Slide 5.

Speech and emotional conversations of behavior in the situation of frustration

An adequately loyal apologizes, if he was wrong, fearlessly, but with respect he looks into his eyes to the opponent, reaches this top of adaptive behavior, in certain, favorable situations. Inadequately loyal in a hurry to apologize without analyzing the situation, obeys the opposite side, the readiness for the adoption of aggression is a child, respects him. Adequately disloyal, aggressive "Fool itself!" Open aggression in response to aggression puts the child to the position of equality, the struggle of ambitions will determine the winner through the ability to provide volitional resistance, without the use of physical force. Adequately disloyal, ignoring open ignoring in response to aggression can put a child over the situation. Such a position helps to preserve self-esteem, a sense of personality. It is important to have sufficient intuition and reflection so as not to overdo it. Passive, not included any communication does not happen, the child avoids communication, closes (pulls his head in the shoulders, looks into a certain space in front of him, turns away, lowers his eyes, etc.). The situation is dangerous because the child may lose his own dignity and self-confidence .

Slide 6.

Styles of communication offered by adults in the family and school

Family Authoritarian Style Liberal-Branched Style Hyper Making Style Value Style Aliented Style School Imperative (Authoritarian) Democratic Style Liberal-Branched (Anti-Authority).

Slide 7.

Mental development oral and written speech

Slide 8.

The correctness of the speech

Correctness of oral speech grammatical correctness; Orphoepic correctness; Pronunciation correctness. The correctness of the written speech is grammatical (building proposals, the formation of morphological forms); Spelling; Punctuation.

Slide 9.

Development of mental functions thinking

Features of the development of thinking in the younger school age thinking becomes a dominant function; The transition from visual-shaped to verbally logical thinking is completed; The emergence of logically loyal reasoning; Use of specific operations; Formation of scientific concepts; Development of the foundations of conceptual (theoretical) thinking; The appearance of reflection; Manifestation of individual differences in kinds of thinking: theorists; practice; artists.

Slide 10.

Development of mental functions

Features of the development of attention in the younger school age the predominance of involuntary attention; Distractions; A small amount of attention; Small attention stability (younger schoolchildren 10-20 minutes, teens 40-45 minutes, high school students 45-50 min); The switching and distribution of attention is difficult; Development of random attention; Individual attention options.

Slide 11.

Development of mental functions Memory

Features of the development of memory in the younger school age: developed mechanical memory; Development of semantic memory; Developed involuntary memory; Development of arbitrary memory; The development of meaningfulness of memorization; Ability to use mnemotechnics.

Slide 12.

Development of mental functions

Mnemonic techniques for younger schoolchildren dividing text into semantic parts; inventing headlines to different parts; planning. Tracking of basic semantic lines; Allocation of semantic reference points or words; Return to the already read parts of the text to clarify their content; Mental recall read part and reproduction out loud or about themselves all material; Rational teaching techniques by heart. Consequence of applications of mnemoniums younger schoolchildren understanding educational material; Binding of educational material with passed; Inclusion in the general system of knowledge existing in a child; Significant material is easily "extracted" from the system of connections and values; Training material is much easier reproduced by the student.

Slide 13.

Development of mental functions

Features of perception in younger students, perception at the beginning of the period is not differentially differentiated (confused 6 and 9); Allocation of the most vivid properties of objects (color, shape, quantity); Observation is developing; The emergence of synthesizing perception (analyzing preschoolers);

Slide 14.

Crisis 7 years

Slide 15.

General characteristics of educational activities

Structure of training activities (D.B. Elkonin): The educational task is that the student must assimilate the method of action; Educational actions - the fact that the student should do to form a sample of the appropriate action and reproduce this sample; Control action - comparison of the reproduced action with the sample; The evaluation action is determined how much the student has reached the result, the degree of change that occurred in the child itself.

Slide 16.

Readiness for school learning

Personal readiness for school learning The desire of a child to a new social situation: Initially, the attractiveness of external attributes (portfolio, form, etc.); The need for new social contacts. Formation of the internal position of the schoolchild: the effect of close adults; The impact and attitude of other children; the opportunity to rise to a new age step in the eyes of younger; the ability to be equal in the position with the elders; Attitude towards teaching, as a more significant type of activity than the game of the preschooler.

Slide 17.

Personal readiness (continued) Formation of an incoming personal form of communication with adults (according to M.I. Lisina): an adult is an indisputable authority, a sample for imitation; The comments of the adult are not offended, and on the contrary, try to correct errors; adequate understanding of the position of the teacher, his professional role; Understanding the convention of school communication, adequate subordination to the rules of the school. Cooperative communication with peers prevails over competitive communication; Availability of a certain attitude towards yourself: the adequate attitude of the child to its abilities, the results of work, behavior; a certain level of self-consciousness development; self-esteem should not be overestimated and undifferentiated; Motivational readiness for learning (cognitive need is stronger than the need for the game (Methodology N.I. Gutkin: Listening to a fairy tale or game with toys)); Specific development of the field of arbitrariness: the ability to fulfill the teacher's learning claims, verbally; work on a visually perceived pattern; The ability to focus on a complex system of requirements (simultaneous following the sample in its work and take into account some additional rules).

Slide 18.

Intellectual readiness for school learning a certain development of the level of mental processes: the ability to generalize, compare objects; classify, identify essential signs; determine causal dependencies; Ability to draw conclusions. The presence of a certain latitude of representations: figurative views; Spatial views. Relevant speech development; Cognitive activity.

Slide 19.

Training from 6 years

Features of 6-year-old children (from the point of view of school learning) features of thinking, correspond to preschool age: the prevalence of involuntary memory; short-term productive attention (10-15 minutes); The predominance of visual-shaped thinking; Cognitive motifs, adequate learning tasks, unstable and situational; Illuminated self-esteem: misunderstanding of the criteria for pedagogical assessment; Assessing his work, the teacher is perceived as an assessment of his personality; A negative assessment does not cause desire to remake, but causes anxiety, the state of discomfort. General instability of behavior; dependence on the emotional state; social instability; Acute need for direct emotional contacts (in formalized school conditions, this need is not satisfied); fast fatiguability; high distractions;

Slide 20.

Slide 21.

Diagnosis of children from 6-7 to 10-11 years

Methodical material

Slide 22.

Methodology L.Ya. Yasyukov

Appointment of methodology Determination of readiness for school. Forecast and prevention of training problems in elementary school. The technique diagnoses: information processing speed, random attention, short-term hearing and visual memory, speech development, conceptual and abstract thinking, features of the prevailing emotional background, the energy balance of the child's body and adaptation opportunities, personal training potential (self-esteem, emotional attitudes towards school, Foreign situation and others).

Slide 23.

Factor personal questionnaireKethella (children's) (from 7 to 12)

Appointment of techniques Factor personal questionnaire R. kettella is widely used in management, tradeborne and vocational education, in power structures, in the practice of clinical psychologists and education. Category Methods: Personal Questionnaire Application Methods Children's option (CPQ) - from 7 to 12 years old Teenage option (HSPQ) - from 12 to 16 years old Adult (16PF) - From 16 years Test time: 40-50 minutes Form: individual, Group, Computer Individual Results Processing: Manual, Computer

Slide 24.

Frustration of Rosenzweig Test

The appointment of the test methodology is designed to identify emotional stereotypes of response in stressful situations and predicting behavior in interpersonal interaction. Application of the Age Range methodology: Children's option - C 7 to 14 years old Adult - from 14 years Test time: 25-30 minutes Form of carrying out: Individual results processing: Manual, Computer L. Ya. Ya Yoryukov Adapted Adult and Children's Variant Methodology "Frustration Test S. Rosenzweig. "

Slide 25.

Test Waxer (Children's Option)

Purpose of the methodology Category of the method: Cognitive test The technique allows you to measure the level of development of general, verbal and non-verbal intelligence, frequent intellectual abilities; identify the potential of the trainee; Determine the level of safety of the inttel-lect. Application of the Age Range methodology: Children's option - C 5 to 16 years old Adult - from 16 years Test time: 90-100 minutes Form of carrying: Individual results processing: manual

Slide 26.

Diagnostics of differentiation of the emotional sphere of the child "houses" (Methods O.A. Orekhova)

PURPOSE OF METHODS CATEGORY OF METHOLOGY: PSYCHOSEMANTIC METHODOL can be used in psychological consulting and psychotherapy for predicting difficulties in the development of the emotional sphere and the development of correctional programs of personal characteristics of children. Application of the Age Range Methods: from 4 to 12 years Test time: 20 minutes Form of carrying: individual, group, Computer Individual results Processing: Manual, Computer

Slide 27.

Slide 28.

Bibliography

Slide 29.

M.V. Gamesezo, E.A. Petrov, L.M. Oorlova

Age and pedagogical psychology Mikhail Viktorovich Gamezo - Professor, Doctor of Psychological Sciences, author of about 100 scientific works, one of the founders of the psycho-formyotic approach of modern domestic psychology. The most famous books - "Atlas on Psychology" and "Course of Psychology" (in 3 parts). Mikhail Viktorovich Gamezo was awarded the signs of the "Excellence of Education of the USSR", "Excellence of Education of the RSFSR", K.D. Medal Ushinsky and Silver Medal VDNH. For a long time he headed the Department of Psychology MGOU. MA Sholokhov, where he continues to work as a professor consultant. Elena Alekseevna Petrova - Professor, Doctor of Psychological Sciences, author of more than 120 scientific and popular work, the most famous of which are "gestures in pedagogical process"," Signs of communication "and others. Elena Alekseevna Petrova - Honorary Worker of the Higher System vocational education Russian Federation, Heads the Department of Social Psychology MGSU, Professor of the Department of Psychology MGOU. Love Mikhailovna Orlova - Associate Professor, Candidate of Psychological Sciences, a specialist in the history of psychology, the psychology of communication, the author of many scientific and educational works, the most famous of which are "psychodiagnostics of a preschooler and a younger schoolboy", " Age-related psychology: Personality from youth to old age. " Veteran of labour.

Slide 30.

