Kindergarten counting and cardinality activities

Thus, if the set A is given by the characteristic property P , then this means that it consists of all objects that have this property, and only of them. If any a has the property P , then it belongs to the set A , and, conversely, if the object a belongs to the set A , then it has the property P .
An infinite set of objects can have some property, only a finite set can have another. Therefore, the sets are subdivided into end and end .
A finite set can be specified by direct enumeration of all its elements in an arbitrary order. For example, the set of children of a given group living on Sadovaya Street can be described using a characteristic property: { x | x - lives on Sadovaya Street) or listing all its elements in an arbitrary order: {Lena, Sasha, Vitya, Ira, Kolya}.
It is quite clear that an infinite set cannot be specified by listing all of its elements .

Mathematics mostly deals with infinite sets (numbers, points, figures and other objects), but the basic mathematical ideas and logical structures can be modeled on finite sets.

Naturally, in pre-mathematical preparation usually deals with finite sets .
COUNT - first and main mathematical activity , based on element-by-element comparison of finite sets.

NUMBER is the general invariant category of the set , which is an indicator of the cardinality of the set. This is just a sound designation.
Theoretical foundations for the formation of elementary mathematical representations in preschoolers include a detailed study of only the system of natural numbers . Therefore, when we say "numbers", we mean natural numbers.
DIGITS - a system of characters (“letters”) for writing numbers (“words”) (numerical characters). The word “digit” without specification usually means one of the following ten characters: 0 1 2 3 4 5 6 7 8 9 (so-called “Arabic numerals”). Combinations of these numbers generate two (or more) digit numbers.
The number has 2 values ​​ : cardinal and ordinal.
With a quantitative value of , we are interested in the number of elements in the set. We use the question HOW MUCH? and we start counting with the cardinal number ONE.
With the ordinal value of a number, we are interested in the place of the number among others or the ordinal number of the element in the set. The question is WHICH COUNTS? and set the direction of the account. Ordinal numbers are used, counting begins with the word FIRST.
When we talk about quantity, it does not matter the direction of the count, the object from which the count began. The final number does not change. With an ordinal count, the final number may change.
COUNTING ACTIVITY is considered as an activity with specific elements of the set, in which a relationship is established between objects and numerals. The study of numerals and sets of objects leads to the assimilation of counting activities.
CALCULATION ACTIVITY - is an activity with abstract numbers, carried out by means of addition and subtraction. The simple naming of numerals will not be called counting activity. The system of computational actions is formed on the basis of quantitative knowledge.
VALUE is the quality and property of an object, with the help of which we compare objects with each other and establish a quantitative characteristic of the compared objects.

Concept the value in mathematics is considered as the main one.
A direct answer to the question "what is a quantity?" no, since the general concept of magnitude is a direct generalization of more specific concepts: length, area, volume, mass, speed, etc.
The size of an object is its relative characteristic , emphasizing the length of individual parts and determining its place among homogeneous ones. The value is a property of an object perceived by various analyzers: visual, tactile and motor. In this case, the size of an object is most often perceived simultaneously by several analyzers: visual-motor, tactile-motor, etc.
Size of item, i.e. Item size is determined by based on comparison of only. It is impossible to say whether it is a large or a small object, it can only be compared with another.

The perception of magnitude depends on the distance from which the object is perceived, and also on the magnitude of the object with which it is compared . The farther an object is from the one who perceives it, the smaller it seems, and vice versa, the closer it is, the larger it seems.

The characteristic of the size of the object also depends on its location in the space . The same object can be characterized either as high (low), or as long (short). This depends on whether it is in a horizontal or vertical position. So, for example, in the figure, objects are located in a vertical position and are characterized as high and low, , and in another figure (in a horizontal position), these same objects are characterized as long and short.

The value of an item is always relative , it depends on which item it is compared to . Comparing an object with a smaller one, we characterize it as a larger one, and comparing the same object with a larger one, we call it a smaller one.

So, the magnitude of a particular object is characterized by the following features : comparability, variability and relativity.

1) comparability , carried out:

- overlay,

- application,

- measurement using a conditional measure,

- comparison by eye.

2) relativity - depends on the object with which we are comparing, on the distance at which we are comparing, on the location in space.

3) variability . Size is closely related to size. And size is a property of magnitude variability. Each item has its own generic purpose. It can change its dimensions without changing its essence.
GEOMETRIC FIGURE is an abstract concept with the help of which we personify all the objects around us in form.

