Active methods of teaching mathematics in primary school. Features of teaching mathematics to junior schoolchildren

The problem of the formation and development of mathematical abilities of younger schoolchildren is relevant at the present time, but at the same time it receives insufficient attention among the problems of pedagogy. Mathematical abilities refer to special abilities that manifest themselves only in a separate type of human activity.

Teachers often try to understand why children studying in the same school, with the same teachers, in the same class, achieve different successes in mastering this discipline. Scientists explain this by the presence or absence of certain abilities.

Abilities are formed and developed in the process of learning, mastering relevant activities, therefore it is necessary to form, develop, educate and improve the abilities of children. In the period from 3-4 years to 8-9 years, rapid development of intelligence occurs. Therefore, during primary school age the opportunities for developing abilities are the highest. The development of the mathematical abilities of a junior schoolchild is understood as the purposeful, didactically and methodically organized formation and development of a set of interrelated properties and qualities of the child’s mathematical thinking style and his abilities for mathematical knowledge of reality.

The first place among academic subjects that pose particular difficulties in learning is given to mathematics, as one of the abstract sciences. For children of primary school age, it is extremely difficult to perceive this science. An explanation for this can be found in the works of L.S. Vygotsky. He argued that in order “to understand the meaning of a word, you need to create a semantic field around it. To build a semantic field, a projection of meaning into a real situation must be carried out.” It follows from this that mathematics is complex, because it is an abstract science, for example, it is impossible to transfer a number series to reality, because it does not exist in nature.

From the above it follows that it is necessary to develop the child’s abilities, and this problem must be approached individually.

The problem of mathematical abilities was considered by the following authors: Krutetsky V.A. “Psychology of mathematical abilities”, Leites N.S. “Age giftedness and individual differences”, Leontyev A.N. "Chapter on Abilities" by Zach Z.A. “Development of intellectual abilities in children” and others.

Today, the problem of developing the mathematical abilities of younger schoolchildren is one of the least developed problems, both methodological and scientific. This determines the relevance of this work.

The purpose of this work: systematization of scientific points of view on this problem and identification of direct and indirect factors influencing the development of mathematical abilities.

When writing this work, the following questions were set: tasks:

1. Studying psychological and pedagogical literature in order to clarify the essence of the concept of ability in the broad sense of the word, and the concept of mathematical ability in the narrow sense.

2. Analysis of psychological and pedagogical literature, periodical materials devoted to the problem of studying mathematical abilities in historical development and at the present stage.

ChapterI. The essence of the concept of ability.

1.1 General concept of abilities.

The problem of abilities is one of the most complex and least developed in psychology. When considering it, first of all, it should be taken into account that the real subject of psychological research is human activity and behavior. There is no doubt that the source of the concept of abilities is the indisputable fact that people differ in the quantity and quality of productivity of their activities. The variety of human activities and the quantitative and qualitative differences in productivity make it possible to distinguish between types and degrees of abilities. A person who does something well and quickly is said to be capable of this task. Judgment about abilities is always comparative in nature, that is, it is based on a comparison of productivity, the skill of one person with the skill of others. The criterion of ability is the level (result) of activity that some people manage to achieve and others do not. The history of social and individual development teaches that any skillful skill is achieved as a result of more or less intense work, various, sometimes gigantic, “superhuman” efforts. On the other hand, some achieve high mastery of activity, skill and skill with less effort and faster, others do not go beyond average achievements, others find themselves below this level, even if they try hard, study and have favorable external conditions. It is the representatives of the first group that are called capable.

Human abilities, their different types and degrees, are among the most important and complex problems of psychology. However, the scientific development of the issue of abilities is still insufficient. Therefore, in psychology there is no single definition of abilities.

V.G. Belinsky understood abilities as the potential natural forces of the individual, or its capabilities.

According to B.M. Teplov, abilities are individual psychological characteristics that distinguish one person from another.

S.L. Rubinstein understands ability as suitability for a particular activity.

The psychological dictionary defines ability as quality, opportunity, ability, experience, skill, talent. Abilities allow you to perform certain actions at a given time.

Ability is an individual's readiness to perform an action; suitability is the existing potential to perform any activity or the ability to achieve a certain level of development of ability.

Based on the above, we can give a general definition of abilities:

Ability is an expression of the correspondence between the requirements of activity and the complex of neuropsychological properties of a person, ensuring high qualitative and quantitative productivity and growth of his activity, which is manifested in a high and rapidly growing (compared to the average person) ability to master this activity and master it.

1.2 The problem of developing the concept of mathematical abilities abroad and in Russia.

A wide variety of directions also determined a wide variety in the approach to the study of mathematical abilities, in methodological tools and theoretical generalizations.

The study of mathematical abilities should begin with defining the subject of research. The only thing that all researchers agree on is the opinion that it is necessary to distinguish between ordinary, “school” abilities for assimilation of mathematical knowledge, for their reproduction and independent application, and creative mathematical abilities associated with the independent creation of an original and socially valuable product.

Back in 1918, Rogers' work noted two sides of mathematical abilities, reproductive (related to the memory function) and productive (related to the thinking function). In accordance with this, the author built a well-known system of mathematical tests.

The famous psychologist Revesh, in his book “Talent and Genius,” published in 1952, considers two main forms of mathematical abilities - applicative (as the ability to quickly discover mathematical relationships without preliminary tests and apply the corresponding knowledge in similar cases) and productive (as the ability to discover relationships, not directly arising from existing knowledge).

Foreign researchers show great unity of views on the issue of innate or acquired mathematical abilities. If here we distinguish between two different aspects of these abilities - “school” and creative abilities, then in relation to the latter there is complete unity - the creative abilities of a scientist - mathematics are an innate education, a favorable environment is necessary only for their manifestation and development. This is, for example, the point of view of mathematicians who were interested in questions of mathematical creativity - Poincaré and Hadamard. Betz also wrote about the innateness of mathematical talent, emphasizing that we are talking about the ability to independently discover mathematical truths, “for probably everyone can understand someone else’s thought.” The thesis about the innate and hereditary nature of mathematical talent was vigorously promoted by Revesh.

Regarding “school” (learning) abilities, foreign psychologists do not speak so unanimously. Here, perhaps, the dominant theory is the parallel action of two factors - biological potential and environment. Until recently, even in relation to school mathematical abilities, the ideas of innateness dominated.

Back in 1909-1910. Stone and independently Curtis, studying achievements in arithmetic and abilities in this subject, came to the conclusion that it is hardly possible to talk about mathematical abilities as a single whole, even in relation to arithmetic. Stone pointed out that children who are skilled at calculations often lag behind in the area of arithmetic reasoning. Curtis also showed that it is possible to combine a child's success in one branch of arithmetic and his failure in another. From this they both concluded that each operation required its own special and relatively independent ability. Some time later, Davis conducted a similar study and came to the same conclusions.

One of the significant studies of mathematical abilities must be recognized as the study of the Swedish psychologist Ingvar Werdelin in his book “Mathematical Abilities”. The author’s main intention was to, based on the multifactor theory of intelligence, analyze the structure of schoolchildren’s mathematical abilities and identify the relative role of each factor in this structure. Werdelin takes as a starting point the following definition of mathematical abilities: “Mathematical ability is the ability to understand the essence of mathematical (and similar) systems, symbols, methods and proofs, to memorize, retain them in memory and reproduce, combine them with other systems, symbols, methods and proofs, use them in solving mathematical (and similar) problems.” The author examines the question of the comparative value and objectivity of measuring mathematical abilities using teachers' grades and special tests and notes that school grades are unreliable, subjective and far from a real measurement of abilities.

The famous American psychologist Thorndike made a great contribution to the study of mathematical abilities. In his work “The Psychology of Algebra” he gives a lot of all kinds of algebraic tests to determine and measure abilities.

Mitchell, in his book on the nature of mathematical thinking, lists several processes that, in his opinion, characterize mathematical thinking, in particular:

1. classification;

2. ability to understand and use symbols;

3. deduction;

4. manipulation of ideas and concepts in an abstract form, without reference to the concrete.

Brown and Johnson, in the article “Ways to Identify and Educate Students with Potential in the Sciences,” indicate that practicing teachers have identified those features that characterize students with potential in mathematics, namely:

1. extraordinary memory;

2. intellectual curiosity;

3. ability for abstract thinking;

4. ability to apply knowledge in a new situation;

5. the ability to quickly “see” the answer when solving problems.

Concluding a review of the works of foreign psychologists, it should be noted that they do not give a more or less clear and distinct idea of the structure of mathematical abilities. In addition, we must also keep in mind that in some works the data were obtained using a less objective introspective method, while others are characterized by a purely quantitative approach, ignoring the qualitative features of thinking. Summarizing the results of all the studies mentioned above, we will obtain the most general characteristics of mathematical thinking, such as the ability for abstraction, the ability for logical reasoning, good memory, the ability for spatial representations, etc.

In Russian pedagogy and psychology, only a few works are devoted to the psychology of abilities in general and the psychology of mathematical abilities in particular. It is necessary to mention the original article by D. Mordukhai-Boltovsky “Psychology of Mathematical Thinking”. The author wrote the article from an idealistic position, attaching, for example, special importance to the “unconscious thought process,” arguing that “the thinking of a mathematician ... is deeply embedded in the unconscious sphere.” The mathematician is not aware of every step of his thought “the sudden appearance in the consciousness of a ready-made solution to a problem that we could not solve for a long time,” the author writes, “we explain by unconscious thinking, which ... continued to engage in the task, ... and the result floats beyond the threshold of consciousness.” .

The author notes the specific nature of mathematical talent and mathematical thinking. He argues that the ability for mathematics is not always inherent even in brilliant people, that there is a difference between a mathematical and a non-mathematical mind.

Of great interest is Mordecai-Boltovsky’s attempt to isolate the components of mathematical abilities. He refers to such components, in particular:

1. “strong memory”, it was stipulated that this meant “mathematical memory”, memory for “a subject of the type with which mathematics deals”;

2. “wit,” which is understood as the ability to “embrace in one judgment” concepts from two poorly connected areas of thought, to find similarities with the given in what is already known;

3. speed of thought (speed of thought is explained by the work that unconscious thinking does in favor of conscious thinking).

D. Mordecai-Boltovsky also expresses his thoughts on the types of mathematical imagination that underlie different types of mathematicians - “geometers” and “algebraists”. “Arithmeticians, algebraists and analysts in general, whose discovery is made in the most abstract form of discontinuous quantitative symbols and their relationships, cannot express it like a geometer.” He also expressed valuable thoughts about the peculiarities of the memory of “geometers” and “algebraists.”

The theory of abilities was created over a long period of time by the joint work of the most prominent psychologists of that time: B.M. Teplov, L.S. Vygotsky, A.N. Leontyev, S.L. Rubinstein, B.G. Anafiev and others.

In addition to general theoretical studies of the problem of abilities, B.M. Teplov, with his monograph “Psychology of Musical Abilities,” laid the foundation for an experimental analysis of the structure of abilities for specific types of activities. The significance of this work goes beyond the narrow question of the essence and structure of musical abilities; it found a solution to the basic, fundamental questions of research into the problem of abilities for specific types of activities.

This work was followed by studies of abilities similar in idea: to visual activity - V.I. Kireenko and E.I. Ignatov, literary abilities - A.G. Kovalev, pedagogical abilities - N.V. Kuzmina and F.N. Gonobolin, design and technical abilities - P.M. Jacobson, N.D. Levitov, V.N. Kolbanovsky and mathematical abilities - V.A. Krutetsky.

A number of experimental studies of thinking were carried out under the leadership of A.N. Leontyev. Some issues of creative thinking were clarified, in particular, how a person comes to the idea of solving a problem, the method of solving which does not directly follow from its conditions. An interesting pattern was established: the effectiveness of exercises leading to the correct solution varies depending on at what stage of solving the main problem auxiliary exercises are presented, i.e. the role of guiding exercises was shown.