Elkonin Daniel Borisovich

Soviet psychologist who was in the backbone of scientific school LS Vygotsky author owns remarkable periodization theories children's Development and children's games, as well as methods of learning children to read. He studied at the Leningrad Pedagogical Institute. A. I. Herzen. From 1929 he worked in this institute; For several years, in collaboration with L. S. Vygotsky studied the problems of the children's game. Peru D. B. Elkonin belongs to several monographs and many scientific articles on the problems of the theory and history of childhood, its periodization, mental development of children of different ages, psychology of game and educational activities, psychodiagnostics, as well as questions of the development of the speech of a child and learning children to read. List of basic scientific Labors D. B. Elkonina: Thinking of the younger schoolboy / Essays of the psychology of children. M., 1951; Child psychology. M., 1960; The letter (experimental). M., 1961; Questions of psychology of educational activities of younger students / Ed. D. B. Elkonina, V. V. Davydova. M., 1962; Intellectual opportunities Junior schoolchildren and learning content. Age ability to learn knowledge. M., 1966; Psychology of learning younger schoolboy. M., 1974; How to learn children to read. M., 1976;

Slide 31.

Vygotsky hp

The cultural and historical concept of the development of the psyche. He introduced a new - experimental and genetic research method mental phenomenaSince it believed that "the problem of the method is the beginning and the basis, Alfa and Omega of the whole history of the cultural development of the child." L.S. Vygotsky has developed a doctrine of age as a unit for analyzing children's development. He proposed a different understanding of the course, conditions, source, forms, specifics and the driving forces of the psychic development of the child; described the epochs, stage and phases of children's development, as well as transitions between them during ontogenesis; He revealed and formulated the basic laws of the psychic development of the child. By hp Vygotsky, the driving force of mental development - training. It is important to note that the development and training is different processes the concept of the nearest development zone has important theoretical significance and is associated with such fundamental problems of children's and pedagogical psychology, as the emergence and development of higher mental functions, the ratio of training and mental development, driving forces and the mechanisms of the psychic development of the child. 1935 Mental development of children in the learning process. [Sat. articles] state-studies. Pedagogue, ed., Moscow. 1982-1984 Collected Works in 6t. (T. 1: Questions of the theory and history of psychology; T. 2: Problems of general psychology; T. 3: Problems of the development of the psyche; T. 4: Child psychology; t. 5: Basics of defectology; t. 6: Scientific inheritance). Pedagogy, Moscow. 1956 Thinking and speech. Problems of the psychological development of the child. Selected pedagogical studies, Publishing House of APN RSFSR. Moscow.

Slide 32.

Leontyev A.N.

Developed in the 20s. together with hp Vygotsky and A.R.Luria cultural and historical theory, conducted a cycle of experimental studies that reveal the mechanism for the formation of higher mental functions (arbitrary attention, memory) as the process of "rotation", the interiorization of external forms of instrument-mediated actions into internal mental processes. Experimental and theoretical works are devoted to the problems of the development of the psyche (its genesis, biological evolution and socio-historical development, the development of the psyche of the child), the problems of engineering psychology, as well as the psychology of perception, thinking. The concept of Leontheva's activities was developed in various sectors of psychology (general, children's, age, pedagogical, medical, social), in turn enriched by its new data. The situation formulated by Leontya's leading activity and its defining effect on the development of the child's psyche served as the basis for the concept of periodization of the mental development of children, extended by D.B. Elkonin. Op.: Select. Psychological works, t.1-2.- M., 1983; Feeling, perception and attention of children of younger school age // Essays of the psychology of children (ml. Sk. Age). - M., 1950; Mental child development. - M., 1950; Category of activity in modern psychology // Questions of psychology, 1979, № 3.

Slide 33.

Kudryavtsev V.T.

doctor of Psychological Sciences, Professor, Head of the Laboratory of the Psychological and Pedagogical Foundations of the Developing Education of the Russian Academy of Education. Raises questions about developing education, the continuity of pre-school and initial school steps. Special sharpness The problem of the continuity of educational steps acquires at a breakfast of preschool and younger school age. The fact is that there is a radical change of social situations of children's development - from communicatively-game to the study. In the context of this contradiction, the problem of continuity of pre-school and primary education is considered in the works of L.S. Vygotsky, D.B. Elkonin. Under the leadership of V.V. Dodavdov and V.T. Kudryavtseva, a special design work was deployed to create an appropriate continuity model. Since 1992, this work has been conducted on the basis of the Moscow School Laboratory "Losina Island" No. 368, including the pre-school and school steps (the latter uses in its activities the technology of developing training on the system D. B. Elkonin - V.V. Dodavdov). Currently, similar experimental sites are created in a number of regions of Russia. Program "Record-Start". The goal of the project is to create conditions that ensure the general mental development of children 3 - 6 years to the means of developing their imagination and other creative abilities, in particular, as the conditions of formation, they began to study the future ability. The following project tasks are given: Initiation and psychological and pedagogical support of the processes of creative development of culture by children within the framework of various types of their activities (games, artistic and aesthetic activities, exercises, etc.); Development of creative imagination of preschoolers based on it system creative abilities of a child (productive thinking, reflection, etc.), creativity as the leading property of his personality; development and maintenance of specific cognitive motivation and intellectual emotions; expanding the prospects for children's development by incorporating preschoolers in the developing forms of joint activities with adults and each other; Cultivation in children of a creative value attitude to their own physical and spiritual health.

Slide 34.

Literature

Vygotsky hp Cathedral cit. At 6 t. T. 5. M.: Pedagogy, 1983. P. 153-165 Vygotsky LS.1982-1984 Collected Works in 6T. (T. 1: Questions of the theory and history of psychology; T. 2: Problems of general psychology; T. 3: Problems of the development of the psyche; T. 4: Child psychology; t. 5: Basics of defectology; t. 6: Scientific inheritance). Pedagogy, Moscow. Gamezo M.V., Petrova E.A., Orlova L.M.Rezhda and Pedagogical Psychology: studies. Manual for students of all specialties of pedagogical universities. - M.: Pedagogical society of Russia, 2003. - 512 p. Korig, D. Brown "Psychology of Development" 9th edition, PETER Publishing House Questions of psychology of educational activities of younger students / Ed. D. B. Elkonina, V. V. Davydova. M., 1962; Thinking of the younger schoolboy / Essays of the psychology of children. M., 1951; Child psychology. M., 1960; The letter (experimental). M., 1961;

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Organization of teaching children in junior school classes. The objective nature of the difficulties faced by a child at the beginning of school training. The main problems of the adaptation period: inclusion in new activities, entering a new system of relations, addictive to the unusual regime of the day and work, the emergence of new duties, the need for such qualities of personality, such as discipline, responsibility, perseverance, perfection, efficiency and hard work. Ways to overcome the difficulties of the adaptation period to school. Additional moral stimulation of a child for progress. Formation of the main components of training activities: academic actions, actions to control and evaluate the results of work. Causes of intellectual passivity and lagging children in primary classes, ways to eliminate them. Group forms of classes in the first months of school training.

Training of younger schoolchildren at home. Special importance of home academic work with first graders. Formation of independent learning activities. The development of speech and thinking through the improvement of the letter. The presentation, retelling of the read, seen or heard, writing letters and small essays is the fixed assets of speech development. Two main directions of improving theoretical and practical thinking of younger students. The role of mathematical, linguistic exercises, household problems in improving the thinking of the child. Different types of creative activity: design, drawing, modeling - as means of improving practical and visual-shaped thinking.

Game and labor activity from younger students. Changing the nature of children's games in the younger school age. The emergence and dissemination of competitions and design games contributing to the development of business intellectual qualities in children. Accustomed child to work. Developing the importance of children's sports games. Developing labor activities. Organization of child labor at school and at home. Labor as an initiative, independent and creative work. The need for child labor and how to stimulate it.

Sources of mental development of children of primary school age. Print, radio, television, various types of art as sources of intellectual development of children of younger school age. art As a means of developing and enriching the perception of the world, as a method of getting rid of an egocentric point of view. Development of a child's ability to proper understanding and making a foreign point of view. The art of film and television as means of expansion and deepening the world's vision. Developing theater capabilities. The role of literature and periodic press in the intellectual development of children. The need to read as a means of improving speech thinking. Causes of lagging in the teaching of children of younger school age. The learner and the level of mental development of the child. Age capability training. The weakness of memory as one of the reasons for the lagging of children in the teaching. Symbolic coding and cognitive material organization in order to improve memory. Psychological and pedagogical analysis of the reasons for the backlog in the teaching of children of younger school age.

Organization of teaching children in junior school classes

Regardless of how much effort and time is spent on ensuring the readiness of children for school training in the preschool, in the initial period of study with certain difficulties, almost all children are facing. Therefore, there is a transitional period from preschool childhood to school, which can be called the child's adaptation period to school. For the general psychological characteristics of this and subsequent periods in the life of a child related to the radical changes of his psychology of cognitive, it is useful to take advantage of social development situation and internal position. The first of these concepts relates to the social conditions in which the process of mental development of the child is underway. It also includes an idea of \u200b\u200ba child's place in society in the system of division of labor associated with this rights and obligations. The second concept characterizes the inner world of the child, the changes that should occur in it so that the child can adapt well to the new social situation and use it for its further psychological growth. These changes are usually associated with the formation of new relations, a new meaning and goal of life, affect the needs, interests and values, forms of behavior and attitudes towards people. In general, they are also associated with the beginning of major changes in personal and interpersonal plan in the child's psychology.

Such moments in a person's life, when deep changes in the social situation of development occur, relatively a bit. This arrival at school, her end, receiving the profession and the beginning of independent work, family formation, transitions from one age to another: from 20-25 to 40-50 years, from 40-50 years to age for 60 years, step for The limits of the 70 years of age. It is clear that such radical changes in the life of a person without internal and external problems do not fail, and this concerns any age. If such a fracture comes in childhood, the task of teachers and parents is to make it easier to make it easier for a child, skillfully and effectively help him overcome the difficulties that have arisen.

How best to do it? First of all, it is necessary to pay attention to the formation of full-fledged training activities among first-graders. The main parameters, signs and methods for evaluating the degree of development of this activity were described in previous section tutorial. We will add what is directly graders. Psychological and pedagogical analysis shows that they most often meet two types of difficulties: the implementation of the regime and entry into new relationships with adults. The most common phenomenon of a negative nature at this time is the suggestion of classes, quickly coming from many children shortly after their arrival in school. Externally, it is usually manifested in the impossibility of maintaining an initial natural interest in school and to educational subjects.

