Lecture
What methods of teaching mathematics used in antiquity is not known for sure, but there is reason to believe that these methods were dogmatic, unsubstantiated. The Egyptian manuscripts contained the following instructions: “Do it this way or do it, as is customary ...” in ancient India: “Look, look,” Greece: “What was required to prove.”
Arithmetic collections of the time provide a list of practical instructions on how certain arithmetic calculations are made.
In Russia of the 17th and 19th centuries, ideas about the methods of teaching mathematics can be obtained from the first Russian “Arithmetic” by Leonty Filippovich Magnitsky, written in 1702.
Questions of content, methods of teaching preschool children to arithmetic and the formation of ideas about dimensions, measures of measurement, time and space, are reflected in the pedagogical educational systems of Ya. A. Komensky, IG Pestalozzi, F. Frobel, M. Montessori, K. D. Ushinsky, L. N. Tolstoy and others.
They came to the conclusion that children need to be taught mathematics, suggested assumptions about the methods of teaching and education in the family, developed books and manuals.
Czech educator J. A. Komensky (1592-1670) in the guide "Parent School" in the program on arithmetic included:
- account within the first two dozen (for 4-6 years old children);
- discrimination of numbers;
- definition of the larger and smaller of them;
- comparison of items;
- familiarity with geometric shapes;
- introduced measurement measures (inch, span, pitch, pound).
Johann Heinrich Pestalozzi (1746-1827), a Swiss teacher, suggested: 1) to teach children counting on specific subjects; 2) understanding of the operations on numbers; 3) the ability to determine the time, widely used clarity.
K. D. Ushinsky (1824-1871) proposed:
1) to teach the account of individual subjects and groups;
2) to teach addition and subtraction;
3) to form an understanding of a dozen as a unit of account.
L.N. Tolstoy published the Alphabet in 1872, one of the parts of which is the “Account”, proposed to teach children counting back and forth within a hundred.
F. Frebel (1782-1852), an outstanding German educator, theorist of preschool education, who developed the idea of kindergarten and the basis of the methodology in it, the ideas of F. Frebel on education and organization of kindergartens brought him worldwide fame.
Developed games and gaming tools. In the first place among the gaming facilities, Fröbel displays a felt ball of various colors - blue, yellow, purple, and golden. Holding such a ball on a string with one hand, the child demonstrates different types and directions of movements: right, left, up and down, circular, oscillating, the vocabulary of children is enriched.
The teacher raised questions about familiarizing children with geometric shapes, size, learning to count, measuring, drawing rows of objects in size, weight. Teaching Mathematics F. Frebel proposed building through a sensory system.
M. Montessori (1870-1952), an Italian teacher, through sensory upbringing, revealed questions of familiarizing children with forms, sizes, drawing rows of subjects in size, weight, etc. She considered it necessary to create a special environment for the development of ideas about the number, form, values , as well as the study of written and oral numbering. For this, she suggested using counting boxes, bundles of colored beads, bills, coins; numerical bars with plates of numbers, figures from a rough paper, figures circles, turrets. This material introduces children to the mathematical knowledge of the world. Hence it is clear why Montessori called them "basic mathematical materials." (Pink pinnacle, brown ladder, red rods, blocks with numbers, liners, etc., indirectly prepare children for the assimilation of mathematical knowledge - children develop mathematical thinking - children measure, compare). The children's mind simultaneously absorbs diverse sensory and motor experiences, naturally developing mathematical skills.
Elizaveta Ivanovna Tiheeva, in her books “Counting in the lives of young children”, “Modern kindergarten” (1920), speaks out against the systematic teaching of preschoolers. She believes that up to seven years children should learn how to count in the process of everyday life and play. At the same time, she objects to the complete spontaneity of training. In the teaching of children to the account, E. I. Tiheeva included:
1. Score up to 10 (developed 60 tasks for game activities, to consolidate quantitative and spatial representations; determined the amount of knowledge that children must master; emphasized the importance of mastering the children of the top ten).
2. Acquaintance of children with numbers (for this, games with paired pictures were offered, counting boxes).
3. Acquaintance of children with addition and subtraction, (through the solution of problems - from practical life).
4. Acquaintance of children with the value (more, less, higher-lower, wider-narrower, etc.).
5. Acquaintance of children with measurement in the game.
6. Acquaintance of children with the volume, measuring the capacity of the vessel. For familiarity with the mass used scales.
EI Tiheeva was for the free training of children in the game, in a relaxed atmosphere, in everyday life.
Faina Nikolaevna Bleher - a representative of the theory of auto-didactism.
The main ideas about the content and teaching methods were stated in the book “Mathematics in Kindergarten and Zero Group”, published in 1934, and which became the first textbook and program in mathematics in kindergarten.
F. N. Blekher suggested teaching children the elements of mathematics from 3-4 years of age and highlighting the concepts of “many” and “one”, forming ideas about the numbers 1, 2, 3.
On average, pre-school age to learn to determine the quantitative characteristics of objects in the range of 10. On the basis of the account to compare the numbers, use the ordinal score.
In the older group to teach children the composition of numbers, numbers, make up practically the number of smaller groups; perform addition, subtraction; master the second dozen; solve simple problems.
The training offered to lead in the games, teaching the account - to use more natural material. In games, children learn to compare objects in size, get acquainted with geometric shapes, spatial directions.
Children should be involved in practical life situations. The method of teaching FN Bleher's account reflected the ideas of the monographic method — to go in learning from number to number. (It is not permissible to teach the account, but the child must know the number, grasp the number with his eyes, and not teach the account), developed educational games, advised to use more natural material.