A series of studies by L.N. is directly related to the problem of abilities. Landes. In one of the first works in this series - “On some shortcomings of studying students’ thinking” - he raises the question of the need to reveal the psychological nature, the internal mechanism of the “ability to think.” To cultivate abilities, according to L.N. Landa means “to teach the technique of thinking”, to form the skills of analytical and synthetic activity. In his other work - “Some Data on the Development of Mental Abilities” - L. N. Landa discovered significant individual differences in schoolchildren’s mastery of a new method of reasoning when solving geometric proof problems - differences in the number of exercises required to master this method, differences in the pace of work, differences in the formation of the ability to differentiate the use of operations depending on the nature of the task conditions and differences in the assimilation of operations.

Of great importance for the theory of mental abilities in general and mathematical abilities in particular are the studies of D.B. Elkonin and V.V. Davydova, L.V. Zankova, A.V. Skripchenko.

It is usually believed that the thinking of children 7-10 years old is figurative in nature and has a low ability for distraction and abstraction. Experiential learning conducted under the guidance of D.B. Elkonin and V.V. Davydov, showed that already in the first grade, with a special teaching method, it is possible to give students in alphabetic symbolism, i.e. in general form, a system of knowledge about the relationships of quantities, dependencies between them, to introduce them to the field of formal sign operations. A.V. Skripchenko showed that, under appropriate conditions, third- and fourth-grade students can develop the ability to solve arithmetic problems by composing an equation with one unknown.

1.3 Mathematical ability and personality

First of all, it should be noted that what characterizes capable mathematicians and is necessary for successful work in the field of mathematics is the “unity of inclinations and abilities in vocation”, expressed in a selective positive attitude towards mathematics, the presence of deep and effective interests in the relevant field, the desire and need to engage in it, passionate passion for the business.

Without a penchant for mathematics, there can be no genuine aptitude for it. If a student does not feel any inclination towards mathematics, then even good abilities are unlikely to ensure a completely successful mastery of mathematics. The role played here by inclination and interest boils down to the fact that a person interested in mathematics is intensively engaged in it, and, consequently, vigorously exercises and develops his abilities.

Numerous studies and characteristics of gifted children in the field of mathematics indicate that abilities develop only if there are inclinations or even a unique need for mathematical activity. The problem is that often students are capable of mathematics, but have little interest in it, and therefore do not have much success in mastering this subject. But if the teacher can arouse their interest in mathematics and the desire to do it, then such a student can achieve great success.

At school, such cases often occur: a student capable of mathematics has little interest in it, and does not show much success in mastering this subject. But if the teacher is able to awaken his interest in mathematics and the inclination to engage in it, then such a student, “captured” by mathematics, can quickly achieve great success.

From this follows the first rule of teaching mathematics: the ability to get students interested in science and encourage them to independently develop their abilities. The emotions experienced by a person are also an important factor in the development of abilities in any activity, not excluding mathematical activity. The joy of creativity, the feeling of satisfaction from intense mental work, mobilize his strength and force him to overcome difficulties. All children with an aptitude for mathematics are distinguished by a deep emotional attitude towards mathematical activity and experience real joy caused by each new achievement. Awakening the creative spirit in a student and teaching him to love mathematics is the second rule of a mathematics teacher.

Many teachers point out that the ability to quickly and deeply generalize can manifest itself in one subject without characterizing the student’s educational activity in other subjects. An example is that a child who is able to generalize and systematize material in literature does not show similar abilities in the field of mathematics.

Unfortunately, teachers sometimes forget that mental abilities, which are general in nature, in some cases act as specific abilities. Many teachers tend to use objective assessment, i.e. if a student is weak in reading, then in principle he cannot achieve heights in the field of mathematics. This opinion is typical for primary school teachers who teach a range of subjects. This leads to an incorrect assessment of the child's abilities, which in turn leads to a lag in mathematics.

1.4 Development of mathematical abilities in younger schoolchildren.

The problem of ability is a problem of individual differences. With the best organization of teaching methods, the student will progress more successfully and faster in one area than in another.

Naturally, success in learning is determined not only by the student’s abilities. In this sense, the content and methods of teaching, as well as the student’s attitude to the subject, are of key importance. Therefore, success and failure in learning do not always provide grounds for making judgments about the nature of the student’s abilities.

The presence of weak abilities in students does not relieve the teacher from the need, as far as possible, to develop the abilities of these students in this area. At the same time, there is an equally important task - to fully develop his abilities in the area in which he demonstrates them.

It is necessary to educate the capable and select the capable, while not forgetting about all schoolchildren, and to raise the overall level of their training in every possible way. In this regard, various collective and individual working methods are needed in their work in order to intensify the activities of students.

The learning process should be comprehensive, both in terms of organizing the learning process itself, and in terms of developing in students a deep interest in mathematics, problem-solving skills, understanding the system of mathematical knowledge, solving with students a special system of non-standard problems that should be offered not only lessons, but also on tests. Thus, a special organization of the presentation of educational material and a well-thought-out system of tasks help to increase the role of meaningful motives for studying mathematics. The number of result-oriented students is decreasing.

In the lesson, not just problem solving, but the unusual way of solving problems used by students should be encouraged in every possible way; in this regard, special importance is placed not only on the result in solving the problem, but on the beauty and rationality of the method.

Teachers successfully use the “problem formulation” technique to determine the direction of motivation. Each task is assessed according to a system of the following indicators: the nature of the task, its correctness and relation to the source text. The same method is sometimes used in a different version: after solving the problem, students were asked to create any problems that were somehow related to the original problem.

To create psycho-pedagogical conditions for increasing the efficiency of organizing the learning process system, the principle of organizing the learning process in the form of substantive communication using cooperative forms of student work is used. This is group problem solving and collective discussion of grading, pair and team forms of work.

Chapter II. The development of mathematical abilities in primary schoolchildren as a methodological problem.

2.1 General characteristics of capable and talented children

The problem of developing children's mathematical abilities is one of the least developed methodological problems of teaching mathematics in primary school today.

The extreme heterogeneity of views on the very concept of mathematical abilities determines the absence of any conceptually sound methods, which in turn creates difficulties in the work of teachers. Perhaps this is why there is a widespread opinion not only among parents, but also among teachers: mathematical abilities are either given or not given. And there’s nothing you can do about it.

Of course, abilities for one or another type of activity are determined by individual differences in the human psyche, which are based on genetic combinations of biological (neurophysiological) components. However, today there is no evidence that certain properties of nerve tissue directly affect the manifestation or absence of certain abilities.

Moreover, targeted compensation for unfavorable natural inclinations can lead to the formation of a personality with pronounced abilities, of which there are many examples in history. Mathematical abilities belong to the group of so-called special abilities (as well as musical, visual, etc.). For their manifestation and further development, the assimilation of a certain stock of knowledge and the presence of certain skills are required, including the ability to apply existing knowledge in mental activity.

Mathematics is one of those subjects where the individual mental characteristics (attention, perception, memory, thinking, imagination) of a child are crucial for its mastery. Behind important characteristics of behavior, behind the success (or failure) of educational activities, those natural dynamic features mentioned above are often hidden. They often give rise to differences in knowledge—its depth, strength, and generality. Based on these qualities of knowledge, which relate (along with value orientations, beliefs, and skills) to the content side of a person’s mental life, children’s giftedness is usually judged.

Individuality and talent are interrelated concepts. Researchers dealing with the problem of mathematical abilities, the problem of the formation and development of mathematical thinking, despite all the differences in opinions, note, first of all, the specific features of the psyche of a mathematically capable child (as well as a professional mathematician), in particular, flexibility of thinking, i.e. unconventionality, originality, the ability to vary ways of solving a cognitive problem, ease of transition from one solution path to another, the ability to go beyond the usual way of activity and find new ways to solve a problem under changed conditions. It is obvious that these features of thinking directly depend on the special organization of memory (free and connected associations), imagination and perception.

Researchers identify such a concept as depth of thinking, i.e. the ability to penetrate into the essence of each fact and phenomenon being studied, the ability to see their relationships with other facts and phenomena, to identify specific, hidden features in the material being studied, as well as purposeful thinking, combined with breadth, i.e. the ability to form generalized methods of action, the ability to cover the whole problem without missing out on details. Psychological analysis of these categories shows that they should be based on a specially formed or natural inclination towards a structural approach to the problem and extremely high stability, concentration and a large amount of attention.

Thus, the individual typological characteristics of the personality of each student separately, by which we mean temperament, character, inclinations, and somatic organization of the personality as a whole, etc., have a significant (and maybe even decisive!) influence on the formation and the development of the child’s mathematical thinking style, which, of course, is a necessary condition for preserving the child’s natural potential (inclinations) in mathematics and its further development into pronounced mathematical abilities.

Experienced subject teachers know that mathematical abilities are a “piecemeal commodity,” and if such a child is not dealt with individually (individually, and not as part of a club or elective), then the abilities may not develop further.

That is why we often see how a first-grader with outstanding abilities “levels off” by the third grade, and in the fifth grade completely ceases to differ from other children. What is this? Research by psychologists shows that there may be different types of age-related mental development:

. “Early rise” (in preschool or primary school age) is due to the presence of bright natural abilities and inclinations of the corresponding type. In the future, consolidation and enrichment of mental qualities may occur, which will serve as a start for the development of outstanding mental abilities.

Moreover, the facts show that almost all scientists who distinguished themselves before the age of 20 were mathematicians.

But “alignment” with peers can also occur. We believe that this “leveling off” is largely due to the lack of a competent and methodologically active individual approach to the child in the early period.

“Slow and extended rise”, i.e. gradual accumulation of intelligence. The absence of early achievements in this case does not mean that the prerequisites for great or outstanding abilities will not emerge in the future. Such a possible “rise” is the age of 16-17 years, when the factor of “intellectual explosion” is the social reorientation of the individual, directing his activity in this direction. However, such a “rise” can also occur in more mature years.

For a primary school teacher, the most pressing problem is “early rise”, which occurs at the age of 6-9 years. It is no secret that one such brightly capable child in the class, who also has a strong type of nervous system, is capable, literally, of preventing any of the children from opening their mouths in class. And as a result, instead of maximally stimulating and developing the little “prodigy,” the teacher is forced to teach him to remain silent (!) and “keep his brilliant thoughts to himself until asked.” After all, there are 25 other children in the class! Such “slowing down,” if it occurs systematically, can lead to the fact that after 3-4 years the child “evens out” with his peers. And since mathematical abilities belong to the group of “early abilities,” then perhaps it is precisely the mathematically capable children that we lose in the process of this “slowing down” and “leveling off.”

Psychological research has shown that although the development of educational abilities and creative talent in typologically different children proceeds differently, children with opposite characteristics of the nervous system can achieve (achieve) an equally high degree of development of these abilities. In this regard, it may be more useful for the teacher to focus not on the typological characteristics of the nervous system of children, but on some general characteristics of capable and talented children, which are noted by most researchers of this problem.

Different authors identify a different “set” of general characteristics of capable children within the framework of the types of activities in which these abilities were studied (mathematics, music, painting, etc.). We believe that it is more convenient for a teacher to rely on some purely procedural characteristics of the activity of capable children, which, as shown by a comparison of a number of special psychological and pedagogical studies on this topic, turn out to be the same for children with different types of abilities and giftedness. Researchers note that most capable children have:

Increased propensity for mental action and a positive emotional response to any new mental challenge. These children don't know what boredom is - they always have something to do. Some psychologists generally interpret this trait as an age-related factor in giftedness.

The constant need to renew and complicate mental workload, which entails a constant increase in the level of achievement. If this child is not burdened, then he finds his own activity and can master chess, a musical instrument, radio, etc., study encyclopedias and reference books, read specialized literature, etc.

The desire to independently choose things to do and plan your activities. This child has his own opinion about everything, stubbornly defends the unlimited initiative of his activities, has high (almost always adequate) self-esteem and is very persistent in self-affirmation in his chosen field.

