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.
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.”
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.
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 thinking techniques 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 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.
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 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.
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 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).
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 - 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 characteristics of an object, you can use the game “Find it using the specified characteristics”: “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.
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 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.
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 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.
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 corners.).” 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 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.”
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.
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Mathematics ability is one of the talents given by nature, which manifests itself from an early age and is directly related to the development of creative potential and the desire to understand the world around the child. But why is learning maths so difficult for some children, and can these abilities be improved?
The opinion that only gifted children can master mathematics is wrong. Mathematical abilities, like other talents, are the result of a child’s harmonious development, and must begin from a very early age.
In the modern computer world with its digital technologies, the ability to “make friends” with numbers is extremely necessary. Many professions are based on mathematics, which develops thinking and is one of the most important factors influencing the intellectual growth of children. This exact science, whose role in the upbringing and education of a child is undeniable, develops logic, teaches one to think consistently, determine the similarities, connections and differences of objects and phenomena, makes the child’s mind fast, attentive and flexible.
For mathematics classes for children five to seven years old to be effective, a serious approach is needed, and the first step is to diagnose their knowledge and skills - to assess at what level the child’s logical thinking and basic mathematical concepts are.
Diagnostics of mathematical abilities of children 5-7 years old using the method of Beloshistaya A.V.
If a child with a mathematical mind has mastered mental calculation at an early age, this is not yet a basis for one hundred percent confidence in his future as a mathematical genius. Mental arithmetic skills are only a small element of an exact science and are far from the most complex. A child’s ability for mathematics is evidenced by a special way of thinking, which is characterized by logic and abstract thinking, understanding of diagrams, tables and formulas, the ability to analyze, and the ability to see figures in space (volume).
To determine whether children from primary preschool (4-5 years old) to primary school age have these abilities, there is an effective diagnostic system created by Doctor of Pedagogical Sciences Anna Vitalievna Beloshista. It is based on the creation by a teacher or parent of certain situations in which the child must apply this or that skill.
Diagnostic stages:
- Testing a 5-6 year old child for analysis and synthesis skills. At this stage, you can evaluate how the child can compare objects of different shapes, separate them and generalize them according to certain characteristics.
- Testing figurative analysis skills in children aged 5-6 years.
- Testing the ability to analyze and synthesize information, the results of which reveal the ability of a preschooler (first grader) to determine the shapes of various figures and notice them in complex pictures with figures superimposed on each other.
- Testing to determine the child’s understanding of the basic concepts of mathematics - we are talking about the concepts of “more” and “less”, ordinal counting, the shape of the simplest geometric figures.
The first two stages of such diagnostics are carried out at the beginning of the school year, the rest - at the end, which makes it possible to assess the dynamics of the child’s mathematical development.
The material used for testing should be understandable and interesting for children - age-appropriate, bright and with pictures.
Diagnosis of a child’s mathematical abilities using the method of Kolesnikova E.V.
Elena Vladimirovna has created many educational and methodological aids for the development of mathematical abilities in preschoolers. Her method of testing children 6 and 7 years old has become widespread among teachers and parents in different countries and meets the requirements of the Federal State Educational Standard (FSES) (Russia).
Thanks to Kolesnikova’s method, it is possible to determine as accurately as possible the level of key indicators of the development of children’s mathematical skills, find out their readiness for school, and identify weaknesses in order to fill gaps in a timely manner. This diagnosis helps to find ways to improve the child’s mathematical abilities.
Developing a child’s mathematical abilities: tips for parents
It is better to introduce a child to any science, even something as serious as mathematics, in a playful way - this will be the best teaching method that parents should choose. Listen to the words of famous scientist Albert Einstein: “Play is the highest form of exploration.” After all, with the help of the game you can get amazing results:
– knowledge of yourself and the world around you;
– formation of a mathematical knowledge base;
– development of thinking:
– personality formation;
– development of communication skills.