Anna Mikhailovna Leushina - a teacher who created the methodology for the formation of elementary mathematical concepts in children of preschool age. Thanks to her work, the method has received a theoretical, scientific, psychological and pedagogical substantiation, and the laws governing the development of quantitative ideas in children under conditions of purposeful instruction in classes in kindergarten were revealed. A. M. Leushina, revealing the patterns of formation and development in children of different ages, ideas about the set, number and operation of the account, developed methods and methods for teaching children counting activities in different age groups, ensuring continuity between them.
Quantitative ideas in children of preschool age are formed through the understanding of the set - this is the so-called pre-literal period. The task of this period is to bring the child to an understanding of quantitative relations.
The child is surrounded by various sets, expressed not only by objects, but also by sounds. The child perceives these sets by various analyzers. The resulting sensations are transmitted to the cerebral cortex and serve as the basis for the formation of ideas about the indefinite multiplicity of different phenomena. This leads to the conclusion that it is necessary for younger preschoolers to form an idea of the set as a structurally integrated unity and to teach each element of the set to be seen and clearly perceived.
The transition from the perception of an uncertain multiplicity to the perception of a set has several stages.
At the first stage it is necessary that children perceive all intermediate elements of the set between the extremes.
At the second stage, children form the idea of a plurality as a structurally integrated unity.
At the third stage - they form and expand the children's ideas about the homogeneous composition of the elements, introducing generic concepts.
At the fourth stage, it is necessary to teach children to act with different groups, to unite them according to different signs.
At the fifth stage - in a timely manner, develop in children the ability to differentiate the elements of a set, not limited to only perception of it, to make a comparison of the number of a set by the practical establishment of its elements. To do this, use the techniques of overlay and application.
The idea of numbers, their sequence, relationships, place in the natural series is formed in preschool children under the influence of the account - a long and complex process. The origins of counting activity are seen in the manipulations of young children with objects. Account how the activity is formed in stages:
Stage 1 - 1.5-2 years. Children are attracted to diverse types of multiplicity: objects, sounds, movements. All movements with objects are accompanied by the repetition of the same word: "here", "here" ..., "here" .., or "more ...", "more ...", or "on ... on ... on ". It is important that each word relates to one thing or one movement. The word helps to distinguish elements from a plurality of homogeneous objects, movements, to more clearly separate one element from another. The child uses this technique spontaneously; it serves as a well-known preparation of the child for counting activity in the future.
Stage 2 - 2-3 years. There is an interest in the comparison of sets (overlay, application). All these facts indicate the desire of children to determine the number of a particular set or size of objects - more, less, equally. These are the first attempts to know the number by comparison.
Stage 3 - 4 years. In the development of counting activity when comparing the elements of sets, the sequential name of words - numerals begins to be included. Through learning, children learn to count operations up to five, correlating numerals with objects. At this time, children often make mistakes miss the elements of the sets, or vice versa, they correlate one number with several objects, and, as a rule, they do not know how to generalize all of these sets.
Stage 4 - 5 years. Children have already clearly learned the sequence in the name of the numerals, more accurately correlate the numeral with each element of the set, master the law of the natural series of numbers n +, - 1, i.e. learn the reciprocal relationship between adjacent numbers.
Stage 5 - 6-7 years. Children learn to score with a different base units, not counting individual items, but groups consisting of several items. Children learn that a whole group can be a unit of account, and not just a separate item.
Stage 6 - school, development of counting activities in the first grade.
The counting process consists of two components: motor and speech.
Motor component:
- the child moves objects;
- touches them;
- indicates objects at a distance;
- allocates each object only with eyes.
- Speech component:
- loudly pronounces the words numerals out loud in the process of counting activities;
- considers a whisper;
- considers only moving his lips;
- considers himself.
Teaching a quantitative account should help children understand the purpose of the account and master the means (rules of the account). Gradually, children are taught ordinal score. In order for the children to learn the regularity of the formation of numbers, the ending is added to the quantitative numeral five - the fifth. Visual material is taken such that each unit is highlighted by something. Children should be taught to distinguish between the questions: “How much?”, “What?”, “Which?” - and correctly answer them.
Teaching children how to count items comes in the following sequence:
- sample count;
- counting by the named number;
- on the basis of the account the establishment of equality (inequality) of groups of objects in a situation when objects are located at different distances from each other, when they differ in size, in the form of their location in space.
Familiarity with the quantitative composition of the number of units within 5 on a particular material:
5 is one, one more, one more, one more and one more.
The formation in children of the concept that the object (sheet of paper, tape, circle, square) can be divided into several equal parts (two, four). Learning to name parts derived from division; to compare the whole and the parts, to understand that the whole object is larger than each of its parts, and the part is smaller than the whole. Children provide the opportunity to exercise themselves in the division of objects.
To continue improving the skills of quantitative and ordinal counting of subjects, consolidating an understanding of the relationship between the numbers of the natural series (7 is greater than 6 by 1, and 6 is less than 7 by 1).
Learning to lay out numbers into two smaller ones and make up of two smaller ones is greater than 10 (it is convenient to use double-sided circles for the first acquaintance).
In the older preschool age, children compose and solve simple addition tasks (a smaller number is added to a larger number) and subtraction (a deductible less residue) on a visual basis; When solving problems, children use the signs of actions: plus (+), minus (-) and the sign of the relationship is (=).
To consolidate counting skills, different types of tasks are used according to the nature of visual material: dramatization, pictures, illustrations, models and verbal ones. There are the following stages in teaching problem solving:
- preparatory stage (children perform operations with sets);
- familiarity with the structure of the problem (condition and question, decision and answer);
- Record arithmetic operations using cards;
- computing activity (children count and count by 1, and then 2, 3).
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Pedagogy and didactics
Terms: Pedagogy and didactics