Perfect self-regulation. This child is capable of fully mobilizing his strength to achieve a goal; able to repeatedly renew mental efforts in an effort to achieve a goal; has, as it were, an “initial” attitude towards overcoming any difficulties, and failures only force him to strive to overcome them with enviable tenacity.

Increased performance. Long-term intellectual stress does not tire this child; on the contrary, he feels good precisely in the situation of having a problem that requires a solution. Purely instinctively, he knows how to use all the reserves of his psyche and his brain, mobilizing and switching them at the right moment.

It is clearly seen that these general procedural characteristics of the activity of capable children, recognized by psychologists as statistically significant, are not uniquely inherent in any one type of human nervous system. Therefore, pedagogically and methodologically, the general tactics and strategy of an individual approach to a capable child should obviously be built on such psychological and didactic principles that ensure that the above-mentioned procedural characteristics of the activities of these children are taken into account.

From a pedagogical point of view, a capable child most of all needs an instructive style of relationship with a teacher, which requires greater information content and validity of the requirements put forward on the part of the teacher. The instructive style, as opposed to the imperative style that dominates in elementary school, involves appealing to the student’s personality, taking into account his individual characteristics and focusing on them. This style of relationship contributes to the development of independence, initiative and creative potential, which is noted by many teacher-researchers. It is equally obvious that, from a didactic point of view, capable children need, at a minimum, to ensure an optimal pace of progress in content and an optimal volume of learning load. Moreover, what is optimal for yourself, for your abilities, i.e. higher than for ordinary children. If we take into account the need for constant complication of mental workload, the persistent craving for self-regulation of their activities and the increased performance of these children, we can say with sufficient confidence that at school these children are by no means “prosperous” students, since their educational activities are constantly not carried out in zone of proximal development (!), and far behind this zone! Thus, in relation to these students, we (wittingly or unwittingly) constantly violate our proclaimed credo, the basic principle of developmental education, which requires teaching the child taking into account his zone of proximal development.

Working with capable children in primary school today is no less a “sick” problem than working with unsuccessful ones.

Its lesser “popularity” in special pedagogical and methodological publications is explained by its lesser “conspicuousness,” since a poor student is an eternal source of trouble for a teacher, and only the teacher (and not always), but Petya’s parents (if they deal with this issue specifically). At the same time, the constant “underload” of a capable child (and the norm for everyone is an underload for a capable child) will contribute to insufficient stimulation of the development of abilities, not only to the “non-use” of the potential of such a child (see points above), but also to the possible extinction of these abilities as unclaimed in educational activities (leading during this period of the child’s life).

There is also a more serious and unpleasant consequence of this: it is too easy for such a child to learn at the initial stage, as a result, he does not sufficiently develop the ability to overcome difficulties, does not develop immunity to failure, which largely explains the massive “collapse” in the performance of such children when transition from primary to secondary level.

In order for a public school teacher to successfully cope with working with a capable child in mathematics, it is not enough to identify the pedagogical and methodological aspects of the problem. As thirty years of practice in implementing a developmental education system have shown, in order for this problem to be solved in the conditions of teaching in a mass primary school, a specific and fundamentally new methodological solution is needed, fully presented to the teacher.

Unfortunately, today there are practically no special teaching aids for primary school teachers intended for working with capable and gifted children in mathematics lessons. We cannot cite a single such manual or methodological development, except for various collections such as the “Mathematical Box”. To work with capable and gifted children, you do not need entertaining tasks; this is too poor food for their minds! We need a special system and special “parallel” teaching aids to existing ones. The lack of methodological support for individual work with a capable child in mathematics leads to the fact that primary school teachers do not do this work at all (club or extracurricular work, where a group of children solves entertaining tasks with the teacher, which, as a rule, are not systematically selected, cannot be considered individual). One can understand the problems of a young teacher who does not have enough time or knowledge to select and systematize appropriate materials. But even an experienced teacher is not always ready to solve such a problem. Another (and, perhaps, the main!) limiting factor here is the presence of a single textbook for the entire class. Working according to a single textbook for all children, according to a single calendar plan, simply does not allow the teacher to implement the requirement of individualizing the pace of learning of a capable child, and the same content volume of the textbook for all children does not allow implementing the requirement of individualizing the volume of the educational load (not to mention the requirement of self-regulation and independent activity planning).

We believe that the creation of special teaching materials in mathematics for working with capable children is the only possible way to implement the principle of individualization of education for these children in the context of teaching a whole class.

2.2 Methodology for long-term assignments

The methodology for using the system of long-term assignments was considered by E.S. Rabunsky when organizing work with high school students in the process of teaching German at school.

A number of pedagogical studies have considered the possibility of creating systems of such tasks in various subjects for high school students, both to master new material and to eliminate knowledge gaps. In the course of research, it was noted that the vast majority of students prefer to perform both types of work in the form of “long-term tasks” or “delayed work.” This type of organization of educational activities, traditionally recommended mainly for labor-intensive creative work (essays, abstracts, etc.), turned out to be the most preferable for the majority of schoolchildren surveyed. It turned out that such “deferred work” satisfies the student more than individual lessons and assignments, since the main criterion for student satisfaction at any age is success at work. The absence of a sharp time limit (as happens in a lesson) and the possibility of freely returning to the content of the work many times allows you to cope with it much more successfully. Thus, tasks designed for long-term preparation can also be considered as a means of cultivating a positive attitude towards the subject.

For many years, it was believed that everything said applies only to older students, but does not correspond to the characteristics of the educational activities of primary school students. Analysis of the procedural characteristics of the activities of capable children of primary school age and the work experience of Beloshista A.V. and teachers who took part in the experimental testing of this methodology, showed the high efficiency of the proposed system when working with capable children. Initially, to develop a system of tasks (hereinafter we will call them sheets in connection with the form of their graphic design, convenient for working with a child), topics related to the formation of computational skills were selected, which are traditionally considered by teachers and methodologists as topics that require constant guidance at the stage acquaintance and constant monitoring at the consolidation stage.

During the experimental work, a large number of printed sheets were developed, combined into blocks covering an entire topic. Each block contains 12-20 sheets. The worksheet is a large system of tasks (up to fifty tasks), methodically and graphically organized in such a way that as they are completed, the student can independently approach the understanding of the essence and method of performing a new computational technique, and then consolidate the new way of activity. A worksheet (or a system of sheets, i.e. a thematic block) is a “long-term task”, the deadlines for which are individualized in accordance with the desires and capabilities of the student working on this system. Such a sheet can be offered in class or instead of homework in the form of a task with a “delayed deadline” for completion, which the teacher either sets individually or allows the student (this path is more productive) to set a deadline for himself (this is a way to form self-discipline, since independent planning of activities in connection with independently determined goals and deadlines is the basis of human self-education).

The teacher determines tactics for working with worksheets for the student individually. At first, they can be offered to the student as homework (instead of a regular assignment), individually agreeing on the timing of its completion (2-4 days). As you master this system, you can move on to the preliminary or parallel method of work, i.e. give the student a sheet before learning the topic (on the eve of the lesson) or during the lesson itself for independent mastery of the material. Attentive and friendly observation of the student in the process of activity, “contractual style” of relationships (let the child decide for himself when he wants to receive this sheet), perhaps even exemption from other lessons on this or the next day to concentrate on the task, advisory assistance (on one question can always be answered immediately when passing a child in class) - all this will help the teacher to fully individualize the learning process of a capable child without spending a lot of time.

Children should not be forced to copy assignments from the sheet. The student works with a pencil on a sheet of paper, writing down answers or completing actions. This organization of learning evokes positive emotions in the child - he likes to work on a printed basis. Freed from the need for tedious copying, the child works with greater productivity. Practice shows that although the worksheets contain up to fifty tasks (the usual homework norm is 6-10 examples), the student enjoys working with them. Many children ask for a new sheet every day! In other words, they exceed the work quota for the lesson and homework several times over, while experiencing positive emotions and working at their own discretion.

During the experiment, such sheets were developed on the topics: “Oral and written calculation techniques”, “Numbering”, “Quantities”, “Fractions”, “Equations”.

Methodological principles for constructing the proposed system:

1. The principle of compliance with the mathematics program for primary grades. The content of the sheets is tied to a stable (standard) mathematics program for primary grades. Thus, we believe it is possible to implement the concept of individualizing mathematics teaching for a capable child in accordance with the procedural features of his educational activities when working with any textbook that corresponds to the standard program.

2. Methodically, the principle of dosage is implemented in each sheet, i.e. in one sheet only one technique or one concept is introduced, or one connection, but essential for a given concept, is revealed. This, on the one hand, helps the child clearly understand the purpose of the work, and on the other hand, helps the teacher to easily monitor the quality of mastery of this technique or concept.

3. Structurally, the sheet represents a detailed methodological solution to the problem of introducing or introducing and consolidating one or another technique, concept, connections of this concept with other concepts. The tasks are selected and grouped (i.e., the order in which they are placed on the sheet matters) in such a way that the child can “move” along the sheet independently, starting from the simplest methods of action already familiar to him, and gradually master a new method, which in the first steps fully revealed in smaller actions that are the basis of this technique. As you move through the sheet, these small actions are gradually arranged into larger blocks. This allows the student to master the technique as a whole, which is the logical conclusion of the entire methodological “construction”. This structure of the sheet allows you to fully implement the principle of a gradual increase in the level of complexity at all stages.

4. This structure of the worksheet also makes it possible to implement the principle of accessibility, and to a much deeper extent than can be done today when working only with a textbook, since the systematic use of sheets allows you to learn the material at an individual pace that is convenient for the student, which the child can regulate independently.

5. The system of sheets (thematic block) allows you to implement the principle of perspective, i.e. gradual inclusion of the student in the activities of planning the educational process. Tasks designed for long-term (delayed) preparation require long-term planning. The ability to organize your work, planning it for a certain period of time, is the most important educational skill.

6. The system of worksheets on the topic also makes it possible to implement the principle of individualization of testing and assessing students’ knowledge, not on the basis of differentiating the level of difficulty of tasks, but on the basis of the unity of requirements for the level of knowledge, skills and abilities. Individualized deadlines and methods for completing tasks make it possible to present all children with tasks of the same level of complexity, corresponding to the program requirements for the norm. This does not mean that talented children should not be held to higher standards. Worksheets at a certain stage allow such children to use material that is more intellectually rich, which in a propaedeutic way will introduce them to the following mathematical concepts of a higher level of complexity.

Conclusion

An analysis of psychological and pedagogical literature on the problem of the formation and development of mathematical abilities shows: without exception, all researchers (both domestic and foreign) connect it not with the content side of the subject, but with the procedural side of mental activity.

Thus, many teachers believe that the development of a child’s mathematical abilities is only possible if there are significant natural abilities for this, i.e. Most often in teaching practice it is believed that abilities need to be developed only in those children who already have them. But experimental research by Beloshistaya A.V. showed that work on the development of mathematical abilities is necessary for every child, regardless of his natural talent. It’s just that the results of this work will be expressed in different degrees of development of these abilities: for some children this will be a significant advance in the level of development of mathematical abilities, for others it will be a correction of natural deficiencies in their development.

The great difficulty for a teacher when organizing work on the development of mathematical abilities is that today there is no specific and fundamentally new methodological solution that can be presented to the teacher in full. The lack of methodological support for individual work with capable children leads to the fact that primary school teachers do not do this work at all.

With my work, I wanted to draw attention to this problem and emphasize that the individual characteristics of each gifted child are not only his characteristics, but, perhaps, the source of his giftedness. And the individualization of such a child’s education is not only a way of his development, but also the basis for his preservation in the status of “capable, gifted.”

Bibliographic list.