In order not to happen, it is necessary to include additional incentives for educational activities. With regard to six-seven-year children, such incentives can be both moral and material. Moral incentives It is not by chance that they are put in the first place, since in stimulating children of younger school age to teaching, they often turn out to be more effective than material. These include, for example, approval, praise, formulation of a child in an example of other children. It is important that carefully observing the behavior of the child, to note on time, to which it reacts best, and more often contact the forms of moral incentive, associated with this at the first time in school, it is advisable to exclude or minimize any punishment for poor studies. What concerns material rewards For progress, they, as practice shows, pedagogically and psychologically unfortunately and operate mainly situationally. They can be used, but can not be abused. At the same time, a combination of material with the moral methods of stimulating the teachings of the child.

Initially, the teaching process in junior school classes is based on the acquaintance of children with the main components of educational activities. These components, according to V. V. Davydov, the following: Educational situations, training actions, control and evaluation. In detail and slowly, it is necessary to demonstrate to children a certain sequence of training actions, highlighting among them those that must be carried out in the subject, stern and mental plans. It is important to create favorable conditions so that the subject actions acquired a mental form with their due to their generalization, abbreviation and development. If, when performing tasks, schoolchildren allow errors, it indicates either the incompleteness of the academic actions by them, as well as actions related to control and assessment, or on weak efforts of these actions. The child's ability to independently compare the results of actions performed with the peculiarities of the Action itself indicates that the initial types of self-control in its study activities have already been formed.

In training situations, children master the general ways to solve some class tasks, and the reproduction of these methods, acts as the main purpose of the study work. Having mastered them, the children immediately apply the found ways to solve the specific tasks with which they are found.

Actions aimed at mastering a general sample - a method for solving the problem, are accordingly motivated. The child is explained why you need to learn exactly the material.

Work on the development of general samples of action should precede the practice of their application when solving specific tasks and stand out as special in the educational process. One of the main requirements of psychology is to organize initial training so that the teaching of most and sections of the Program occurred on the basis of training situations that guide children to master the general ways to allocate the properties of some concept or general samples of solving a specific class task. Studies show that a number of significant deficiencies in mastering individual concepts and methods for solving problems are related to the fact that in the formation of these concepts and methods for solving problems, children were not trained to fulfill all the necessary training actions.

In the ability to convert specific-practical tasks in educational and theoretical manifests the most high level development of school educational activities. If in the younger school age this ability is not properly formed, then subsequently neither adjacent, nor good faith can be a psychological source of successful teaching. The need to control and self-control in training activities create favorable conditions for the formation of the younger schoolchildren with the ability to plan and implement actions to themselves, in the internal plan, as well as to arbitrary regulation.

In the development of thinking and speech, children are very helpful to spontaneous arguments out loud. In one of the experiments, a group of children of 9-10 years old were taught to reason out loud during the fulfillment of the task. The control group did not receive such experience. Children from the experimental group with the implementation of the intellectual task coped much faster and more efficiently than children from the control group. The need to reasoning out loud and the rationale for their decisions leads to the development of reflexivity as an important quality of the mind that allows a person to analyze and realize its judgments and actions. There is a development of arbitrary attention, transformation of memory processes on an arbitrary and meaningful basis. At the same time, arbitrary and involuntary types of memory interact and contribute to the development of each other.

Mental abilities and ability to learn the learning material with younger schoolchildren are quite high. With properly organized training, children perceive and assimilate more than the traditional school traditionally gives. The first thing you need to teach the younger student when performing homework is the allocation of an educational task. The child must clearly imagine how the task is to fulfill the task, he needs to master, for which it is necessary for something or another task as an educational, which it can teach.

Good results in teaching young children are given by group forms of classes, resembling plot role-playing games, which children are accustomed to even in preschool age and in which they are happy to participate. At first, school education can be recommended to organize joint, group learning activities. However, such a form of reference occupied, especially in the first months of learning children at school, requires careful training. One of the main tasks that must be resolved, starting to group training, is that it is properly distributed roles, to establish the atmosphere of benevolent interpersonal relations based on mutual assistance in the educational group.

Features of learning in the younger school age. The younger school age is the period in the life of the child about six to ten years, when it is undergoing training in primary classes.

During this period, the teaching is the main type of activity in which a person is formed. In primary grades, children start knowledge of the start of science. At this stage, the intellectual and informative sphere of the psyche is predominantly developed. At this stage there are many mental neoplasms, older are improved and developed. The school period is characterized by the intensive development of cognitive functions, sensory-perceptual, mental, mnemic and others. Usually student elementary school With hunting goes to this school.

For students of the first - third classes, the desire for the position of the schoolboy is characteristic. From the moment of entering the school, the central place is a social motive - the desire for a new social position of the schoolchild. In the first days of studies at school great importance Has experience acquired by the child at home. Previously, a small preschooler was the only and unique being, but with admission to school, it falls on Wednesday, where the same unique and only one around him. In addition to the need to adapt to the rhythm of school life and new requirements, master the space of the school, master the methods of self-organization and the organization of their time, the younger schoolboy must learn to interact with classmates.

But the main task of the younger schoolboy is to successfully study at school. It is also important to note that at the stage of the younger school age, the child is experiencing the so-called crisis of seven years. The child changes the perception of their place in the system of relationships.

The social situation of development is changing, and the child turns out to be on the border of the new age period. The child is aware of his place in the world of public relations and acquires a new social position of a schoolboy, which is directly related to educational activities. This process radically changes its self-consciousness, which leads to reassessment of values. Study becomes enormous for a schoolboy, therefore, for example, a circuit of the child's failures in this leading activity can lead to the formation of sustainable complexes or even chronic failure syndrome.

Of course, the teaching becomes the leading activity, it must also be organized in particular. An important element Curriculum is a game in the process of which the child learns to interact with peers, mastering social roles, requirements and rules adopted in human society. The game that takes social painting is developing feelings of rivalry and cooperation.

During the game, younger schoolchildren assimilate such concepts as equality, submission, justice, injustice. Usually younger schoolchildren prefer the company of their peers of one with them gender. The assimilation of the norms of behavior inherent in their semi and approved by society continues. In addition, younger students can not sit in one place for a long time. They need movement.

The lesson must contain not only the explanation of the new material, its fixation and repetition of the old one. But the time must also be given to various motor actions, games, mobile activities. Given that preschoolers, the game was leading activities, training activities, which becomes the lead at this stage of development, is directly related to the game. Therefore, training activities may occur only at a certain stage of the game. Thanks to the educational activities of the framework of perception, the child of the world is expanding.

Unconscious and fictional fears of past years are replaced by more conscious lessons, natural phenomena, injections. The most important personal characteristics of the younger schoolchildren include gullible subordination to authority, increased susceptibility, attentiveness, naive, playing relationships to a lot of what he faces. In the behavior of the student of primary classes, obedience, conformism and imitation is visible. Education at school is a new enough for children and therefore interesting activities, while they face a number of difficulties.

Schoolchildren originally, naturally, do not know how to independently formulate training tasks and perform actions to solve them. For the time being, a teacher helps them until time, but they are gradually the corresponding skills they acquire themselves in this process they have independent training activities, the ability to learn. Children at this age have a fraction of impulsiveness, capriciousness, stubbornness.

Volval processes are still not developed enough from younger students. But gradually the ability to show volitional efforts appears in mental activity and the behavior of schoolchildren. Schoolchildren have arbitrary mental actions, for example, intentional memorization, volitional attention, aimed and persistent observation, perseverance in solving a variety of tasks. Therefore, the value of the evaluation of the results of the schoolboy from adults increases. The educational and cognitive activity of the schoolchild as socially and individually meaning essentially has a dual stimulation inner when the schoolboy receives satisfaction, acquiring new knowledge and skills, and external when its achievements in knowledge is evaluated by the teacher.

Assessment from the teacher is an incentive for a student. This assessment strongly affects the student's self-esteem. Moreover, the need for evaluation and the power of experiences is much higher in weaker students. The assessment acts as promotion.

The assessment of the teacher helps the child over time to learn how to evaluate their work. Moreover, it should be not just an assessment of the result, but also the actions of the schoolchildren chosen by him to solve any particular task. A teacher in primary school schools cannot limit the magazine as an assessment of the student's activities. A meaningful assessment is important here, that is, the teacher needs to explain to the schoolboy, why this estimate is set, to highlight the positive and negative sides of the child's work. Subsequently, the teacher, assessing the training activities of children, its results and the process, forms the evaluation criteria in children.

Learning activities prompted by various motifs. The child appears a desire for self-development and cognitive need. This is an interest in the meaningful side of educational activities, to what is being studied, and interest in the activity process - how, in which results are achieved, training tasks are solved.

But not only the result of educational activities, the assessment is motivated by a small schoolboy, as well as the process of training activities and improving itself as a person, their talents, abilities. Schoolboy, becoming a subject of cognitive activity in the general system of educational influences, at the same time it acquires personal properties and personal attitude to what he does, and the learning process as a whole. The originality and complexity of the educational and cognitive activity of the school period is that it is carried out mainly in conditions of direct communication with teachers and students and school students.

At the beginning, the younger students are entirely relying on the opinion of the teacher. They look at the attitude of the teacher to various students and can even be taken through this attitude. But in the process of communicating with his classmates and educational activities, younger students are already more critical. They begin to evaluate both bad and good deeds.

Although the student with a teacher is still central to the educational process. In the younger school age, the most favorable opportunities for the formation of moral and social qualities, positive personalities of the person. Support and famous suggestibility of schoolchildren, their credulity, a tendency to imitation, a huge authority, which teacher uses, create favorable prerequisites for the formation of an highly moral personality.

The prevailing type of thinking is visual-shaped, and the process of holistic perception is not yet enough formed, attention is often involuntary. First-graders pay attention to the fact that the magnitude, form, color or color is distinguished brighter. The child has a long and thorny way of studying at school, during which he will master new items, new skills, new skills. It will be self-improvement, and develop its abilities, but the foundations for their further formation are laid in exactly the first years of study.

End of work -

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1.1 Features of learning younger schoolchildren

The boundaries of younger school age coinciding with a period of training in elementary school are currently being established from 6-7 to 9-10 years. During this period, the child's further physical and psychophysiological development of the child, which ensures the possibility of systematic learning in school. First of all, the work of the brain and nervous system. According to physiologists, by 7 years, the bark of large hemispheres is already in the characteristics of the behavior, organization and emotional sphere for children: younger schoolchildren are easily distracted, are not capable of long-term concentration, excitable, emotional. In the younger school age, the unevenness of psycho-physiological development among different children is noted. Differences are preserved in the pace of development of boys and girls: girls are still ahead of the boys. Pointing to it, some authors come to the conclusion that in fact children of different ages sit in the one and the same desk: on average boys younger girls for a year and a half, although this is a difference and not at calendar age. "

The beginning of training at school leads to a fundamental change in the social situation of the child's development. It becomes a "public" subject and now has socially significant duties, the execution of which receives a public assessment.

Leading in the younger school age is becoming training activities. It defines the most important changes in the development of the psyche of children at the age stage. As part of the training activities, psychological neoplasms are developing, characterizing the most significant achievements in the development of younger students and are the foundation that provides development in the next age stage.

According to L.S. Vygotsky, the specificity of younger school age is that the objectives of the activity are asked for children mostly adults. Teachers and parents define that it is possible and that it is impossible to do the child, what tasks to perform, what rules to obey. Even among those schoolchildren who are willing to fulfill an adult order, are quite frequent of cases when children do not cope with the tasks, since they did not care if it was quickly lost their initial interest in the task or simply forgotten to fulfill it on time. These difficulties can be avoided if, giving children any order, comply with certain rules.

Junior school age is the most responsible stage of school childhood. The high sensitivity of this age period determines the large potential capabilities of the diversified child's development. The main achievements of this age are due to the leading nature of the training activities and are largely defining training for the next years: by the end of the younger school age, the child should want to learn, be able to learn and believe in their strength.

Each age stage is characterized by the special position of the child in the system of relations adopted in this society. In line with this, the lives of children of different ages are filled with a specific content: a special relationship with the surrounding people and special, leading for this stage of development activities. Recall that L.S. Vygotsky allocated the following types of leading activities:

· Infants are directly emotional communication.

· Early childhood - manipulative activity.

· Preschoolers - game activities.

· Junior schoolchildren - training activities.

· Teenagers are socially educated and socially furnished activities.

· High school students - educational and professional activities.

Features of arbitrary memory of younger schoolchildren. The intention to remember this or that material does not yet determine the content of the mnemic task, which is to be solved by the subject. To do this, it should allocate a specific subject of memorization in the text, which represents a special task. Some schoolchildren as such a matter of memorization allocate cognitive text content (about 20% of schoolchildren), its other plot (23%), others do not allocate a certain subject of memorization. Thus, the task is transformed into different molemic tasks, which can be explained by differences in learning motivation and the level of formation of mechanisms for goalkeepers.

Thinking children of younger school age is significantly different from thinking of preschoolers. So if the preschooler thinking is characterized by such quality as involuntary, small handling and in the formulation of a mental task, and in its solution, they are more and easier and easier to think about what they are interested that they are fascinated. That junior schoolchildren as a result of training at school, when it is necessary to regularly fulfill the tasks on mandatory, learn how to manage your thinking, think when necessary.

In many ways, the formation of such arbitrary, manageable thinking contributes to the instructions of the teacher in a lesson, encouraging children to think. When communicating in primary classes, children are formed conscious critical thinking. This is due to the fact that in the class it is discussed ways to solve problems, various solutions are considered. The teacher constantly requires schoolchildren to justify, tell, prove the correctness of his judgment, requires children to solve the tasks on their own.

Thus, the presence of a particular type of thinking from the child can be judged by how it solves the corresponding this kind tasks. If the child successfully solves light challenges designed to apply one or another type of thinking, but it is difficult to solve more complex, then in this case it is believed that he has a second level of development in the appropriate form of thinking.

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The development of younger students in the process of learning mathematics

What is educational training?

The term "educational training" is actively used in psychological, pedagogical and methodical literature. However, the content of this concept remains very problematic, and the answers to the question: "What kind of training can be called developing?" Pretty contradictory. This, on the one hand, is due to the multidimensionality of the concept of "educational training", and on the other hand, some contradictory of the term itself, because It is unlikely that you can talk about "undeveloping learning." Undoubtedly, any training is developing a child.

However, it is impossible to disagree with the fact that in one case, training seems to be discharged over development, as L. S. Vygotsky, "Woven in the tail" at the development, providing a natural influence on him, in the other - purposefully ensures it (leads to development) and actively uses knowledge, skills, skills for learning. In the first case, we have priority information function Learning, in the second - the priority of the developing function, which radically changes the construction of the learning process.

As writes D.B. Elkonin - the answer to the question, in which relationship these two processes are, "complicated by the fact that the categories of learning and development are different.

The effectiveness of training is usually measured by the number and quality of acquired knowledge, and the development efficiency is measured by the level of which they reach the ability of students, i.e., how developed the basic forms of their mental activity are achieved, allowing you to quickly, deeply and correctly navigate in the phenomena of the surrounding reality.

It has long been noticed that you can know much, but at the same time not to show any creative abilities, i.e., do not be able to independently understand the new phenomenon, even from the relatively well-known sphere of science " .

It is not by chance that the term "educational training" methodists are used with great care. Complex dynamic relations between the processes of training and mental development of the child are not subject to the study of the methodological science, in which real, practical learning results are customary to describe in the language of knowledge, skills and skills.

Since psychology is engaged in the study of the psychic development of the child, then when building a developing learning, the technique must undoubtedly be based on the results of studies of this science. As V.V.Davdov writes, "the mental development of a person is, first of all, the formation of his activities, consciousness and, of course, all" serving "them mental processes (cognitive processes, emotions, etc.) " . It follows that the development of students largely depends on the activities they perform in the learning process.

From the course didactics, you know that this activity can be reproductive and productive. They are closely related to each other, but depending on what type of activity prevails, training has a different impact on the development of children.

Reproductive activities are characterized by the fact that the student receives the finished information, perceives it, understands, remembers, then reproduces. The main purpose of such activities is the formation of a schoolboy with knowledge, skills and skills, the development of attention and memory.

Productive activities are related to the active work of thinking and finds its expression in such mental operations as analysis and synthesis, comparison, classification, analogy, generalization. These thinking operations in psychological and pedagogical literature are customary called logical techniques of thinking or techniques of mental actions.

The inclusion of these operations into the process of mastering mathematical content is one of the important conditions for building educational training, as the productive (creative) activity has positive influence on the development of all mental functions. "... The organization of developing learning implies the creation of conditions for mastering schoolchildren techniques of mental activity. Mastering them not only provides a new level of assimilation, but gives substantial shifts in the mental development of the child. Miscellaneous by these techniques, students become more independent in solving educational tasks, can rationally build their activities on learning knowledge " .

Consider the possibilities of active inclusion in the process of learning the mathematics of various techniques of mental actions.

3.2. Analysis and synthesis

The most important thinking operations are the analysis and synthesis.

The analysis is associated with the allocation of elements of this object, its signs or properties. Synthesis is a compound of various elements, the sides of the object in a single whole.

In the mental activity of man, the analysis and synthesis complement each other, since the analysis is carried out through synthesis, synthesis - through the analysis.

The ability to analytical synthetic activity finds its expression not only in the ability to allocate elements of a particular object, its various features or to connect elements into a single whole, but also in the ability to include them in new connections, see their new features.

The formation of these skills may contribute to: a) consideration of this object in terms of various concepts; b) setting various tasks to this mathematical object.

To consider this object in terms of various concepts of younger schoolchildren, such tasks are usually proposed when teaching mathematics:

Read differently expressions 16 - 5 (16 decreased by 5; the difference of numbers 16 and 5; out of 16 subtracts 5).

Read in different ways 15-5 \u003d 10 (15 decrease by 5, we obtain 10; 15 more than 10 to 5; the difference of numbers 15 and 5 is 10;

15 - reduced, 5 - subtracted, 10 - difference; If the difference (10) add subtracted (5), then we obtain the reduced (15); Number 5 less than 15 to 10).

How can you call a square differently? (Rectangle, quadrangle, polygon.)

Tell me all you know about the number 325. (This is a three-digit number; it is written in numbers 3, 2, 5; in it 325 units, 32 dozen, 3 hundred; it can be written in the form of the sum of the discharge terms as follows: 300 + 20 + 5 ; It is 1 unit greater than the number 324 and 1 unit less than the number 326; it can be represented as the sum of the two terms, three, four, etc.)

Of course, it should not strive for each student to uttered this monologue, but, focusing on it, you can offer children questions and tasks, when performing which they will consider this object from various points of view.

Most often it is tasks to classify or identify various patterns (rules).

For example:

What features can you decompose buttons in two boxes?

Considering buttons from the point of view of their size, we put 4 buttons in one box, and in another 3,

– from the point of view of color: 1 and 6,

– from the point of view of form: 4 and 3.

Sold down the rule by which the table is drawn up, and fill the missed cells:

Seeing that in this table two lines, students try to identify a certain rule in each of them, find out how much one number is less (more) of another. For this, they perform addition and subtraction. Without finding a pattern either in the top, nor in the bottom line, they try to analyze this table from another point of view, comparing each number of the top row with the corresponding (standing) number of the bottom, string. Get: 4 8 to 1; 3\u003e 2 at 1. If at the number of 8 to write a number 9, and at number 6 - number 7, we have:

8 p per 1, p\u003e 4 to 1.

Similarly, you can compare each number of the bottom line with the corresponding (standing above) the number of the top row.

Such assignments with geometric material are possible.

Find a segment of the sun. What can you tell about him? (Sun - the side of the triangle everything; Sun - the side of the triangleDBC.; Sun smaller thanDC; Sun is less than AV; Sun - side of the cornerBCD. And the corner is all).

How many segments on this drawing? How many triangles? How many polygons?

Consideration of mathematical objects in terms of various concepts is a way to compile variable tasks. Take, for example, such a task: "We write out all even numbers from 2 to 20 and all odd numbers from 1 to 19. " The result of its execution is the recording of two rows of numbers:

2, 4, 6, 8, 10,12,14,16,18,20 1,3,5,7,9, 11, 13, 15, 17, 19

We now use these mathematical objects to draw up tasks:

Spacing the number of each row into two groups so that each has numbers similar to each other.

What rule is the first row recorded? Continue it.

What numbers need to be deleted in the first row so that each next is 4 more than the previous one?

Is it possible to perform this task for the second row?

Pick up from the first row of a couple of numbers, the difference of which is equal to 10

(2 and 12, 4 and 14, 6 and 16, 8 and 18, 10 and 20).

From the second row of a couple of numbers, the difference of which is 10 (1 and 11.3 and 13, 5 and 15, 7 and 17, 9 and 19).

What a pair of "extra"? (10 and 20, in it two double digits, in all other pairs there is a two-digit number and unambiguous).

Find in the first row the sum of the first and last number, the sum of the second numbers from the beginning and on the end of the row, the sum of the third numbers from the beginning and on the end of the row. What are these amounts like?

Take the same task for the second row. What are the amounts given?

Task 80. Come up with tasks in the process of executing which students will consider the data in them objects from different points of view.

3.3. Receive comparison

A special role in organizing the productive activities of junior schoolchildren in the process of learning mathematics plays a comparison. The formation of the ability to use this technique should be carried out in stages, in close connection with the study of specific content. It is advisable, for example, to focus on the steps:

selection of features or properties of one object;

establishing similarities and differences between features of two objects;

identification of similarities between features of three, four or more objects.

Since the work on the formation of the children in the children of logical comparison, it is better to start from the first lessons of mathematics, then as objects you can first use objects or pictures with the image of items, well to them, in which they can highlight certain signs, based on them representation.

To organize students aimed at identifying signs of one or another object, one must first offer such a question:

– What can you tell about the subject? (Round apple, large, red; pumpkin - yellow, large, striped, with a tail; round-big, green; square, small, yellow).

In the process of work, the teacher introduces children with the concepts of "size", "Form" and offers them the following questions:

– What can you say about the sizes (forms) of these items? (Large, small, round, like a triangle, like a square, etc.)

To identify signs or properties of some kind of subject, the teacher usually refers to children with questions:

– What is the similarity and distinction of these items? - What changed?

It is possible to introduce them with the term "sign" and use it when performing tasks: "Name signs of the subject", "call similar and various signs of objects."

Task 81. Pick up various pairs of items and images that you can offer first-graders to establish similarities and distinction between them. Come up with the illustration for the task "What has changed ...".

The ability to allocate signs and, focusing on them, compare the items of students are transferred to mathematical objects.

V Name signs:

a) expressions 3 + 2 (numbers 3, 2 and the sign "+");

b) expressions 6-1 (numbers 6, 1 and the sign "-");

c) equality x + 5 \u003d 9 (x - unknown number, numbers 5, 9, signs "+" and "\u003d").

According to these external features accessible to perception, children can establish similarities and the difference between mathematical objects and comprehend these signs from the point of view of various concepts.

For example:

What is the similarity and difference:

a) expressions: 6 + 2 and 6-2; 9 4 and 9 5; 6+ (7 + 3) and (6 + 7) +3;

b) numbers: 32 and 45; 32 and 42; 32 and 23; 1 and 11; 2 and 12; 111 and 11; 112 and 12, etc;

c) equalities: 4 + 5 \u003d 9 and 5 + 4 \u003d 9; 3 8 \u003d 24 and 8 3 \u003d 24; 4 (5 + 3) \u003d 32 and 4 5 + 4 3 \u003d 32; 3 (7 10) \u003d 210 and (3 7) 10 \u003d 210;

d) texts of tasks:

Kolya caught 2 fish, Petya - 6. As far as she caught Petya fish more than Kohl?

Kolya caught 2 fish, Peter - b. How many times did the fish caught Petya than Kohl? e) geometric shapes:

e) equations: 3 + x \u003d 5 and x + 3 \u003d 5; 10-x \u003d 6 and (7 + 3) -x \u003d 6;

12-x \u003d 4 and (10 + 2) -h \u003d 3 + 1;

g) computing receptions:

9 + 6 \u003d (9 + 1) +5 and 6 + 3 \u003d (6 + 2) +1

L L.

1+5 2+1

Reception of comparison can be used when meeting students with new concepts. For example:

What are all like each other:

a) numbers: 50, 70, 20, 10, 90 (discharge tens);

b) geometric shapes (quadrangles);

c) Mathematical recordings: 3 + 2, 13 + 7, 12 + 25 (expressions called sum).

Task 82. Make out of these mathematical expressions:

9 + 4, 520-1.9 4, 4 + 9, 371, 520 1, 371, 520 1, 33, 13 1,520: 1,333, 173, 9 + 1, 520 + 1, 222, 13: 1 Various pairs in which children can reveal Signs of similarities and differences. When studying what questions of the course of the Mathematics of the initial classes, you can offer each of your tasks?

In training younger schoolchildren, a large role is given to the exercises that are associated with the transfer of "subject actions" into the language of mathematics. In these exercises, they usually relate subject objects and symbolic. For example:

a) What picture corresponds to records 2 * 3, 2 + 3?

b) What picture corresponds to recording 3 5? If there is no such pattern, then draw it.

c) Pictures corresponding to this posts: 3 * 7, 4 2 + 4 * 3, 3 + 7.

Task 83. Come up with various exercises on the correlation of subject and symbolic objects that can be offered by students when studying the meaning of addition, division, multiplication tables, division with the residue.

An indicator formed ™ receiving a comparison is the ability of children to independently use it to solve various tasks, without specifying: "Compare ..., indicate signs .., what is the similarity and distinction ...".

We present specific examples of such tasks:

a) remove the lipstick item ... (When fulfilling his schoolchildren focus on the similarity and difference of signs.)

b) Around the number in Ascending order: 12, 9, 7, 15, 24, 2. (To perform this task, students should identify signs of differences in these numbers.)

c) The amount of numbers in the first column is 74. How, without performing addition in the second and third columns, find the amounts of numbers:

21 22 23

30 31 32

11 12 13

12 13 14 74

d)) continue the ranks of numbers: 2, 4, 6, 8, ...; 1, 5, 9, 13, ... (The basis for establishing patterns (rules) of the number of numbers is also a comparison operation.)

Task 84. Show the possibility of applying comparison when studying the addition of unambiguous numbers within 20, addition and subtraction within 100, rules of the procedure for performing actions, as well as when you meet younger students with a rectangle and square.

3.4. Accepting classification

The ability to allocate signs of objects and establish similarity between them and the difference is the basis for accepting classification.

From the course of mathematics it is known that when splitting the set to classes, the following conditions must be performed: 1) None of the subsets are empty; 2) subsets in pairs do not intersect;

3) The combination of all subsets is this set. By offering children to the classification for children, these conditions must be considered. Just as in the formation of comparison, children first fulfill tasks to the classification of well-known objects and geometric shapes. For example:

Students consider items: cucumber, tomato, cabbage, hammer, onions, beets, radish. Focusing on the concept of "vegetable", they can break a lot of items into two classes: vegetables are not vegetables.

Task 85. Invent the exercises of various content with the instructions "Elecess the extra object" or "name of an excess subject" that you could offer students of the 1st, 2nd, 3rd grade.

The ability to perform classification is formed in schoolchildren in close connection with the study of specific content. For example, for exercises in the score, they are often offered illustrations to which you can put questions starting with the words "how much ...". Consider the drawing to which you can put the following questions:

- How many big circles? Little? Blue? Red? Big red? Little blue?

Exercising in the account, students are seduced by a logical admission of classification.

The tasks associated with the reception of the classification are usually formulated in this form: "Emboss (spread) all the circles into two groups on some kind of sign."

Most children successfully cope with this task, focusing on such signs as color and size. As the various concepts of the task, the classification may include numbers, expressions, equality, equations, geometric shapes. For example, when studying numbers of numbers within 100, it is possible to offer such a task:

Split the data of the number into two groups so that each turns out to be similar:

a) 33, 84, 75, 22, 13, 11, 44, 53 (in one group included numbers recorded by two identical numbers to another - different);

b) 91, 81, 82, 95, 87, 94, 85 (the basis of the classification is the number of tens, in one group of numbers it is 8, in the other - 9);

c) 45, 36, 25, 52, 54, 61, 16, 63, 43, 27, 72, 34 (base classification "numbers", which recorded the number of numbers in one group is 9, in the other - 7 ).

If the task does not indicate the number of split groups, then various options are possible. For example: 37, 61, 57, 34, 81, 64, 27 (these numbers can be divided into three groups, if you focus on the numbers recorded in the discharge of units, and into two groups, if you focus on the numbers recorded in the discharge of dozens. Possible And another grouping).

Task 86. Make exercises to the classification that you could offer children to assign the numbering of five-digit and six-digit numbers.

When studying the addition and subtraction of numbers within 10, such assignments are possible:

Spread these expressions into groups on some basis:

a) 3 + 1, 4-1, 5 + 1, 6-1, 7 + 1, 8 - 1. (In this case, the base for splitting into two groups, children are easily found, since the sign is clearly submitted in the recording of the expression.)

But you can pick up other expressions:

b) 3 + 2, 6-3, 4 + 5, 9-2, 4 + 1, 7 - 2, 10 - 1, 6 + 1, 3 + 4. (Bringing into groups given many expressions, students can navigate not only on the sign of arithmetic action, but also on the result.)

Getting Started with new tasks, children usually first focus on the signs that took place when performing preceding tasks. In this case, it is useful to indicate the number of split groups. For example, to expressions: 3 + 2, 4 + 1, 6 + 1, 3 + 4, 5 + 2, you can propose a task in this wording: "Over the expression on three groups on some basis." Pupils naturally first focus on the sign of arithmetic action, but then there is no partition into three groups. They begin to navigate the result, but only two groups are also obtained. In the search process, it turns out that it is possible to smash into three groups, focusing on the meaning of the second term (2, 1, 4).

As a basis for splitting expressions on groups, a computing reception can also be. To this end, you can use the task of this type: "According to what a feature you can split the expression data into two groups: 57 + 4, 23 + 4, 36 + 2, 75 + 2, 68 + 4, 52 + 7.76 + 7.44 + 3,88 + 6, 82 + 6? "

If students cannot see the necessary grounds for classification, then the teacher helps them as follows: "In one group, I will write such an expression: 57 + 4," he says, to another: 23 + 4. What group will you record expression 36 + 9? " If, in this case, children are hampered, the teacher can tell them the base: "What computing reception do you use to find the meaning of each expression?".

The tasks for the classification can be applied not only for productive consolidation of knowledge, skills and skills, but also when meeting students with new concepts. For example, to determine the concept of "rectangle" to the set of geometric figures located on the flannelhemph, one can offer such a sequence of tasks and questions:

Remove the "excess" figure. (Children remove the triangle and actually split the many figures into two groups, focusing on the number of sides and corners in each figure.)

What are all the other figures like? (They have 4 angle and 4 sides) v How can all these figures call? (Quadrangles.)

Show the quadrangles with one straight angle (6 and 5). (To verify its assumption, the students use a direct angle model, appropriately applying it to the specified figure.)

Show the quadrangles: a) with two straight corners (3 and 10);

b) with three straight corners (no such); c) with four straight corners (2, 4, 7, 8, 9).

Speakers of quadrangles on the group by the number of direct angles (1st group - 5 and 6, 2nd group - 3 and 10, 3rd group - 2, 4, 7, 8, 9).

The quadrangles are appropriately laid out on the flannelhemph. The third group includes quadrangles, who have all the corners are direct. These are rectangles.

Thus, when teaching mathematics, you can use tasks for the classification of various types:

1. Preparatory tasks. These include: "Element (name)" Excess "subject", "Draw objects of the same color (forms, size)", "give a name to a group of subjects." This can also include tasks for the development of attention and observation:

"What subject removed?" And "What has changed?".

2. Tasks in which the teacher indicates on the basis of the classification.

3. Tasks, when executing which children themselves identify the basis of the classification.

Task 87. Make up different kinds Tasks for the classification that you could offer students when studying geometric material, division with the residue, computational techniques of oral multiplication and division within 100, as well as when meeting a square.

3.5. Acceptance of analogy

The concept of "similar" translated from the Greek language means "similar", "appropriate", the concept of an analogy is similarity in any respect between objects, phenomena, concepts, methods of action.

In the process of learning mathematics, the teacher often says to children: "Make analogy" or "This is a similar task." Typically, such instructions are given to consolidate certain actions (operations). For example, after considering the multiplication properties of the amount, various expressions are offered:

(3 + 5) 2, (5 + 7) 3, (9 + 2) * 4, etc., with which actions similar to this sample are performed.

But another option is also possible when, using an analogy, students find new ways of activity and check their guess. In this case, they themselves should see the similarities between the objects in some respects and independently express a guessed about similarity in other respects, i.e., make a conclusion by analogy. But so that students can express "guess", it is necessary to organize their activities in a certain way. For example, students learned the translation algorithm of two-digit numbers. Turning to the written addition of three-digit numbers, the teacher invites them to find the values \u200b\u200bof expressions: 74 + 35, 68 + 13, 54 + 29, etc. After that, asks: "Who can guess how to make the addition of such numbers: 254 + 129?". It turns out that there were two numbers in the examined cases, the same is offered in the new case. With the addition of two-digit numbers, they were recorded one under the other, focusing on their discharge composition, and folded downwardly. Cancellation occurs - it is probably possible to add three-digit numbers. The conclusion of the correctness of the guessed may give a teacher or to offer children to compare performed actions with the sample.

The conclusion is also possible to apply during the transition to a written addition and subtraction of multivalued numbers, comparing it with addition and subtraction of three-digit.

The conclusion can be used by analogy when studying the properties of arithmetic action. In particular, the transition property of multiplication. For this purpose, students first are invited to find values \u200b\u200bof expressions:

6+3 7+4 8+4 3+6 4+7 4+8

– What property did you use when performing a task? (By the prolonged property of addition).

– Think: how to install whether a moving property is performed for multiplication?

Students write down the pairs of works and find the value of each, replacing the product amount.

For proper conclusions, by analogy, it is necessary to highlight essential features of objects, otherwise the conclusion may be incorrect. For example, some students are trying to apply the method of multiplying the number in the amount of multiplying the number on the work. This suggests that the essential property of this expression - multiplication of the amount turned out to be outside their field of view.

Forming in the younger schoolchildren, the ability to fulfill conclusions by analogy, it is necessary to keep in mind the following:

The analogy is based on comparison, so the success of its application depends on how many students know how to identify the signs of objects and establish similarities and the difference between them.

To use analogy, you must have two objects, one of which is known, the second is compared with it for any signs. Hence, the use of analogue reception contributes to the repetition of the studied and systematization of knowledge and skills.

For the orientation of schoolchildren to use analogy, it is necessary in an accessible form to clarify them the essence of this reception, turning their attention to the fact that in mathematics it is often new way Actions can be discussed by guessing, remembering and analyzing a well-known method of action and this new task.

For the correct actions, by analogy, signs of objects substantial in this situation are compared. Otherwise, the conclusion may be incorrect.

Task 88. Give examples of conclusions by analogy that can be used in the study of written multiplication and division algorithms.

3.6. Reception of generalization

The allocation of essential signs of mathematical objects, their properties and relationships is the main characteristic of such a technique of mental actions as a generalization.

The result and the process of generalization should be distinguished. The result is fixed in concepts, judgments, rules. The process of generalization can be organized in different ways. Depending on this, they are talking about two types of generalization - theoretical and empirical.

In the course of initial mathematics, an empirical type is most often used, in which the generalization of knowledge is the result of inductive reasoning (conclusions).

Translated into Russian, "induction" means "guidance", therefore, using inductive conclusions, students can independently "open" mathematical properties and ways of action (rules), which in mathematics are strictly proved.

For proper generalization, an inductive way is necessary:

1) Think over the selection of mathematical objects and the sequence of issues for targeted observation and comparison;

2) consider as many private objects as possible in which the regularity is repeated that students should notice;

3) vary the types of private objects, i.e. use subject situations, schemes, tables, expressions, reflecting in each form of the object and the same pattern;

4) Helping children verbally to formulate their observations by asking leading questions, specifying and correcting the formulations they offer.

Consider on a specific example, how to implement the recommendations you can implement. In order to bring students to the wording of the Multiplication Properties, the teacher offers them such tasks:

Consider the drawing and try to quickly calculate how many windows in the house.

Children can offer the following ways: 3 + 3 + 3 + 3, 4 + 4 + 4 or 3 * 4 \u003d 12; 4 * 3 \u003d 12.

The teacher proposes to compare the obtained equality, i.e., to identify their similarity and difference. It is noted that both works are the same, and multipliers are rearranged.

A similar task Students are performed with a rectangle that is divided into squares. As a result, 9 * 3 \u003d 27 is obtained; 3 * 9 \u003d 27 and verbally describe those similarities and differences that exist between recorded equalities.

Pupils are offered independent work: Find the values \u200b\u200bof the following expressions, replacing multiplication by adding:

3*2 4*2 3*6 4*5 5*3 8*4 2*3 2*4 6*3 5*4 3*5 4*8

It turns out what are similar and what is the difference between equality in each column. The answers may be such: "The factors are the same, they are no longer", "the works are the same" or "the factors are the same, they are rearranged, works are the same."

The teacher helps to formulate a property with the help of a leading question: "If multipliers rearrange, what can I say about the work?"

Conclusion: "If the multipliers are rearranged, the work will not change" or "the value of the multipliers will change the value of the work."

Task 89. Pick up the tasks sequence that can be used to perform inductive conclusions when learning:

a) rules "If the work of two numbers is divided into one multiplier, then we get another":

b) the recessing properties of the addition;

c) the principle of formation of a natural range of numbers (if to add a unit to the number, then we obtain the following at the score; if deductible 1, then we get the previous number);

d) interrelations between divisible, divisory and private;

e) conclusions: "The sum of two consecutive numbers is odd"; "If the subsequent number of subtraction is the previous one, then I will succeed; "The product of two consecutive numbers is divided into 2"; "If you add to any number, and then subtract out of it the same number, then we get the original number."

Describe work with these tasks, taking into account the methodological requirements for the use of inductive reasoning when studying a new material.

Forming in the younger schoolchildren, the ability to summarize the observed facts in an inductive way is useful to offer tasks, when performing which they can make incorrect generalizations.

Consider several such examples:

Compare expressions, find a common in the received inequalities and

make appropriate conclusions:

2+3 ...2*3 4+5...4*5 3+4...3*4 5+6...5*6

By comparing these expressions and noting patterns: the amount is recorded on the left, on the right, the work of two consecutive numbers; The amount is always less than the work, most children conclude: "The sum of two consecutive numbers is always less than the work." But the generalization expressed is erroneously, since cases are not taken into account:

0+1 ...0*1

1+2... 1*2

You can try to make the right generalization, in which certain conditions will be taken into account: "The sum of two consecutive numbers, starting with Numbers 2, is always less than the work of the same numbers."

Find the amount. Compare it with each allegation. Make the appropriate conclusion.

Speed

Based on the analysis of the considered special cases, students come to the conclusion that: "The amount is always greater than each of the terms". But it can be refuted, as: 1 + 0 \u003d 1, 2 + 0 \u003d 2. In these cases, the amount is equal to one of the terms.

V Verify whether each term will be divided into number 2, and make a conclusion.

(2+4):2=3 (4+4):2=4 (6+2):2=4 (6+8):2=7 (8+10):2=9

Analyzing the proposed private cases, children may come to the conclusion that: "If the amount of numbers is divided into 2, then each term is divided into 2". But this conclusion is erroneous, as it can be refuted: (1 + 3): 2. Here the amount is divided into 2, every term is not divided.

Task 90. \u200b\u200bUsing the content of the initial mathematics course, come up with tasks when performing students can make incorrect inductive conclusions.

Most psychologists, teachers and methodologists believe that empirical generalization is based on the comparison, for younger schoolchildren is most accessible. This, in fact, is due to the construction of a math course in primary classes.

Comparing mathematical objects or action methods, the child highlights their external general propertieswhich can be the content of the concept. However, the external benchmark, available for perception, the properties of compared mathematical objects does not always make it possible to reveal the essence of the concept being studied or assimilate the general way of action. In case of empirical generalization, students often focus on the insignificant properties of objects and on specific situations. This adversely affects the formation of concepts and general methods of action. For example, forming the concept of "more on", the teacher usually offers a series of specific situations that differ from each other only with numerical characteristics. In practice, it looks like this: the children are invited to put three red mug in a row, under them put the same blue, then it turns out - how to make it so that in the bottom row there are more circles on 2 (add 2 mug). The teacher then proposes to put in the first row 5 (4,6,7 ...) circles, in the second row on 3 (2.5,4 ...) more. It is assumed that as a result of the fulfillment of such tasks, the child will be formed by the concept of "more on", which will find its expression in the method of action: "Take the same and more ...". But, as practice shows, the focus of students in this case, first of all, there are various numerical characteristics, and not a general method of action. Indeed, by completing the first task, the student can conclude only about how to "make more on 2" by following the following tasks - "How to do more on 3 (by 4, by 5)", etc. As a result, generalized verbal The wording of the method of action: "You need to take the same amount and also" is given by the teacher, and most children assimilate the concept of "more on" only as a result of the implementation of monotonous training exercises. Therefore, they are able to perform certain arguments only within the framework of this particular situation and on a limited area of \u200b\u200bnumbers.

Unlike empirical, theoretical generalization is carried out by analyzing data on any one object or situation in order to identify significant internal connections. These bonds are immediately recorded abstractly (theoretically - with the help of the word, signs, schemes) and become the basis on which private (specific) actions are performed.

Prerequisite Formation of the younger schoolchildren's ability to theoretical generalization is the focus of training for the formation of general ways of activity. To fulfill this condition, we need to consider such actions with mathematical objects, as a result of which children will be able to "open" the essential properties of the concepts and general methods of action with them.

The development of this issue at the methodological level represents a certain complexity. Currently, this is one of the most relevant initial learning issues, the solution of which is associated both with a change in content and a change in the organization of the learning activities of younger students aimed at assimilation.

To the course of initial mathematics (V.V. Davydov), the purpose of which is the development of ability to theoretical generalization in children, significant changes have been made. They concern and its contents and ways to organize activities. The basis of theoretical generalizations in this course is subject actions with values \u200b\u200b(length, volume), as well as various methods of modeling these actions using geometric shapes and symbols. This creates certain conditions for the implementation of theoretical generalizations. Consider a specific situation that is associated with the formation of the concept of "more on". Two banks are offered students. In one (first) nanita water, the other (second) is empty. The teacher proposes to find a way to solve the following problem: how to make it so that in the second bank of water would be to this cup (shows a cup with water) more than in the first? As a result of the discussion of various proposals, a conclusion is concluded: it is necessary to pour water from the first bank to the second, that is, pour into the second the same water as it is poured into the first jar, and then pour into the second more cup of water. The created situation allows children to find the necessary way of action to find the necessary way, and the teacher focus on the essential sign of the concept of "more on", that is, to identify students for mastering the general way: "as much as yet."

The use of values \u200b\u200bfor the formation of generalized methods from schoolchildren is one of possible options Building the initial course of mathematics. But the same task can be solved by performing various actions and with sets of objects. Examples of such situations were reflected in Articles G. Mikulina .

It advises to form the concept of "more on" to use the situation with sets of items: the children are offered a pack of red cards. It is necessary to fold a pack of green cards so that it was here so much (a bundle of blue cards is shown) more than in a pack of red. Condition: Card can not be recalculated.

Taking advantage of the method of establishing mutually unequivocal compliance, the students lay as many cards in a green pack as they are in red, and add another third bundle to it (from blue cards).

Along with empirical and theoretical generalizations in the course of mathematics, there is a generalization agreement. Examples of such generalizations are the rules of multiplication by 1 and by 0, are valid for any number. They are usually accompanied by explanations:

"Mathematics agreed ...", "It is considered in mathematics ...".

Task 91. Using the contents of the initial mathematics, come up with situations for theoretical and empirical generalization when studying any concept, properties or method of action.

3.7. Ways to substantiate the truth of judgment

An indispensable condition for developing learning is the formation of students to justify (prove) the judgments that they express. In practice, this ability is usually associated with the ability to reason, prove its point of view.

The judgments are isolated: something is approved or denied by one subject. For example: "Number 12; The ABSD square does not have sharp corners; Equation 23-x \u003d 30 has no solution (as part of the initial classes), etc..

In addition to single judgments, the judgments are private and common. In private, something is approved or denied relative to a certain set of objects from this class or relative to a certain subset of this set of objects. For example: "Equation X - 7 \u003d 10 is solved on the basis of the relationship between the reduced, subtracted and difference." In this judgment, we are talking about the partial view equation, which is a subset of the set of all equations studied in primary classes.

In general judgments, something is approved or denied with respect to all items of this population. For example:

"In the rectangle, the opposite directions are equal." Here we are talking about any, i.e. About all rectangles. Therefore, judgment is general, although in this proposal the word "all" is missing. Any elementary equation is solved on the basis of the relationship between the results and components of arithmetic action. It is also a general judgment, since it covers all kinds of equations encountered in the course of the math of primary classes.

Offers expressing judgments may be different in shape: affirmative, negative, conditional (for example: "If the number ends with zero, it is divided into 10").

As you know, in mathematics, all offers, with the exception of the initial, as a rule, are proved deductive. The essence of deductive reasoning is reduced to the fact that on the basis of some general judgment on the subjects of this class and some isolated judgment about this object, a new unit judgment is expressed about the same object. A general judgment is customary to be called a common parcel, a first unit judgment - a private parcel, a new unit judgment - the conclusion. Let, for example, it is required to solve the equation: 7 * x \u003d 14. To find an unknown multiplier, a rule is used: "If the value of the product is divided into one multiplier (known), we will get another (the value of an unknown multiplier)".

This rule (general judgment) is a common parcel. In this equation, the work is 14, the famous multiplier 7. This is a private parcel.

Conclusion: "You need to separate on 7, we get 2". The peculiarity of deductive reasoning in primary grades is that they are used in an implicit form, that is, the total and private parcels are lowered in most cases (not primarily pronounced), students immediately begin the action that matches the conclusion.

Therefore, in fact, it seems that deductive arguments are absent in the course of the mathematics of primary classes.

For conscious implementation of deductive conclusions, a large preparatory work is necessary to assimilate the withdrawal, patterns, properties in general, associated with the development of mathematical speech of students. For example, quite a long work on the assimilation of the principle of building a natural range of numbers allows students to master the rule:

"If you add 1 to any number, then we get the following number followed; If one of any number will be subtracted 1, then we obtain the number preceding it. "

Making up the table P + 1 and P - 1, the student actually uses this rule as a general premise, thereby performing deductive arguments. An example of deductive conclusions in primary learning mathematics is that reasoning:

"four

Deductive arguments take place in the initial course of mathematics and when calculating the values \u200b\u200bof expressions. As a common parcel, the rules of the procedure for performing actions in expressions, as a private parcel, is a specific numerical expression, when the meaning of which students are guided by the rule of the procedure for performing actions.

An analysis of school practice allows us to conclude that for the formation of schoolchildren of skills, all methodological capabilities are not always used to reason. For example, when performing a task:

Compare expression by putting a sign<.> or \u003d to make a correct entry:

6+3 ... 6+2 6+4 ... 4+6

students prefer to replace the reasoning with computing:

"6 + 2 . She offered two sheets for children, on one of which general parcels were written, on the other - private. You need to set which general premise is each private. Pupils are given instructions: "You must perform each task on a sheet 2, without resorting to calculations, but only using one of the rules recorded on the sheet 1.

Task 92. Following the above instructions, do this task.

Sheet 1.

1. If a diminished increase by several units, without changing the subtractable, then the difference will increase the same units.

2. If the divider is reduced several times without changing the division, then the private will increase the same time.

3. If one of the components increase by several units, without changing the other, then the amount will increase the same units.

4. If each term is divided into this number, then the amount is also divided into this number.

5. If the number of one precedes the number preceding it, then we get ...

Leaf 2.

Tasks are located in another sequence than parcels.

1. Find the difference 84 - 84, 32 - 31, 54 - 53.

2. Name the amounts that are divided by 3: 9 + 27, 6 + 9, 5 + 18, 12 + 24, 3 + 4, "+6.

3. Compare expressions and put signs<.> or \u003d:

125–87 ... 127–87 246–93 ... 249–93 584–121... 588– 121

4. Compare the expression and put signs or \u003d:

304:8 ... 3044 243:9 ... 243:3 1088:4 . . 1088:2

5. How to quickly find the amount in each column:

9999 12 15 12 16 30 30 32 32 40 40 40 40 Reply: 91.

Thus, deductive arguments may be one of the ways to substantiate the truth of judgments in the initial course of mathematics. Considering that they are not available to all younger students, in primary classes are used other ways to substantiate the truth of judgments, which in a strict sense cannot be attributed to evidence. These include experiment, calculations and measurements.

The experiment is usually associated with the use of visibility and subject action. For example, a child can justify the judgment 7\u003e 6, posted in one row of 7 circles, under it - 6. By setting between the circles of the first and second row, it actually substantiates its judgment (in the first row one circle without a pair, "Excess "So 7\u003e 6). The child can refer to the subject action and to substantiate the truth of the result obtained when adding, subtracting, multiplying and dividing, when answering questions: "How much is one number (less) of another?", "How many times one number (less) of another ? ". Subsection can be replaced with graphic patterns and drawings. For example, to substantiate the result of dividing 7: 3 \u003d 2 (OST.1), it can use the drawing:

For the formation of students, the ability to justify their judgments is useful to offer them tasks for choosing a method of action (while both methods may be: a) faithful, b) incorrect, c) one faithful, other incorrect). In this case, each proposed way to fulfill the task can be viewed as a judgment, to justify the students should use various methods evidence.

For example, when studying the topic "Unit of Square", students are proposed to task (M2I):

How many times is the area of \u200b\u200bthe AVD rectangle more than the KMOO rectangle? Write an answer to a numerical equality.

Masha recorded such equality: 15: 3 \u003d 5, 30: 6 \u003d 5.

Misha is equality: 60: 12 \u003d 5.

Which of them is right? How did Misha and Masha reasoned?

To justify the judgments made by Misha and Masha, students can use as a method of deductive arguments, where the rule of multiple comparison of numbers and practical one acts as a common parcel. In this case, they are based on the drawing.

By proposing a way to solve the problem, students also express judgments using the mathematical content given in the plot of tasks to proof. Reception of the choice of finished judgments activates this activity. As an example, such tasks can be given:

Tourists on the first day passed 18 km, on the second day, moving at the same speed, they passed 27 km. What speed did tourists go if they spent 9 hours for all way?

Misha recorded the problem of the problem like this:

1) 18: 9 \u003d 2 (km / h)

2) 27: 9 \u003d 3 (km / h)

3) 2 + 3 \u003d 5 (km / h) Masha - so:

1) 18 + 27 \u003d 45 (km)

2) 45: 9 \u003d 5 (km / h) Which of them is right: Misha or Masha?

How many potatoes have collected from 10 bushes if there are 7 potatoes from three, from four to 9, from six to 8, and from seven 4 potatoes? Masha decided the task like this:

1) 7 * 3 \u003d 21 (k.)

2) 4 * 7 \u003d 28 (k.)

3) 21 + 28 \u003d 49 (k.) Answer: 49 potatoes collected from 10 bushes. And Misha solved this task:

1) 9 4 \u003d 36 (k.)

2) 8 * 6 \u003d 48 (k.)

3) 36 + 48 \u003d 84 (k.) Answer: 84 potatoes collected from 10 bushes. Which of them is right?

The process of performing any task should always represent a chain of judgment (general, private, single), to substantiate the truth of which students use various ways.

Show it on the example example:

V Insert the numbers in the "windows" to come true equality:

P: 6 \u003d 27054 P: 7 \u003d 4083 (OST. 4)

Students express a general judgment: "If the value of private multiply on a divider, we will get divisible." Private judgment: "Private value - 27054, divider - b". Conclusion:

"27054 * 6".

Now, as a common parcel, a written multiplication algorithm is the result: 162324. A judgment is expressed: 162324: 6 \u003d 27054.

The truth of this judgment can be checked by performing the division of the "corner" or using the calculator.

Similarly come with the second record.

Make sure equalities using numbers: 6, 7, 8, 48, 56.

Students express judgment:

6 * 8 \u003d 48 (Rationale - Calculations) 56 - 48 \u003d 8 (Rationale - Calculations)

8 * 6 \u003d 48 (To justify the judgment, you can use the overall premise: "From the permutation of multipliers, the value of the work will not change").

48: 8 \u003d 6 (general package is also possible, etc.) "Thus, in most cases, students turn to computations and deductive arguments to substantiate the truth of judgments in the initial course of mathematics. So, justifying the result in solving an example on the order of action, They use the general premise in the form of the rule rule rule, then perform calculations.

Measurement as a way of substantiating the truth of judgments is usually used in the study of values \u200b\u200band geometric material. For example, judgments: "The blue cut is longer than the red", "the sides of the quadrangle are equal to", "one side of the rectangle is more different" children can justify the measurement.

Task 93. Describe the ways of justifying the truth of judgment. expressed by students when performing the following tasks. When studying what questions of the course of the Mathematics of the initial classes, it is advisable to offer these tasks 9

9*7+9+5 8*6+8+3 7*9+9+5 8*7+3 9*8+5 7*8+3

Is it possible to argue that the values \u200b\u200bof expressions in each column "are the same:

12*5 16*4 (8+4)*5 (8+8)*4 (7+5)*5 (9+7)*4 (10+2)*5 (10+6)*4

Insert signs or \u003d to get faithful records:

(14+8)*3 ... 14*3+8*3 (27+8)*6 ...27*6+8 (36+4)*18 ...40*18 .

What signs of action need to be inserted into the "windows" to get faithful equality

8 * 8 \u003d 8p7p8 8 * 3 \u003d 8p4p8 8 * 6 \u003d 6p8p0 8 * 5 \u003d 8p0p32

It is possible to assert that the values \u200b\u200bof expressions in each column are the same:

8*(4*6) (9*3)*3 8*24 2*27 (8*4)*6 9*(3*2) 6*32 (2*3)*9

3.8. The relationship of logical and algorithmic thinking of schoolchildren

The ability to consistently, clearly and consistently express their thoughts is closely related to the ability to represent a complex effect in the form of an organized sequence of simple. Such skill is called algorithmic. It finds his expression that a person, seeing the final goal, can compile an algorithmic prescription or algorithm (if it exists), as a result of the execution of which the goal will be achieved.

Drawing up algorithmic prescriptions (algorithms) -The referring task, so the initial course of mathematics does not aim as its purpose. But it can and should take on a certain preparation for its achievement, thereby contributing to the development of the logical thinking of schoolchildren.

To do this, starting from the 1st grade, it is necessary, first of all, to teach children to "see" algorithms and to realize the algorithmic essence of the actions that they perform. Start this work follows from the simplest algorithms available and understandable to them. You can make a street transition algorithm with an unregulated and adjustable intersection, algorithms for using various household appliances, cooking any dish (recipe for cooking), to present in the form of consecutive operations path from home to school, from school to the nearest bus stop, etc.

The method of making a coffee drink is written on the box and represents the following algorithm:

1. Pour a glass of hot water in a saucepan.

2. Take a teaspoon of drink.

3. Fill (pour) a coffee drink in a saucepan with water.

4. Heat the contents of the pan to boil.

5. Give a drink to stand.

6. Pour a drink into a glass.

Considering such instructions, the term "algorithm" itself can not be administered, but to talk about the rules in which items indicating certain actions, as a result of which the task is solved.

It should be noted that the term "algorithm" itself can only be consumed conditionally, since those rules and regulations, which are considered in the course of the mathematics of primary classes, do not have all the properties that are characterized. Algorithms in primary classes describe the sequence of actions on a specific example, not in general form, they are reflected not all operations included in the action performed, so their sequence is strictly defined. For example, a sequence of actions when multiplying the numbers ending with zeros, an unambiguous number (800 * 4) is performed as follows:

1. Imagine the first factor in the form of a work of a unique number and a unit, finishing zeros: (8 * 100) 4;

2. We use the combination property of multiplication:

(8*100)*4 =8 *(100*4);

3. We use the moving property of multiplication:

8*(100*4)=8*(4*100);

4. We use the combination property of multiplication:

8*(4*100)=(8*4)*100;

5. Replace the work in brackets by its value:

(8*4)*100 =32*100;

6. When multiplying a number of 1 with zeros, you need to attribute as much zeros as they are in the second multiplier:

32*100=3200.

Of course, younger schoolchildren cannot learn the sequence of actions in this form, but, presenting clearly all operations, the teacher may offer children various exercises, the execution of which will allow children to realize the way of activity. For example:

It is possible, without performing calculations, to argue that the values \u200b\u200bof expressions in each column are the same:

9*(8*100) 800*7 (9*8)*100 (8*7)*100 (9*100)*8 8*(7*100) 9*100 8*700 72*100 56*100

Explain how the expression recorded on the right is obtained:

4*6*10=40*6 2*8*10=20*8 8*5*10=8*50 5*7*10=7*50

It is possible to argue that the values \u200b\u200bof the works in each pair are the same:

45*10 54*10 32*10 9*50 60*9 8*40

For the awareness of the children's algorithmic essence, they need to reformulate these mathematical tasks in the form of a specific program.

For example, the task "Find 5 numbers, the first of which is 3, each next to 2 more previous" can be represented as an algorithmic prescription as follows:

1. Write the number 3.

2. Increase it to 2.

3. The result is increased by 2.

4. Repeat the operation 3 until you write 5 numbers. A verbal algorithmic prescription can be replaced with a schematic:

This will allow students more clearly present each operation and the sequence of their execution.

Task 94. Word the following mathematical tasks in the form of algorithmic prescriptions and imagine them as a scheme.

actions:

a) write 4 numbers, the first of which is 1, each next

2 times more than the previous one;

b) Write 4 numbers, the first of which is 0, the second most of the first one is 1 third more second to 2, the fourth more than third to 3;

c) write 6 numbers: if the first is 9, second 1, and each next one is equal to the sum of the previous two.

Along with verbal and schematic regulations, you can specify the algorithm in the form of a table.

For example, task: "write a number from 1 to 6. Each increase:

a) by 2; b) on 3 "can be submitted to such a table:

+

Thus, algorithmic prescriptions can be set as a verbal way, a scheme and table.

Acting with specific mathematical objects and generalizations in the form of rules, children master the ability to allocate elementary steps of their actions and determine their sequence.

For example, the rule checking rule can be formulated as an algorithmic prescription as follows. In order to check the addition of subtraction, you need:

1) out of the amount of deducting one of the terms;

2) compare the result obtained with another term;

3) if the result obtained is equal to another term, then the addition is done correctly;

4) Otherwise, look for a mistake.

Task 95. Make algorithmic prescriptions that younger students will be able to use at: a) the addition of unambiguous numbers with the transition through a category; b) comparing multivalued numbers; c) solving equations; d) writing multiplication on an unambiguous number.

To form the ability to draw up algorithms to teach children: find a general way of action; allocate basic, elementary actions, of which this is this; plan a sequence of dedicated actions; Put the algorithm correctly.

Consider the tasks whose purpose is to identify the method of action:

There are numbers (see Figure). Make an expression and find their meanings. How many examples of addition can be made up? How to reason at the same time not to miss a single case?

When performing this task, students are aware of the need for common way actions. For example, fix the first term 31, as a second to add all the numbers of the second column, then as the first component to be fixed, for example, the number 41 and again choose all the numbers from the second column, etc. You can fix the second term and sort out all the numbers of the first Point. It is important that the child understands that, adhering to a certain method of action, he will not miss a single case and none of the cases will write twice.

There are three chandeliers in the hall and 6 windows. For the holiday to decorate from each chandelier to each window stretched the garland. How much did the garlands hang out? (When solving, you can use a schematic pattern.)

For the formation of students, combinatorial tasks are useful to identify the method of action. Their peculiarity is that they have not one, but many decisions, and when they are fulfilled, it is necessary to exercise brute force in the rational sequence. For example:

How many different five-digit numbers can be written using the numbers 55522 (the number 5 can be repeated three times, 2 - two times).

To solve this combinatorial task, you can take advantage of the construction of "wood". First, one digit is discharged from which you can start recording a number. The further algorithm of actions comes down to record numbers that can be put after each digit until we get a five-digit number. Following this algorithm, it is necessary to combine and count how many times the numbers 5 and 2 repeated.

Opened "branches" with different numbers: 55522, 55252, 55225, 52552, 52525, 52255. Then the number 2 is written out.

We write down the numbers, moving on the "twigs": 22555, 25525, 25552, 25255. Answer: You can write 10 numbers.

Task 96. Select combinatorial tasks that you would suggest students of the first, second and third grade when studying the various concepts of the initial courses of mathematics.

Chapter 4. As a study of junior schoolchildren to solving problems

4.1. The concept of "task" in the initial course of mathematics

Any mathematical task can be considered as a task, having allocated a condition in it, that is, that part, where information about the known and unknown values \u200b\u200bof magnitude, about relations between them, and the requirement (i.e., an indication of what to find) . Consider examples of mathematical tasks from the elementary classes:

\u003e Put signs, \u003d so that the right records are: 3 ... 5, 8 ... 4.

The condition of the problem is the numbers 3 and 5, 8 and 4. The requirement is to compare these numbers.

*\u003e Share Equation: X + 4 \u003d 9.

The condition is given equation. The requirement is to solve it, that is, to substitute instead of x such a true equality.

Here the condition is given triangles. Requirement - fold the rectangle.

To perform each requirement, a certain method or method of action is applied, depending on which various types of mathematical tasks are distinguished: to build, test