of cells. For example, a fly moves three squares up. The third participant has a graphical grid diagram and represents each move (each movement of the fly). Next he says “Stop” and the other players must say where they think the fly is at the moment. The winner is the one who correctly named the square where the fly stopped (checked according to the diagram drawn up by the third participant).

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

Graphic exercises

They are performed by eye without the use of any auxiliary objects (ruler, pen, compass, etc.).

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

2. Which of the figures will be able to pass between object A and object B?

Picture from the book by Postalovsky I.Z. “Imaginative thinking training”

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

Picture from the book by Postalovsky I.Z. “Imaginative thinking training”

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

Picture from the book by Postalovsky I.Z. “Imaginative thinking training”

5. Determine where the ball will fall.

Picture from the book by Postalovsky I.Z. “Imaginative thinking training”

Natalia Smetanskaya
Formation of mathematical abilities in older preschoolers

Consultation for parents

Formation of mathematical abilities in older preschoolers

Mathematical development of preschool children age is carried out both as a result of the child’s acquisition of knowledge in everyday life, and through targeted learning in classes formation of elementary mathematical knowledge in kindergarten.

During the learning process, children develop ability perceive the world around us more accurately and completely, highlight the signs of objects and phenomena, reveal their connections, notice properties; are being formed mental actions, methods of mental activity, internal conditions are created for the transition to new forms of memory, thinking and imagination.

There is a reciprocal relationship between learning and development. Education actively contributes to the child's development, but also depends significantly on his level of development.

It is known that mathematics- is a powerful factor in the intellectual development of a child, formation his educational and creative abilities. From efficiency mathematical development of a child in preschool educational success depends on age mathematics in elementary school.

Why is it so difficult for many children? mathematics not only in elementary school, but now, in preparation for educational activities?

In modern primary school educational programs, important importance is attached to the logical component.

The development of a child’s logical thinking involves formation logical techniques of mental activity, as well as the ability to understand and trace the cause-and-effect relationships of phenomena and the ability to build simple conclusions based on cause-and-effect relationships.

However, when training mathematics these skills help a child in class for a very short time mathematicians. The stock of memorized knowledge ends very quickly (in a month or two, and lack of formation own ability to think productively very quickly leads to the emergence of “problems with mathematics".

At the same time, a child with developed logical thinking always has a greater chance of being successful in mathematics, even if he was not previously taught the elements of the school curriculum (counting, calculations, etc.).

The school curriculum is structured in such a way that already in the first lessons the child must use the skills to compare, classify, analyze and generalize the results of his activities.

Development of logical thinking

Logical thinking is being formed, based on the figurative and is the highest stage of development of children's thinking.

Achieving this stage is an active and complex process, since the full development of logical thinking requires not only high activity of mental activity, but also generalized knowledge about the general and essential features of objects and phenomena.

At approximately 14 years of age, the child reaches the stage formal logical operations when his thinking acquires features characteristic of the mental activity of adults. The development of logical thinking should begin in preschool childhood. For example, at the age of 5-7 years, a child is already able to master at an elementary level such techniques of logical thinking as comparison, generalization, classification, systematization and semantic correlation. In the first stages formation these techniques should be carried out based on visual, specific material and, as it were, with the participation of visual-figurative thinking.

How to teach a child to compare?

Comparison is a technique aimed at establishing signs of similarity and difference between objects and phenomena.

By the age of 5-6, a child usually already knows how to compare different objects with each other, but does this, as a rule, on the basis of only a few signs (for example, colors, forms, magnitude and some others). In addition, the selection of these features is often random and does not involve a comprehensive analysis of the object.

Children 6 years old usually identify only two or three properties in an object, while there are an infinite number of them. In order for a child to be able to see this many properties, he must learn to analyze an object from different sides, compare this object with another object that has different properties. By selecting objects for comparison in advance, you can gradually teach the child to see in them qualities that were previously hidden from him. At the same time, to master this skill well means to learn not only to identify the properties of an object, but also to name them.

When the child has learned to identify properties and compare one object with another, one should begin formation the ability to identify common and distinctive features of objects. First of all, you need to teach the ability to conduct a comparative analysis of the selected properties. Then you should move on to general properties. In this case, it is first important to teach the child to see common properties in two objects, and then in several.

