Methods of teaching children elements of mathematics
Anna Fadyunina
Methods of teaching children elements of mathematics
Methods of teaching children elements of mathematics
In pedagogy, the method is characterized as a purposeful system of actions of the teacher and children that correspond to the goals of learning , the content of educational material , the very essence of the subject, and the level of mental development of the child.
In the theory and methodology of children's mathematical development, the term method is used in a broad and narrow sense. The method can denote a historically established approach to the mathematical preparation of children in kindergarten (monographic, computational and the method of mutually inverse actions).
When choosing methods, the following are taken into account : goals, learning ; the content of the knowledge being formed at this stage; age and individual characteristics of children ; availability of necessary teaching aids; the teacher's personal attitude towards certain methods ; specific conditions in which the learning , etc.
I. G. Pestalozzi,
F. Froebel,
M. Montessori I. is considered the founder of the theory of primary education . G. Pestalozzi.
He proposed teaching children to count based on understanding operations with numbers, and not on simply memorizing the results of calculations, and sharply criticized the dogmatic teaching methods . the methodology developed by I. G. Pestalozzi was the transition from simple elements to more complex ones. Particular importance was attached to visual methods that facilitate the assimilation of numbers.
F. Frebel and M. Montessori paid great attention to visual and practical methods . Developed special manuals ( “Gifts”
F. Froebel and M. Montessori's didactic sets) ensured the acquisition of sufficiently conscious knowledge in
children . In the method of F. used a game as his main , in which the child received sufficient freedom.
According to F. Frebel and M. Montessori, the child’s freedom should be active and based on independence. The role of the teacher in this case comes down to creating favorable conditions. Ya. A. Komensky Currently, there are several different classifications of didactic methods .
One of the first was classification, which was dominated by verbal methods .
Y. A. Komensky, along with verbal ones, began to use another method based on acquiring information not from words, but “from the ground, from oaks and beeches”
, that is, through knowledge of
the objects . The main thing in this technique was the reliance on the practical activities of children . At the beginning of the 20th century, the classification of methods was mainly carried out according to the source of knowledge: verbal, visual, practical.
E. I. Tikheyeva The theory and practice of teaching have accumulated some experience in using different methods in working with preschool children. During the period of the formation of public preschool education, the development of methods for the formation of elementary mathematical concepts was influenced by methods of teaching mathematics in elementary school . Working with preschoolers. E.I. Tikheyeva contributed a lot of new things to the development of methods for teaching children ; the games she compiled and the games she created combined words, actions and visuals. In her opinion, children under seven years old should learn to count through play and everyday life. Game as a teaching method E. I. Tikheyeva proposed introducing as one or another numerical representation has already been “extracted by children from life itself”
.
F. N. Blecher. Proposed the idea of using games in teaching preschoolers (30s-40s)
A. M. Leushina She considered practical methods in the system of verbal and visual methods . It is with practical actions with objective sets that children begin to become acquainted with elementary mathematics . (from the 50s)
Practical methods (exercises, experiments, productive activities)
are most consistent with the age characteristics and level of development of thinking of preschoolers.
The essence of these methods is that children perform actions consisting of a number of operations. For example, counting objects : name numerals in order, correlate each numeral with a separate object , pointing at it with a finger or fixing your gaze on it, correlate the last numeral with the entire quantity, remember the total number.
However, excessive use of practical methods and delays at the level of practical actions can negatively affect the development of the child.
Practical methods are characterized primarily by independent performance of actions and the use of didactic material . On the basis of practical actions, the child develops the first ideas about the knowledge being formed. Practical methods ensure the development of skills and abilities and allow the widespread use of acquired skills in other types of activities.
Visual and verbal methods in teaching mathematics are not independent. They accompany practical and playful methods . But this does not at all detract from their importance in the mathematical development of children .
Visual teaching methods include : demonstration of objects and illustrations, observation, display, examination of tables and models. Verbal methods include storytelling, conversation, explanation, explanations, and verbal didactic games. Often in one lesson different methods in different combinations.
The components of the method are called methodological techniques . The main ones used in mathematics are: overlay, application, didactic games, comparison, instructions, questions for children, examination, etc.
As is known, mutual transitions are possible between methods and methodological techniques Thus, a didactic game can be used as a method , especially in working with younger children, if the teacher develops knowledge and skills through the game, but it can also be used as a didactic technique when the game is used, for example, to increase the activity of children ( “Who is faster?” ?