You can use various games:
- Counting sticks. Thanks to them, the baby remembers the shapes of objects, develops his attention, memory, ingenuity, and develops comparison skills and perseverance.
- Puzzles that develop logic and ingenuity, attention and memory. Logic puzzles help children learn better spatial awareness, thoughtful planning, simple and backward counting, and ordinal counting.
- Mathematical riddles are a great way to develop the basic aspects of thinking: logic, analysis and synthesis, comparison and generalization. While searching for a solution, children learn to draw their own conclusions, cope with difficulties and defend their point of view.
The development of mathematical abilities through play creates learning excitement, adds vivid emotions, and helps the child fall in love with the subject of study that interests him. It is also worth noting that gaming activities also contribute to the development of creative abilities.
The role of fairy tales in the development of mathematical abilities of preschool children
Children's memory has its own characteristics: it records vivid emotional moments, that is, the child remembers information that is associated with surprise, joy, and admiration. And learning “from under pressure” is an extremely ineffective way. In the search for effective teaching methods, adults should remember such a simple and ordinary element as a fairy tale. A fairy tale is one of the first means of introducing a child to the world around him.
For children, fairy tales and reality are closely connected, magical characters are real and alive. Thanks to fairy tales, a child’s speech, imagination and ingenuity develop; they give the concept of goodness, honesty, broaden horizons, and also provide an opportunity to develop mathematical skills.
For example, in the fairy tale “The Three Bears,” the child unobtrusively gets acquainted with counting to three, the concepts of “small,” “medium,” and “large.” “Turnip”, “Teremok”, “The Little Goat Who Could Count to 10”, “The Wolf and the Seven Little Kids” - in these tales you can learn simple and ordinal counting.
When discussing fairy-tale characters, you can invite your child to compare them in width and height, to “hide” them in geometric shapes that are suitable in size or shape, which contributes to the development of abstract thinking.
You can use fairy tales not only at home, but also in school. Children really love lessons based on the plots of their favorite fairy tales, using riddles, labyrinths, and fingering. Such classes will become a real adventure in which the kids will take personal part, which means the material will be learned better. The main thing is to involve children in the game process and arouse their interest.
Mathematical abilities in children are classified as innate talents. Children take their first steps towards learning mathematics in preschool age. Mathematical thinking is closely related to creativity and the level of development of mental abilities. But not all children easily master exact science. Why is this happening? Is it possible to develop mathematical abilities in a child?
It is wrong to think that children's minds are limited and cannot understand mathematics. Like any other natural gift, mathematical abilities will open only as a result of correct, systematic development. This means that in teaching children it is not only possible, but it is very important from early preschool age to pay attention to the development of these inclinations.
It is all the more important to do this because a new generation of children will look for their calling in a world ruled by digital technologies. Any profession is related to mathematics, even the most humanitarian or creative ones. Thanks to mathematics, a child learns holistic and quick thinking, analysis, and makes informed conclusions.
How to develop the mathematical abilities of a child under 7 years old? The results depend not only on the age at which you started training, but also on the methods chosen. Diagnosis of the mathematical abilities of children aged 5, 6 and 7 years will help determine the course and load in teaching preschoolers. It will allow you to assess the presence and level of development of children’s mathematical thinking and basic knowledge in mathematics.
Diagnosis of mathematical abilities in a child according to A. V. Beloshistaya
If a child quickly learns numbers and learns to count, this does not mean that a mathematician is growing up in the family. Mental arithmetic is the simplest topic in exact science. Mathematical abilities are judged by mental qualities such as:
- analysis and logic;
- ability to read diagrams and formulas;
- understanding of abstract concepts;
- the ability to accurately perceive the shapes of objects in space.
Doctor of Sciences V. A. Beloshistaya has been working on the issue of diagnosing and developing mathematical abilities in preschool children (younger - 5 and 6 years old, older - 6 and 7 years old). Her method of assessing children's mathematical talents has several courses:
- Diagnostics for children 5-6 years old. It is carried out in two stages in order to assess the ability of synthesis and analysis. Individual testing. Based on its results, one can judge whether the child understands the difference between figures and shapes of objects, whether he can divide things into groups according to an independently chosen criterion, and whether he has the skills of generalization and comparison.