You can try to show with simple examples how the concepts of “general” attribute and “essential” attribute relate to each other. It is important to draw the child’s attention to the fact that a “general” feature is not always “essential,” but “essential” is always “general.” For example, show your child two objects, where their “common” but “insignificant” feature is color, and their “common” and “essential” feature is form.

The ability to find essential features of an object is one of the important prerequisites for mastering the technique of generalization.

Publications on the topic:

The use of didactic games in the formation of elementary mathematical concepts of older preschoolers Prepared by: Antonets E.V. “Game is the spark that ignites the flame of inquisitiveness and curiosity” V.A. Sukhomlinsky Introduction Kindergarten.

“Development of mathematical abilities of preschoolers through games by V.V. Voskobovich.” Presentation of experience Slide 1. Everyone knows that for children, and especially for preschoolers, the best form of learning is learning through play. Very important.

“Development of mathematical abilities in preschoolers” Logical thinking is formed on the basis of figurative thinking and is the highest stage of development of children's thinking. Reaching this stage is a long process.

Formation and development of mathematical abilities, development of logical thinking in preschool children The development of science and technology, universal computerization determine the increasing role of mathematical training of the younger generation. Mathematics.

Explanatory note The relevance of the program lies in the fact that the mathematical development of preschool children is one of.

Consultation for parents: “Development of mathematical abilities in preschoolers through gaming activities” Development of mental abilities.

Pedagogical Council No. 4 “Formation of mathematical abilities: ways and forms” Goals: To increase the level of knowledge of teachers on the FEMP methodology; Master the methods of developing mental activity in children in the classroom.

Self-education project “Formation of elementary mathematical concepts in older preschoolers” Stages of development Implementation deadlines Study literature on this topic September Develop a card index of didactic games November Create a center.

The development of mathematical abilities in preschool children begins... Conduct a diagnosis of a preschooler in order to select an individual...

Mathematical ability is the ability to think logically. Is it possible to develop mathematical abilities in preschool children? Yes, it's possible. A person is born with an underdeveloped left hemisphere of the brain. It is responsible for logic and is activated gradually, along with the acquisition of new skills. The success of this process largely depends on the baby’s environment. With the right approach, you can achieve good results in the development of his intellect, and therefore his mathematical abilities.

Modern theories and technologies of mathematical development of preschool children suggest:

formation of elementary mathematical concepts in preschoolers;
development of their logical thinking;
use of modern teaching tools and methods.

It is advisable to first diagnose the development of each preschooler in order to select an individual educational program for him.

Mathematical representations

The development of mathematical abilities in preschool children begins with their immersion in a mathematical environment. In order to later feel comfortable among mathematical formulas and problems, they must in preschool age;

find out what a figure and a number are;
learn ordinal and quantitative calculations;
learn to add and subtract within tens;
find out what the shape of an object and volume are;
learn to measure the width, height and length of objects;
distinguish between temporal concepts “earlier”, “later”, “today”, “tomorrow”, etc.;
navigate in space, understanding the concepts of “further”, “closer”, “ahead”, “behind”, etc.;
be able to compare: “narrower - wider”, “lower - higher”, “less - more”.

Don't be scared! Mathematical concepts can be mastered at home, casually, in a playful way. How to do it?

Count objects out loud whenever possible or involve your child in doing so. (How many flowers do we have in the vase?, How many plates do we need to put?) Ask your child to follow your instructions: “Please bring me two pencils.”

Thematic material:

Are you walking down the street together? Count to ten and back: in a duet, alternately, then let him count alone.

Teach your child to find the next and previous numbers. (Do you know which number is greater than 3 and less than 5?)

Help him understand addition and subtraction operations. In elementary school, there are children who find it difficult to solve problems because they do not understand the meaning of these mathematical operations. If in one problem the boxes were folded, then in all other problems about boxes these students try to fold them, regardless of the conditions of the problem. Prepare your child before school. Take candy, apples, cups and use a clear example to explain to him what addition means and what subtraction means.

Teach him to compare objects. (Look, a magpie! Is it bigger than a sparrow or smaller?) Draw his attention to the fact that there can be different numbers of objects. (There are a lot of apples and few pears in the vase. What can you do to make the fruits equal?)