,
“Get things in order”
, etc.).
widely used methodological technique is demonstration . This technique is a demonstration; it can be characterized as visually practical and effective. Certain requirements are imposed on the display: clarity and dissection; consistency of action and word; accuracy, brevity, expressiveness of speech.
One of the essential verbal techniques in teaching children mathematics is instruction , which reflects the essence of the activity that the children have to perform. In the senior group, the instructions are holistic in nature and are given before completing the task. In the younger group, the instructions should be short, often given as the actions are performed.
Questions for children occupy a special place in the methodology of teaching mathematics They can be reproductive-mnemonic, reproductive-cognitive, productive-cognitive. In this case, the questions must be accurate, specific, and concise. They are characterized by logical consistency and variety of formulations. In the learning there should be an optimal combination of reproductive and productive issues depending on the age of the children the material being studied . Questions are valuable because they enable the development of thinking. Prompt and alternative questions should be avoided.
children's questions and answers is called a conversation. During the conversation, the teacher monitors the children’s correct use of mathematical terminology and the literacy of their speech, accompanying it with various explanations. children's immediate perceptions are clarified . For example, a teacher teaches children to examine a geometric figure and explains: “Take the figure in your left hand - like this, trace it with the index finger of your right hand, show the sides of the square, they are the same. A square has corners. Show me the corners." Or another example. The teacher teaches children to measure , showing practical actions with explanations of how to apply a measure, mark its end, remove it, and apply it again. Then he shows and tells how measures are calculated.
The older the children, the more important problematic issues and problematic situations are learning Problem situations arise when:
— the connection between fact and result is not revealed immediately, but gradually. This raises the question, “Why does this happen?”
(we lower different objects into the water: some drown, others don’t)
;
- after presenting some part of the material, the child needs to make an assumption (experiment with warm water, melting ice, problem solving)
;
- use of words and phrases “sometimes”
,
“some”
,
“only in certain cases”
serves as a kind of identifying signs or signals of facts or results
(games with hoops)
;
- for the concept of a fact, it is necessary to compare it with other facts, create a system of reasoning, i.e., perform some mental operations (measurement with different measures, counting in groups, etc.)
.
Numerous experimental studies have proven that when choosing a method, it is important to take into account the content of the knowledge being generated. Thus, in the formation of spatial and temporal concepts, the leading methods are didactic games and exercises (T. D. Richterman, O. A. Funtikova, etc.)
.
When introducing children to shape and size, along with various play methods and techniques, visual and practical ones are used.
The place of the game method in the learning process is assessed differently. In recent years, the idea of the simplest logical training of preschoolers has been developed, introducing them to the field of logical and mathematical representations (properties, operations with sets)
based on usage
special series of " educational "
games
(A. A. Stolyar)
.
These games are valuable because they actualize the hidden intellectual capabilities of children and develop them (B. P. Nikitin)
.
to ensure comprehensive mathematical training for children with a skillful combination of game methods and direct teaching methods . Although it is clear that the game captivates children , it does not overload them mentally and physically. children's interest in play to interest in learning is completely natural.
Methods and techniques for forming mathematical concepts in preschoolers.
Methods and techniques for forming mathematical concepts in preschoolers.
In the process of forming elementary mathematical concepts in preschoolers, the teacher uses a variety of teaching methods: practical, visual, verbal, and playful.
When choosing a method, a number of factors are taken into account: program tasks being solved at this stage, age and individual characteristics of children, availability of necessary didactic tools, etc.
The teacher’s constant attention to the informed choice of methods and techniques and their rational use in each specific case ensures:
— successful formation of elementary mathematical concepts and their reflection in speech;
- the ability to perceive and highlight relations of equality and inequality (in number, size, shape), sequential dependence (decrease or increase in size, number), highlight quantity, shape, value as a common feature of the analyzed objects, determine connections and dependencies;
- orientation of children to the use of mastered methods of practical actions (for example, comparison by comparison, counting, measurement) in new conditions and an independent search for practical ways to identify, detect signs, properties, connections that are significant in a given situation. For example, in a game, identify the sequence order, the pattern of alternation of features, the commonality of properties.