1. Beloshistaya, A.V. Development of a schoolchild’s mathematical abilities as a methodological problem [Text] / A.V. White-haired // Elementary school. - 2003. - No. 1. - P. 45 - 53

2. Vygotsky, L.S. Collection of essays in 6 volumes (volume 3) [Text] / L.S. Vygotsky. - M, 1983. - P. 368

3. Dorofeev, G.V. Mathematics and intellectual development of schoolchildren [Text] / G.V. Dorofeev // World of education in the world. - 2008. - No. 1. - P. 68 - 78

4. Zaitseva, S.A. Activation of mathematical activity of junior schoolchildren [Text] / S.A. Zaitseva // Primary education. - 2009. - No. 1. - pp. 12 - 19

5. Zak, A.Z. Development of intellectual abilities in children aged 8 - 9 years [Text] / A.Z. Zach. - M.: New School, 1996. - P. 278

6. Krutetsky, V.A. Fundamentals of educational psychology [Text] / V.A. Krutetsky - M., 1972. - P. 256

7. Leontiev, A.N. Chapter on abilities [Text] / A.N. Leontiev // Questions of psychology. - 2003. - No. 2. - P.7

8. Morduchai-Boltovskoy, D. Philosophy. Psychology. Mathematics[Text] / D. Mordukhai-Boltovskoy. - M., 1988. - P. 560

9. Nemov, R.S. Psychology: in 3 books (volume 1) [Text] / R.S. Nemov. - M.: VLADOS, 2006. - P. 688

10. Ozhegov, S.I. Explanatory dictionary of the Russian language [Text] / S.I. Ozhegov. - Onyx, 2008. - P. 736

11.Reversh, J.. Talent and Genius [Text] / J. Reversh. - M., 1982. - P. 512

12.Teplov, B.M. The problem of individual abilities [Text] / B.M. Teplov. - M.: APN RSFSR, 1961. - P. 535

13. Thorndike, E.L. Principles of learning based on psychology [electronic resource]. - Access mode. - http://metodolog.ru/vigotskiy40.html

14.Psychology [Text]/ ed. A.A. Krylova. - M.: Science, 2008. - P. 752

15.Shadrikov V.D. Development of abilities [Text] / V.D.Shadrikov //Primary school. - 2004. - No. 5. - p18-25

16.Volkov, I.P. Is there a lot of talent at the school? [Text] / I.P. Volkov. - M.: Knowledge, 1989. - P.78

17. Dorofeev, G.V. Does teaching mathematics improve the level of intellectual development of schoolchildren? [Text] /G.V. Dorofeev // Mathematics at school. - 2007. - No. 4. - pp. 24 - 29

18. Istomina, N.V. Methods of teaching mathematics in primary classes [Text] / N.V. Istomina. - M.: Academy, 2002. - P. 288

19. Savenkov, A.I. Gifted child in a public school [Text] / ed. M.A. Ushakova. - M.: September, 2001. - P. 201

20. Elkonin, D.B. Questions of psychology of educational activities of junior schoolchildren [Text] / Ed. V. V. Davydova, V. P. Zinchenko. - M.: Education, 2001. - P. 574

Development of mathematical abilities

among younger schoolchildren

The development of the mathematical abilities of a junior schoolchild is understood as the purposeful, didactically and methodically organized formation and development of a set of interrelated properties and qualities of the child’s mathematical thinking style and his abilities for mathematical knowledge of reality.

The problem of ability is a problem of individual differences. With the best organization of teaching methods, the student will progress more successfully and faster in one area than in another.

The learning process should be comprehensive, both in terms of organizing the learning process itself, and in terms of developing in students a deep interest in mathematics, problem-solving skills, understanding the system of mathematical knowledge, solving with students a special system of non-standard problems, which should be offered not only in lessons, but also on tests. Thus, a special organization of the presentation of educational material and a well-thought-out system of tasks help to increase the role of meaningful motives for studying mathematics. The number of result-oriented students is decreasing.

Teachers successfully use the method of “composing tasks” to determine the direction of motivation. Each task is assessed according to a system of the following indicators: the nature of the task, its correctness and relation to the source text. The same method is sometimes used in a different version: after solving the problem, students were asked to create any problems that were somehow related to the original problem.

To create psychological and pedagogical conditions for increasing the efficiency of organizing the learning process system, the principle of organizing the learning process in the form of substantive communication using cooperative forms of student work is used. This is group problem solving and collective discussion of grading, pair and team forms of work.

The methodology for using the system of long-term assignments was considered by E.S. Rabunsky when organizing work with high school students in the process of teaching German at school.

During the experiment, such sheets were developed on the topics: “Oral and written calculation techniques”, “Numbering”, “Quantities”, “Fractions”, “Equations”.

Methodological principles for constructing the proposed system:

The principle of compliance with the mathematics program for primary grades. The content of the sheets is tied to a stable (standard) mathematics program for primary grades. Thus, we believe it is possible to implement the concept of individualizing mathematics teaching for a capable child in accordance with the procedural features of his educational activities when working with any textbook that corresponds to the standard program.
Methodically, each sheet implements the principle of dosage, i.e. in one sheet only one technique or one concept is introduced, or one connection, but essential for a given concept, is revealed. This, on the one hand, helps the child clearly understand the purpose of the work, and on the other hand, helps the teacher to easily monitor the quality of mastery of this technique or concept.
Structurally, the sheet represents a detailed methodological solution to the problem of introducing or introducing and consolidating one or another technique, concept, connections of this concept with other concepts. The tasks are selected and grouped (i.e., the order in which they are placed on the sheet matters) in such a way that the child can “move” along the sheet independently, starting from the simplest methods of action already familiar to him, and gradually master a new method, which in the first steps fully revealed in smaller actions that are the basis of this technique. As you move through the sheet, these small actions are gradually arranged into larger blocks. This allows the student to master the technique as a whole, which is the logical conclusion of the entire methodological “construction”. This structure of the sheet allows you to fully implement the principle of a gradual increase in the level of complexity at all stages.
This structure of the worksheet also makes it possible to implement the principle of accessibility, and to a much deeper extent than can be done today when working only with a textbook, since the systematic use of sheets allows you to learn the material at an individual pace that is convenient for the student, which the child can regulate independently.
The system of sheets (thematic block) allows you to implement the principle of perspective, i.e. gradual inclusion of the student in the activities of planning the educational process. Tasks designed for long-term (delayed) preparation require long-term planning. The ability to organize your work, planning it for a certain period of time, is the most important educational skill.
The system of worksheets on the topic also makes it possible to implement the principle of individualization of testing and assessing students’ knowledge, not on the basis of differentiating the level of difficulty of tasks, but on the basis of the unity of requirements for the level of knowledge, skills and abilities. Individualized deadlines and methods for completing tasks make it possible to present all children with tasks of the same level of complexity, corresponding to the program requirements for the norm. This does not mean that talented children should not be held to higher standards. Worksheets at a certain stage allow such children to use material that is more intellectually rich, which in a propaedeutic way will introduce them to the following mathematical concepts of a higher level of complexity.

Teaching mathematics in primary school is very important. It is this subject that, if successfully studied, will create the prerequisites for the mental activity of a student in middle and senior education.

Mathematics as a subject forms stable cognitive interest and logical thinking skills. Mathematical tasks contribute to the development of a child's thinking, attention, observation, strict consistency of reasoning and creative imagination.

Today's world is undergoing significant changes that place new demands on people. If a student in the future wants to actively participate in all spheres of society, then he needs to be creative, continuously improve himself and develop his individual abilities. But this is exactly what school should teach a child.

Unfortunately, the teaching of younger schoolchildren is most often carried out according to the traditional system, when the most common way in the lesson is to organize the actions of students according to a model, that is, most mathematical tasks are training exercises that do not require the initiative and creativity of children. The priority tendency is for the student to memorize educational material, memorize calculation techniques and solve problems using a ready-made algorithm.

It must be said that many teachers are already developing technologies for teaching mathematics to schoolchildren, which involve children solving non-standard problems, that is, those that form independent thinking and cognitive activity. The main goal of school education at this stage is the development of children’s searching, investigative thinking.

Accordingly, the tasks of modern education today have changed greatly. Now the school focuses not only on giving the student a set of certain knowledge, but also on the development of the child’s personality. All education is aimed at realizing two main goals: educational and educational.

Educational includes the formation of basic mathematical skills, abilities and knowledge.

The developmental function of education is aimed at the development of the student, and the educational function is aimed at the formation of moral values in him.

What is the peculiarity of mathematical teaching? At the very beginning of his studies, the child thinks in specific categories. At the end of primary school, he should learn to reason, compare, see simple patterns and draw conclusions. That is, at first he has a general abstract idea of the concept, and at the end of training this general idea is concretized, supplemented with facts and examples, and, therefore, turns into a truly scientific concept.

Teaching methods and techniques must fully develop the child’s mental activity. This is possible only when the child finds attractive aspects during the learning process. That is, technologies for teaching younger schoolchildren should affect the formation of mental qualities - perception, memory, attention, thinking. Only then will learning be successful.

At the present stage, methods are of primary importance for the implementation of these tasks. Here is an overview of some of them.

Based on the methodology according to L.V. Zankov, learning is based on the mental functions of the child, which have not yet matured. The method assumes three lines of development of the student’s psyche - mind, feelings and will.

The idea of L.V. Zankov was embodied in the curriculum for studying mathematics, the author of which was I.I. Arginskaya. The educational material here involves significant independent activity of the student in acquiring and mastering new knowledge. Particular importance is attached to tasks with different forms of comparison. They are given systematically and taking into account the increasing complexity of the material.

The emphasis of teaching is on the classroom activities of the students themselves. Moreover, schoolchildren do not just solve and discuss tasks, but compare, classify, generalize, and find patterns. It is precisely this kind of activity that strains the mind, awakens intellectual feelings, and, therefore, gives children pleasure from the work done. In such lessons, it becomes possible to reach a point where students learn not for grades, but to gain new knowledge.

A feature of I. I. Arginskaya’s methodology is its flexibility, that is, the teacher uses every thought expressed by the student in the lesson, even if it was not planned by the teacher. In addition, it is expected to actively include weak schoolchildren in productive activities, providing them with measured assistance.

N.B. Istomina’s methodological concept is also based on the principles of developmental education. The course is based on systematic work to develop in schoolchildren such techniques for studying mathematics as analysis and comparison, synthesis and classification, and generalization.

N.B. Istomina’s technique is aimed not only at developing the necessary knowledge, skills and abilities, but also at improving logical thinking. A special feature of the program is the use of special methodological techniques to develop general methods of mathematical operations, which will take into account the individual abilities of the individual student.

The use of this educational and methodological complex allows you to create a favorable atmosphere in the lesson in which children freely express their opinions, participate in discussions and receive, if necessary, help from the teacher. For the development of the child, the textbook includes tasks of a creative and exploratory nature, the implementation of which is associated with the child’s experience, previously acquired knowledge, and, possibly, with a guess.

In the methodology of N. B. Istomina, work is systematically and purposefully carried out to develop the student’s mental activity.

One of the traditional methods is the course of teaching mathematics to junior schoolchildren by M. I. Moro. The leading principle of the course is a skillful combination of training and education, the practical orientation of the material, and the development of the necessary skills and abilities. The methodology is based on the assertion that in order to successfully master mathematics, it is necessary to create a solid foundation for learning in the elementary grades.

The traditional method develops in students conscious, sometimes even automatic, computational skills. Much attention in the program is paid to the systematic use of comparison, comparison, and generalization of educational material.

A special feature of M.I. Moro’s course is that the concepts, relationships, and patterns studied are applied in solving specific problems. After all, solving word problems is a powerful tool for developing children’s imagination, speech, and logical thinking.

Many experts highlight the advantage of this technique - it is the prevention of student mistakes by performing numerous training exercises with the same techniques.

But a lot is said about its shortcomings - the program does not fully ensure the activation of schoolchildren’s thinking in the classroom.