Introduce your child to scales. It's great if you have a mechanical kitchen scale with weights. Let the child weigh the apple, an empty mug, or a mug of water.

Explain how to tell the time using a clock with hands.

Place toys on the table. Teach your child to distinguish which toy is closer to him, which is further away, which is in between.

Draw a quadrilateral, triangle, circle, oval. Let him try to explain how the first two figures differ from the second two. Show him where the angle is in the triangle. Count the angles, and the child himself will guess why the triangle has such a name.

Teach your preschooler easily, unobtrusively, and he will become friends with mathematics.

Formation of logical thinking

To successfully master mathematical science, you must be able to perform operations on given objects: find similarities or differences, regroup them according to a given criterion. Start mastering these wisdom before your child enters school. This will help him both in solving mathematical problems and in everyday life.

Techniques for developing mathematical abilities in preschool children:

The ability to identify an object or group of objects based on a given characteristic (analysis).
Bringing together some elements, properties or characteristics into a single whole (synthesis).
Arranging any objects in ascending or descending order according to a given criterion.
Comparison with the aim of finding similarities or differences between objects (comparison).
Distribution of objects into groups by name, color, size, shape, etc. (classification).
Conclusion, comparison result (generalization). This technique is given special significance.

Analysis tasks for children 5-7 years old

Mathematical development of preschool children with the help of simple exercises.

Exercise 1

In Figure 1, find the extra figure. (This is a red square)

Picture 1

Task 2

In Figure 1, distribute the circles into two groups. Explain your decision. (You can distribute by color, or by size).

Task 3

In Figure 2, show three triangles. (Two small and one along the outer contour)

Synthesis problems

Combining elements and aspects of an object into a single system.

Exercise 1

Do what I do. In this task, an adult and a child construct identical objects. The child repeats the actions of the adult.

Task 2

Repeat the same from memory.

Task 3

Build a tower, design a scooter, etc. This is a creative activity. It is done without a sample.

Figure 2

Organizing tasks

Collecting and sorting items from smallest to largest or vice versa.

Exercise 1

Build the nesting dolls according to height, starting with the smallest one.

Task 2

Put on the pyramid rings, starting from the largest to the smallest.

Analysis tasks for children 2-4 years old

Performed with toys or pictures.

Exercise 1

Choose a blue car. Choose a car, but not a blue one.

Task 2

Select all the small cars. Select all the cars, but not the small ones.

Task 3

Choose the little blue car.

Comparison tasks for children 2-4 years old

The difference and similarity of elements according to some characteristic.

Exercise 1

What's round like a ball? (Apple, orange)

Task 2

Play with your child: first you describe the characteristics of the object, and the child guesses, then vice versa.

Example: Small, gray, can fly. Who is this? (Sparrow)

Comparison problems for older children

Same as the previous task, only for older children.

Exercise 1

In Figure 3, find a figure similar to the sun. (Circle)

Task 2

In Figure 3, show all the red shapes. What number corresponds to them? (Number 2)

Figure 3

Task 3

What else corresponds to the number 2 in Figure 3? (Number of yellow pieces)

A task on the ability to classify objects for children 2-4 years old

The adult names the animals, and the child says which of them can swim and which cannot. Then the child chooses what to ask about (about fruits, cars, etc.), and the adult answers.

Task for a child 5-7 years old

In Figure 3, select the polygons into a separate group and divide them by color. (All shapes except the circle. The square and triangle will be in one group, and the rectangle in the other)

Generalization task

Figure 4 shows geometric shapes. What do they have in common? (These are quadrilaterals)

Figure 4

Entertaining games and tasks

Modern construction sets - puzzles - have been invented for preschoolers to play independently. These are flat construction sets “Pythagoras”, “Magic Circle” and others, as well as volumetric construction sets “Snake”, “Magic Balls”, “Pyramid”. All of them teach the child to think geometrically.

Fun tasks like:

There were 3 pears on the table. One was cut in half. How many pears are left on the table? (3)
A team of dogs ran 4 km. How far did each dog run? (4)

By offering your child such tasks, you will teach him to listen carefully to the condition and find the catch. The child will understand that mathematics can be very interesting.

Read and tell your child something from the history of mathematics: how ancient people believed, who invented the numbers we use, where geometric figures came from...