The leading method in the formation of elementary mathematical concepts is the practical method. Its essence lies in organizing the practical activities of children, aimed at mastering strictly defined methods of acting with objects or their substitutes (images, graphic drawings, models, etc.).
Characteristic features of the practical method in the formation of elementary mathematical concepts:
— performing various practical actions;
- widespread use of didactic material;
- the emergence of ideas as a result of practical actions with didactic material:
- development of counting skills, measurement and calculations in the most elementary form;
- widespread use of formed ideas and mastered actions in everyday life, play, work, i.e. in various types of activities.
This method involves the organization of special exercises, which can be offered in the form of a task, organized as actions with demonstration material, or proceed in the form of independent work with handout didactic material.
Exercises can be collective - performed by all children at the same time - and individual - performed by an individual child at the board or teacher’s table. Collective exercises, in addition to assimilation and consolidation of knowledge, can be used for control.
Individuals, performing the same functions, also serve as a model by which children are guided in collective activities.
Game elements are included in exercises in all age groups: younger ones - in the form of a surprise moment, imitation movements, a fairy-tale character, etc.; in older children they take on the character of search and competition.
As children age, the exercises become more complex: they consist of a larger number of links, the educational and cognitive content in them is not masked by a practical or game task, in many cases, their implementation requires performance actions, the manifestation of ingenuity, and ingenuity. So, in the younger group, the teacher invites the children to take carrots and treat each hare; in the senior class - determine the number of circles on a card posted on the board, find the same number of objects in the group room, prove the equality of the circles on the card and the group of objects. If in the first case the exercise consists of a conventionally selected one link, then in the second - of three.
The most effective are complex exercises that make it possible to simultaneously solve program problems from different sections, organically combining them with each other, for example: “quantity and counting” and “magnitude”, “quantity and counting” and “geometric figures”, “geometric figures”, “magnitude” and “quantity and counting”, etc. Such exercises increase the efficiency of the lesson.
In kindergarten, exercises of the same type (i.e., pursuing the same goal and carried out with the same content) are widely used, thanks to which the necessary methods of action are developed; mastery of counting, measurement, and simple calculations; a circle of elementary mathematical concepts is formed.
The currently existing system of exercises in all age groups is based on the following principle; each previous and subsequent exercise has common elements - material, methods of action, results, etc.
From the point of view of children’s manifestation of activity, independence, and creativity in the process of execution, reproductive (imitative) and productive exercises can be distinguished.
Reproductive ones are based on simple reproduction of the method of action. At the same time, the actions of children are completely regulated by adults in the form of a sample, explanation, requirement, rule that determines what and how to do. Strict adherence to them gives a positive result, ensures the correct completion of the task, and prevents possible mistakes.
Productive exercises are characterized by the fact that children must fully or partially discover the method of action themselves. This develops independent thinking, requires a creative approach, and develops focus and dedication. The teacher usually says what needs to be done, but does not tell or demonstrate the method of action. When performing exercises, the child resorts to mental and practical tests, shows intelligence, ingenuity, etc. When performing such exercises, the teacher provides help not directly, but indirectly, invites the children to think and try again, and approves of the correct actions.
When forming elementary mathematical concepts, the game acts as an independent teaching method. But it can also be classified as a group of practical methods, bearing in mind the special significance of different types of games in mastering various practical actions, such as composing a whole from parts, rows of figures, counting, superposition and application, grouping, generalization, comparison, etc.
Didactic games are the most widely used. Thanks to the learning task, presented in a game form (game concept), game actions and rules, the child unintentionally learns certain cognitive content. All types of didactic games (subject, board-printed, verbal) are an effective means and method of forming elementary mathematical concepts. Subject and word games are carried out in and outside of mathematics classes. Desktop - printed, as a rule - in free time from classes. All of them perform the basic functions of education: educational, educational and developmental. There are didactic games for the formation of quantitative ideas, ideas about size, shape, figures, space, time. Thus, it is very promising to present each section of the “mathematics” program in kindergarten with a system of didactic games that serve to exercise children in applying knowledge.
Game as a method of teaching and forming elementary mathematical concepts involves the use in classes of individual elements of different types of games (story, movement, etc.), gaming techniques (surprise moment, competition, search, etc. Currently, a system of so-called educational games.
All didactic games for the formation of elementary mathematical concepts are divided into several groups:
1. Games with numbers and numbers
2. Time travel games
3. Games for orientation in space
4. Games with geometric shapes
5. Logical thinking games
Visual and verbal methods in the formation of “elementary” mathematical concepts are not independent; they accompany practical and game methods.
Techniques for forming mathematical representations.
In kindergarten, techniques related to visual, verbal and practical methods and used in close unity with each other are widely used:
1. Showing (demonstration) of the method of action in combination with an explanation or example from the teacher. This is the main method of teaching, it is visual, practical and effective in nature, carried out using a variety of didactic means, and makes it possible to develop skills and abilities in children. The following requirements apply to it:
— clarity, dissection of the demonstration of methods of action;
— consistency of actions with verbal explanations;
— accuracy, brevity and expressiveness of speech accompanying the show:
- activation of perception, thinking and speech of children.
2. Instructions for performing independent exercises. This technique is associated with the teacher’s demonstration of methods of action and follows from it. The instructions reflect what and how to do to get the desired result. In older groups, the instructions are given in full before the task begins; in younger groups, they precede each new action.
3. Explanations, clarifications, instructions. These verbal techniques are used by the teacher when demonstrating a method of action or while children are performing a task in order to prevent mistakes, overcome difficulties, etc. They must be specific, short and figurative.
Demonstration is appropriate in all age groups when familiarizing with new actions (application, measurement), but it requires activation of mental activity, excluding direct imitation. In the course of mastering a new action, developing the ability to count and measure, it is advisable to avoid repeated demonstrations.
Mastering an action and improving it is carried out under the influence of verbal techniques: explanations, instructions, questions. At the same time, the verbal expression of the method of action is being mastered.
4. One of the main methods of forming elementary mathematical concepts in all age groups is asking questions to children. In pedagogy, the following classification of issues is accepted:
- reproductive-mnemonic (How many? What is it? What is the name of this figure? What is the difference between a square and a triangle?);
- reproductive-cognitive (How many cubes will be on the shelf if I put one more? Which number is greater (smaller): nine or seven?);
-productive-cognitive (What needs to be done so that there are 9 circles? How to divide the strip into equal parts? How can you determine which flag in the row is red?).
Questions activate children’s perception, memory, thinking, and speech, ensuring comprehension and assimilation of the material. When forming elementary mathematical concepts, the most significant series of questions is: from simpler ones, aimed at describing specific features, properties of an object, results of practical actions, i.e., stating, to more complex ones, requiring the establishment of connections, relationships, dependencies, their justification and explanation, use the simplest evidence. Most often, such questions are asked after the teacher demonstrates a sample or the children perform exercises. For example, after the children have divided a paper rectangle into two equal parts, the teacher asks: “What did you do? What are these parts called? Why can each of these two parts be called a half? What shape did the parts turn out to be? How to prove that the result is squares? What must be done to divide the rectangle into four equal parts?
Questions of different nature cause different types of cognitive activity: from reproductive, reproducing the studied material, to productive, aimed at solving problematic problems.
Basic requirements for questions as a methodological technique:
— accuracy, specificity, laconicism:
— logical sequence;
- variety of wording, i.e. the same thing should be asked in different ways.
— the optimal balance between reproductive and productive issues depending on the age of the children and the material being studied;
- questions should develop the child’s thinking, make him think, highlight what is required, carry out analysis, comparison, juxtaposition, generalization;
-the number of questions should be small, but sufficient to achieve the set didactic goal;
- Prompt and alternative questions should be avoided.
The teacher usually asks a question to the whole group, and the called child answers it. In some cases, choral responses are possible, especially in younger groups. Children need to be given the opportunity to think about their answer.
Older preschoolers should be taught to formulate questions independently. In a specific situation, using didactic material, the teacher invites children to ask about the number of objects, their ordinal place, size, shape, method of measurement, etc. The teacher teaches them to ask questions based on the results of direct comparison: “Kolya compared a square and a rectangle. What can I ask him?”, following the practical action performed at the board: “Ask Galya what she learned by arranging the objects in two rows? Look what I did. What can you ask me?”, based on the action performed by the child sitting next to him: “What can you ask Anya?” Children successfully master the ability to ask questions if they are addressed to a specific person - a teacher, a friend.
Children's answers should be:
- short or complete, depending on the nature of the question;
- independent, conscious;
— accurate, clear, loud enough;
- grammatically correct (observance of word order, rules of their agreement, use of special terminology).
When working with preschoolers, an adult often has to resort to the technique of reformulating an answer, giving it the correct sample and asking them to repeat it. For example: “There are four mushrooms on the shelf,” says the kid. “There are four mushrooms on the shelf,” the teacher clarifies.
5. Control and evaluation. These techniques are interrelated. Control is carried out through monitoring the process of children completing tasks, the results of their actions, and answers. These techniques are combined with instructions, explanations, clarifications, demonstration of methods of action to adults as a model, direct assistance, and include correction of errors.
The teacher corrects errors during individual and collective work with children. Practical and speech errors are subject to correction. The adult explains their reasons, gives an example, or uses the actions and responses of other children as an example. Gradually, the teacher begins to combine control with self- and mutual control. Knowing the typical mistakes that children make when counting, measuring, simple calculations, etc., the teacher carries out preventive work.
The methods and results of actions and the behavior of the children are subject to evaluation. The assessment of an adult who teaches one to be guided by a model begins to be combined with the assessment of comrades and self-esteem. This technique is used during and at the end of an exercise, game, or lesson.
The use of control and assessment has its own specifics depending on the age of the children and the degree to which they have mastered knowledge and methods of action. Control is gradually transferred to the result, the assessment becomes more differentiated and meaningful. These techniques, in addition to teaching, also perform an educational function: they help to cultivate a friendly attitude towards comrades, the desire and ability to help them, etc.
6. During the formation of elementary mathematical concepts in preschoolers, comparison, analysis, synthesis, and generalization act not only as cognitive processes (operations), but also as methodological techniques that determine the path along which the child’s thought moves in the learning process.
Comparison is based on establishing similarities and differences between objects. Children compare objects by quantity, shape, size, spatial location, time intervals by duration, etc. First, they are taught to compare the minimum number of objects. Then the number of objects is gradually increased, and the degree of contrast of the compared features is correspondingly reduced.
Analysis and synthesis as methodological techniques appear in unity. An example of their use is the formation in children of ideas about “many” and “one”, which arise under the influence of observation and practical actions with objects.
The teacher brings a large number of identical toys into the group at once - as many as there are children. Gives one toy to each child, and then collects them together. Before the children's eyes, a group of objects is split into parts, and the whole is recreated from them.
Based on analysis and synthesis, children are led to a generalization, which usually summarizes the results of all observations and actions. These techniques are aimed at understanding quantitative, spatial and temporal relationships, at highlighting the main, essential. A summary is made at the end of each part and the entire lesson. First, the teacher generalizes, and then the children.
Comparison, analysis, synthesis, generalization are carried out on a visual basis using a variety of didactic means. Observations, practical actions with objects, reflection of their results in speech, questions to children are the external expression of these methodological techniques, which are closely related to each other and are most often used in combination.
7. In the methodology for the formation of elementary mathematical concepts, some special methods of action leading to the formation of concepts and the development of mathematical relations act as methodological techniques. These are techniques of applying and applying, examining the shape of an object, “weighing” an object “on the hand”, introducing counters - equivalents, counting and counting by unit, etc. Children master these techniques in the process of showing, explaining, performing exercises and then resort to to them for the purpose of verification, proof, in explanations and answers, in games and other activities.
8. Modeling is a visual and practical technique that includes the creation of models and their use in order to form elementary mathematical concepts in children. The technique is extremely promising due to the following factors:
— the use of models and modeling puts the child in an active position and stimulates his cognitive activity;
- a preschooler has some psychological prerequisites for the introduction of individual models and elements of modeling: the development of visual-effective and visual-figurative thinking.
Models can perform different roles: some reproduce external connections, help the child see those that he does not notice on his own, others reproduce the sought-after but hidden connections, the directly not perceived properties of things.
Models are widely used in the formation
· temporary representations: model of parts of the day, week, year, calendar;
· quantitative; numerical ladder, numerical figure, etc.), spatial: (models of geometric figures), etc.
· when forming elementary mathematical concepts, subject-specific, subject-schematic, and graphical models are used.
9. Experimentation is a method of mental education that ensures the child’s independent identification through trial and error of connections and dependencies hidden from direct observation. For example, experimentation in measurement (size, measurement, volume).
10. Training is a method of familiarization with social reality (the world of money).
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