Teaching mathematics to primary schoolchildren assumes that each teacher has the right to independently choose the program in which he will work. And yet, we must take into account that today’s education requires increased active thinking of students. But not every task requires thinking. If the student has mastered the solution method, then memory and perception are sufficient to cope with the proposed task. It’s another matter if a student is given a non-standard task that requires a creative approach, when the accumulated knowledge must be applied in new conditions. Then mental activity will be fully realized.

Thus, one of the important factors ensuring mental activity is the use of non-standard, entertaining tasks.

Another way to awaken a child’s thoughts is to use interactive learning in mathematics lessons. Dialogue teaches a student to defend his opinion, pose questions to a teacher or classmate, review peers’ answers, explain incomprehensible points to weaker students, and find several different ways to solve a cognitive problem.

A very important condition for activating thought and developing cognitive interest is the creation of a problem situation in a mathematics lesson. It helps to attract the student to the educational material, confront him with some complexity, which can be overcome, while activating mental activity.

Activation of students' mental work will also occur if such developmental operations as analysis, comparison, synthesis, analogy, and generalization are included in the learning process.

Primary school students find it easier to find differences between objects than to determine what they have in common. This is due to their predominantly visual and figurative thinking. In order to compare and find commonality between objects, the child must move from visual methods of thinking to verbal-logical ones.

Comparison and comparison will lead to the discovery of differences and similarities. This means that it will be possible to classify according to some criteria.

Thus, for a successful result in teaching mathematics, the teacher needs to include a number of techniques in the process, the most important of which are solving entertaining problems, analyzing various types of educational tasks, using a problem situation and using the “teacher-student-student” dialogue. Based on this, we can highlight the main task of teaching mathematics - to teach children to think, reason, and identify patterns. The lesson should create an atmosphere of search in which every student can become a pioneer.

Homework plays a very important role in children's mathematical development. Many teachers are of the opinion that the number of homework should be reduced to a minimum or even abolished. Thus, the student’s workload, which has a negative impact on health, is reduced.

On the other hand, deep research and creativity require leisurely reflection, which should be carried out outside the lesson. And, if a student’s homework involves not only educational functions, but also developmental ones, then the quality of learning the material will significantly increase. Thus, the teacher should design homework so that students can engage in creative and exploratory activities both at school and at home.

When a student completes homework, parents play a big role. Therefore, the main advice to parents is that the child should do his math homework himself. But this does not mean that he should not receive help at all. If a student cannot cope with solving a task, then you can help him find the rule with which the example is solved, give a similar task, give him the opportunity to independently find the error and correct it. Under no circumstances should you complete the task for your child. The main educational goal of both the teacher and the parent is the same - to teach the child to obtain knowledge himself, and not to receive ready-made ones.

Parents need to remember that the purchased book “Ready Homework” should not be in the hands of the student. The purpose of this book is to help parents check the accuracy of homework, and not to give the student the opportunity, using it, to rewrite ready-made solutions. In such cases, you can completely forget about the child’s good performance in the subject.

The formation of general educational skills is also facilitated by the correct organization of the student’s work at home. The role of parents is to create conditions for their child to work. The student must do homework in a room where the TV is not on and there are no other distractions. You need to help him plan his time correctly, for example, specifically choose an hour to do his homework and never put off this work until the very last moment. Helping your child with homework is sometimes simply necessary. And skillful help will show him the relationship between school and home.

Thus, parents also play an important role for the successful education of the student. In no case should they reduce the child’s independence in learning, but at the same time skillfully come to his aid if necessary.

ACTIVE METHODS OF TEACHING JUNIOR SCHOOL CHILDREN MATHEMATICS.

Kuznetsova Nadezhda Vladimirovna primary school teacher

MBOU BGO Secondary School No. 4, Borisoglebsk

The problem of choosing working methods has always arisen for teachers. But in new conditions, new methods are needed that allow us to organize the learning process and the relationship between teacher and student in a new way.

In the overall volume of knowledge, skills and abilities acquired by students in primary school, mathematics plays an important place, which is widely used in the study of other subjects. The main task of every teacher is not only to give students a certain amount of knowledge, but to develop their interest in learning and teach them how to learn.

A lesson is the main form of organizing the educational process, and the quality of teaching is, first of all, the quality of the lesson. Without well-thought-out teaching methods, it is difficult to organize the assimilation of program material. Methods and means of teaching should be improved in order to involve students in cognitive search, in the work of learning: they help teach students to actively acquire knowledge independently, and develop interest in the subject.

To better remember the studied material, as well as to control the assimilation of knowledge, didactic games are used in lessons:

Math Domino;

Feedback cards;

Crosswords.

The effectiveness of teaching mathematics to schoolchildren largely depends on the choice of methods for organizing the educational process. Active learning methods are a set of ways to organize and manage the educational and cognitive activities of teachers.

When using active teaching methods, the effectiveness of the lesson increases noticeably. Students willingly complete the tasks assigned to them and become teacher assistants in conducting the lesson. Activation of the educational process promotes the use of heuristic and search methods. Leading questions encourage students to get to the bottom of things and together determine which of them and how deeply prepared they are for the new lesson.

Active learning methods also provide targeted activation of students’ mental processes, i.e. stimulate thinking when using specific problem situations and conducting business games, facilitate memorization when highlighting the main thing in practical classes, arouse interest in mathematics and develop the need for independent acquisition of knowledge.

The teacher’s task is to make maximum use of active learning methods to develop the mental abilities of each child. The game “Yes” - “No” is successfully used to reinforce new material. The question is read once, you cannot ask again; while reading the question you must write down the answer “yes” or “no”. The main thing here is to involve even the most passive students in the work.

The educational process includes integrated lessons, mathematical dictations, business games, olympiads, competition lessons, quizzes, KVN, press conferences, brainstorming sessions, and auctions of ideas.

The main methods of teaching schoolchildren: conversation, games, creative activities are included in the structure of the BIT lesson. Students do not have time to get tired; their attention is maintained and developed all the time. Such a lesson, due to its emotional intensity and elements of competition, has a deep educational effect. The children see in practice the opportunities that creative teamwork presents.

Let me give you a few examples.

"Auction of Ideas".

Before the “auction” begins, experts determine the “sale value” of the ideas. Then the ideas are “sold”, the author of the idea who received the highest price is recognized as the winner. The idea passes to the developers, who justify their options. The auction can be extended in two rounds. Ideas that make it to the second round can be tested in practical problems.

"Brain attack".

The lesson is similar to an “auction”. The group is divided into “generators” and “experts”. Generators are offered a situation (of a creative nature). For a certain time, students are offered various options for solving the proposed problem, recorded on the board. At the end of the allotted time, the “experts” enter the battle. During the discussion, the best proposals are accepted and the teams change roles. Providing students in the classroom with the opportunity to propose, discuss, and exchange ideas not only develops their creative thinking and increases confidence in the teacher, but also makes learning “comfortable.”

It is more convenient to conduct a business game when repeating and generalizing the topic. The class is divided into groups. Each group is given a task and then their solution is shared. There is an exchange of tasks.

The use of active methods involves a departure from the authoritarian teaching style, the inclusion of students in educational activities, stimulate and activate, and also provides for improving the quality of education.

Literature.

1. Antsibor M.M. Active forms and methods of teaching. Tula, 2002

2. Brushmensky A.V. Psychology of thinking and problem-based learning. - M, 2003.

Ministry of Education, Science and Youth Policy of the Republic of Dagestan

GBOUSPO "Republican Pedagogical College" named after. Z.N. Batymurzaeva.

Course work

on TONKM with teaching methods

on the topic of: " Active methods of teaching mathematics in primary school"

Completed by: St. 3 "v" course

Ezerkhanova Zalina

Scientific adviser:

Adilkhanova S.A.

Khasavyurt 2014

Introduction

Chapter I.

Chapter II

Conclusion

Literature

Introduction

“The mathematician takes pleasure in the knowledge he has already mastered and always strives for new knowledge.”

The effectiveness of teaching mathematics to schoolchildren largely depends on the choice of forms of organizing the educational process. In my work I give preference to active learning methods. Active learning methods are a set of methods for organizing and managing the educational and cognitive activities of students, which have the following main features:

forced learning activity;

independent development of solutions by students;

high degree of involvement of students in the educational process;

constant processing of communication between students and teachers, and control of independent learning.

The main point of developing federal state educational standards, solving the strategic task of the development of Russian education - improving the quality of education, achieving new educational results. In other words, the Federal State Educational Standard is not intended to fix the state of education achieved at the previous stages of its development, but orients education towards achieving a new quality that is adequate to the modern (and even predictable) needs of the individual, society and the state.

The methodological basis of the standards for primary general education of the new generation is the system-activity approach.

The system-activity approach is aimed at personal development and the formation of civic identity. Training must be organized in such a way as to purposefully lead development. Since the main form of organization of learning is the lesson, it is necessary to know the principles of lesson construction, an approximate typology of lessons and criteria for assessing a lesson within the framework of a systemic activity approach and active methods of work used in the lesson.

Currently, the student has great difficulty setting goals and drawing conclusions, synthesizing material and connecting complex structures, generalizing knowledge, and even more so finding connections in it. Teachers, noting students’ indifference to knowledge, reluctance to learn, and low level of development of cognitive interests, try to design more effective forms, models, methods, and learning conditions.

Creating didactic and psychological conditions for the meaningfulness of learning and the inclusion of students in it at the level of not only intellectual, but personal and social activity is possible with the use of active teaching methods. The emergence and development of active methods is due to the fact that learning faced new tasks: not only to give students knowledge, but also to ensure the formation and development of cognitive interests and abilities, skills and abilities of independent mental work, the development of creative and communicative abilities of the individual.

A chain of failures can turn talented children away from mathematics; on the other hand, learning should proceed close to the ceiling of the student’s capabilities: a feeling of success is created by the understanding that significant difficulties have been overcome. Therefore, for each lesson you need to carefully select and prepare individual knowledge, cards, based on an adequate assessment of the student’s capabilities at the moment, taking into account his individual abilities.

active method of teaching mathematics

To organize active cognitive activity of students in the classroom, the optimal combination of active learning methods is crucial. It is very important for me to evaluate the work and psychological climate in my lessons. Therefore, we need to try to ensure that children are not only actively engaged in their studies, but also feel confident and comfortable.

The problem of individual activity in learning is one of the most pressing in educational practice.

Taking this into account, I chose the research topic: “Active methods of teaching mathematics in primary school.”

Purpose of the study: to identify and theoretically substantiate the effectiveness of using active teaching methods for primary schoolchildren with learning difficulties in mathematics lessons.

Research problem: what methods contribute to the activation of cognitive activity in students during the learning process.

Object of study: the process of teaching mathematics to junior schoolchildren.

Subject of research: studying active methods of teaching mathematics in primary school.

Research hypothesis: the process of teaching mathematics to junior schoolchildren will be more successful under the following conditions if:

During mathematics lessons, active teaching methods will be used for younger students.

Research objectives:

)study the literature on the problem of using active methods of teaching mathematics in primary school;

2)Identify and reveal the features of active methods of teaching mathematics in elementary school;

)Consider active methods of teaching mathematics in elementary school.

Research methods:

analysis of psychological and pedagogical literature on the problem of studying active methods of teaching mathematics in primary school;

observation of younger schoolchildren.

Structure of the work: the work consists of an introduction, 2 chapters, a conclusion, and a list of references.

Chapter I

1.1 Introduction to active learning methods

Method (from the Greek methodos - path of research) - a way to achieve.

Active teaching methods are a system of methods that ensure activity and diversity in the mental and practical activities of students in the process of mastering educational material.

Active methods provide solutions to educational problems in various aspects:

A teaching method is an ordered set of didactic techniques and means by which the goals of teaching and education are realized. Teaching methods include interconnected, sequentially alternating methods of purposeful activity of the teacher and students.

Any teaching method presupposes a goal, a system of actions, learning tools and an intended result. The object and subject of the teaching method is the student.

Any one teaching method is used in its pure form only for specially planned educational or research purposes. Usually the teacher combines various teaching methods.

Today there are different approaches to the modern theory of teaching methods.

Active learning methods are methods that encourage students to engage in active mental and practical activity in the process of mastering educational material. Active learning involves the use of a system of methods that is aimed primarily not at the teacher presenting ready-made knowledge, memorizing and reproducing it, but at students’ independent acquisition of knowledge and skills in the process of active mental and practical activity. The use of active methods in mathematics lessons helps to develop not just reproduction knowledge, but the skills and needs to apply this knowledge to analyze, assess the situation and make the right decision.

Active methods ensure interaction between participants in the educational process. When using them, the distribution of “responsibilities” is carried out when receiving, processing and applying information between the teacher and the student, between the students themselves. It is clear that a large developmental load is borne by the learning process, which is active on the part of the student.

When choosing active learning methods, you should be guided by a number of criteria, namely:

· compliance with goals and objectives, principles of training;

· compliance with the content of the topic being studied;

· compliance with the capabilities of the trainees: age, psychological development, level of education and upbringing, etc.

· compliance with the conditions and time allocated for training;

· compliance with the teacher’s capabilities: his experience, desires, level of professional skill, personal qualities.

· Student activity can be ensured if the teacher purposefully and makes maximum use of tasks in the lesson: formulate a concept, prove, explain, develop an alternative point of view, etc. In addition, the teacher can use techniques for correcting “intentionally made” errors, formulating and developing tasks for friends.

· An important role is played by developing the skill of asking questions. Analytical and problematic questions like “Why? What does it follow from? What does it depend on? require constant updating in work and special training in their production. The methods of this training are varied: from tasks to pose a question to a text in class to the game “Who can ask the most questions on a certain topic in a minute.

· Active methods provide solutions to educational problems in various aspects:

· formation of positive learning motivation;

· increasing the cognitive activity of students;

· active involvement of students in the educational process;

· stimulation of independent activity;

· development of cognitive processes - speech, memory, thinking;

· effective assimilation of a large volume of educational information;

· development of creative abilities and innovative thinking;

· development of the communicative-emotional sphere of the student’s personality;

· revealing the personal and individual capabilities of each student and determining the conditions for their manifestation and development;

· development of independent mental work skills;

· development of universal skills.

Let's talk about the effectiveness of teaching methods in more detail.

Active learning methods place the student in a new position. Previously, the student was completely subordinate to the teacher, now active actions, thoughts, ideas and doubts are expected from him.

The quality of teaching and upbringing is directly related to the interaction of thinking processes and the formation of a student’s conscious knowledge, strong skills, and active learning methods.

The activity of students is possible only if there are incentives. Therefore, among the principles of activation, the motivation of educational and cognitive activity acquires a special place. An important factor of motivation is encouragement. Primary school children have unstable learning motives, especially cognitive ones, so positive emotions accompany the formation of cognitive activity.

1.2 Application of active teaching methods in primary school

One of the problems that worries teachers is how to develop a child’s sustainable interest in learning, knowledge and the need for independent search, in other words, how to intensify cognitive activity in the learning process.

If a habitual and desirable form of activity for a child is a game, then it is necessary to use this form of organizing activities for learning, combining the game and the educational process, or more precisely, using a game form of organizing the activities of students to achieve educational goals. Thus, the motivational potential of the game will be aimed at more effective development of the educational program by schoolchildren. And the role of motivation in successful learning can hardly be overestimated. Conducted studies of student motivation have revealed interesting patterns. It turned out that the importance of motivation for successful study is higher than the importance of the student’s intelligence. High positive motivation can play the role of a compensating factor in the case of a student’s insufficiently high abilities, but this principle does not work in the opposite direction - no abilities can compensate for the absence of a learning motive or its low expression and ensure significant academic success.

The goals of school education, which are set for the school by the state, society and family, in addition to acquiring a certain set of knowledge and skills, are to reveal and develop the child’s potential, to create favorable conditions for the realization of his natural abilities. A natural play environment, in which there is no coercion and there is an opportunity for each child to find his place, show initiative and independence, and freely realize his abilities and educational needs, is optimal for achieving these goals.

To create such an environment in the classroom, I use active learning methods.

Using active learning methods in the classroom allows you to:

provide positive motivation for learning;

conduct a lesson at a high aesthetic and emotional level;

ensure a high degree of differentiation of training;

increase the volume of work performed in class by 1.5 - 2 times;

improve knowledge control;

rationally organize the educational process, increase the effectiveness of the lesson.

Active learning methods can be used at various stages of the educational process:

stage - primary acquisition of knowledge. This could be a problem lecture, a heuristic conversation, an educational discussion, etc.

stage - knowledge control (consolidation). Methods such as collective mental activity, testing, etc. can be used.

stage - the formation of skills based on knowledge and the development of creative abilities; It is possible to use simulated learning, game and non-game methods.

In addition to intensifying the development of educational information, active teaching methods make it possible to carry out the educational process just as effectively during the lesson and in extracurricular activities. Team work, joint project and research activities, defending one’s position and a tolerant attitude towards other people’s opinions, taking responsibility for oneself and the team form the personality traits, moral attitudes and value guidelines of the student that meet the modern needs of society. But this is not all the possibilities of active learning methods. In parallel with training and education, the use of active teaching methods in the educational process ensures the formation and development of so-called soft or universal skills in students. These usually include the ability to make decisions and solve problems, communication skills and qualities, the ability to clearly formulate messages and clearly set tasks, the ability to listen and take into account different points of view and opinions of other people, leadership skills and qualities, the ability to work in a team and etc. And today many already understand that, despite their softness, these skills in modern life play a key role both in achieving success in professional and social activities, and in ensuring harmony in personal life.

Innovation is an important feature of modern education. Education changes in content, forms, methods, responds to changes in society, and takes into account global trends.

Educational innovation is the result of the creative search of teachers and scientists: new ideas, technologies, approaches, teaching methods, as well as individual elements of the educational process.

The wisdom of the desert dwellers says: “You can lead a camel to water, but you cannot force him to drink.” This proverb reflects the basic principle of learning - you can create all the necessary conditions for learning, but knowledge itself will happen only when the student wants to know. How can we make sure that the student feels needed at every stage of the lesson and is a full-fledged member of the class team? Another wisdom teaches: “Tell me - I will forget. Show me - I will remember. Let me act on my own - and I will learn.” According to this principle, one’s own active activity is the basis for learning. And therefore, one of the ways to increase effectiveness in studying school subjects is to introduce active forms of work at different stages of the lesson.

Based on the degree of activity of students in the educational process, teaching methods are conventionally divided into two classes: traditional and active. The fundamental difference between these methods is that when they are used, students are created conditions under which they cannot remain passive and have the opportunity for active exchange of knowledge and work experience.

The goal of using active learning methods in elementary school is to develop curiosity.Therefore, for students you can create a journey into the world of knowledge with fairy-tale characters.

In the course of his research, the outstanding Swiss psychologist Jean Piaget expressed the opinion that logic is not innate, but develops gradually with the development of the child. Therefore, in lessons in grades 2-4, it is necessary to use more logical problems related to mathematics, language, knowledge of the world around us, etc. Tasks require the performance of specific operations: intuitive thinking based on detailed ideas about objects, simple operations (classification, generalization, one-to-one correspondence).

Let's consider several examples of the use of active methods in the educational process.

Conversation is a dialogical method of presenting educational material (from the Greek dialogos - a conversation between two or more persons), which in itself speaks of the essential specificity of this method. The essence of the conversation is that the teacher, through skillfully posed questions, encourages students to reason, to analyze the facts and phenomena being studied in a certain logical sequence, and to independently formulate appropriate theoretical conclusions and generalizations.

A conversation is not a reporting, but a question-and-answer method of educational work to comprehend new material. The main point of the conversation is to encourage students, with the help of questions, to reason, analyze the material and generalize, to independently “discover” conclusions, ideas, laws, etc. that are new to them. Therefore, when conducting a conversation to comprehend new material, it is necessary to pose questions so that they require not monosyllabic affirmative or negative answers, but detailed reasoning, certain arguments and comparisons, as a result of which students isolate the essential features and properties of the objects and phenomena being studied and in this way acquire new ones. knowledge. It is equally important that the questions have a clear sequence and focus, allowing students to deeply comprehend the internal logic of the knowledge they are learning.

These specific features of conversation make it a very active learning method. However, the use of this method also has its limitations, because not all material can be presented through conversation. This method is most often used when the topic being studied is relatively simple and when students have a certain stock of ideas or life observations on it that allow them to comprehend and assimilate knowledge in a heuristic (from the Greek heurisko - I find) way.

Active methods involve conducting classes through the organization of gaming activities for students. The pedagogy of the game collects ideas that facilitate contacts in the group, the exchange of thoughts and feelings, the understanding of specific problems and the search for ways to solve them. It has an auxiliary function in the entire learning process. The purpose of play pedagogy is to provide techniques that support group work and create an atmosphere that makes participants feel safe and good.

The pedagogy of the game helps the presenter realize the various needs of the participants: the need for movement, experiences, overcoming fear, the desire to be with other people. It also helps to overcome timidity, shyness, as well as existing social stereotypes.

For active teaching methods, a special place is occupied by forms of organizing the educational process - non-standard lessons: a lesson - a fairy tale, a game, a journey, a scenario, a quiz, lessons - knowledge reviews.

During such lessons, children's activity increases; they are happy to help Kolobok escape from the fox, save ships from attacks by pirates, and store food for the squirrel for the winter. In such lessons, children are in for a surprise, so they try to work fruitfully and complete as many different tasks as possible. The very beginning of such lessons captivates children from the first minutes: “We’re going to the forest for science today” or “The floorboard is creaking about something...” Books from the series “I’m going to a lesson in elementary school” and, of course, the creativity of the student himself help to teach such lessons. teachers. They help the teacher prepare for lessons in less time and conduct them in a more meaningful, modern, and interesting way.

In my work, feedback tools have acquired particular importance, which make it possible to quickly obtain information about the movement of each student’s thoughts, about the correctness of his actions at any moment of the lesson. Feedback tools are used to monitor the quality of acquisition of knowledge, skills and abilities. Every student has feedback tools (we make them ourselves during labor lessons or purchase them in stores), they are an essential logical component of his cognitive activity. These are signal circles, cards, number and letter fans, traffic lights. The use of feedback tools makes it possible to make the work of the class more rhythmic, forcing each student to study. It is important that such work is carried out systematically.

One of the new means of checking the quality of training is tests. This is a qualitative way of checking learning outcomes, characterized by such parameters as reliability and objectivity. Tests test theoretical knowledge and practical skills. With the arrival of a computer at school, new methods for intensifying educational activities open up for teachers.

Modern teaching methods are mainly focused on teaching not ready-made knowledge, but activities for independent acquisition of new knowledge, i.e. cognitive activity.

In the practice of many teachers, independent work of students is widely used. It is carried out in almost every lesson within 7-15 minutes. The first independent works on the topic are mainly educational and corrective in nature. With their help, prompt feedback in teaching is provided: the teacher sees all the shortcomings in the students’ knowledge and eliminates them in a timely manner. You can refrain from recording grades “2” and “3” in the class journal for now (by posting them in the student’s notebook or diary). This assessment system is quite humane, mobilizes students well, helps them better understand their difficulties and overcome them, and helps improve the quality of knowledge. Students find themselves better prepared for the test; their fear of such work and the fear of getting a bad mark disappear. The number of unsatisfactory grades, as a rule, is sharply reduced. Students develop a positive attitude towards business-like, rhythmic work, and rational use of lesson time.

Don't forget the restorative power of relaxation in the classroom. After all, sometimes a few minutes are enough to shake yourself up, relax cheerfully and actively, and restore energy. Active methods - "physical minutes" "Earth, air, fire and water", "Bunnies" and many others will allow you to do this without leaving the classroom.

If the teacher himself takes part in this exercise, in addition to benefiting himself, he will also help insecure and shy students to participate more actively in the exercise.

1.3 Features of active methods of teaching mathematics in primary school

· using an activity-based approach to learning;

· practical orientation of the activities of participants in the educational process;

· playful and creative nature of learning;

· interactivity of the educational process;

· inclusion of various communications, dialogue and polylogue in the work;

· using the knowledge and experience of students;

· reflection of the learning process by its participants

Another necessary quality of a mathematician is an interest in patterns. Regularity is the most stable characteristic of a constantly changing world. Today cannot be like yesterday. You cannot see the same face twice from the same angle. Regularities are found already at the very beginning of arithmetic. The multiplication table contains many elementary examples of patterns. Here's one of them. Typically, children like to multiply by 2 and 5, because the last digits of the answer are easy to remember: when multiplied by 2, even numbers are always obtained, and when multiplied by 5, even simpler, it is always 0 or 5. But even multiplying by 7 has its own patterns . If we look at the last digits of the products 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, i.e. by 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, then we will see that the difference between the next and previous digits is: - 3; +7; - 3; - 3; +7; - 3; - 3, - 3. There is a very definite rhythm in this row.

If we read the final digits of the answers when multiplying by 7 in reverse order, then we get the final digits from multiplying by 3. Even in elementary school, you can develop the skill of observing mathematical patterns.

During the adaptation period of first-graders, you must try to be attentive to the little person, support her, worry about her, try to interest her in learning, help so that further education for the child is successful and brings mutual joy to the teacher and student. The quality of teaching and upbringing is directly related to the interaction of thinking processes and the formation of a student’s conscious knowledge, strong skills, and active learning methods.

The key to quality education is love for children and constant search.

The direct involvement of students in educational and cognitive activities during the educational process is associated with the use of appropriate methods, which have received the general name of active learning methods. For active learning, the principle of individuality is important - the organization of educational and cognitive activities taking into account individual abilities and capabilities. This includes pedagogical techniques and special forms of classes. Active methods help make the learning process easy and accessible to every child. The activity of students is possible only if there are incentives. Therefore, among the principles of activation, the motivation of educational and cognitive activity acquires a special place. An important factor of motivation is encouragement. Primary school children have unstable learning motives, especially cognitive ones, so positive emotions accompany the formation of cognitive activity.

The age and psychological characteristics of younger schoolchildren indicate the need to use incentives to achieve activation of the educational process. Encouragement not only evaluates the positive results visible at the moment, but in itself it encourages further fruitful work. Encouragement involves the factor of recognition and assessment of the child’s achievements, if necessary, correction of knowledge, statement of success, stimulating further achievements. Encouragement promotes the development of memory, thinking, and creates cognitive interest.

The success of learning also depends on visual aids. These are tables, supporting diagrams, didactic and handouts, individual teaching aids that help make the lesson interesting, joyful, and ensure deep assimilation of the program material.

Individual teaching aids (mathematical pencil cases, letter boxes, abaci) ensure that children are involved in the active learning process, they become active participants in the educational process, and activate children’s attention and thinking.

1Using information technology in a mathematics lesson in elementary school .

In elementary school, it is impossible to conduct a lesson without using visual aids, and problems often arise. Where can I find the material I need and how best to demonstrate it? The computer came to the rescue.

1.2The most effective means of including a child in the creative process in the classroom are:

· play activities;

· creating positive emotional situations;

· work in pairs;

· problem-based learning.

Over the past 10 years, there has been a radical change in the role and place of personal computers and information technology in the life of society. Proficiency in information technology is ranked in the modern world on a par with such qualities as the ability to read and write. A person who skillfully and effectively masters technology and information has a different, new style of thinking and has a fundamentally different approach to assessing the problem that has arisen and to organizing his activities. As practice shows, it is no longer possible to imagine a modern school without new information technologies. It is obvious that in the coming decades the role of personal computers will increase and, in accordance with this, the requirements for computer literacy of entry-level students will increase. The use of ICT in primary school lessons helps students navigate the information flows of the world around them, master practical ways of working with information, and develop skills that allow them to exchange information using modern technical means. In the process of studying, diverse application and use of ICT tools, a person is formed who can act not only according to a model, but also independently, receiving the necessary information from as many sources as possible; able to analyze it, put forward hypotheses, build models, experiment and draw conclusions, make decisions in difficult situations. In the process of using ICT, the student develops, prepares students for a free and comfortable life in the information society, including:

development of visual-figurative, visual-effective, theoretical, intuitive, creative types of thinking; - aesthetic education through the use of computer graphics and multimedia technology;

development of communication abilities;

developing the skills to make the optimal decision or propose solutions in a difficult situation (the use of situational computer games aimed at optimizing decision-making activities);

formation of information culture, skills to process information.

ICT leads to the intensification of all levels of the educational process, providing:

increasing the efficiency and quality of the learning process through the implementation of ICT tools;

providing incentives (stimuli) that determine the activation of cognitive activity;

deepening interdisciplinary connections through the use of modern information processing tools, including audiovisual, when solving problems from various subject areas.

Using information technology in primary school lessonsis one of the most modern means of developing the personality of a junior schoolchild and forming his information culture.

Teachers are increasingly beginning to use computer capabilities in preparing and conducting lessons in primary school.Modern computer programs make it possible to demonstrate vivid clarity, offer various interesting dynamic types of work, and identify the level of knowledge and skills of students.

The role of the teacher in culture is also changing - he must become a coordinator of information flow.

Today, when information becomes a strategic resource for the development of society, and knowledge becomes a relative and unreliable subject, as it quickly becomes outdated and requires constant updating in the information society, it becomes obvious that modern education is a continuous process.

The rapid development of new information technologies and their implementation in our country have left their mark on the development of the personality of the modern child. Today, a new link is being introduced into the traditional scheme “teacher - student - textbook” - a computer, and computer education is being introduced into school consciousness. One of the main parts of informatization of education is the use of information technologies in educational disciplines.

For primary schools, this means a change in priorities in setting educational goals: one of the results of training and education in a first-level school should be the readiness of children to master modern computer technologies and the ability to update the information obtained with their help for further self-education. To achieve these goals, there is a need to apply different strategies for teaching younger schoolchildren in the practice of primary school teachers, and, first of all, the use of information and communication technologies in the teaching and educational process.

Lessons using computer technology make them more interesting, thoughtful, and mobile. Almost any material is used, there is no need to prepare a lot of encyclopedias, reproductions, audio accompaniments for the lesson - all this is already prepared in advance and is contained on a small CD or flash card. Lessons using ICT are especially relevant in elementary school. Students in grades 1-4 have visual-figurative thinking, so it is very important to build their education using as much high-quality illustrative material as possible, involving not only vision, but also hearing, emotions, and imagination in the process of perceiving new things. Here, the brightness and entertainment of computer slides and animation comes in handy.

The organization of the educational process in primary school, first of all, should contribute to the activation of the cognitive sphere of students, the successful assimilation of educational material and contribute to the mental development of the child. Consequently, ICT should perform a certain educational function, help the child understand the flow of information, perceive it, remember it, and, in no case, undermine their health. ICT should act as an auxiliary element of the educational process, and not the main one. Taking into account the psychological characteristics of a primary school student, work using ICT should be clearly thought out and dosed. Thus, the use of ITC in the classroom should be gentle. When planning a lesson (work) in primary school, the teacher must carefully consider the purpose, place and method of using ICT. Consequently, the teacher needs to master modern methods and new educational technologies in order to communicate in the same language with the child.

Chapter II

2.1 Classification of active methods of teaching mathematics in primary school on various grounds

By the nature of cognitive activity:

explanatory and illustrative (story, lecture, conversation, demonstration, etc.);

reproductive (solving problems, repeating experiments, etc.);

problematic (problematic tasks, cognitive tasks, etc.);

partially search - heuristic;

research.

By activity components:

organizational-effective - methods of organizing and implementing educational and cognitive activities;

stimulating - methods of stimulating and motivating educational and cognitive activity;

control and evaluation - methods of monitoring and self-control of the effectiveness of educational and cognitive activities.

For didactic purposes:

methods of studying new knowledge;

methods of consolidating knowledge;

control methods.

By way of presenting educational material:

monologue - informational and informative (story, lecture, explanation);

dialogical (problem presentation, conversation, debate).

By sources of knowledge transfer:

verbal (story, lecture, conversation, instruction, discussion);

visual (demonstration, illustration, diagram, display of material, graph);

practical (exercise, laboratory work, workshop).

Taking into account the personality structure:

consciousness (story, conversation, instruction, illustration, etc.);

behavior (exercise, training, etc.);

feelings - stimulation (approval, praise, blame, control, etc.).

The choice of teaching methods is a creative matter, but it is based on knowledge of learning theory. Teaching methods cannot be divided, universalized or considered in isolation. In addition, the same teaching method may be effective or ineffective depending on the conditions under which it is applied. New content of education gives rise to new methods in teaching mathematics. An integrated approach to the application of teaching methods, their flexibility and dynamism are required.

The main methods of mathematical research are: observation and experience; comparison; analysis and synthesis; generalization and specialization; abstraction and concretization.

Modern methods of teaching mathematics: problem-based (prospective), laboratory, programmed learning, heuristic, building mathematical models, axiomatic, etc.

Let's consider the classification of teaching methods:

Information and development methods are divided into two classes:

Transmission of information in finished form (lecture, explanation, demonstration of educational films and videos, listening to tape recordings, etc.);

Independent acquisition of knowledge (independent work with a book, with a training program, with information databases - the use of information technologies).

Problem-based search methods: problematic presentation of educational material (heuristic conversation), educational discussion, laboratory search work (preceding the study of the material), organization of collective mental activity in small groups, organizational activity game, research work.

Reproductive methods: retelling educational material, performing exercises according to a model, laboratory work according to instructions, exercises on simulators.

Creative and reproductive methods: essays, variable exercises, analysis of production situations, business games and other types of imitation of professional activities.

An integral part of teaching methods are the methods of educational activity of the teacher and students. Methodological techniques - actions, methods of work aimed at solving a specific problem. Hidden behind the methods of educational work are the methods of mental activity (analysis and synthesis, comparison and generalization, proof, abstraction, concretization, identification of the essential, formulation of conclusions, concepts, techniques of imagination and memorization).

2.2 Heuristic method of teaching mathematics

One of the main methods that allows students to be creative in the process of learning mathematics is the heuristic method. Roughly speaking, this method consists in the fact that the teacher poses a certain educational problem to the class, and then, through sequentially assigned tasks, “guides” students to independently discover this or that mathematical fact. Students gradually, step by step, overcome difficulties in solving the problem and “discover” its solution themselves.

It is known that in the process of studying mathematics, schoolchildren often encounter various difficulties. However, in heuristically structured learning, these difficulties often become a kind of stimulus for learning. So, for example, if schoolchildren are found to have an insufficient supply of knowledge to solve a problem or prove a theorem, then they themselves strive to fill this gap by independently “discovering” this or that property and thereby immediately discovering the usefulness of studying it. In this case, the teacher’s role comes down to organizing and directing the student’s work so that the difficulties that the student overcomes are within his capabilities. Often the heuristic method appears in teaching practice in the form of a so-called heuristic conversation. The experience of many teachers who widely use the heuristic method has shown that it influences students' attitudes towards learning activities. Having acquired a “taste” for heuristics, students begin to regard working according to “ready-made instructions” as uninteresting and boring work. The most significant moments of their learning activities in the classroom and at home are the independent “discoveries” of one or another way to solve a problem. Students' interest in those types of work in which heuristic methods and techniques are used is clearly increasing.

Modern experimental studies conducted in Soviet and foreign schools indicate the usefulness of the widespread use of the heuristic method in the study of mathematics by secondary school students, starting from primary school age. Naturally, in this case, students can be presented with only those educational problems that can be understood and resolved by students at this stage of training.