Don't neglect simple riddles. They also teach you to think.

Tools to help parents of young mathematicians

First of all, this is visual didactic material:

images of objects drawn on cards;
household items, toys, etc.;
cards with numbers and arithmetic signs, geometric figures;
magnetic board;
regular and hourglass;
scales;
counting sticks.

Buy educational games, construction sets, puzzles, counting materials, checkers and chess.

Everyone knows board games with dice, chips and a playing field. This is a useful and interesting game. She teaches the child to count and perform tasks carefully. In addition, the whole family can take part in it.

Buy children's educational books with good illustrations.

Encourage your child's curiosity.
Look for answers to his questions together. Reason with him.
Don't complain about lack of time. Talk and play while walking together, before bed.
Trusting relationships between an adult and a preschooler are of great importance. Never laugh at your child's mistakes.
Do not overload your baby with activities. This will harm his health and discourage him from learning.
Pay attention not only to the development of mathematical abilities in preschool children, but also to their spiritual and physical development. Only then will your child become a harmonious personality.

Summary: Development of mathematical abilities in children. More than twenty exercises for the development of logical and mathematical thinking in a child. Training in the ability to compare, classify, analyze and summarize the results of one’s activities.

At the same time, a child with developed logical thinking always has a greater chance of being successful in mathematics, even if he was not previously taught the elements of the school curriculum (counting, calculations, etc.). It is no coincidence that in recent years, many schools working on developmental programs have conducted interviews with children entering first grade, the main content of which is questions and tasks of a logical, and not just arithmetic, nature. Is this approach to selecting children for education logical? Yes, it is natural, since the mathematics textbooks of these systems are structured in such a way that already in the first lessons the child must use the ability to compare, classify, analyze and generalize the results of his activities.

It is most advisable to develop the logical thinking of a preschooler in line with mathematical development. The process of a child’s assimilation of knowledge in this area is further enhanced by the use of tasks that actively develop fine motor skills, that is, tasks of a logical and constructive nature. In addition, there are various methods of mental action that help enhance the effectiveness of using logical-constructive tasks.

Seriation is the construction of ordered increasing or decreasing series based on a selected characteristic. A classic example of seriation: nesting dolls, pyramids, insert bowls, etc.

Analysis is the selection of the properties of an object, or the selection of an object from a group, or the selection of a group of objects according to a certain criterion.

Synthesis is the combination of various elements (signs, properties) into a single whole. In psychology, analysis and synthesis are considered as mutually complementary processes (analysis is carried out through synthesis, and synthesis is carried out through analysis).

1. A task to select an object from a group based on any criterion: “Take the red ball”; “Take the red one, but not the ball”; "Take the ball, but not the red one."

2. A task to select several objects according to the specified criterion: “Choose all the balls”; “Choose round balls, but not balls.”

As an example of organizing activities that develop a child’s ability to analyze and synthesize, we will give several exercises for children five to six years old.

Exercise 1

Material: set of figures - five circles (blue: large and two small, green: large and small), small red square.

Assignment: “Determine which of the figures in this set is extra. (Square.) Explain why. (All the rest are circles.).”

Exercise 2

Material: the same as for Exercise 1, but without the square.
Assignment: “Divide the remaining circles into two groups. Explain why you divided them this way. (By color, by size.).”

Exercise 3

Material: the same and cards with numbers 2 and 3.
Assignment: “What does the number 2 mean on the circles? (Two large circles, two green circles.) The number 3? (Three blue circles, three small circles.).”

Exercise 4

Material: the same didactic set (a set of plastic figures: colored squares, circles and triangles).
Assignment: “Remember what color was the square that we removed? (Red.) Open the box, Didactic set.” Find the red square. What other colors are there squares? Take as many squares as there are circles (see exercises 2, 3). How many squares? (Five.) Can you make one big square out of them? (No.) Add as many squares as needed. How many squares did you add? (Four.) How many are there now? (Nine.)".

The traditional form of tasks for the development of visual analysis are tasks for choosing an “extra” figure (object). Here are a few tasks for children five to six years old.

Exercise 5

Material: drawing of figurines-faces.

Assignment: “One of the figures is different from all the others. Which one? (The fourth one.) How is it different?”