A geometric figure is the presence of points on a plane, limited by space.
Figures are flat (circle, square, triangle, polygon…) and spatial (ball, cube, parallelepiped, cone...), which are also called geometric bodies.
GEOMETRIC BODY is a closed part of space bounded by flat and curved surfaces.
If the surface bounding the body consists of planes, then the body is called a polyhedron . These planes intersect along straight lines, which are called edges, and form the faces of the body. Each of the faces is a polygon whose sides are the edges of the polyhedron; the vertices of this polygon are called the vertices of the polyhedron.
Some polyhedra with a certain number of faces have special names : tetrahedron - tetrahedron, hexahedron - exahedron, octahedron - octahedron, dodecahedron - dodecahedron, twenty-sided - icosahedron.
What is a geometric FORM?

SHAPE is the outline, the outer appearance of an object.

Form (lat. forma - form, appearance) - the mutual arrangement of the boundaries (contours) of the subject, object, as well as the relative position of the points of the line.
TIME is a philosophical concept, which is characterized by the change of events and phenomena and the duration of their existence.
Time has properties :

- fluidity (time cannot be stopped)

- irreversibility and uniqueness

- duration .
SPACE is a quality by which relations such as neighborhoods and distances are established.

Orientation in space implies orientation towards oneself, away from oneself, from other objects, orientation on the plane and orientation on the ground.
2. Subject and tasks of the course "Methods of mathematical development and teaching of mathematics". Connection of the methodology of mathematical development with other sciences.
The method of forming elementary mathematical concepts in the system of pedagogical sciences is designed to assist in preparing preschool children for the perception and assimilation of mathematics - one of the most important subjects in school, to contribute to the education of a comprehensively developed personality.
Having stood out from preschool pedagogy, the methodology for the formation of elementary mathematical representations has become an independent scientific and educational area.
The subject of her research is the study of the basic patterns of the process of formation of elementary mathematical representations in preschoolers in the context of public education.
The range of tasks solved by the methodology is quite extensive:

- scientific substantiation of program requirements for the level of development of quantitative, spatial, temporal and other mathematical representations of children in each age group;

- determination of the content of factual material to prepare a child in kindergarten for mastering mathematics at school;

- improvement of the material on the formation of mathematical representations in the kindergarten curriculum;

- development and implementation in practice of effective didactic tools, methods and various forms of organizing the process of developing elementary mathematical concepts;

- implementation of continuity in the formation of basic mathematical concepts in kindergarten and the corresponding concepts in school;

- development of the content for the training of highly qualified personnel capable of carrying out pedagogical and methodological work on the formation and development of mathematical concepts in children at all levels of the preschool education system;

- scientifically based development of guidelines for parents on the development of mathematical concepts in children in a family setting.
The general task of methodology is to study and develop the practical foundations for the process of forming elementary mathematical representations in preschool children. It is solved from the standpoint of the Marxist-Leninist theory, which will develop a unified view of the world, having discovered the laws of development of nature, society, and the individual, and serves as the methodological, ideological basis of science.
The formation of elementary mathematical concepts is a purposeful and organized process of transferring and mastering knowledge, techniques and methods of mental activity provided for by the program requirements Its main goal is not only preparation for the successful mastery of mathematics at school, but also the comprehensive development of children.
The methodology for the formation of elementary mathematical representations in children in kindergarten is associated with many sciences , and above all with those whose subject matter is different aspects of the personality and activity of a preschool child, the process of upbringing and learning.

She has the closest connection with preschool pedagogy . The methodology for the formation of elementary mathematical representations is based on the tasks of teaching and mental education of the younger generation developed by preschool pedagogy and didactics: principles, conditions, ways, content, means, methods, forms of organization, etc. This connection is mutual in nature: research and development of problems the formation of elementary mathematical concepts in children, in turn, improve the pedagogical theory, enriching it with new factual material.
Multilateral contacts exist between private methods that study the specific patterns of the process of raising and teaching young children: the method of forming elementary mathematical representations, speech development, theory and methods of physical education, etc.
Preparing children for learning mathematics at school cannot be carried out successfully without connection with the methodology of elementary teaching of mathematics and those aspects of mathematics itself , which are the theoretical basis for teaching preschoolers and primary school students .

Reliance on these sciences allows, firstly, to determine the volume and content of knowledge that should be mastered by children in kindergarten, and serve as the foundation of mathematical education; secondly, to use teaching methods and means that fully meet the age characteristics of preschoolers, the requirements of the principle of continuity.

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

Psychological features and patterns of perception by a child of a variety of objects, numbers, space, time serve as the basis for developing a methodology for the formation of elementary mathematical representations. Psychology determines the age capabilities of children in acquiring knowledge and skills that are not something frozen and vary depending on the type of education.
The rational construction of the learning process is associated with the creation of optimal conditions based on the anatomical and physiological characteristics of young children. The regularities of the course of physiological processes in preschoolers serve as the basis for determining the duration of classes for the formation of elementary mathematical representations for each age group of the kindergarten, determine their very structure, the combination and alternation of various methods and means of teaching, different types of activities (inclusion of physical education minutes, dosing of educational cognitive tasks, etc.).
Communication with various sciences creates a theoretical basis for the methodology for the formation of mathematical representations in children in kindergarten.

3. Stages of development of the methodology of mathematical development: empirical, classical, modern.
Issues of mathematical development of preschool children are rooted in classical and folk pedagogy. Various counting rhymes, proverbs, sayings, riddles, nursery rhymes were good material in teaching children to count, they made it possible for the child to form concepts about numbers, shape, size, space.

In the course of mastering them, children not only mastered the recalculation of objects, but also the ability to perceive and realize changes occurring in the reality around them: natural, color, spatial and temporal; quantitative, changes in shape, size, location, proportions. This ensured the natural development in children of certain ideas, ingenuity and ingenuity.
The first printed educational book by I. Fedorov "Primer" (1574) included thoughts about the need to teach children to count in the process of various exercises.
In the XIII-XIX centuries. questions of the content and methods of teaching mathematics to preschool children and the formation of their ideas about size, measurement, time and space can be found in the pedagogical works of Ya. A. Comenius, M.G. Pestalozzi, K.D. Ushinsky, L.N. Tolstoy and others.
Views of teachers of the XIII—XIX centuries. on the content and methods of developing mathematical concepts in children - this is the first stage in the development of the methodology - empirical.
Teachers of that era, under the influence of the requirements of developing practice, came to the conclusion that it was necessary to prepare children for learning mathematics at school. They made certain proposals about the content and methods of teaching children, mainly in a family setting. It must be said that they did not develop special manuals for preparing children for school, but included their main ideas in books on education and training.

Thus, the Czech humanist thinker and teacher Ya.A. Comenius (1592-1670) in the book "Mother School" (1632) recommends even before school to teach the child to count within twenty, the ability to distinguish numbers greater-smaller, even-odd, to compare objects by size, to recognize and name some geometric shapes, to use in practice with units of measurement: inch, span, step, pound, etc.

I. G. Pestalozzi (1746-1827), a Swiss democratic teacher, pointed out the shortcomings of the teaching methods existing at that time, which were based on cramming, and recommended teaching children to count specific objects, understand operations on numbers, and the ability to tell time . The teaching methods proposed by him presuppose the transition from simple elements to more complex ones, the widespread use of visualization, which facilitates the assimilation of numbers by children. The ideas of J. G. Pestalozzi later (mid-19th century) served as the basis for reform in the field of teaching mathematics at school.
Progressive ideas in teaching children arithmetic before school were expressed by the Russian democrat teacher, the founder of scientific pedagogy in Russia, K. D. Ushinsky (1824-1871). He considered it important to teach the child to count individual objects and their groups, to perform addition and subtraction, to form the concept of ten as a unit of account. However, all this was just wishes, without any scientific justification.
Writer and teacher Leo Tolstoy published in 1872 "ABC", one of the parts of which was called "Account". Criticizing the existing teaching methods, L.N. Tolstoy suggested teaching children to count "forward" and "backward" within a hundred and numbering, based on children's practical experience gained in the game.

Methods for developing ideas about number and form in children were reflected and further developed in the systems of sensory education of the German teacher F. Fröbel (1782-1852), the Italian teacher Maria Montessori (1870-1952), etc.
In the classical systems of sensory learning F. Fröbel (1782-1852) and M. Montessori (1870-1952) presented a methodology for introducing children to geometric shapes, sizes, measurement and counting, drawing up rows of objects by size, weight, etc.
F. Fröbel saw the tasks of teaching counting in the assimilation of a number of numbers by children of preschool age. He created the famous "Gifts" - a special manual for the development of constructive skills in unity with the knowledge of numbers, shapes, sizes, spatial relationships. F. Fröbel was convinced that the development of "spatial" imagination and thinking at preschool age creates the conditions for the transition to the assimilation of geometry at school. The "gifts" created by F. Fröbel are still being used as didactic material to familiarize children with number, shape, size and spatial relationships.
M. Montessori, relying on the ideas of self-development and self-learning, recognized the need to create a special environment for mastering numbers, shapes, sizes, as well as written and oral numbering. She suggested using special material for this: counting boxes, bundles of colored beads strung in dozens, abacus, coins, and much more.

M. Montessori's pedagogical activity proceeded most effectively in the first half of the 20th century. The use of materials for the development of mathematical representation in children in the education and upbringing of a child was based on a certain style of interaction between an adult and a child; the need to monitor the behavior of children in a specially created environment; organizing joint free work with the child, etc. The M. Montessori system provides for the development of the child's sensorimotor sphere and, in the future, intellect. The "golden" mathematical material, which is especially distinguished by its significance, is first mastered by the child as a set of beads in different quantities, then in symbols (numbers), after that - as a means of mastering the ability to compare numbers. Thus, the decimal number system is presented to the child visibly and tangibly, which leads to the successful mastery of arithmetic.
The section “Logic and Counting” is extensively represented in the M. Montessori system: the study of figures, sizes, methods of measurement, projection, modeling of sets. The following benefits are most interesting: “Figures from carnations”, “Mathematical sun”, “Fold the pattern”, “Combine the sets”.
In general, teaching mathematics according to the M. Montessori system began with a sensory impression, then a transition was made to understanding the symbol (i.e., from the concrete to the abstract), which made mathematics attractive and accessible even for 3-4-year-old children.
So, advanced teachers of the past , Russian and foreign, recognized the role and necessity of primary mathematical knowledge in the development and education of children before school , at the same time singled out account as a means of mental development and strongly recommended that children be taught it as early as possible from about three years of age. Learning was understood by them as "exercise" in performing practical, playful actions using visual material, using the experience accumulated by children in distinguishing numbers, time, space, measures in a variety of children's activities.

The questions of the methodology of mathematical development are of particular importance in the pedagogical literature of elementary school at the turn The authors of the methodological recommendations at that time were advanced teachers and methodologists. The experience of practical workers was not always scientifically substantiated, but it was tested in practice. Over time, he improved, stronger and more fully revealed progressive pedagogical thought.
In the late 19th - early 20th century methodologists needed to develop a scientific foundation for the arithmetic technique . A significant contribution to the development of the methodology was made by advanced Russian teachers and methodologists P.S. Guriev, A.I. Goldenberg, D.F. Egorov, V.A. Evtushevsky, D.D. Galanin and others.
Formation of the methodology for the development of elementary mathematical concepts in the 19th — early 20th centuries. was also under the direct influence of ideas for reforming school teaching methods arithmetic. Two directions stand out in particular : the so-called method of studying numbers, or the monographic method, is associated with one of them, and the method of studying actions, which is called computational, is associated with the other.

According to the method of studying numbers, in the development of the German methodologist A. V. Grube the teaching of arithmetic was carried out "from number to number." Each of the numbers allegedly accessible to "direct contemplation" was compared with each of the previous numbers by establishing a difference and multiple ratio between them. Actions seemed to follow from knowledge by heart of the composition of numbers. Monographic method received the definition of a method that describes a number.

In the process of studying each number , fingers, strokes on a blackboard or in a notebook, sticks served as the material for counting. For example, when studying the number 6, it was proposed to decompose the sticks one by one. Questions were asked: “How many sticks made up the number?”, “Count one stick at a time to get six, etc. After each group of such exercises, the actions were recorded in the form of a table, the results of which were memorized in order to subsequently perform arithmetic operations from memory without resorting to calculations.

In the 70s 19th century Opponents of the monographic method began to appear. Dissatisfaction with the method grew, and in the 80-90s. Russian mathematicians sharply criticized it, contrasting it with the method for studying actions , or, otherwise, computational method .. Method for studying actions (computational) - involved teaching children to calculate and understand the meaning of arithmetic operations. The training was based on decimal concentrations. Within each concentration, not individual numbers were studied, but counting and operations with numbers.

Both methods (both monographic and computational) played a positive role in the further development of the methodology, which absorbed the techniques, exercises, didactic means of one and the other methods.
At the end of the 19th - beginning of the 20th centuries. the ideas of teaching mathematics without coercion and didacticity, amusing, but without excessive entertainment were widely disseminated. Mathematicians, psychologists, teachers developed mathematical games and entertainment, compiled collections of tasks for ingenuity, transformation of figures, solving puzzles (V. A. Latyshev, N. N. Amenitsky, I. P. Sakharov, A. P. Domoryad, V. Arene and etc.).
The authors tried to give a clear logic of construction, unusualness to joke problems, arithmetic rebuses, puzzle problems, problems of dividing a whole into parts, etc. In the course of solving such problems, the ability to think correctly, the logic and consistency of thought, a sharp mind and savvy. Tasks for ingenuity, ingenuity teach children to apply their knowledge to various cases of life, teach them to self-control, and most importantly, they contribute to the development in children of the ability to independently look for a solution.
A number of books were published specifically with the aim of developing the abilities of children, in particular, “Funny Arithmetic” by N. N. Amenitsky and I. P. Sakharov. It offered a lively and amusing solution to various practical problems and issues, which stimulated manifestations of children's amateur performance.

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