Unfortunately, the frequent use of the heuristic method in the process of teaching posed educational problems requires much more educational time than studying the same issue by the method of the teacher communicating a ready-made solution (proof, result). Therefore, the teacher cannot use the heuristic teaching method in every lesson. In addition, long-term use of only one (even a very effective method) is contraindicated in training. However, it should be noted that “time spent on fundamental issues, worked out with the personal participation of students, is not wasted time: new knowledge is acquired almost effortlessly thanks to previous deep thinking experience.” Heuristic activity or heuristic processes, although they include mental operations as an important component, at the same time have some specificity. That is why heuristic activity should be considered as a type of human thinking that creates a new system of actions or discovers previously unknown patterns of objects surrounding a person (or objects of the science being studied).

The beginning of the use of the heuristic method as a method of teaching mathematics can be found in the book of the famous French teacher and mathematician Lezan “Development of mathematical initiative”. In this book, the heuristic method does not yet have a modern name and appears in the form of advice to the teacher. Here are some of them:

The basic principle of teaching is “to maintain the appearance of play, respect the child’s freedom, maintaining the illusion (if there is one) of his own discovery of the truth”; “to avoid in the initial upbringing of a child the dangerous temptation of abusing memory exercises,” because this kills his innate qualities; teach based on interest in what is being studied.

The famous methodologist-mathematician V.M. Bradis defines the heuristic method as follows: “A teaching method is called heuristic when the teacher does not inform students of ready-made information to be learned, but leads students to independently rediscover the relevant proposals and rules.”

But the essence of these definitions is the same - an independent, planned only in general terms, search for a solution to the problem posed.

The role of heuristic activity in science and in the practice of teaching mathematics is covered in detail in the books of the American mathematician D. Polya. The purpose of heuristics is to explore the rules and methods that lead to discoveries and inventions. Interestingly, the main method by which one can study the structure of the creative thought process is, in his opinion, the study of personal experience in solving problems and observing how others solve problems. The author tries to derive some rules, following which one can come to discoveries, without analyzing the mental activity in relation to which these rules are proposed. “The first rule is that you must have ability, and along with it, luck. The second rule is to hold firm and not give up until a happy idea appears.” The problem solving diagram given at the end of the book is interesting. The diagram indicates the sequence in which actions must be taken to achieve success. It includes four stages:

Understanding the problem statement.

Drawing up a solution plan.

Implementation of the plan.

Looking back (studying the resulting solution).

During these steps, the problem solver must answer the following questions: What is unknown? What is given? What is the condition? Haven't I encountered this problem before, at least in a slightly different form? Is there any related task to this one? Is it possible to use it?

The book “Prelude to Mathematics” by the American teacher W. Sawyer is very interesting from the point of view of using the heuristic method in school.

“All mathematicians,” writes Sawyer, “are characterized by audacity of mind. A mathematician does not like to be told about something; he wants to figure it out himself.”

This “boldness of mind,” according to Sawyer, is especially pronounced in children.

2.3 Special methods of teaching mathematics

These are the basic methods of cognition adapted for teaching, used in mathematics itself, methods of studying reality characteristic of mathematics.

PROBLEM-BASED LEARNING Problem-based learning is a didactic system based on the patterns of creative assimilation of knowledge and methods of activity, including a combination of techniques and methods of teaching and learning, which have the main features of scientific research.

Problem-based teaching method is training that takes place in the form of removing (resolving) problem situations that are consistently created for educational purposes.

A problematic situation is a conscious difficulty generated by a discrepancy between existing knowledge and the knowledge that is necessary to solve the proposed problem.

A task that creates a problematic situation is called a problem, or a problematic task.

The problem should be understandable to students, and its formulation should arouse students’ interest and desire to solve it.

It is necessary to distinguish between a problematic task and a problem. The problem is broader; it breaks down into a sequential or branched set of problematic tasks. A problematic task can be considered as the simplest, special case of a problem consisting of one task. Problem-based learning is focused on the formation and development of students’ ability for creative activity and the need for it. It is advisable to start problem-based learning with problematic tasks, thereby preparing the ground for setting educational goals.

PROGRAMMED TRAINING

Programmed training is such training when the solution to a problem is presented in the form of a strict sequence of elementary operations; in training programs, the material being studied is presented in the form of a strict sequence of frames. In the era of computerization, programmed learning is carried out using training programs that determine not only the content, but also the learning process. There are two different systems for programming educational material - linear and branched.

The advantages of programmed training include: dosage of educational material, which is absorbed accurately, which leads to high learning results; individual assimilation; constant monitoring of assimilation; possibility of using technical automated teaching devices.

Significant disadvantages of using this method: not all educational material is amenable to programmed processing; the method limits the mental development of students to reproductive operations; when using it, there is a lack of communication between the teacher and students; there is no emotional and sensory component of learning.

2.4 Interactive methods of teaching mathematics and their advantages

The learning process is inextricably linked with such a concept as teaching methodology. Methodology is not what books we use, but how our training is organized. In other words, teaching methodology is a form of interaction between students and teachers in the learning process. Within the current learning conditions, the learning process is considered as a process of interaction between the teacher and students, the purpose of which is to familiarize the latter with certain knowledge, skills, abilities and values. Generally speaking, from the first days of the existence of education as such until today, only three forms of interaction between teacher and students have developed, established themselves and become widespread. Methodological approaches to teaching can be divided into three groups:

.Passive methods.

2.Active methods.

.Interactive methods.

A passive methodological approach is a form of interaction between students and teachers in which the teacher is the main active figure in the lesson, and students act as passive listeners. Feedback in passive lessons is carried out through surveys, independent work, tests, tests, etc. The passive method is considered the most ineffective from the point of view of students’ assimilation of educational material, but its advantages are the relatively easy preparation of a lesson and the ability to present a relatively large amount of educational material in a limited time frame. Given these advantages, many teachers prefer it to other methods. Indeed, in some cases this approach works successfully in the hands of a skillful and experienced teacher, especially if students already have clear goals aimed at thorough learning of the subject.

An active methodological approach is a form of interaction between students and teachers, in which the teacher and students interact with each other during the lesson and students are no longer passive listeners, but active participants in the lesson. If in a passive lesson the main character was the teacher, then here the teacher and students are on equal terms. If passive lessons assumed an authoritarian teaching style, then active ones assumed a democratic style. Active and interactive methodological approaches have much in common. In general, the interactive method can be considered as the most modern form of active methods. It’s just that, unlike active methods, interactive ones are focused on broader interaction of students not only with the teacher, but also with each other and on the dominance of student activity in the learning process.

Interactive (“Inter” is mutual, “act” is to act) - means to interact or is in the mode of conversation, dialogue with someone. In other words, interactive teaching methods are a special form of organizing cognitive and communicative activities in which students are involved in the process of cognition, have the opportunity to engage and reflect on what they know and think. The teacher's place in interactive lessons often comes down to directing students' activities to achieve the lesson's goals. He also develops a lesson plan (as a rule, this is a set of interactive exercises and tasks, during which the student learns the material).

Thus, the main components of interactive lessons are interactive exercises and tasks that students complete.

The fundamental difference between interactive exercises and tasks is that during their implementation, not only and not so much the already learned material is consolidated, but new material is learned. And then interactive exercises and tasks are designed for so-called interactive approaches. Modern pedagogy has accumulated a rich arsenal of interactive approaches, among which the following can be distinguished:

Creative tasks;

Work in small groups;

Educational games (role-playing games, simulations, business games and educational games);

Use of public resources (invitation of a specialist, excursions);

Social projects, classroom teaching methods (social projects, competitions, radio and newspapers, films, performances, exhibitions, performances, songs and fairy tales);

Warm-ups;

Studying and consolidating new material (interactive lecture, working with visual video and audio materials, “student in the role of teacher”, everyone teaches everyone, mosaic (openwork saw), use of questions, Socratic dialogue);

Discussion of complex and debatable issues and problems (“Take a position”, “opinion scale”, POPS - formula, projective techniques, “One - two - all together”, “Change position”, “Carousel”, “Discussion in the style of television talk - show, debate);

Problem solving (“Decision tree”, “Brainstorming”, “Case analysis”)

Creative tasks should be understood as such educational tasks that require students not to simply reproduce information, but to create creativity, since tasks contain a greater or lesser element of uncertainty and, as a rule, have several approaches.

The creative task constitutes the content, the basis of any interactive method. An atmosphere of openness and search is created around him. A creative task, especially a practical one, gives meaning to learning and motivates students. The choice of a creative task in itself is a creative task for the teacher, since it is necessary to find a task that would meet the following criteria: does not have an unambiguous and monosyllabic answer or solution; is practical and useful for students; related to students' lives; arouses interest among students; serves learning purposes as best as possible. If students are not used to working creatively, then they should gradually introduce simple exercises first, and then more and more complex tasks.

Small group work - This is one of the most popular strategies, as it gives all students (including shy ones) the opportunity to participate in work, practice cooperation and interpersonal communication skills (in particular, the ability to listen, develop a common opinion, resolve disagreements). All this is often impossible in a large team. Small group work is an integral part of many interactive methods, such as mosaics, debates, public hearings, almost all types of simulations, etc.

At the same time, working in small groups requires a lot of time; this strategy should not be overused. Group work should be used when there is a problem to be solved that students cannot solve on their own. You should start group work slowly. You can organize pairs first. Pay special attention to students who have difficulty adjusting to small group work. When students learn to work in pairs, move on to working in a group of three students. Once we are confident that this group is able to function independently, we gradually add new students.

Students spend more time presenting their point of view, are able to discuss an issue in more detail, and learn to look at an issue from multiple perspectives. In such groups, more constructive relationships between participants are built.

Interactive learning helps a child not only learn, but also live. Thus, interactive learning is undoubtedly an interesting, creative, promising direction in our pedagogy.

Conclusion

Lessons using active learning methods are interesting not only for students, but also for teachers. But their unsystematic, ill-considered use does not give good results. Therefore, it is very important to actively develop and implement your own gaming methods into the lesson in accordance with the individual characteristics of your class.

It is not necessary to use these techniques all in one lesson.

In the classroom, quite acceptable work noise is created when discussing problems: sometimes, due to their psychological age characteristics, elementary school children cannot cope with their emotions. Therefore, it is better to introduce these methods gradually, cultivating a culture of discussion and cooperation among students.

The use of active methods strengthens motivation to learn and develops the best sides of the student. At the same time, there is no need to use these methods without searching for an answer to the question: why are we using them and what consequences may result from this (both for the teacher and for the students).

Without well-thought-out teaching methods, it is difficult to organize the assimilation of program material. That is why it is necessary to improve those methods and means of teaching that help to involve students in cognitive search, in the work of learning: they help teach students to actively, independently obtain knowledge, stimulate their thoughts and develop interest in the subject. There are many different formulas in a mathematics course. In order for students to be able to operate them freely when solving problems and exercises, they must know the most common ones, often encountered in practice, by heart. Thus, the teacher’s task is to create conditions for the practical application of abilities for each student, to choose teaching methods that would allow each student to show their activity, and also to intensify the student’s cognitive activity in the process of learning mathematics. Correct selection of types of educational activities, various forms and methods of work, search for various resources to increase students’ motivation to study mathematics, orient students towards acquiring competencies necessary for life and

activities in a multicultural world will provide the required

learning result.

The use of active teaching methods not only increases the effectiveness of the lesson, but also harmonizes personal development, which is possible only through active activity.

Thus, active teaching methods are ways of activating the educational and cognitive activity of students, which encourage them to active mental and practical activity in the process of mastering the material, when not only the teacher is active, but the students are also active.

To summarize, I will note that each student is interesting for his uniqueness, and my task is to preserve this uniqueness, grow a self-valued personality, develop inclinations and talents, and expand the capabilities of each self.

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