Exercise 6

Material: drawing of human figures.

Task: “Among these figures there is an extra one. Find it. (Fifth figure.) Why is it extra?”

A more complex form of such a task is the task of isolating a figure from a composition formed by superimposing some forms on others. Such tasks can be offered to children five to seven years old.

Exercise 7

Material: drawing of two small triangles forming one large one.

Assignment: “There are three triangles hidden in this picture. Find and show them.”

Note. You need to help the child show the triangles correctly (circle with a small pointer or finger).

As preparatory tasks, it is useful to use tasks that require the child to synthesize compositions from geometric shapes at the material level (from material material).

Exercise 8

Material: 4 identical triangles.

Psychologically, the ability to synthesize is formed in a child earlier than the ability to analyze. That is, if a child knows how it was assembled (folded, designed), it is easier for him to analyze and identify its component parts. That is why such serious importance is given in preschool age to activities that actively form synthesis - construction.

At first, this is a patterned activity, that is, performing tasks of the “do as I do” type. At first, the child learns to reproduce the object, repeating the entire construction process after the adult; then - repeating the process of construction from memory, and finally moves on to the third stage: independently restores the method of constructing a ready-made object (tasks like “make the same one”). The fourth stage of tasks of this kind is creative: “build a tall house”, “build a garage for this car”, “build a rooster”. The tasks are given without a sample, the child works according to the idea, but must adhere to the given parameters: a garage specifically for this car.

For construction, any mosaics, construction sets, cubes, cut-out pictures are used that are suitable for this age and make the child want to tinker with them. An adult plays the role of an unobtrusive assistant; his goal is to help bring the work to completion, that is, until the intended or required whole object is obtained.

Comparison is a logical method of mental action that requires identifying similarities and differences between the characteristics of an object (object, phenomenon, group of objects).

Performing a comparison requires the ability to identify some features of an object (or group of objects) and abstract from others. To highlight various features of an object, you can use the game “Find it using the specified features”: “Which (of these objects) is big yellow? (Ball and bear.) What is big yellow and round? (Ball),” etc.

The child should use the role of the leader as often as the answerer, this will prepare him for the next stage - the ability to answer the question: “What can you tell about him? (The watermelon is large, round, green. The sun is round, yellow, hot.)” . Or: “Who will tell you more about this? (The ribbon is long, blue, shiny, silk.).” Or: “What is this: white, cold, crumbly?” etc.

Types of comparison tasks:

1. Tasks to separate a group of objects according to some criteria (large and small, red and blue, etc.).

2. All games of the “Find the same” type. For a child two to four years old, the set of characteristics by which similarities are sought should be clearly defined. For older children, exercises are offered in which the number and nature of similarities can vary widely.

Let us give examples of tasks for children five to six years old, in which the child is required to compare the same objects according to various criteria.

Exercise 9

Material: images of two apples, a small yellow one and a large red one. The child has a set of shapes: a blue triangle, a red square, a small green circle, a large yellow circle, a red triangle, a yellow square.

Exercise 10

Material: the same set of cards with numbers from 1 to 9.
Assignment: “Put all the yellow figures to the right. What number fits this group? Why 2? (Two figures.) What other group can be matched to this number? (A blue and red triangle - there are two of them; two red figures, two circles; two square - all options are analyzed.)". The child makes groups, uses a stencil frame to sketch and paint them, then signs the number 2 under each group. “Take all the blue figures. How many are there? (One.) How many colors are there in total? (Four.) Figures? (Six.) ".

The ability to identify the characteristics of an object and, focusing on them, to compare objects is universal, applicable to any class of objects. Once formed and well developed, this skill will then be transferred by the child to any situations requiring its use.

An indicator of the maturity of the comparison technique will be the child’s ability to independently apply it in activities without special instructions from an adult on the signs by which objects need to be compared.

Classification is the division of a set into groups according to some criterion, which is called the basis of classification. Classification can be carried out either according to a given basis, or with the task of searching for the basis itself (this option is more often used with children six to seven years old, as it requires a certain level of formation of the operations of analysis, comparison and generalization).

It should be taken into account that when classifying a set, the resulting subsets should not intersect in pairs and the union of all subsets should form this set. In other words, each object must be included in only one set, and with a correctly defined basis for classification, not a single object will remain outside the groups defined by this basis.