- Diagnostics for figurative analysis in preschoolers 5 and 6 years old.
- Testing older preschoolers (5-7 years old) to determine the level of development of analysis and synthesis skills. In the task, children need to identify specific figures in complex images from many intersecting figures.
- Diagnostics of basic mathematical concepts: counting, comparison, knowledge of the concepts of “more” and “less”, “wider” and “narrower”, etc.
For a more complete picture of the development of mathematical abilities in preschoolers in dynamics, the first two types of diagnostics are carried out at the beginning of the school year, and the second two - in May (at the end of the year).
The material at hand for tests should be bright, easy to use, and understandable for the child. Different tasks are used for each age.
Method of Kolesnikova E.V. to diagnose a child’s mathematical abilities
The well-known teacher and scientist E.V. Kolesnikova in Russia has more than a dozen books and manuals on the preparation of primary and secondary preschoolers. One of the main courses of her work is diagnosing mathematical abilities in children 6-7 years old. Kolesnikova’s method has been approved by the Federal State Educational Standard, as one that meets the standards of pedagogical diagnostics in Russia. However, the method is successfully used to assess the level of mathematical abilities of preschoolers in different countries.
The purpose of the methodology: assessing the child’s level of readiness for school, searching for gaps in the study of basic mathematical knowledge to correct learning deficiencies at the stage of preparation for school. The advantage of the method is the accurate and most complete diagnosis of the child’s knowledge.
Tips for parents on developing their child’s mathematical abilities
Albert Einstein called play the highest form of exploration. When choosing methods for developing children, it is useful for parents to use play activities.
Developing science abilities in children in this way helps:
- better understand the world around us;
- assess your capabilities;
- become sociable;
- train thinking;
- gain basic understanding of mathematics as a science;
- become more confident and independent.
The following games are used in training:
- Counting sticks. With their help, children learn to distinguish the shapes of objects, compare, develop attention, memory, intelligence and perseverance.
- Puzzles. They perfectly develop logic and analytical thinking, teach how to synthesize information, summarize and classify data. That is, mathematical riddles comprehensively develop mathematical intelligence, and also cultivate perseverance and strong-willed qualities that help solve assigned problems despite difficulties.
- Puzzles. They train spatial thinking, develop memory and logic, observation and ingenuity. In solving them, the child learns to calculate his steps and masters counting (simple, ordinal).
Developing math skills through play activities is beneficial for several reasons:
- it is easier for a child to perceive knowledge;
- a positive attitude towards the subject is formed, and therefore internal interest;
- the game provides an opportunity to apply a creative approach to solving problems (develops creative potential);
- the game is interesting, which means the child sees meaning in learning (motivation).
Is it possible to develop the mathematical abilities of preschoolers with the help of fairy tales?
You cannot force anything into a child’s memory - through cramming and many repetitions. If the knowledge is associated with a very real emotion, it will probably settle in the child’s memory for a long time. Therefore, the task of parents is to delight, surprise and delight their little students during the lessons. How to do it? It is unlikely that I will reveal a secret if I say that a fairy tale is ideal for this matter - the first guide in getting to know the peculiarities of the surrounding world, relationships between people.
For children, a fairy tale plot is no less real than the events of real life. Fairy tales develop imagination, speech, flexibility of thinking, create a special vision of the world, and teach good qualities (honesty, kindness, loyalty). Developing mathematical abilities through fairy tales is easy if you show a little imagination:
- It’s fun to learn simple counting with a fairy tale about a little goat who could count to ten, “The Wolf and the Seven Little Goats.”
- Ordinal counting will help you master “Teremok” and even “Turnip”.
- In “Three Bears” the child gets acquainted with the concepts of “big”, “small” and “medium”, and learns to count to three.
Activities with fairy tales can be endlessly changed and complicated. For example, invite your child to compare animals with geometric shapes. Finding similarities between fairy tale characters and figures develops the ability to think abstractly.
It is convenient to develop mathematical abilities with the help of fairy tales, since parents can do this at any time outside of class (at home, on a walk, on a trip). A fairy tale can also become part of the curriculum in kindergarten or school. Based on a plot well known to children, teachers create riddles and labyrinths, using them as the basis for numerical problems and counting rhymes for exercising their fingers. But the most important thing is that children like such activities.
How does mental arithmetic Soroban develop thinking?
By the time they enter school, children should have acquired a relatively wide range of interrelated knowledge about set and number, shape and size, and learn to navigate in space and time.
Practice shows that the difficulties of first-graders are associated, as a rule, with the need to assimilate abstract knowledge, to move from acting with concrete objects and their images to acting with numbers and other abstract concepts. Such a transition requires the child’s developed mental activity. Therefore, in the preparatory group for school, special attention is paid to the development in children of the ability to navigate in some hidden essential mathematical connections, relationships, dependencies: “equal”, “more”, “less”, “whole and part”, dependencies between quantities, dependencies of the measurement result on the size of a measure, etc. Children master ways of establishing various kinds of mathematical connections and relationships, for example, a method of establishing correspondence between elements of sets (practical comparison of elements of sets one to one, using superposition techniques, applications for clarifying relationships of quantities). They begin to understand that the most accurate ways to establish quantitative relationships are by counting objects and measuring quantities. Their counting and measurement skills become quite strong and conscious. The ability to navigate essential mathematical connections and dependencies and mastery of the corresponding actions make it possible to raise the visual-figurative thinking of preschoolers to a new level and create the prerequisites for the development of their mental activity in general. Children learn to count with their eyes alone, silently, they develop an eye and a quick reaction to form.
No less important at this age is the development of mental abilities, independence of thinking, mental operations of analysis, synthesis, comparison, the ability to abstract and generalize, and spatial imagination. Children should develop a strong interest in mathematical knowledge, the ability to use it, and the desire to acquire it independently. The program for the development of elementary mathematical concepts of the preparatory group for school provides for the generalization, systematization, expansion and deepening of the knowledge acquired by children in previous groups. Work on the development of mathematical concepts is mainly carried out in the classroom. How should they be structured to ensure children’s solid learning?
In the pre-school mathematics group, 2 classes are held per week, during the year - 72 classes. Duration of classes: - 30 min.
Structure of classes.
The structure of each lesson is determined by its content: whether it is devoted to learning new things, repeating and consolidating what has been learned, testing children’s knowledge acquisition. The first lesson on a new topic is almost entirely devoted to working on new material. Introduction to new material is organized when children are most productive, i.e. at 3-5 minutes. from the beginning of the lesson, and ends at 15-18 minutes. Repetition of what has been covered is given to 3-4 minutes. at the beginning and 4-8 min. at the end of the lesson. Why is it advisable to organize work this way? Learning new things tires children out, and repeating material gives them some relief. Therefore, where possible, it is useful to repeat the material covered while working on new ones, since it is very important to introduce new knowledge into the system of previously acquired ones. In the second and third lessons on this topic, approximately 50% of the time is devoted to it, and in the second part of the lessons they repeat (or continue to study) the immediately preceding material, in the third part they repeat what the children have already learned. When conducting a lesson, it is important to organically connect its individual parts, ensure the correct distribution of mental load, and alternation of types and forms of organizing educational activities.
Methodological techniques for the formation of elementary mathematical knowledge, by section:
Quantity and count
At the beginning of the school year, it is advisable to check whether all children, and especially those who have come to kindergarten for the first time, can count objects, compare the number of different objects and determine which are more (less) or equal; what method is used to do this: counting, one-to-one correlation, identification by eye or comparison of numbers? Do children know how to compare the numbers of aggregates, distracting from the size of objects and the area they occupy? Sample tasks and questions: “How many big nesting dolls are there?” Count out how many small nesting dolls there are. Find out which squares are more numerous: blue or red. (There are 5 large blue squares and 6 small red ones lying randomly on the table.) Find out which cubes are more: yellow or green.” (There are 2 rows of cubes on the table; 6 yellow ones stand at large intervals from one another, and 7 blue ones stand close to each other.) The test will tell you to what extent the children have mastered counting and what questions should be paid special attention to. A similar test can be repeated after 2-3 months in order to identify the children’s progress in mastering knowledge.
Formation of numbers.
During the first lessons, it is advisable to remind children how the numbers of the second heel are formed. In one lesson, the formation of two numbers is sequentially considered and they are compared with each other. This helps children learn the general principle of forming a subsequent number by adding one to the previous one, as well as obtaining the previous number by removing one from the subsequent one (6 - 1 = 5). The latter is especially important because children are much more difficult to obtain a smaller number, and therefore highlight the inverse relationship.
Children practice counting and counting objects within 10 throughout the school year. They must firmly remember the order of the numerals and be able to correctly correlate the numerals with the items being counted, and understand that the last number named when counting indicates the total number of items in the collection. If children make mistakes when counting, it is necessary to show and explain their actions. By the time children enter school, they should have developed the habit of counting and arranging objects from left to right using their right hand. But, answering the question how many?, children can count objects in any direction: from left to right and from right to left, as well as from top to bottom and from bottom to top. They are convinced that they can count in any direction, but it is important not to miss a single object and not to count a single object twice.
Independence of the number of objects from their size and shape of arrangement.
The formation of the concepts of “equally”, “more”, “less”, conscious and strong numeracy skills involves the use of a large variety of exercises and visual aids. Particular attention is paid to comparing the numbers of many objects of different sizes (long and short, wide and narrow, large and small), differently located and occupying different areas. Children compare collections of objects, for example, groups of circles arranged in different ways: they find cards with a certain number of circles in accordance with the sample, but arranged differently, forming a different figure. Children count the same number of objects as circles on the card, or 1 more (less), etc. Children are encouraged to look for ways to count objects more conveniently and quickly, depending on the nature of their location. Grouping objects according to different criteria (formation of groups of objects). From comparing the numbers of 2 groups of objects that differ in one characteristic, for example, size, we move on to comparing the numbers of groups of objects that differ in 2, 3 characteristics, for example, size, shape, location, etc.
Equality and inequality of numbers of sets.
Children should ensure that any collections containing the same number of elements are denoted by the same number. Exercises in establishing equality between the numbers of sets of different or homogeneous objects that differ in qualitative characteristics are performed in different ways. Children must understand that there can be an equal number of any objects: 3, 4, 5, and 6. Useful exercises require indirect equalization of the number of elements of 2-3 sets, when children are asked to immediately bring the missing number of objects, for example , so many pens and notebooks so that all the students have enough, so many ribbons so that they can tie bows for all the girls.
The book meets federal state requirements for the structure of the basic general education program of preschool education. It presents the planned results of mastering the “Mathematical Steps” program. The methods used for diagnosis make it possible to obtain the required amount of information in the optimal time frame. The tasks proposed in the book are designed to assess a child’s mathematical preparation for school and promptly identify and fill gaps in his mathematical development.
Diagnostics of mathematical abilities of children 6-7 years old. Kolesnikova E.V.
Description of the textbook
Ability to generalize mathematical material
Quantity and count
Connect rectangles with the same number of objects.
Tell me, which rectangles did you connect? Circle the birds that are the most numerous.
Which birds did you circle? Why?
Quantity and count
Color only the math symbols.
Ability to generalize mathematical material
Geometric figures
Draw as many leaves on each branch as there are circles on the left.
How many leaves did you draw on the top branch? Why? On the middle one? Why? On the bottom branch? Why?
Connect each twig with a card that has as many circles as there are leaves on the twig.
Which card did you connect with which branch?
Ability to generalize mathematical material
Write the numbers from 0 to 9 in order in the squares.
Color only the numbers.
Name the numbers you shaded.
Ability to generalize mathematical material
Color only the geometric shapes.
Name the geometric shapes that you shaded. Color only the quadrilaterals.
Name the geometric shapes that you shaded.
Ability to generalize mathematical material
Trace the shapes with the fewest corners.
What shapes did you circle and why? Color in geometric shapes that have no corners.
What geometric shapes did you paint?
Ability to generalize mathematical material
Magnitude
Circle the houses of the same height.
How many houses did you circle and why? Connect trees with trunks of the same thickness.
Which trees did you connect and why?
Ability to generalize mathematical material
Time orientation
Color the pictures of morning
How many pictures did you color and why?
Ability to generalize mathematical material
Listen to an excerpt from P. Bashmakov’s poem “Days of the Week.” Under each picture, write a number indicating what day of the week the girl did.
On Monday I did the laundry, on Tuesday I swept the floor, on Wednesday I baked kalach, all Thursday I looked for the ball,
I washed the cups on Friday, and bought a cake on Saturday. I invited all my girlfriends to my birthday party on Sunday.
Name the days of the week in order.
Ability to generalize mathematical material
Which picture did you connect with which and why?
Ability to generalize mathematical material
Time orientation
Match the clocks that show the same time.
What time does the clock you connected show?
Draw the hands on the clock so that they show the time that is written in the squares below them.
What time does the first clock show? Second? Third? Fourth?
Under each square, write a number corresponding to the number of circles in them.
Name the numbers in the first row, in the second. Write the “greater than” (^or “less than” signs) in the circles
Match each card with the example it matches.
Tell me which card you paired with which example.
Divide the squares into 2, 3, 4, 5 triangles.
Divide the squares into 5, 4, 3, 2 triangles.
Color the triangles so that they are all different colors.
Color in the fish, which consists of the geometric shapes drawn on the right.
Why did you paint over this fish?
Color only those geometric shapes on the right that make up the fish.
What shapes did you paint?
Write the numbers from 1 to 6 in the squares, starting from the largest nesting doll.
Write the numbers from 1 to 6 in the squares, starting from the smallest ball.
Circle the objects to the left of the bear and color the objects to the right of it.
What objects have you painted? What objects did you circle?
Color in the objects to the left of the bear and circle the objects to the right of it.
What items did you circle? What objects did you color?
Draw as many objects on the right as possible from the geometric shapes on the left.
Show with an arrow which floor each funny little person lives on. To find out, you need to solve the example he is holding in his hand.
Write the numbers in the empty squares so that when you add them you get the answer that is written at the top.
Seven children played football. One was called home. He looks out the window and counts: How many friends are playing?
Guess a riddle. Write your answer in the square.
Seven tiny kittens, Everyone eats what they are given, And one asks for sour cream. How many kittens are there?
Guess a riddle. Write your answer in the square.
The hedgehog gave the ducklings Eight leather boots. Which of the guys will answer, How many ducklings were there?
Five crows landed on the roof, two more flew to them. Answer quickly, boldly, How many of them have arrived?
Listen and complete the task from Dunno I made beads from different numbers, And in those circles where there are no numbers, Arrange the minuses and pluses, To get the given answer.
Write greater than or less than signs in the empty squares.
Write in the circle the number indicating the number that the bunny wished for. And he thought of a number that is one less than seven, but one more than five.
Answer the questions. How many ears do two mice have?
How many paws do two cubs have?
How many days are there in a week?
How many parts are there in a day?
How many months are there in a year?
Who is bigger: a small hippopotamus or a big hare?
Which is longer: a snake or a caterpillar?
Can summer come immediately after winter?
What is the name of the fifth day of the week?
Which geometric figure has the fewest angles?
Diagnostics of mathematical abilities of children 6-7 years old.