Classification with preschool children can be carried out:

By name (cups and plates, shells and pebbles, skittles and balls, etc.);
- by size (large balls in one group, small ones in another, long pencils in one box, short pencils in another, etc.);
- by color (this box has red buttons, this one has green buttons);
- in shape (this box contains squares, and this box contains circles; this box contains cubes, this box contains bricks, etc.);
- based on other non-mathematical characteristics: what can and cannot be eaten; who flies, who runs, who swims; who lives in the house and who in the forest; what happens in summer and what happens in winter; what grows in the garden and what in the forest, etc.

All of the examples listed above are classifications based on a given basis: the adult communicates it to the child, and the child carries out the division. In another case, classification is performed on a basis determined by the child independently. Here, the adult sets the number of groups into which many objects (objects) should be divided, and the child independently looks for the appropriate basis. Moreover, such a basis can be determined in more than one way.

For example, tasks for children five to seven years old.

Exercise 11

Material: several circles of the same size, but different colors (two colors).
Assignment: “Divide the circles into two groups. By what criteria can this be done? (By color.).”

Exercise 12

Material: several squares of the same colors are added to the previous set (two colors). The figures are mixed.
Assignment: “Try to divide the figures into two groups again.” There are two options for separation: by shape and by color. An adult helps the child clarify the wording. The child usually says: “These are circles, these are squares.” The adult generalizes: “So, they divided it according to shape.”

In exercise 11, the classification was unambiguously specified by the corresponding set of figures on only one basis, and in exercise 12, the addition of a set of figures was deliberately made in such a way that classification on two different grounds became possible.

Generalization is the presentation in verbal form of the results of the comparison process.

Generalization is formed in preschool age as the identification and fixation of a common feature of two or more objects. A generalization is well understood by a child if it is the result of an activity carried out by him independently, for example, classification: these are all big, these are all small; these are all red, these are all blue; these all fly, these all run, etc.

All of the above examples of comparisons and classifications ended with generalizations. For preschoolers, empirical types of generalization are possible, that is, generalization of the results of their activities. To lead children to this kind of generalization, the adult organizes work on the task accordingly: selects objects of activity, asks questions in a specially designed sequence to lead the child to the desired generalization. When formulating a generalization, you should help the child construct it correctly, use the necessary terms and verbiage.

Here are examples of generalization tasks for children five to seven years old.

Exercise 14

Material: set of six figures of different shapes.

When selecting material for a task, an adult must ensure that the child does not end up with a set that focuses the child on unimportant features of objects, which will encourage incorrect generalizations. It should be remembered that when making empirical generalizations, the child relies on external visible signs of objects, which does not always help to correctly reveal their essence and define the concept.

For example, in exercise 14, figure 4, in general, is also a quadrilateral, but non-convex. A child will become acquainted with figures of this kind only in the ninth grade of high school, where the definition of the concept “convex flat figure” is formulated in a geometry textbook. In this case, the first part of the task was focused on the operation of comparing and identifying a figure that differs in external shape from other figures in a given group. But the generalization is made based on a group of figures with characteristic features, frequently occurring quadrangles. If a child becomes interested in figure 4, an adult can note that it is also a quadrangle, but of an unusual shape. Forming in children the ability to independently make generalizations is extremely important from a general developmental point of view.

Next, we give an example of several interrelated exercises (tasks) of a logical and constructive nature on the formation of an idea of a triangle for five-year-old children. For modeling constructive activities, children use counting sticks, a stencil frame with slots in the shape of geometric shapes, paper, and colored pencils. The adult also uses sticks and figures.

Exercise 15

The purpose of the exercise is to prepare the child for subsequent modeling activities through simple constructive actions, to update counting skills, and to organize attention.

Assignment: “Take from the box as many sticks as I have (two). Place them in front of you the same way (vertically side by side). How many sticks? (Two.) What color sticks do you have (the sticks in the box are of two colors: red and green)? Make them different colors. What color are your sticks? (One is red, one is green.) How many are there together?

Exercise 16

The purpose of the exercise is to organize constructive activities according to the model. Counting exercises, development of imagination, speech activity.

Material: counting sticks of two colors.
Assignment: “Take another stick and put it on top. How many sticks are there? Let’s count. (Three.) What does the figure look like? (Like a gate, the letter “P.”) What words start with “P”?”

Exercise 17

The purpose of the exercise is to develop observation, imagination and speech activity. Formation of the ability to evaluate the quantitative characteristics of a changing structure (without changing the number of elements).

Material: counting sticks of two colors.
Note: the first task of the exercise is also preparatory to the correct perception of the meaning of arithmetic operations. Assignment: “Move the top stick like this (the adult moves the stick down so that it is in the middle of the vertical sticks). Has the number of sticks changed? Why hasn’t it changed? (The stick has been rearranged, but not removed or added.) What does the figure look like now? ( Starting with the letter "N".) Name the words starting with "N".

Exercise 18

The purpose of the exercise is to develop design skills, imagination, memory and attention.

Material: counting sticks of two colors.
Assignment: “What else can be put together from three sticks? (The child puts together figures and letters. Names them, comes up with words.).”

Exercise 19

The purpose of the exercise is to form an image of a triangle, a primary examination of the triangle model.

Material: counting sticks of two colors, a triangle drawn by an adult.

Exercise 20

The purpose of the exercise is to consolidate the image of the triangle on the kinesthetic (tactile sensations) and visual level. Recognition of triangles among other figures (volume and stability of perception). Outlining and shading triangles (development of small muscles of the hand).

Note: the task is problematic because the frame used has several triangles and figures similar to them with sharp corners (rhombus, trapezoid).

Material: stencil frame with figures of different shapes.
Assignment: “Find a triangle on the frame. Circle it. Color in the triangle along the frame.” The shading is done inside the frame, the brush moves freely, the pencil “knocks” on the frame.

Exercise 21

The purpose of the exercise is to consolidate the visual image of a triangle. Recognition of the desired triangles among other triangles (perceptual accuracy). Development of imagination and attention. Development of fine motor skills.

Assignment: “Look at this drawing: here is a mother cat, a father cat and a kitten. What shapes are they made of? (Circles and triangles.) What triangle is needed for a kitten? For a mother cat? For a father cat? Draw your cat ". Then the child completes the drawings of the remaining cats, focusing on the sample, but independently. The adult draws attention to the fact that the father cat is the tallest. “Place the frame correctly so that the daddy cat turns out to be the tallest.”

It is obvious that the child’s constructive activity in the process of performing these exercises develops not only the child’s mathematical abilities and logical thinking, but also his attention, imagination, trains motor skills, eye, spatial concepts, accuracy, etc.

Each of the above exercises is aimed at developing logical thinking techniques. For example, exercise 15 teaches the child to compare; exercise 16 - compare and generalize, as well as analyze; Exercise 17 teaches analysis and comparison; exercise 18 - synthesis; exercise 19 - analysis, synthesis and generalization; exercise 20 - actual classification by attribute; exercise 21 teaches comparison, synthesis and elementary seriation.

The logical development of a child also presupposes the formation of the ability to understand and trace the cause-and-effect relationships of phenomena and the ability to build simple conclusions based on cause-and-effect relationships. It is easy to see that when completing all the above examples of tasks and task systems, the child practices these skills, since they are also based on mental actions: analysis, synthesis, generalization, etc.

Thus, two years before school it is possible to have a significant impact on the development of a preschooler’s mathematical abilities. Even if your child does not become an indispensable winner of mathematical Olympiads, he will not have problems with mathematics in elementary school, and if he does not have them in elementary school, then there is every reason to expect that he will not have them in the future.

Dear parents and teachers! If you do not yet know about the existence of the games-for-kids.ru website, then we highly recommend that you visit it right now. This is the best site on the Internet with an incredibly large number of free educational games and exercises for children. Here you will find games to develop thinking, attention, memory in preschoolers, exercises for learning to count and read, crafts, drawing lessons and much more. All tasks were developed with the participation of experienced child psychologists and preschool teachers. If you are interested in the topic “Teaching numeracy and mathematics to preschoolers,” be sure to look at the special section of the site “Entertaining mathematics for preschoolers.” Here you will find computer and paper versions of tasks for teaching numeracy, familiarity with numbers and the development of logical and mathematical abilities in preschool children. Here are screenshots of some tasks for your reference: