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 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 in a broad sense words and concepts of mathematical abilities 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 on modern 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, skill, 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 general definition 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 "an object 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 musical abilities" marked the beginning of 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 special technique teaching, 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

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.

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.

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 teachers primary classes, which teach a complex 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 methods of work 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 in teaching mathematics today. primary school.

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, in the literal sense of the word, 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 activities of capable children, which, as shown by a comparison of a number of special psychological and pedagogical research on this topic, turn out to be the same for children with various types abilities and talent. 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 forces 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 is to the greatest extent needs an instructive style of relationship with the teacher, which requires more information 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 be provided optimal pace advancement in content and optimal volume of teaching 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 methodological manuals for primary school teachers, designed to work 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 creating special teaching materials in mathematics for working with capable children is the only possible way to implement the principle of individualization of learning 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 based on various subjects for high school students both in mastering new material and in eliminating 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 sheets were developed on printed basis, combined into blocks covering the whole 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 new way activities. 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. Simply, the results of this work will be expressed in to varying degrees 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. Basics 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

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 show creative activity, continuously improve yourself and develop your 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 mathematics 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 mental activity child. 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 achieve 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 methodology develops in students conscious, sometimes even automatic, computational skills. Much attention The program focuses on 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, find several different ways solving 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 to rewrite it using it. 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.

The new paradigm of education in the Russian Federation is characterized personally oriented approach, the idea of ​​developmental education, the creation of conditions for self-organization and self-development of the individual, the subjectivity of education, the focus on designing the content, forms and methods of teaching and upbringing that ensure the development of each student, his cognitive abilities and personal qualities.

The concept of school mathematical education highlights its main goals - teaching students the techniques and methods of mathematical knowledge, developing in them the qualities of mathematical thinking, corresponding mental abilities and skills. The importance of this area of ​​work is enhanced by the increasing importance and application of mathematics in various fields of science, economics and industry.

The need for mathematical development of younger schoolchildren in educational activities is noted by many leading Russian scientists (V.A. Gusev, G.V. Dorofeev, N.B. Istomina, Yu.M. Kolyagin, L.G. Peterson, etc.). This is due to the fact that during the preschool and primary school period, the child not only intensively develops all mental functions, but also lays the general foundation of cognitive abilities and intellectual potential personality. Numerous facts indicate that if the corresponding intellectual or emotional qualities for one reason or another do not receive proper development in early childhood, then subsequently overcoming this kind of shortcomings turns out to be difficult and sometimes impossible (P.Ya. Galperin, A.V. Zaporozhets, S.N. Karpova).

Thus, new paradigm education, on the one hand, presupposes the maximum possible individualization of the educational process, and on the other hand, it requires solving the problem of creating educational technologies that ensure the implementation of the main provisions of the Concept of school mathematics education.

In psychology, the term “development” is understood as consistent, progressive significant changes in the psyche and personality of a person, manifesting themselves as certain new formations. The position on the possibility and feasibility of education focused on the development of the child was substantiated back in the 1930s. outstanding Russian psychologist L.S. Vygotsky.

One of the first attempts to practically implement the ideas of L.S. Vygotsky in our country was undertaken by L.V. Zankov, who in the 1950-1960s. developed in principle new system primary education, which has found a large number of followers. In the L.V. system Zankov, for the effective development of students’ cognitive abilities, the following five basic principles are implemented: learning at a high level of difficulty; the leading role of theoretical knowledge; moving forward at a fast pace; conscious participation of schoolchildren in the educational process; systematic work on the development of all students.

Theoretical (rather than traditional empirical) knowledge and thinking, educational activity were placed at the forefront by the authors of another theory of developmental education - D.B. Elkonin and V.V. Davydov. They considered the most important thing to change the student's position in the learning process. Unlike traditional education, where the student is the object of the teacher’s pedagogical influences, in developmental education conditions are created under which he becomes the subject of learning. Today, this theory of educational activity is recognized throughout the world as one of the most promising and consistent in terms of implementing the well-known provisions of L.S. Vygotsky about the developmental and anticipatory nature of learning.

In domestic pedagogy, in addition to these two systems, the concepts of developmental education by Z.I. Kalmykova, E.N. Kabanova-Meller, G.A. Tsukerman, S.A. Smirnova and others. It should also be noted the extremely interesting psychological searches of P.Ya. Galperin and N.F. Talyzina based on the theory of gradual formation they created mental actions. However, as noted by V.A. Tests, in most of the mentioned pedagogical systems, the development of the student is still the responsibility of the teacher, and the role of the former is reduced to following the developmental influence of the latter.

In line with developmental education, many different programs and teaching aids in mathematics have appeared, both for primary grades (textbooks by E.N. Alexandrova, I.I. Arginskaya, N.B. Istomina, L.G. Peterson, etc.), and for secondary school (textbooks by G.V. Dorofeev, A.G. Mordkovich, S.M. Reshetnikov, L.N. Shevrin, etc.). Textbook authors have different understandings of personality development in the process of learning mathematics. Some emphasize the development of observation, thinking and practical actions, others - on the formation of certain mental actions, others - on creating conditions that ensure the formation of educational activities, the development of theoretical thinking.

It is clear that the problem of developing mathematical thinking in teaching mathematics at school cannot be solved only by improving the content of education (even if there is good textbooks), since the implementation of different levels in practice requires the teacher to have a fundamentally new approach to organizing the educational activities of students in the classroom, in home and extracurricular work, which allows him to take into account the typological and individual characteristics of students.

It is known that the younger school age sensitive, most favorable for the development of cognitive mental processes and intelligence. Developing students' thinking is one of the main tasks of primary school. It is on this psychological feature that we concentrated our efforts, relying on the psychological and pedagogical concept of the development of thinking by D.B. Elkonin, position of V.V. Davydov on the transition from empirical to theoretical thinking in the process of specially organized educational activities, based on the works of R. Atakhanov, L.K. Maksimova, A.A. Stolyara, P. - H. van Hiele, related to identifying the levels of development of mathematical thinking and their psychological characteristics.

The idea of ​​L.S. Vygotsky’s idea that learning should be carried out in the zone of proximal development of students, and its effectiveness is determined by which zone (large or small) it prepares, is well known to everyone. At the theoretical (conceptual) level, it is shared almost throughout the world. The problem is her practical implementation: how to define (measure) this zone and what should be the teaching technology so that the process of learning the scientific foundations and mastering (“appropriating”) human culture takes place exactly in it, providing the maximum developmental effect?

Thus, psychological and pedagogical science has substantiated the expediency of mathematical development of younger schoolchildren, but the mechanisms for its implementation have not been sufficiently developed. Consideration of the concept of “development” as a result of learning from a methodological point of view shows that it is an integral continuous process, the driving force of which is the resolution of contradictions that arise in the process of change. Psychologists argue that the process of overcoming contradictions creates conditions for development, as a result of which individual knowledge and skills develop into a new holistic formation, into a new ability. Therefore the problem of constructing new concept mathematical development of primary schoolchildren is determined by contradictions.

Belarusian State Pedagogical University named after Maxim Tank

Faculty of Pedagogy and Primary Education Methods

Department of Mathematics and Methods of Its Teaching

USING EDUCATIONAL TECHNOLOGY “SCHOOL 2100” IN TEACHING MATHEMATICS TO JUNIOR SCHOOL CHILDREN

Graduate work

INTRODUCTION… 3

CHAPTER 1. Features of the mathematics course of the general education program “School 2100” and its technology... 5

1.1. Prerequisites for the emergence of an alternative program... 5

2.2. The essence of educational technology... 9

1.3. Humanitarian-oriented teaching of mathematics using educational technology “School 2100”… 12

1.4. Modern goals of education and didactic principles of organizing educational activities in mathematics lessons... 15

CHAPTER 2. Features of working on educational technology “School 2100” in mathematics lessons... 20

2.1. Using the activity method in teaching mathematics to primary schoolchildren... 20

2.1.1. Staging learning task… 21

2.1.2. “Discovery” of new knowledge by children... 21

2.1.3. Primary consolidation… 22

2.1.4. Independent work with testing in class... 22

2.1.5. Training exercises... 23

2.1.6. Delayed control of knowledge… 23

2.2. Training lesson… 25

2.2.1. Structure of training lessons… 25

2.2.2. Model of a training lesson... 28

2.3. Oral exercises in mathematics lessons... 28

2.4. Knowledge control… 29

Chapter 3. Analysis of the experiment... 36

3.1. Ascertaining experiment... 36

3.2. Educational experiment... 37

3.3. Control experiment... 40

Conclusion... 43

Literature… 46

Appendix 1… 48

Appendix 2… 69

2.2. The essence of educational technology

Before defining educational technology, it is necessary to reveal the etymology of the word “technology” (the science of skill, art, because from the Greek - techne- craftsmanship, art and logos- the science). The concept of technology in its modern meaning is used primarily in production (industrial, agricultural), various types of scientific and production human activities and presupposes a body of knowledge about methods (a set of methods, operations, actions) of carrying out production processes that guarantee obtaining a certain result.

Thus, the leading features and characteristics of the technology are:

· A set (combination, connection) of any components.

· Logic, sequence of components.

· Methods (methods), techniques, actions, operations (as components).

· Guaranteed results.

The essence of educational activity is the internalization (transfer of social ideas into the consciousness of an individual) by the student of a certain amount of information corresponding cultural norms and the ethical expectations of the society in which the student grows and develops.

The controlled process of transferring elements of the spiritual culture of previous generations to a new generation (controlled educational activity) is called education, and the transmitted elements of culture themselves - content of education .

The interiorized content of education (the result of educational activity) in relation to the subject of internalization is also called education(Sometimes - education).

Thus, the concept of “education” has three meanings: a social institution of society, the activities of this institution and the result of its activities.

There is a two-level nature of interiorization: interiorization that does not affect the subconscious will be called assimilation, and internalization, affecting the subconscious (forming automatisms of actions), - assignment .

It is logical to name the learned facts representations, assigned- knowledge, learned methods of activity - skills, assigned - skills, and learned value orientations and emotional-personal relationships - standards, assigned - beliefs or meanings .

In a specific educational process, the object of internalization is the target group. The relationship of power in the target group corresponds to the internalization of the corresponding components by the subject of the study: primary elements must be appropriated, secondary elements must be assimilated. We will call the pedagogical target groups interpreted in the described way targets. For example, a target group with the primary elements of “facts and ways of doing things” and the secondary element of “values” sets the target for knowledge, skills and norms. The assignment of primary goals occurs explicitly as a result of specially organized and controlled educational activities (education), and the assimilation of secondary goals occurs implicitly, as a result of uncontrolled educational activities and a by-product of education.

In each specific case, the educational process is regulated by a certain system of rules for its organization and management. This system of rules can be obtained empirically (observation and generalization) or theoretically (designed based on known scientific laws and tested experimentally). In the first case, it may relate to the transmission of some specific content or be generalized to various types of content. In the second case, it is contentless by definition and can be adjusted to various specific content options.

An empirically derived system of rules for transmitting specific content is called teaching methodology .

An empirically derived or theoretically designed system of rules for educational activities that is not related to specific content is a educational technology .

A set of rules of educational activity that do not have signs of systematicity is called pedagogical experience, if obtained empirically, and methodological developments or recommendations, if it is obtained theoretically (designed).

We are only interested in educational technology. The goals of educational activity are a system-forming factor in relation to educational technologies, considered as systems of rules for this activity.

Classification of educational technologies according to technological targets, that is, in a pedagogical sense, according to objects of appropriation:

· Informational.

· Information and value.

· Activity.

· Activity-value.

· Value-based.

· Value-informational.

· Value-based activity.

Unfortunately, the first of these names has been assigned to technologies that are not related to educational activities. Information It is customary to call technologies in which information is not a source of the target group, but an object of activity. Therefore, educational technologies in which facts are the primary element of activity goals, that is, knowledge constitutes the technological target setting, are usually called information-perceptual .

The final classification of educational technologies according to technological targets (objects of assignment) looks like this:

· Information-perceptual.

· Information and activity.

· Information and value.

· Activity.

· Activity and information.

· Activity-value.

· Value-based.

· Value-informational.

· Value-based activity.

Really existing educational technologies have yet to be sorted into classes. Apparently some classrooms are currently empty. The choice of classes of educational technologies used by one or another society (one or another humanitarian system) in a specific historical situation depends on what components of the accumulated spiritual culture of the society in this situation considers the most important for its survival and development. They define goals external to educational technology that make up the pedagogical paradigm of a given society (a given humanitarian system). This essential question is philosophical and cannot be the subject of a formal theory of educational technology.

The primary elements of technological targets when designing educational technology set a set of explicit (explicitly formulated) goals, secondary elements form the basis of implicit goals (which are not explicitly formulated). The main paradox of didactics is that implicit goals are achieved involuntarily, through subconscious acts, and therefore secondary goals are learned almost effortlessly. Hence the main paradox of educational technology: the procedures of educational technology are set by primary goals, and its effectiveness is determined by secondary ones. This can be considered a design principle for educational technology.

1.3. Humanitarian-oriented teaching of mathematics using educational technology “School 2100”

Modern approaches to organizing the school education system, including mathematics education, are determined, first of all, by the rejection of a uniform, unitary secondary school. The guiding vectors of this approach are humanization and humanitarization school education.

This determines the transition from the principle of “all mathematics for everyone” to careful consideration of individual personality parameters - why a particular student needs and will need mathematics in the future, to what extent and on what level he wants and/or can master it, to design a course of “mathematics for everyone,” or, more precisely, “mathematics for everyone.”

One of the main goals of the academic subject “Mathematics” as a component of general secondary education related to to each for the student, is the development of thinking, first of all, the formation abstract thinking, the ability to abstract and the ability to “work” with abstract, “intangible” objects. In the process of studying mathematics, logical and algorithmic thinking, many qualities of thinking, such as strength and flexibility, constructiveness and criticality, etc., can be formed in its purest form.

These qualities of thinking in themselves are not associated with any mathematical content and with mathematics in general, but teaching mathematics introduces an important and specific component into their formation, which currently cannot be effectively implemented even by the entire set of individual school subjects.

At the same time, specific mathematical knowledge that lies beyond, relatively speaking, the arithmetic of natural numbers and the primary foundations of geometry, are not“a subject of basic necessity” for the vast majority of people and cannot, therefore, form the target basis for teaching mathematics as a subject of general education.

That is why, as a fundamental principle of the educational technology “School 2100” in the aspect of “mathematics for everyone,” the principle of priority of the developmental function in teaching mathematics comes to the fore. In other words, teaching mathematics is focused not so much on mathematics education itself, in in the narrow sense of the word, how much for education with using mathematics.

According to this principle, the main task of teaching mathematics is not to learn the basics. mathematical science as such, and general intellectual development is the formation in students, in the process of studying mathematics, of the qualities of thinking necessary for the full functioning of a person in modern society, for the dynamic adaptation of a person to this society.

The formation of conditions for individual human activity, based on acquired specific mathematical knowledge, for knowledge and awareness of the surrounding world by means of mathematics remains, naturally, an equally essential component of school mathematical education.

From the point of view of the priority of the developmental function, specific mathematical knowledge in “mathematics for everyone” is considered not so much as a goal of learning, but as a base, a “testing ground” for organizing intellectually valuable activities of students. For the formation of a student’s personality, for achieving a high level of his development, it is precisely this activity, if we talk about a mass school, that, as a rule, turns out to be more significant than the specific mathematical knowledge that served as its basis.

The humanitarian orientation of teaching mathematics as a subject of general education and the resulting idea of ​​priority in “mathematics for everyone” of the developmental function of teaching in relation to its purely educational function requires a reorientation of the methodological system of teaching mathematics from increasing the amount of information intended for “one hundred percent” assimilation by students to formation of skills to analyze, produce and use information.

Among the general goals of mathematics education in educational technology, “School 2100” occupies a central place development of the abstract thinking, which includes not only the ability to perceive specific abstract objects and structures inherent in mathematics, but also the ability to operate with such objects and structures according to prescribed rules. A necessary component of abstract thinking is logical thinking- both deductive, including axiomatic, and productive - heuristic and algorithmic thinking.

The ability to see mathematical patterns in everyday practice and use them on the basis of mathematical modeling, mastering mathematical terminology as words are also considered as general goals of mathematical education native language and mathematical symbolism as a fragment of a global artificial language that plays a significant role in the process of communication and is currently necessary for everyone educated person.

The humanitarian orientation of teaching mathematics as a general education subject determines the specification of general goals in building a methodological system for teaching mathematics, reflecting the priority of the developmental function of teaching. Taking into account the obvious and unconditional need for all students to acquire a certain amount of specific mathematical knowledge and skills, the goals of teaching mathematics in the educational technology “School 2100” can be formulated as follows:

Mastery of a complex of mathematical knowledge, abilities and skills necessary: ​​a) for everyday life at a high quality level and professional activity, the content of which does not require the use of mathematical knowledge that goes beyond the needs of everyday life; b) to study school subjects in the natural sciences and humanities at a modern level; c) to continue studying mathematics in any form of continuous education (including, at the appropriate stage of education, upon transition to training in any profile at the senior level of school);

Formation and development of the qualities of thinking necessary for an educated person to function fully in modern society, in particular heuristic (creative) and algorithmic (performing) thinking in their unity and internally contradictory relationship;

Formation and development of students' abstract thinking and, above all, logical thinking, its deductive component as a specific characteristic of mathematics;

Increasing the level of students' proficiency in their native language in terms of the correctness and accuracy of expressing thoughts in active and passive speech;

Formation of activity skills and development in students of moral and ethical personality traits adequate to full-fledged mathematical activity;

Realization of the possibilities of mathematics in the formation of the scientific worldview of students, in their mastery of scientific pictures of the world;

Formation of a mathematical language and mathematical apparatus as a means of describing and studying the surrounding world and its patterns, in particular as a basis for computer literacy and culture;

Familiarization with the role of mathematics in the development of human civilization and culture, in the scientific and technological progress of society, in modern science and production;

Familiarization with the nature of scientific knowledge, with the principles of constructing scientific theories in the unity and opposition of mathematics and natural and humanities, with criteria of truth in different forms of human activity.

1.4. Modern goals of education and didactic principles of organizing educational activities in mathematics lessons

The rapid social transformations that our society has been experiencing in recent decades have radically changed not only the living conditions of people, but also educational situation. In this regard, the task of creating a new concept of education that reflects both the interests of society and the interests of each individual has become urgent.

Thus, in last years society has developed a new understanding of the main goal of education: the formation readiness for self-development, ensuring the integration of the individual into national and world culture.

The implementation of this goal requires the implementation of a whole range of tasks, among which the main ones are:

1) activity training - the ability to set goals, organize your activities to achieve them and evaluate the results of your actions;

2) formation of personal qualities - mind, will, feelings and emotions, creative abilities, cognitive motives of activity;

3) formation of a picture of the world, adequate modern level knowledge and level of the educational program.

It should be emphasized that the focus on developmental education is completely does not mean a refusal to develop knowledge, skills and abilities, without which personal self-determination and self-realization are impossible.

That is why the didactic system of Ya.A. Comenius, which has absorbed the centuries-old traditions of the system of transmitting knowledge about the world to students, and today constitutes methodological basis the so-called “traditional” school:

· Didactic principles - clarity, accessibility, scientific character, systematicity, and conscientiousness in mastering educational material.

· Teaching method - explanatory and illustrative.

· Form of study - class lesson.

However, it is obvious to everyone that the existing didactic system, although it has not exhausted its significance, at the same time does not allow for the effective implementation of the developmental function of education. In recent years, in the works of L.V. Zankova, V.V. Davydova, P.Ya. Galperin and many other teacher-scientists and practitioners have formed new didactic requirements that solve modern educational problems taking into account the needs of the future. The main ones:

1. Operating principle

The main conclusion of psychological and pedagogical research in recent years is that The formation of a student’s personality and his advancement in development takes place not when he perceives ready-made knowledge, but in the process of his own activity aimed at “discovering” new knowledge.

Thus, the main mechanism for realizing the goals and objectives of developmental education is inclusion of the child in educational and cognitive activities. IN that's what it's all about operating principle, Education that implements the principle of activity is called an activity approach.

2. The principle of a holistic view of the world

Also Y.A. Comenius noted that phenomena need to be studied in mutual connection, and not separately (not like a “pile of firewood”). Nowadays, this thesis acquires even greater significance. It means that The child must form a generalized, holistic idea of ​​the world (nature - society - himself), about the role and place of each science in the system of sciences. Naturally, the knowledge formed by students should reflect the language and structure of scientific knowledge.

The principle of a unified picture of the world in the activity approach is closely related to the didactic principle of scientificity in the traditional system, but is much deeper than it. Here we are not just talking about the formation scientific picture world, but also about the personal attitude of students to the acquired knowledge, as well as about ability to apply them in their practical activities. For example, if we are talking about environmental knowledge, then the student should not just to know that it is not good to pick certain flowers, leave garbage behind in the forest, etc., and make your own decision don't do that.

3. The principle of continuity

Continuity principle means continuity between all levels of education at the level of methodology, content and technique .

The idea of ​​continuity is also not new for pedagogy, however, until now it is most often limited to the so-called “propaedeutics”, and is not solved systematically. The problem of continuity has acquired particular relevance in connection with the emergence of variable programs.

The implementation of continuity in the content of mathematical education is associated with the names of N.Ya. Vilenkina, G.V. Dorofeeva and others. Management aspects in the “preschool preparation - school - university” model have been developed in recent years by V.N. Prosvirkin.

4. Minimax principle

All children are different, and each of them develops at their own pace. At the same time, education in mass schools is focused on a certain average level, which is too high for weak children and clearly insufficient for stronger ones. This hinders the development of both strong children and weak ones.

To take into account the individual characteristics of students, 2, 4, etc. are often distinguished. level. However, there are exactly as many real levels in a class as there are children! Is it possible to accurately determine them? Not to mention that it is practically difficult to account for even four - after all, for a teacher this means 20 preparations a day!

The solution is simple: select only two levels - maximum, determined by the zone of proximal development of children, and necessary minimum. The minimax principle is as follows: the school must offer the student educational content at the maximum level, and the student must master this content at the minimum level(see Appendix 1) .

The minimax system is apparently optimal for implementing an individual approach, since it self-regulating system. A weak student will limit himself to the minimum, while a strong student will take everything and move on. Everyone else will be placed between these two levels in accordance with their abilities and capabilities - they will choose their level themselves to its maximum possible.

The work is carried out at a high level of difficulty, but only assessed required result, and success. This will allow students to develop an attitude towards achieving success, rather than avoiding getting a bad grade, which is much more important for the development of the motivational sphere.

5. The principle of psychological comfort

The principle of psychological comfort implies removing, if possible, all stress-forming factors of the educational process, creating an atmosphere at school and in the classroom that relaxes children and in which they feel “at home.”

No academic success will be of any use if it is “involved” in fear of adults and suppression of the child’s personality.

However, psychological comfort is necessary not only for the assimilation of knowledge - it depends on physiological state children. Adaptation to specific conditions, creating an atmosphere of goodwill will help relieve tension and neuroses that destroy health children.

6. The principle of variability

Modern life requires a person to be able to make a choice - from choosing goods and services to choosing friends and choosing a life path. The principle of variability presupposes the development of variable thinking in students, that is understanding the possibility of various options for solving a problem and the ability to systematically enumerate options.

Education, which implements the principle of variability, removes the fear of mistakes in students and teaches them to perceive failure not as a tragedy, but as a signal for its correction. This approach to solving problems, especially in difficult situations, is also necessary in life: in case of failure, do not become discouraged, but look for and find a constructive way.

On the other hand, the principle of variability ensures the teacher’s right to independence in choice educational literature, forms and methods of work, the degree of their adaptation in the educational process. However, this right also gives rise to greater responsibility for the teacher for the final result of his activities - the quality of teaching.

7. The principle of creativity (creativity)

The principle of creativity presupposes maximum orientation towards creativity in the educational activities of schoolchildren, their acquisition own experience creative activity.

We are not talking here about simply “inventing” tasks by analogy, although such tasks should be welcomed in every possible way. Here, first of all, we mean the formation in students of the ability to independently find solutions to problems that have not been encountered before, their independent “discovery” of new ways of action.

The ability to create something new and find a non-standard solution to life’s problems has become an integral part of the real life success of any person today. Therefore, the development of creative abilities is acquiring general educational importance these days.

The teaching principles outlined above, developing the ideas of traditional didactics, integrate useful and non-conflicting ideas from new concepts of education from the standpoint of continuity of scientific views. They don't reject, but continue and develop traditional didactics towards solving modern educational problems.

In fact, it is obvious that the knowledge that the child himself “discovered” is visual for him, accessible and consciously assimilated by him. However, the inclusion of a child in activities, in contrast to traditional visual learning, activates his thinking and forms his readiness for self-development (V.V. Davydov).

Education that implements the principle of the integrity of the picture of the world meets the requirement of being scientific, but at the same time it also implements new approaches, such as humanization and humanitarization of education (G.V. Dorofeev, A.A. Leontyev, L.V. Tarasov).

The minimax system effectively promotes the development of personal qualities and forms the motivational sphere. Here the problem of multi-level teaching is solved, which makes it possible to promote the development of all children, both strong and weak (L.V. Zankov).

The requirements of psychological comfort ensure that the child’s psychophysiological state is taken into account, promotes the development of cognitive interests and the preservation of children’s health (L.V. Zankov, A.A. Leontyev, Sh.A. Amonashvili).

The principle of continuity gives a systemic character to the solution of succession issues (N.Ya. Vilenkin, G.V. Dororfeev, V.N. Prosvirkin, V.F. Purkina).

The principle of variability and the principle of creativity reflect the necessary conditions for the successful integration of the individual into the modern social life.

Thus, the listed didactic principles of educational technology “School 2100” to a certain extent necessary and sufficient to achieve modern educational goals and can already be carried out today in secondary schools.

At the same time, it should be emphasized that the formation of a system of didactic principles cannot be completed, because life itself places accents of significance, and each emphasis is justified by a specific historical, cultural and social application.

CHAPTER 2. Features of working on educational technology “School 2100” in mathematics lessons

2.1. Using the activity method in teaching mathematics to primary schoolchildren

Practical adaptation of the new didactic system requires updating traditional forms and methods of teaching, and developing new educational content.

Indeed, the inclusion of students in activities - the main type of knowledge acquisition in the activity approach - is not included in the technology of the explanatory-illustrative method on which education in a “traditional” school is based today. The main stages of this method are: communication of the topic and purpose of the lesson, updating knowledge, explanation, consolidation, control - do not provide a systematic passage of the necessary stages of educational activity, which are:

· setting a learning task;

· learning activities;

· actions of self-control and self-esteem.

Thus, communicating the topic and purpose of the lesson does not provide a statement of the problem. A teacher’s explanation cannot replace children’s learning activities, as a result of which they independently “discover” new knowledge. The differences between control and self-control of knowledge are also fundamental. Consequently, the explanatory and illustrative method cannot fully achieve the goals of developmental education. A new technology is needed, which, on the one hand, will allow the implementation of the principle of activity, and on the other, will ensure the passage of the necessary stages of knowledge acquisition, namely:

· motivation;

· creation of an indicative basis of action (IBA):

· material or materialized action;

· external speech;

· inner speech;

· automatic mental action(P.Ya. Galperin). These requirements are satisfied by the activity method, the main stages of which are presented in the following diagram:

(Steps included in a lesson on introducing a new concept are marked with a dotted line).

Let us describe in more detail the main stages of working on a concept in this technology.

2.1.1. Setting a learning task

Any process of cognition begins with an impulse that encourages action. Surprise is necessary, coming from the impossibility of momentarily ensuring this or that phenomenon. What is needed is delight, an emotional surge that comes from participation in this phenomenon. In a word, motivation is needed to encourage the student to enter into activity.

The stage of setting a learning task is the stage of motivation and goal-setting of activity. Students complete tasks that update their knowledge. The list of tasks includes a question that creates a “collision,” that is, a problematic situation that is personally significant for the student and shapes his need mastering this or that concept (I don’t know what’s happening. I don’t know how it’s happening. But I can find out - I’m interested in it!). The cognitive target.

2.1.2. “Discovery” of new knowledge by children

The next stage of work on the concept is solving the problem, which is carried out teach yourself taking place during a discussion, discussion based on substantive actions with material or materialized objects. The teacher organizes a leading or stimulating dialogue. Finally, he concludes by introducing common terminology.

This stage includes students in active work in which there are no disinterested people, because the teacher’s dialogue with the class is the teacher’s dialogue with each student, focusing on the degree and speed of mastering the sought concept and adjusting the quantity and quality of tasks that will help ensure a solution to the problem. The dialogical form of searching for truth is the most important aspect of the activity method.

2.1.3. Primary consolidation

Primary consolidation is carried out through commenting on each sought-after situation, speaking out loud the established action algorithms (what I am doing and why, what follows what, what should happen).

At this stage, the effect of mastering the material is enhanced, since the student not only reinforces written speech, but also voices the internal speech through which search work in his mind. The effectiveness of primary reinforcement depends on the completeness of the presentation of essential features, the variation of non-essential ones and the repeated playback of educational material in the independent actions of students.

2.1.4. Independent work with testing in class

Task fourth stage- self-control and self-esteem. Self-control encourages students to take a responsible attitude to the work they do and teaches them to adequately evaluate the results of their actions.

In the process of self-control, the action is not accompanied by loud speech, but moves to the internal plane. The student pronounces the algorithm of action “to himself,” as if conducting a dialogue with his intended opponent. It is important that at this stage a situation is created for each student success(I can, I can do it).

It is better to go through the four stages of working on a concept listed above in one lesson, without separating them over time. This usually takes about 20-25 minutes of a lesson. The remaining time is devoted, on the one hand, to consolidating the knowledge, skills and abilities accumulated earlier and their integration with new material, and on the other hand, to advanced preparation for the following topics. Here, errors on a new topic that could arise at the self-control stage are individually refined: positive self-esteem is important for every student, so we must do everything possible to correct the situation in the same lesson.

You should also pay attention to organizational issues, setting general goals and objectives at the beginning of the lesson and summing up the activities at the end of the lesson.

Thus, lessons for introducing new knowledge in the activity approach have the following structure:

1) Organizing time, overall plan lesson.

2) Statement of the educational task.

3) “Discovery” of new knowledge by children.

4) Primary consolidation.

5) Independent work with testing in class.

6) Repetition and consolidation of previously studied material.

7) Lesson summary.

(See Appendix 2.)

The principle of creativity determines the nature of consolidating new material in homework. Not reproductive, but productive activity is the key to lasting assimilation. Therefore, as often as possible, homework assignments should be offered in which it is necessary to correlate the particular and the general, to identify stable connections and patterns. Only in this case does knowledge become thinking and acquire consistency and dynamics.

2.1.5. Training exercises

In subsequent lessons, the learned material is practiced and consolidated, bringing it to the level of automated mental action. Knowledge undergoes a qualitative change: a revolution occurs in the process of cognition.

According to L.V. Zankov, consolidation of material in the system of developmental education should not be merely reproducing in nature, but should be carried out in parallel with the study of new ideas - deepen the learned properties and relationships, broaden the horizons of children.

Therefore, the activity method, as a rule, does not provide lessons for “pure” consolidation. Even in lessons whose main goal is to practice the studied material, some new elements are included - this can be the expansion and deepening of the material being studied, advanced preparation for the study of subsequent topics, etc. This “layer cake” allows every child move forward at your own pace: children with a low level of preparation have enough time to “slowly” master the material, and more prepared children constantly receive “food for the mind,” which makes the lessons attractive to all children - both strong and weak.

2.1.6. Delayed knowledge control

The final test should be offered to students based on the minimax principle (readiness at the top level of knowledge, control at the bottom). Under this condition, the negative reaction of schoolchildren to grades and the emotional pressure of the expected result in the form of a grade will be minimized. The teacher’s task is to evaluate the mastery of educational material according to the bar necessary for further advancement.

Described teaching technology - activity method- developed and implemented in a mathematics course, but can, in our opinion, be used in the study of any subject. This method creates favorable conditions for multi-level learning and practical implementation of all didactic principles of the activity approach.

The main difference between the activity method and the visual method is that it ensures the inclusion of children in activities :

1) goal setting and motivation are carried out at the stage of setting the educational task;

2) educational activities of children - at the stage of “discovery” of new knowledge;

3) actions of self-control and self-esteem - at the stage of independent work, which children check here in the classroom.

On the other hand, the activity method ensures completion of all necessary stages of mastering concepts, which allows you to significantly increase the strength of knowledge. Indeed, setting a learning task ensures the motivation of the concept and the construction of an indicative basis for action (IBA). The “discovery” of new knowledge by children is carried out through their performance of objective actions with material or materialized objects. Primary consolidation ensures the passage of the stage of external speech - children speak out loud and at the same time carry out established action algorithms in written form. In independent learning work, the action is no longer accompanied by speech; students pronounce the action algorithms “to themselves”, internal speech (see Appendix 3). And finally, in the process of performing the final training exercises, the action moves to the internal plane and becomes automated (mental action).

Thus, The activity method meets the necessary requirements for teaching technologies that implement modern educational goals. It makes it possible to master subject content in accordance with a unified approach, with a unified focus on activating both external and internal factors that determine the development of the child.

New education goals require updating content education and search forms training that will enable their optimal implementation. The entire set of information should be subordinated to an orientation towards life, towards the ability to act in any situations, towards getting out of crises, conflict situations, which also include situations of knowledge seeking. A student at school learns not only to solve mathematical problems, but through them also life problems, not only the rules of spelling, but also the rules social hostel, not only the perception of culture, but also its creation.

The main form of organizing educational and cognitive activity of students in the activity approach is collective dialogue. It is through collective dialogue that “teacher-student”, “student-student” communication takes place, in which learning material is learned at the level of personal adaptation. Dialogue can be built in pairs, in groups and in the whole class under the guidance of a teacher. Thus, the entire range of organizational forms of the lesson, developed today in teaching practice, can be effectively used within the framework of the activity approach.

2.2. Lesson-training

This is a lesson in active mental and verbal activity of students, the form of organization of which is group work. In 1st grade it is work in pairs, from 2nd grade it is work in fours.

Trainings can be used to study new material and consolidate what has been learned. However, it is especially advisable to use them when generalizing and systematizing students’ knowledge.

Conducting training is not an easy task. Special skill is required from the teacher. In such a lesson, the teacher is a conductor, whose task is to skillfully switch and concentrate the attention of students.

Main actor a student appears at a training lesson.

2.2.1. Structure of training lessons

1. Setting a goal

The teacher, together with the students, determines the main goals of the lesson, including the sociocultural position, which is inextricably linked with “revealing the secrets of words.” The fact is that each lesson has an epigraph, the words of which reveal their special meaning for each only at the end of the lesson. To understand them, you need to “live” the lesson.

Motivation to work is reinforced in the resource circle. Children stand in a circle and hold hands. The teacher’s task is to make every child feel supported, good relations to him. A feeling of unity with the class and the teacher helps create an atmosphere of trust and mutual understanding.

2. Independent work. Making your own decision

Each student receives a task card. The question contains a question and three possible answers. One, two, or all three options may be correct. The choice hides possible common mistakes made by students.

Before starting to complete tasks, children pronounce the “rules” of work that will help them organize a dialogue. They may be different in each class. Here is one option: “Everyone should speak out and listen to everyone.” Pronouncing these rules out loud helps create a mindset for all children in the group to participate in the dialogue.

At the stage of independent work, the student must consider all three answer options, comparing and contrasting them, make a choice and prepare to explain his choice to a friend: why he thinks this way and not otherwise. To do this, everyone needs to delve into their knowledge base. The knowledge acquired by students in lessons is built into a system and becomes a means for evidence-based choice. The child learns to systematically search through options, compare them, and find the best option.

In the process of this work, not only systematization, but also generalization of knowledge occurs, since the studied material is separated into separate topics, blocks, and didactic units are enlarged.

3. Work in pairs (fours)

When working in a group, each student must explain which answer option he chose and why. Thus, working in pairs (fours) necessarily requires active speech activity from each child and develops listening and hearing skills. Psychologists say: students retain 90% of what they say out loud and 95% of what they teach themselves. During the training, the child both speaks and explains. The knowledge acquired by students in the classroom becomes in demand.

At the moment of logical comprehension and structuring of speech, concepts are adjusted and knowledge is structured.

An important point at this stage is the adoption of a group decision. The very process of making such a decision contributes to the adjustment of personal qualities and creates conditions for the development of the individual and the group.

4. Listen to different opinions as a class

By giving the floor to different groups of students, the teacher has an excellent opportunity to track how well the concepts are formed, how strong the knowledge is, how well the children have mastered the terminology, and whether they include it in their speech.

It is important to organize the work in such a way that students themselves can hear and highlight the sample of the most convincing speech.

5. Expert assessment

After the discussion, the teacher or students voice the correct choice.

6. Self-esteem

The child learns to evaluate the results of his activities himself. This is facilitated by a system of questions:

Did you listen carefully to your friend?

Were you able to prove the correctness of your choice?

If not, why not?

What happened, what was difficult? Why?

What needs to be done to make the work successful?

Thus, the child learns to evaluate his actions, plan them, realize his understanding or misunderstanding, his progress.

Students open a new card with the task, and the work again proceeds in stages - from 2 to 6.

In total, trainings include from 4 to 7 tasks.

7. Summing up

Summing up takes place in the resource circle. Everyone has the opportunity to express (or not express) their attitude to the epigraph, as they understand it. At this stage, the “secret of words” of the epigraph is revealed. This technique allows the teacher to address problems of morality, the relationship of educational activities with real problems of the surrounding world, and allows students to perceive educational activities as their own social experience.

Trainings should not be confused with practical lessons, where strong skills and abilities are formed through a variety of training exercises. They also differ from testing, although they also provide for a choice of answer. However, during testing, it is difficult for the teacher to monitor how justified the choice was made by the student; a choice at random is not excluded, since the student’s reasoning remains at the level of internal speech.

The essence of training lessons is in the development of a unified conceptual apparatus, in students’ awareness of their achievements and problems.

The success and efficiency of this technology is possible with a high level of lesson organization, the necessary conditions of which are the thoughtfulness of working pairs (fours) and the experience of students working together. Pairs or fours should be formed from children with different types of perception (visual, auditory, motor), taking into account their activity. In this case, joint activities will contribute to a holistic perception of the material and self-development of each child.

The training lessons were developed in accordance with the thematic planning of L.G. Peterson and are conducted through reserve lessons. Subjects of training lessons: numbering, the meaning of arithmetic operations, methods of calculations, order of actions, quantities, solving problems and equations. During the academic year, from 5 to 10 trainings are conducted depending on the class.

Thus, in the 1st grade it is proposed to conduct 5 trainings on the main topics of the course.

November: Addition and subtraction within 9 .

December: Task .

February: Quantities .

March: Solving equations .

April: Problem solving .

In each training, the sequence of tasks is built according to the algorithm of actions that form the knowledge, skills and abilities of students on a given topic.

2.2.2. Lesson-training model

2.3. Oral exercises in mathematics lessons

Changing priorities for the goals of mathematics education have significantly affected the process of teaching mathematics. The main idea is the priority of the developmental function in teaching. Oral exercises are one of the means in the educational and cognitive process that makes it possible to realize the idea of ​​development.

Oral exercises contain enormous potential for developing thinking and activating students’ cognitive activity. They allow you to organize the educational process in such a way that as a result of their implementation, students form a holistic picture of the phenomenon under consideration. This provides the opportunity not only to retain in memory, but also to reproduce exactly those fragments that turn out to be necessary in the process of passing subsequent steps of cognition.

The use of oral exercises reduces the number of tasks in the lesson that require full written documentation, which leads to more effective speech development, mental operations and creative abilities of students.

Oral exercises destroy stereotypical thinking by constantly involving the student in the analysis of initial information and predicting errors. The main thing when working with information is to involve students themselves in creating an indicative basis, which shifts the emphasis of the educational process from the need for memorization to the need for the ability to apply information, and thereby contributes to the transfer of students from the level of reproductive assimilation of knowledge to the level of research activity.

Thus, a well-thought-out system of oral exercises allows not only to carry out systematic work on the formation of computational skills and skills in solving word problems, but also in many other areas, such as:

a) development of attention, memory, mental operations, speech;

b) formation of heuristic techniques;

c) development of combinatorial thinking;

d) formation of spatial representations.

2.4. Knowledge control

Modern technologies training can significantly increase the efficiency of the learning process. At the same time, most of these technologies leave out of the scope of their attention innovations related to such important components of the educational process as knowledge control. The methods of organizing control over the level of students' training currently used at school have not undergone any significant changes over a long period. Until now, many believe that teachers successfully cope with this type of activity and do not experience significant difficulties in their practical implementation. At best, the question of what is advisable to submit for control is discussed. Issues related to the forms of control, and especially the methods of processing and storing the results obtained during control educational information remain without due attention from teachers. At the same time, in modern society, an information revolution has occurred quite a long time ago; new methods of analysis, collection and storage of data have appeared, making this process more efficient in terms of the volume and quality of information retrieved.

Knowledge control is one of the most important components of the educational process. Monitoring students' knowledge can be considered as an element of the control system that implements feedback in the corresponding control loops. How this feedback will be organized, how much information received during this communication reliable, comprehensive and reliable, The effectiveness of the decisions made also depends. The modern system of public education is organized in such a way that the management of the learning process of schoolchildren is carried out at several levels.

The first level is the student, who must consciously manage his activities, directing them to achieve learning goals. If management at this level is absent or is not coordinated with learning goals, then a situation occurs when the student is taught, but he himself does not learn. Accordingly, in order to effectively manage his activities, a student must have all the necessary information about the learning results he achieves. Naturally, at the lower stages of education, the student mainly receives this information from the teacher. finished form.

The second level is the teacher. This is the main figure directly responsible for managing the educational process. He organizes both the activities of each individual student and the class as a whole, directs and corrects the course of the educational process. The objects of control for the teacher are individual students and classes. The teacher himself collects all the information necessary to manage the educational process; in addition, he must prepare and transmit to students the information they need so that they can consciously take part in the educational process.

The third level is public education authorities. This level represents a hierarchical system of institutions for managing public education. Management bodies deal both with information that they receive independently and independently of the teacher, and with information transmitted to them by teachers.

The information that the teacher transmits to students and to higher authorities is the school grade assigned by the teacher based on the results of students’ activities during the educational process. It is advisable to distinguish between two types: current and final grade. The current assessment takes into account, as a rule, the results of students’ performance certain types activity, the final one is, as it were, a derivative of current assessments. Thus, the final grade may not directly reflect the final level of student preparation.

Assessment of students' achievements by the teacher is a necessary component of the educational process, ensuring its successful functioning. Any attempts to ignore knowledge assessment (in one form or another) lead to disruption of the normal course of the educational process. Evaluation, on the one hand serves as a guide For students, showing them how their efforts meet the teacher's requirements. On the other hand, the presence of assessment allows educational authorities, as well as parents of students, to monitor the success of the educational process and the effectiveness of the control actions taken. In general grade - This is a judgment about the quality of an object or process, made on the basis of correlating the identified properties of this object or process with some given criterion. An example of an assessment would be the award of a rank in sports. The category is assigned based on measuring the athlete's performance results by comparing them with given standards. (For example, the running result in seconds is compared with the standards corresponding to a particular category.)

Evaluation is secondary to measurement and Maybe be obtained only after the measurement has been carried out. In modern schools, these two processes are often not distinguished, since the measurement process takes place as if in a compressed form, and the assessment itself has the form of a number. Teachers do not think about the fact that, by recording the number of actions correctly performed by a student (or the number of errors made by him/her) when performing this or that work, they thereby measure the results of the students’ activities, and when giving a grade to the student, they correlate the identified quantitative indicators with those available in the their disposal of evaluation criteria. Thus, teachers themselves, having, as a rule, the results of measurements that they use to grade students, rarely inform other participants in the educational process about them. This significantly narrows the information available to students, their parents and governing bodies.

Knowledge assessment can be in either numerical or verbal form, which in turn creates additional confusion that often exists between measurements and assessments. The measurement results can only be in numerical form, since in general measurement is establishing a correspondence between an object and a number. The form of the assessment is an unimportant characteristic of it. So, for example, a judgment like “student fully has mastered the taught material” may be equivalent to the statement “the student knows the covered material in Great” or “the student has a grade of 5 for the completed course material.” The only thing that researchers and practitioners should remember is that in the latter case the assessment 5 is not a number in the mathematical sense and with it no arithmetic operations are allowed. A score of 5 serves to classify a given student into a certain category, the meaning of which can be deciphered unambiguously only taking into account the adopted assessment system.

The modern school assessment system suffers from a number of significant shortcomings that do not allow it to be fully used as a high-quality source of information about the level of student preparation. Mark, as a rule, is subjective, relative and unreliable. The main flaws of this assessment system are that, on the one hand, the existing assessment criteria are poorly formalized, which allows them to be interpreted ambiguously; on the other hand, there are no clear measurement algorithms, on the basis of which the assessment should be based. normal system assessment.

As measuring instruments The educational process uses standard tests and independent work, common to all students. The results of these tests are assessed by the teacher. In modern methodological literature, much attention is paid to the content of these tests, they are improved and brought into line with the stated learning goals. At the same time, the issues of processing test results, measuring student performance results and their evaluation in most of the methodological literature are studied at an insufficiently high level of development and formalization. This leads to the fact that teachers often give different grades to students for the same work results. There may be even greater differences in the results of assessing the same work by different teachers. The latter occurs due to the fact that in the absence of strictly formalized rules defining algorithm measurement and assessment, different teachers may perceive the measurement algorithms and assessment criteria proposed to them differently, replacing them with their own.

The teachers themselves explain it as follows. When evaluating work, they have in mind first of all student's reaction on the rating he received. The main task of the teacher is to encourage the student to new achievements, and here for them lower value has the function of assessment as an objective and reliable source of information about the level of preparation of students, but to a greater extent teachers are aimed at implementing the control function of assessment.

Modern methods for measuring the level of student preparation, focused on the use of computer technology, fully meeting the realities of our time, provide the teacher with fundamentally new opportunities and increase the efficiency of his activities. A significant advantage of these technologies is that they provide new opportunities not only for the teacher, but also for the student. They enable the student to cease being an object of learning, but to become a subject who consciously participates in the learning process and reasonably makes independent decisions related to this process.

If, with traditional control, information about the level of students’ preparation was owned and completely controlled only by the teacher, then when using new methods of collecting and analyzing information, it becomes available to the student himself and his parents. This allows students and their parents to consciously make decisions related to the course of the educational process, makes the student and teacher comrades in the same important matter, in the results of which they are equally interested.

Traditional control is represented by independent and test work (12 workbooks that make up a set of mathematics for primary school).

When carrying out independent work, the goal is primarily to identify the level of mathematical preparation of children and promptly eliminate existing knowledge gaps. At the end of each independent work there is a space for work on bugs. At first, the teacher should help children choose tasks that allow them to correct their mistakes in a timely manner. Throughout the year, independent work with corrected errors is collected in a folder, which helps students track their path in mastering knowledge.

Tests summarize this work. Unlike independent work, the main function of control work is precisely the control of knowledge. From the very first steps, a child should be taught to be especially attentive and precise in his actions while monitoring knowledge. Test results, as a rule, are not corrected - you need to prepare for knowledge testing before him, and not after. But this is exactly how any competitions, exams, administrative tests are conducted - after they are carried out, the result cannot be corrected, and children need to be gradually psychologically prepared for this. At the same time, preparatory work and timely correction of errors during independent work provide a certain guarantee that the test will be written successfully.

The basic principle of knowledge control is minimizing children's stress. The atmosphere in the classroom should be calm and friendly. Possible errors in independent work should be perceived as nothing more than a signal for their improvement and elimination. A calm atmosphere during tests is determined by the extensive preparatory work that has been done in advance and which removes all reasons for concern. In addition, the child must clearly feel the teacher’s faith in his strength and interest in his success.

The level of difficulty of the work is quite high, but experience shows that children gradually accept it and almost all of them, without exception, cope with the proposed variants of tasks.

Independent work usually takes 7-10 minutes (sometimes up to 15). If the child does not have time to complete the independent work assignment within the allotted time, after checking the work by the teacher, he finalizes these assignments at home.

Grading for independent work is given after the errors have been corrected. What is assessed is not so much what the child managed to do during the lesson, but how he ultimately worked on the material. Therefore, even those independent works that were not written very well in class can be given a good or excellent score. In independent work, the quality of work on oneself is fundamentally important and only success is assessed.

Test work takes from 30 to 45 minutes. If one of the children does not complete the tests within the allotted time, then at the initial stages of training you can allocate some additional time for him to give him the opportunity to calmly finish the work. Such “adding” to work is excluded when carrying out independent work. But in the control work there is no provision for subsequent “revision” - the result is evaluated. The grade for the test is usually corrected in the next test.

When grading, you can rely on the following scale (tasks with an asterisk are not included in the mandatory part and are assessed with an additional mark):

“3” - if at least 50% of the work has been done;

“4” - if at least 75% of the work has been done;

“5” - if the work contains no more than 2 defects.

This scale is very arbitrary, since when giving a grade, the teacher must take into account many different factors, including the level of preparedness of the children, and their mental, physical, and emotional condition. In the end, assessment should not be a sword of pre-Mocles in the hands of a teacher, but a tool that helps a child learn to work on himself, overcome difficulties, and believe in himself. Therefore, first of all, you should be guided by common sense and traditions: “5” is excellent work, “4” is good, “3” is satisfactory. It should also be noted that in 1st grade, grades are given only for works written as “good” and “excellent”. You can say to the rest: “We need to catch up, we will succeed too!”

In most cases, work is carried out on a printed basis. But in some cases, they are offered on cards or can even be written on the board to accustom children to different forms of presentation of material. The teacher can easily determine in what form the work is being carried out by whether there is space left for writing in answers or not.

Independent work is offered approximately 1-2 times a week, and tests are offered 2-3 times a quarter. At the end of the year children first they write the translation work, determining the ability to continue education in the next grade in accordance with the state knowledge standard, and then - the final test.

The final work has a high level of complexity. At the same time, experience shows that with systematic, systematic work throughout the year in the proposed methodological system, almost all children cope with it. However, depending on the specific working conditions, the level of the final test may be reduced. In any case, a child’s failure to complete it cannot serve as a basis for giving him an unsatisfactory grade.

the main objective final work - to identify the real level of knowledge of children, their mastery of general educational skills, to give the children the opportunity to realize the result of their work, to emotionally experience the joy of victory.

The high level of testing proposed in this manual, as well as the high level of work in the classroom, does not means that the level of administrative control of knowledge must increase. Administrative control is carried out in the same way as in classes taught according to any other programs and textbooks. You should only take into account that the material on topics is sometimes distributed differently (for example, the methodology adopted in this textbook assumes a later introduction of the first ten numbers). Therefore, it is advisable to carry out administrative control at the end educational of the year .

Chapter 3. Analysis of the experiment

How do schoolchildren perceive the simplest tasks? Is the approach proposed by the School 2100 program more effective in teaching problem solving compared to the traditional one?

To answer these questions, we conducted an experiment in gymnasium No. 5 and secondary school No. 74 in Minsk. Students took part in the experiment preparatory classes. The experiment consisted of three parts.

Stater. Simple tasks were proposed that needed to be solved according to plan:

1. Condition.

2. Question.

4. Expression.

5. Solution.

A system of exercises was proposed using the activity method in order to develop skills to solve simple problems.

Control. The students were offered problems similar to the problems from the ascertaining experiment, as well as tasks more difficult level.

3.1. Ascertaining experiment

The students were given the following tasks:

1. Dasha has 3 apples and 2 pears. How many fruits does Dasha have in total?

2. The cat Murka has 7 kittens. Of these, 3 are white and the rest are variegated. How many motley kittens does Murka have?

3. There were 5 passengers on the bus. At the stop, some of the passengers got off, only 1 passenger remained. How many passengers got off?

The purpose of the ascertaining experiment: check the initial level of knowledge, skills and abilities of preparatory school students when solving simple problems.

Conclusion. The result of the ascertaining experiment is reflected in the graph.

Decided: 25 problems - students of gymnasium No. 5

24 problems - students of secondary school No. 74

30 people took part in the experiment: 15 people from gymnasium No. 5 and 15 people from school No. 74 in Minsk.

The highest results were achieved when solving problem No. 1. The lowest results were achieved when solving problem No. 3.

The general level of students in the two groups who coped with solving these problems is approximately the same.

Reasons for low results:

1. Not all students have the knowledge, skills and abilities necessary to solve simple problems. Namely:

a) the ability to identify elements of a task (condition, question);

b) the ability to model the text of a problem using segments (building a diagram);

c) the ability to justify the choice of an arithmetic operation;

d) knowledge of tabular cases of addition within 10;

e) the ability to compare numbers within 10.

2. Students experience the greatest difficulties when drawing up a diagram for a problem (“dressing” the diagram) and composing an expression.

3.2. Educational experiment

Purpose of the experiment: continue work on solving problems using the activity method with students from gymnasium No. 5 studying under the “School 2100” program. To develop stronger knowledge, skills and abilities when solving problems Special attention was devoted to drawing up a diagram (“dressing” the scheme) and composing an expression according to the scheme.

The following tasks were offered.

1. Game “Part or whole?”

In another version of the proposed game, the situation is closer to the one in which students will find themselves when modeling the problem. Schemes are drawn up on the board in advance. The teacher asks what is known in each case: the part or the whole? Answering. Students can use the technique noted above or give a written answer using the following conventions:

¾ - whole

The technique of mutual verification and the technique of reconciliation with the correct execution of the task on the board can be used.

2. Game "What changed?"

The diagram is in front of the students:

It turns out what is known: a part or a whole. Then the students close their eyes, the diagram takes the form 2), the students answer the same question, close their eyes again, the diagram is transformed, etc. - as many times as he counts needed teacher.

Similar tasks in a game form can be offered to students with a question mark. Only the task will be formulated somewhat differently: “What unknown: part or whole?”

In previous assignments, students “read” the diagram; It is equally important to be able to “dress” the scheme.

3. Game “Wear the scheme”

Before the start of the lesson, each student receives a small piece of paper with diagrams that are “dressed up” according to the teacher’s instructions. Tasks can be like this:

- A- Part;

- b– whole;

Unknown whole;

Unknown part.

4. Game “Choose a scheme”

The teacher reads the problem, and the students must name the number of the diagram on which the question mark was placed in accordance with the text of the problem. For example: in a group of “a” boys and “b” girls, how many children are in the group?

The rationale for the answer may be as follows. All children of the group (whole) consist of boys (part) and girls (other part). This means that the question mark is correctly placed in the second diagram.

When modeling the text of a problem, the student must clearly imagine what needs to be found in the problem: a part or a whole. For this purpose, the following work can be carried out.

5. Game “What is unknown?”

The teacher reads the text of the problem, and the students answer the question about what is unknown in the problem: part or whole. A card that looks like this can be used as a means of feedback:

on the one hand, on the other: .

For example: in one bunch there are 3 carrots, and in the other there are 5 carrots. How many carrots are there in two bunches? (the whole is unknown).

The work can be done in the form of a mathematical dictation.

At the next stage, along with the question of what needs to be found in the problem: a part or a whole, the question is asked about how to do this (by what action). Students are prepared to make informed choices of arithmetic operations based on the relationship between the whole and its parts.

Show the whole, show the parts. What is known, what is unknown?

I show - do you name what it is: a whole or a part, is it known or not?

What is greater, the part or the whole?

How to find the whole?

How to find a part?

What can you find if you know the whole and the part? How? (What action?).

What can you find if you know the parts of the whole? How? (What action?).

What and what do you need to know to find the whole? How? (What action?).

What and what do you need to know to find the part? How? (What action?).

Write an expression for each diagram?

The reference diagrams used at this stage of working on the task can look like this:

During the experiment, students came up with their own problems, illustrated them, “dressed up” diagrams, used commenting, and worked independently with various types of testing.

3.3. Control experiment

Target: check the effectiveness of the approach to solving simple problems proposed by the educational program “School 2100”.

The following tasks were proposed:

There were 3 books on one shelf and 4 books on the other. How many books were on the two shelves?

9 children were playing in the yard, 5 of them boys. How many girls were there?

6 birds were sitting on a birch tree. Several birds flew away, 4 birds remained. How many birds flew away?

Tanya had 3 red pencils, 2 blue and 4 green. How many pencils did Tanya have?

Dima read 8 pages in three days. On the first day he read 2 pages, on the second - 4 pages. How many pages did Dima read on the third day?

Conclusion. The result of the control experiment is reflected in the graph.

Decided: 63 problems – students of gymnasium No. 5

50 problems – students of school No. 74

As we can see, the results of students from gymnasium No. 5 in solving problems are higher than those of students from secondary school No. 74.

So, the results of the experiment confirm the hypothesis that if the educational program “School 2100” (an activity method) is used when teaching mathematics to primary schoolchildren, then the learning process will be more productive and creative. We see confirmation of this in the results of solving problems No. 4 and No. 5. Students have not previously been offered such problems. When solving such problems, it was necessary, using a certain base of knowledge, skills and abilities, to independently find solutions to more complex problems. Students from gymnasium No. 5 completed them more successfully (21 problems solved) than students from secondary school No. 74 (14 problems solved).

I would like to present the result of a survey of teachers working under this program. 15 teachers were selected as experts. They noted that children who study the new mathematics course (the percentage of affirmative answers is given):

Calmly answer at the board 100%

Able to express their thoughts more clearly and clearly 100%

Not afraid to make a mistake 100%

Became more active and independent 86.7%

93.3% are not afraid to express their point of view

Better justify their answers 100%

Calmer and easier to navigate unusual situations(at school, at home) 66.7%

Teachers also noted that children began to show originality and creativity more often, because:

· students have become more reasonable, cautious and serious in their actions;

· children are at ease and bold in communicating with adults, they easily come into contact with them;

· they have excellent self-control skills, including in the area of ​​relationships and rules of behavior.

Conclusion

Based on personal practice, having studied the concept, we came to the conclusion: the “School 2100” system can be called variable personal activity approach in education, which is based on three groups of principles: personality-oriented, culturally-oriented, activity-oriented. It should be emphasized that the “School 2100” program was created specifically for mass secondary schools. The following can be distinguished benefits of this program:

1. The principle of psychological comfort embedded in the program is based on the fact that each student:

· is an active participant cognitive activity in the classroom, can show their creative abilities;

· progresses while studying the material at a pace convenient for him, gradually assimilating the material;

· masters the material to the extent that is accessible and necessary to him (the minimax principle);

· feels interest in what is happening in each lesson, learns to solve problems that are interesting in content and form, learns new things not only from the mathematics course, but also from other areas of knowledge.

Textbooks L.G. Peterson take into account the age and psychophysiological characteristics of schoolchildren .

2. The teacher in the lesson acts not as an informant, but as an organizer search activity of students. A specially selected system of tasks, during which students analyze the situation, express their suggestions, listen to others and find the right answer, helps the teacher in this.

The teacher often offers tasks during which the children cut out, measure, color, and trace. This allows you not to memorize the material mechanically, but to study it consciously, “passing it through your hands.” Children draw their own conclusions.

The exercise system is designed in such a way that it also contains a sufficient set of exercises that require actions according to a given pattern. In such exercises, skills and abilities are not only developed, but algorithmic thinking is also developed. There are also a sufficient number of creative exercises that contribute to the development of heuristic thinking.

3. Developmental aspect. One cannot fail to mention special exercises aimed at developing the creative abilities of students. The important thing is that these tasks are given in the system, starting from the first lessons. Children come up with their own examples, problems, equations, etc. They really enjoy this activity. It's no coincidence that's why creative works Children, on their own initiative, are usually brightly and colorfully decorated.

Textbooks are multi-level, allow you to organize differentiated work with textbooks in the lesson. Assignments typically include both practice of mathematics education standards and questions that require the application of knowledge at a constructive level. The teacher builds his system of work taking into account the characteristics of the class, the presence in it of groups of poorly prepared students and students who have achieved high performance in studying mathematics.

5. The program provides effective preparation for studying algebra and geometry courses in high school.

From the very beginning of the mathematics course, students are accustomed to working with algebraic expressions. Moreover, the work is carried out in two directions: composing and reading expressions.

The ability to compose letter expressions is honed in an unconventional type of task - blitz tournaments. These tasks arouse great interest in children and are successfully completed by them, despite the fairly high level of complexity.

Early use elements of algebra allows you to lay a solid foundation for the study of mathematical models and for revealing to students at higher levels of education the role and significance of the method of mathematical modeling.

This program provides an opportunity through activities to lay the foundation for further study of geometry. Already in elementary school, children “discover” various geometric patterns: they derive the formula for the area of ​​a right triangle, and put forward a hypothesis about the sum of the angles of a triangle.

6. The program develops interest in the subject. It is impossible to achieve good learning results if students have low interest in mathematics. To develop and consolidate it, the course offers quite a lot of exercises that are interesting in content and form. A large number of numerical crosswords, puzzles, ingenuity tasks, decodings help the teacher make lessons truly exciting and interesting. In the course of completing these tasks, children decipher either a new concept or a riddle... Among the deciphered words are the names of literary characters, titles of works, names of historical figures that are not always familiar to children. This stimulates learning new things; there is a desire to work with additional sources (dictionaries, reference books, encyclopedias, etc.)

7. Textbooks have a multi-linear structure, giving the ability to systematically work on repeating material. It is well known that knowledge that is not included in work for a certain time is forgotten. It is difficult for a teacher to independently work on selecting knowledge for repetition, because searching for them takes considerable time. These textbooks provide the teacher with great assistance in this matter.

8. Printed textbook base in elementary school, it saves time and focuses students on solving problems, which makes the lesson more voluminous and informative. At the same time, the most important task of developing students’ skills is solved self-control.

The work carried out confirmed the hypothesis put forward. The use of an activity-based approach to teaching mathematics to junior schoolchildren has shown that cognitive activity, creativity, and liberation of students increase, and fatigue decreases. The “School 2100” program meets the challenges of modern education and lesson requirements. For several years, children did not have unsatisfactory grades in the entrance exams to the gymnasium - an indicator of the effectiveness of the “School 2100” program in schools of the Republic of Belarus.

Literature

1. Azarov Yu.P. Pedagogy of love and freedom. M.: Politizdat, 1994. - 238 p.

2. Belkin E.L. Theoretical prerequisites for creating effective teaching methods // Primary school. - M., 2001. - No. 4. - P. 11-20.

3. Bespalko V.P. Components of pedagogical technology. M.: Higher School, 1989. - 141 p.

4. Blonsky P.P. Favorites pedagogical works. M.: Academy of Pedagogists. Sciences of the RSFSR, 1961. - 695 p.

5. Vilenkin N.Ya., Peterson L.G. Mathematics. 1 class. Part 3. Textbook for 1st grade. M.: Ballas. - 1996. - 96 p.

6. Vorontsov A.B. The practice of developmental education. M.: Knowledge, 1998. - 316 p.

7. Vygotsky L.S. Pedagogical psychology. M.: Pedagogy, 1996. - 479 p.

8. Grigoryan N.V., Zhigulev L.A., Lukicheva E.Yu., Smykalova E.V. On the problem of continuity in teaching mathematics between primary and secondary schools // Primary school: plus before and after. - M., 2002. - No. 7. P. 17-21.

9. Guzeev V.V. Towards the construction of a formalized theory of educational technology: target groups and target settings // School technologies. – 2002. - No. 2. - P. 3-10.

10. Davydov V.V. Scientific support of education in the light of new pedagogical thinking. M.: 1989.

11. Davydov V.V. Developmental learning theory. M.: INTOR, 1996. - 542 p.

12. Davydov V.V. Principles of teaching in the school of the future // Reader on developmental and pedagogical psychology. - M.: Pedagogy, 1981. - 138 p.

13. Selected psychological works: In 2 volumes. Ed. V.V. Davydova and others - M.: Pedagogika, T. 1. 1983. - 391 p. T. 2. 1983. - 318 p.

14. Kapterev P.F. Selected pedagogical essays. M.: Pedagogy, 1982. - 704 p.

15. Kashlev S.S. Modern technologies pedagogical process. Mn.: Universitetskoe. - 2001. - 95 p.

16. Clarin N.V. Pedagogical technology in the educational process. - M.: Knowledge, 1989. - 75 p.

17. Korosteleva O.A. Methods of working on equations in elementary school. // Elementary school: plus or minus. 2001. - No. 2. - P. 36-42.

18. Kostyukovich N.V., Podgornaya V.V. Methods of teaching solving simple problems. – Mn.: Bestprint. - 2001. - 50 p.

19. Ksenzova G.Yu. Promising school technologies. – M.: Pedagogical Society of Russia. - 2000. - 224 p.

20. Kurevina O.A., Peterson L.G. The concept of education: a modern view. - M., 1999. - 22 p.

21. Leontiev A.A. What is the activity approach in education? // Primary school: plus or minus. - 2001. - No. 1. - P. 3-6.

22. Monakhov V.N. Axiomatic approach to the design of pedagogical technology // Pedagogy. - 1997. - No. 6.

23. Medvedskaya V.N. Methods of teaching mathematics in primary school. - Brest, 2001. - 106 p.

24. Methods of initial teaching of mathematics. Ed. A.A. Stolyara, V.L. Drozda. - Mn.: Higher school. - 1989. - 254 p.

25. Obukhova L.F. Age-related psychology. - M.: Rospedagogika, 1996. - 372 p.

26. Peterson L.G. Program “Mathematics” // Primary school. - M. - 2001. - No. 8. P. 13-14.

27. Peterson L.G., Barzinova E.R., Nevretdinova A.A. Independent and test work in mathematics in elementary school. Issue 2. Options 1, 2. Study guide. - M., 1998. - 112 p.

28. Appendix to the letter of the Ministry of Education of the Russian Federation dated December 17, 2001 No. 957/13-13. Features of kits recommended for general education institutions participating in an experiment to improve the structure and content of general education // Elementary school. - M. - 2002. - No. 5. - P. 3-14.

29. Collection regulatory documents Ministry of Education of the Republic of Belarus. Brest. 1998. - 126 p.

30. Serekurova E.A. Modular lessons in elementary school. // Elementary school: plus or minus. - 2002. - No. 1. - P. 70-72.

31. Modern dictionary of pedagogy / Comp. Rapatsevich E.S. - Mn.: Modern Word, 2001. - 928 p.

32. Talyzina N.F. Formation of cognitive activity of younger schoolchildren. - M. Education, 1988. - 173 p.

33. Ushinsky K.D. Selected pedagogical essays. T. 2. - M.: Pedagogy, 1974. - 568 p.

34. Fradkin F.A. Pedagogical technology in historical perspective. - M.: Knowledge, 1992. - 78 p.

35. “School 2100.” Priority directions for the development of the educational program. Issue 4. M., 2000. - 208 p.

36. Shchurkova N.E. Pedagogical technologies. M.: Pedagogy, 1992. - 249 p.

Annex 1

Topic: SUBTRACTING TWO-DIGITUAL NUMBERS WITH TRANSITION THROUGH THE DIGIT

2nd grade. 1 hour (1 - 4)

Target: 1) Introduce the technique of subtracting two-digit numbers with transition through the digit.

2) Consolidate the learned computational techniques, the ability to independently analyze and solve compound problems.

3) Develop thinking, speech, cognitive interests, creative abilities.

During the classes:

1. Organizational moment.

2. Statement of the educational task.

2.1. Solving subtraction examples with transition through digits within 20.

The teacher asks the children to solve examples:

Children verbally name the answers. The teacher writes the children's answers on the board.

Divide the examples into groups. (By the value of the difference - 8 or 7; examples in which the subtrahend is equal to the difference and not equal to the difference; the subtrahend is equal to 8 and not equal to 8, etc.)

What do all the examples have in common? (The same calculation method is subtraction with transition through the digit.)

What other subtraction examples can you solve? (For subtracting two-digit numbers.)

2.2. Solving examples on subtracting two-digit numbers without jumping through the place value.

Let's see who can solve these examples better! What's interesting about the differences: *9-64, 7*-54, *5-44,

It is better to place examples one below the other. Children should notice that in the minuend one digit is unknown; unknown tens and ones alternate; all known digits in the minuend are odd and are in descending order: in the subtrahend, the number of tens is reduced by 1, but the number of units does not change.

Solve the minuend if you know that the difference between the digits denoting tens and units is 3. (In the 1st example - 6 d., 12 d. cannot be taken, since only one digit can be put in a digit; in the 2nd example - 4 units, since 10 units are not suitable; in the 3rd - 6 units, 3 units cannot be taken, since the minuend must be greater than the subtracted; similarly in the 4th - 6 units, and in the 5th. - 4 days)

The teacher reveals closed numbers and asks children to solve examples:

69 - 64. 74 - 54, 85 - 44. 36 - 34, 41 - 24.

For 2-3 examples, the algorithm for subtracting two-digit numbers is spoken out loud: 69 - 64 =. From 9 units. subtract 4 units, we get 5 units. From 6 d. subtract 6 d., we get O d. Answer: 5.

2.3. Formulation of the problem. Goal setting.

When solving the last example, children experience difficulty (different answers are possible, some will not be able to solve it at all): 41-24 = ?

The goal of our lesson is to invent a subtraction technique that will help us solve this example and examples like it.

Children lay out the example model on the desk and on the demonstration canvas:

How to subtract two-digit numbers? (Subtract tens from tens, and ones from ones.)

Why did the difficulty arise here? (The minuend is missing units.)

Is our minuend less than our subtrahend? (No, the minuend is greater.)

Where are the few hiding? (In the top ten.)

What need to do? (Replace 1 ten with 10 units. - Discovery!)

Well done! Solve the example.

Children replace the tens triangle in the minuend with a triangle on which 10 units are drawn:

11e -4e = 7e, Zd-2d=1d. In total it turned out to be 1 d. and 7 e. or 17.

So. “Sasha” offered us a new method of calculations. It is as follows: split ten and take from his missing units. Therefore, we could write down our example and solve it like this (the entry is commented):

Can you think of what you should always remember when using this technique, where an error is possible? (The number of tens is reduced by 1.)

4. Physical education minute.

5. Primary consolidation.

1) No. 1, page 16.

Comment on the first example using the following example:

32 - 15. From 2 units. You cannot subtract 5 units. Let's split ten. From 12 units. subtract 5 units, and from the remaining 2 tenths. subtract 1 dec. We get 1 dec. and 7 units, that is 17.

Decide following examples with an explanation.

Children complete the graphical models of the examples and at the same time comment on the solution aloud. Lines connect pictures with equalities.

2) No. 2, p. 16

Once again, the solution and commentary on the example are clearly stated in a column:

81 _82 _83 _84 _85 _86

29 29 29 29 29 29

I write: units under units, tens under tens.

I subtract units: from 1 unit. You cannot subtract 9 units. I borrow 1 day and put an end to it. 11-9 = 2 units. I write under units.

I subtract the tens: 7-2 = 5 dec.

Children solve and comment on examples until they notice a pattern (usually 2-3 examples). Based on the established pattern in the remaining examples, they write down the answer without solving them.

3) № 3, p. 16.

Let's play a guessing game:

82 - 6 41 -17 74-39 93-45

82-16 51-17 74-9 63-45

Children write down and solve examples in squared notebooks. Comparing them. they see that the examples are interconnected. Therefore, in each column only the first example is solved, and in the rest the answer is guessed, provided that the correct justification is given and everyone agrees with it.

The teacher invites the children to copy examples from the board in a column. for a new computing technique

98-19, 64-12, 76 - 18, 89 - 14, 54 - 17.

Children write in notebooks in a cage necessary examples, and then check the correctness of their entries using the finished sample:

19 18 17

They then solve the written examples on their own. After 2-3 minutes the teacher shows the correct answers. Children check them themselves, mark correctly solved examples with a plus, and correct mistakes.

Find a pattern. (The numbers in the minuends are written in order from 9 to 4, the subtrahends themselves go in decreasing order, etc.)

Write your own example that would continue this pattern.

7. Repetition tasks.

Children who have completed their independent work come up with and solve problems in their notebooks, and those who have made mistakes refine their mistakes individually together with the teacher or consultants. then they solve 1-2 more examples on a new topic on their own.

Come up with a problem and solve according to the options:

Option 1 Option 2

Perform cross-check. What did you notice? (The answers to the problems are the same. These are mutually inverse problems.)

8. Lesson summary.

What examples did you learn to solve?

Can you now solve the example that caused difficulties at the beginning of the lesson?

Come up with and solve such an example for a new technique!

Children offer several options. One is selected. Children. write it down and solve it in a notebook, and one of the children does it on the board.

9. Homework.

No. 5, p. 16. (Unravel the name of the fairy tale and the author.)

Compose your own example of a new computational technique and solve it graphically and columnarly.

Topic: MULTIPLICATION BY 0 AND 1.

2kl., 2h. (1-4)

Target: 1) Introduce special cases of multiplication with 0 and 1.

2) Reinforce the meaning of multiplication and the commutative property of multiplication, practice computational skills,

3) Develop attention, memory, mental operations, speech, creativity, interest in mathematics.

During the classes:

1. Organizational moment.

2.1. Tasks for the development of attention.

On the board and on the table the children have a two-color picture with numbers:

What's interesting about the numbers written down? (Write in different colors; all “red” numbers are even, and “blue” numbers are odd.)

Which number is the odd one out? (10 is round, and the rest are not; 10 is two-digit, and the rest are single-digit; 5 is repeated twice, and the rest - one at a time.)

I’ll close the number 10. Is there an extra one among the other numbers? (3 - he doesn’t have a pair until 10, but the rest do.)

Find the sum of all the “red” numbers and write it in the red square. (thirty.)

Find the sum of all the “blue” numbers and write it in the blue square. (23.)

How much more is 30 than 23? (On 7.)

How much is 23 less than 30? (Also at 7.)

What action did you use? (By subtraction.)

2.2. Tasks for the development of memory and speech. Updating knowledge.

a) -Repeat in order the words that I will name: addend, addend, sum, minuend, subtrahend, difference. (Children try to reproduce the order of words.)

What components of actions were named? (Addition and subtraction.)

What new action are we introduced to? (Multiplication.)

Name the components of multiplication. (Multiplier, multiplier, product.)

What does the first factor mean? (Equal terms in the sum.)

What does the second factor mean? (The number of such terms.)

Write down the definition of multiplication.

b) -Look at the notes. What task will you be doing?

12 + 12 + 12 + 12 + 12

33 + 33 + 33 + 33

(Replace the sum with the product.)

What will happen? (The first expression has 5 terms, each equal to 12, so it is equal to

12 5. Similarly - 33 4, and 3)

c) - Name the inverse operation. (Replace the product with the sum.)

Replace the product with the sum in the expressions: 99 - 2. 8 4. b 3. (99 + 99, 8 + 8 + 8 + 8, b+b+b).

d) Equalities are written on the board:

21 3 = 21+22 + 23

44 + 44 + 44 + 44 = 44 + 4

17 + 17-17 + 17-17 = 17 5

The teacher places pictures of a chicken, a baby elephant, a frog and a mouse next to each equation, respectively.

The animals from the forest school were completing a task. Did they do it correctly?

Children establish that the baby elephant, frog and mouse made a mistake, and explain what their mistakes were.

e) - Compare the expressions:

8 – 5… 5 – 8 34 – 9… 31 2

5 6… 3 6 a – 3… a 2 + a

(8 5 = 5 8, since the sum does not change from rearranging the terms; 5 6 > 3 6, since there are 6 terms on the left and right, but there are more terms on the left; 34 9 > 31 - 2. since there are more terms on the left and themselves the terms are greater; a 3 = a 2 + a, since there are 3 terms on the left and right, equal to a.)

What property of multiplication was used in the first example? (Commutative.)

2.3. Formulation of the problem. Goal setting.

Look at the picture. Are the equalities true? Why? (Correct, since the sum is 5 + 5 + 5 = 15. Then the sum becomes one more term 5, and the sum increases by 5.)

5 3 = 15 5 5 = 25

5 4 = 20 5 6 = 30

Continue this pattern to the right. (5 7 = 35; 5 8 = 40...)

Continue it now to the left. (5 2 = 10; 5 1=5; 5 0 = 0.)

What does the expression 5 1 mean? 50? (? Problem!) Bottom line discussions:

In our example, it would be convenient to assume that 5 1 = 5, and 5 0 = 0. However, the expressions 5 1 and 5 0 do not make sense. We can agree to consider these equalities true. But to do this, we need to check whether we will violate the commutative property of multiplication. So, the goal of our lesson is determine whether we can count equalities 5 1 = 5 and 5 0 = 0 true? - Lesson problem!

3. “Discovery” of new knowledge by children.

1) No. 1, page 80.

a) - Follow steps: 1 7, 1 4, 1 5.

Children solve examples with comments in a textbook-notebook:

1 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7

1 4 = 1 + 1 + 1 + 1 = 4

1 5 = 1 + 1 + 1 + 1 +1 = 5

Draw a conclusion: 1 a -? (1 a = a.) The teacher puts out a card: 1 a = a

b) - Do the expressions 7 1, 4 1, 5 1 make sense? Why? (No, because the sum cannot have one term.)

What should they be equal to so that the commutative property of multiplication is not violated? (7 1 must also equal 7, so 7 1 = 7.)

4 1 = 4 are considered similarly. 5 1 = 5.

Draw a conclusion: and 1 =? (a 1 = a.)

The card is displayed: a 1 = a. The teacher puts the first card on the second: a 1 = 1 a = a.

Our conclusion coincides with what we got on number line? (Yes.)

Translate this equality into Russian. (When you multiply a number by 1 or 1 by a number, you get the same number.)

a 1 = 1 a = a.

2) The case of multiplication from 0 in No. 4, p. 80 is studied in a similar way. Conclusion - multiplying a number by 0 or 0 by a number produces zero:

a 0 = 0 a = 0.

Compare both equalities: what do 0 and 1 remind you of?

Children express their versions. You can draw their attention to those images that are given in the textbook: 1 - “mirror”, 0 - “terrible beast” or “invisible hat”.

Well done! So, when multiplied by 1, the same number is obtained (1 is a “mirror”), and when multiplied by 0, the result is 0 (0 is an “invisible hat”).

4. Physical education minute.

5. Primary consolidation.

Examples written on the board:

23 1 = 0 925 = 364 1 =

1 89= 156 0 = 0 1 =

Children solve them in a notebook with the resulting rules spoken out loud, for example:

3 1 = 3, since when a number is multiplied by 1, the same number is obtained (1 is a “mirror”), etc.

2) No. 1, p. 80.

a) 145 x = 145; b) x 437 = 437.

When multiplying 145 by an unknown number, the result was 145. This means that they multiplied by 1 x= 1. Etc.

3) No. 6, p. 81.

a) 8 x = 0; b) x 1= 0.

When multiplying 8 by an unknown number, the result was 0. So, multiplied by 0 x = 0. Etc.

6. Independent work with testing in class.

1) No. 2, p. 80.

1 729 = 956 1 = 1 1 =

No. 5, p. 81.

0 294 = 876 0 = 0 0 = 1 0 =

Children independently solve written examples. Then, based on the finished sample, they check their answers with pronunciation in loud speech, mark correctly solved examples with a plus, and correct the mistakes made. Those who made mistakes receive a similar task on a card and refine it individually with the teacher while the class solves repetition problems.

7. Repetition tasks.

a) - We are invited to visit today, but to whom? You will find out by deciphering the recording:

[P] (18 + 2) - 8 [O] (42+ 9) + 8

[A] 14 - (4 + 3) [H] 48 + 26 - 26

[F] 9 + (8 - 1) [T] 15 + 23 - 15

Who are we invited to visit? (To Fortran.)

b) - Professor Fortran is a computer expert. But the thing is, we don't have an address. Cat X - Professor Fortran's best student - left a program for us (A poster like the one on page 56, M-2, part 1 is hung up.) We set off on the journey according to X's program. Which house did you come to?

One student follows the poster on the board, and the rest follow the program in their textbooks and find the Fortran house.

c) - Professor Fortran meets us with his students. His best student, the caterpillar, has prepared a task for you: “I thought of a number, subtracted 7 from it, added 15, then added 4 and got 45. What number did I think of?”

The reverse operations must be done in reverse order: 45-4-15 + 7 = 31.

G) Game-competition.

- The Fortran professor himself invited us to play the game “ Computing machines”.

Table in students' notebooks. They independently perform calculations and fill out the table. The first 5 people who complete the task correctly win.

8. Lesson summary.

Did you do everything you planned in the lesson?

What new rules have you met?

9. Homework.

1) №№ 8, 10, p. 82 - in a squared notebook.

2) Optional: № 9 or № 11 on p.82 - on a printed basis.

Topic: PROBLEM SOLVING.

2nd grade, 4 hours (1 - 3).

Target: 1) Learn to solve problems using sum and difference.

2) Strengthen computational skills, composing letter expressions for word problems.

3) Develop attention, mental operations, speech, communication skills, interest in mathematics.

During the classes:

1. Organizational moment .

2. Statement of the educational task.

2.1. Oral exercises.

The class is divided into 3 groups - “teams”. One representative from each team performs individual task on the board, the rest of the children work from the front.

Front work:

Reduce the number 244 by 2 times (122)

Find the product of 57 and 2 (114)

Reduce the number 350 by 230 (120)

How much is 134 greater than 8? (126)

Reduce the number 1280 by 10 times (128)

What is the quotient of 363 and 3? (121)

How many centimeters are in 1 m 2 dm 4 cm? (124)

Arrange the resulting numbers in ascending order:

Individual work at the blackboard:

- Three The trickster bunnies received gifts on their birthday. See if any of them have the same gifts? (Children find examples with the same answers).

What numbers are left without a pair? (Number 7.)

Describe this number. (Single digit, odd, multiples of 1 and 7.)

2.2. Setting a learning task.

Each team receives 4 “Blitz Tournament” problems, a plate and a diagram.

“Blitz tournament”

a) One hare put on a rings, and the other one put on 2 more rings than the first. How many rings do they both have?

b) The mother hare had rings. She gave her three daughters each b rings How many rings does she have left?

c) There were red rings, b white rings and pink rings. They were distributed equally to 4 bunnies. How many rings did each hare receive?

d) The mother hare had rings. She gave them to her two daughters so that one of them got n more rings than the other. How many rings did each daughter receive?

For the 1st team:

For the 2nd team:

For the III team:

It has become fashionable among rabbits to wear rings in their ears. Read the problems on your sheets of paper and determine which problem your diagram and your expression fit into?

Students discuss problems in groups and find the answer together. One person from the group “defends” the team’s opinion.

For what problem did I not choose a diagram and expression?

Which of these schemes is suitable for the fourth problem?

Write an expression for this problem. (Children offer various solutions, one of them is a: 2.)

Is this decision correct? Why not? Under what conditions could we consider it correct? (If both hare had the same number of rings.)

We encountered a new type of problem: in them the sum and difference of numbers are known, but the numbers themselves are unknown. Our task today is to learn how to solve problems by sum and difference.

3. “Discovery” of new knowledge.

Children's reasoning Necessarily accompanied by objective actions of children with stripes.

Place strips of colored paper in front of you, as shown in the diagram:

Explain what letter indicates the sum of the rings in the diagram? (Letter a.) Difference of rings? (Letter n .)

Is it possible to equalize the number of rings on both hare? How to do it? (Children bend or tear off part of a long strip so that both segments become equal.)

How to write down the expression how many rings there are? (a-n)

This is twice the amount or larger number? (Less.)

How to find smaller number? ((a-n): 2.)

Have we answered the problem question? (No.)

What else should you know? (Larger number.)

How to find a larger number? (Add difference: (a-n): 2 + n)

Tablets with the obtained expressions are recorded on the board:

(a-n): 2 - smaller number,

(a-n): 2 + n - greater number.

We first found twice the smaller number. How else could one reason? (Find twice the number.)

How to do it? (a + n)

How then to answer the questions of the task? ((a + n): 2 is the larger number, (a + n): 2-n is the smaller number.)

Conclusion: So, we have found two ways to solve such problems by sum and difference: first find double the smaller number - by subtraction, or find first double a larger number by addition. Both solutions are compared on the board:

1 way 2 way

(a-n):2 (a + n):2

(a-n):2 + n (a + n):2 – n

4. Physical education minute.

5. Primary consolidation.

Students work with a textbook-notebook. Tasks are solved with comments, the solution is written down on a printed basis.

a) - Read the problem to yourself № 6(a), p. 7.

What do we know about the problem and what do we need to find? (We know that there are 56 people in two classes, and in class 1 there are 2 more people than in class two. We need to find the number of students in each class.)

- “Dress” the diagram and analyze the problem. (We know the sum - 56 people, and the difference - 2 students. First, we will find twice the smaller number: 56 - 2 = 54 people. Then we will find out how many students are in the second grade: 54: 2 = 27 people. Now we will find out how many students are in first class - 27 + 2 = 29 people.)

How else can you find out how many students are in first grade? (56 – 27 = 29 people.)

How to check if a problem has been solved correctly? (Calculate the sum and difference: 27 + 29 = 56, 29 – 27 = 2.)

How could the problem be solved differently? (First find the number of students in the first grade and subtract 2 from it.)

b) - Read the problem to yourself № 6 (b), page 7. Analyze which quantities are known and which are not and come up with a solution plan.

After a minute of discussion in the teams, a representative of the team that was ready first speaks. Both ways of solving the problem are discussed orally. After discussing each method, a ready-made solution recording sample is opened and compared with the student’s answer:

I method II method

1) 18 – 4= 14 (kg) 1) 18 + 4 = 22 (kg)

2) 14:2 = 7 (kg) 2) 22: 2 = 11 (kg)

3) 18 – 7 = 11 (kg) 3) 11 – 4 = 7 (kg)

6. Independent work with testing in class.

Students solve assignment No. 7, p. 7 using the options on a printed basis (I option - No. 7 (a), II option - No. 7 (b)).

No. 7 (a), p. 7.

I method II method

1) 248-8 = 240(m.) 1) 248 +8 = 256(m.)

2) 240:2=120 (m.) 2) 256:2= 128 (m.)

3) 120 + 8= 128 (m.) 3) 128-8= 120 (m.)

Answer: 120 marks; 128 marks.

No. 7(6), p. 7.

I method II method

1) 372+ 12 = 384 (open) 1) 372-12 = 360 (open)

2) 384:2= 192 (open) 2) 360:2= 180 (open)

3) 192 – 12 =180 (open) 3)180+12 = 192 (open)

Answer: 180 postcards; 192 postcards.

Check - according to the finished sample on the board.

Each team receives a sign with the task: “Find a pattern and enter the required numbers instead of question marks.”

1 team:

2 team:

3 team:

Team captains report on the team's performance.

8. Lesson summary.

Explain how you reason when solving problems if the following operations are performed:

9. Homework.

Come up with your own new type of problem and solve it in two ways.

Topic: COMPARISON OF ANGLES.

4th grade, 3 hours (1-4)

Target: 1) Review the concepts: point, ray, angle, vertex of an angle (point), sides of an angle (rays).

2) Introduce students to the method of comparing angles using direct superposition.

3) Repeat problems into parts, practice solving problems to find a part of a number.

4) Develop memory, mental operations, speech, cognitive interest, research abilities.

During the classes:

1. Organizational moment.

2. Statement of the educational task.

a) - Continue the series:

1) 3, 4, 6, 7, 9, 10,...; 2) 2, ½, 3, 1/3,...; 3) 824, 818, 812,...

b) - Calculate and arrange in descending order:

[I] 60-8 [L] 84-28 [F] 240: 40 [A] 15 - 6

[G] 49 + 6 [U] 7 9 [R] 560: 8 [H] 68: 4

Cross out the extra 2 letters. What word did you get? (FIGURE.)

c) - Name the figures that you see in the picture:

Which figures can be extended indefinitely? (Straight line, beam, sides of an angle.)

I connect the center of the circle with a point lying on the circle. What happens? (The segment is called the radius.)

Which of the broken lines is closed and which is not?

What other flat geometric shapes do you know? (Rectangle, square, triangle, pentagon, oval, etc.) Spatial figures? (Parallelepiped, cubic ball, cylinder, cone, pyramid, etc.)

What types of angles are there? (Straight, sharp, blunt.)

Show with pencils a model of an acute angle, a right angle, an obtuse one.

What are the sides of an angle - segments or rays?

If you continue the sides of the angle, will you get the same angle or a different one?

d) No. 1, p. 1.

Children must determine that all corners in the picture have the side formed by the large arrow in common. The more the arrows are “spread apart,” the greater the angle.

e) No. 2, p. 1.

Children's opinions about the relationship between angles usually vary. This serves as the basis for creating a problematic situation.

3. “Discovery” of new knowledge by children.

The teacher and children have models of corners cut out of paper. Children are encouraged to explore the situation and find a way to compare angles.

They must guess that the first two methods are not suitable, since continuation of the sides of the corners none of the corners is inside the other. Then, based on the third method - “which fits”, a rule for comparing angles is derived: the angles must be superimposed on one another so that one side coincides. - Opening!

The teacher summarizes the discussion:

To compare two angles, you can superimpose them so that one side coincides. Then the angle whose side is inside the other angle is smaller.

The resulting output is compared with the textbook text on page 1.

4. Primary consolidation.

Task No. 4, page 2 of the textbook is solved with commentary, aloud the rule for comparing angles is spelled out.

In task No. 4, page 2, the angles must be compared “by eye” and arranged in ascending order. The name of the pharaoh is CHEOPS.

5. Independent work with testing in class.

Students perform independently practical work in No. 3, page 2, then in pairs explain how they overlapped the angles. After this, 2-3 pairs explain the solution to the whole class.

6. Physical education minute.

7. Solving repetition problems.

1) - I have a difficult task. Who wants to try to solve it?

During a mathematical dictation, two volunteers together must come up with a solution to the problem: “Find 35% of 4/7 of the number x” .

2) The mathematical dictation was recorded on a tape recorder. Two write down the task on individual boards, the rest - in a notebook “in a column”:

Find 4/9 of number a. (a: 9 4)

Find a number if 3/8 of it is b. (b: 3 8)

Find 16% of the village. (from: 100 16)

Find a number whose 25% is x . (X : 25 100)

What part of the number 7 is the number y? (7/y)

What part leap year is February? (29/366)

Check - according to the sample solution on portable boards. Errors made while completing a task are analyzed according to the scheme: it is established what is unknown - the whole or the part.

3) Analysis of the solution to the additional task: (x: 7 4): 100 35.

Students recite the rule for finding a part of a number: To find the part of a number expressed as a fraction, you can divide this number by the denominator of the fraction and multiply it by its numerator.

4) No. 9, p. 3 - orally with justification for the decision:

- A greater than 2/3, since 2/3 is a proper fraction;

Bless than 8/5, since 8/5 is an improper fraction;

3/11 of c is less than c, and 11/3 of c is greater than c, so the first number is less than the second.

5) No. 10, page 3. The first line is solved with commentary:

To find 7/8 of 240, divide 240 by the denominator 8 and multiply by the numerator 7. 240: 8 7 = 210

To find 9/7 of 56, you need to divide 56 by the denominator 7 and multiply by the numerator 9. 56: 7 9 = 72.

14% is 14/100. To find 14/100 of 4000, you need to divide 4000 by the denominator 100 and multiply by the numerator 14. 4000: 100 14 = 560.

The second line solves itself. The one who finishes first deciphers the name of the pharaoh in whose honor the very first pyramid was built:

6) No. 12(6), page 3

The camel's mass is 700 kg, and the mass of the load it carries on its back is 40% of the camel's mass. What is the mass of the camel with its load?

Students mark the condition of the problem on the diagram and analyze it independently:

To find the mass of a camel with a load, you need to add the mass of the load to the mass of the camel (we are looking for the whole). The mass of the camel is known - 700 kg, and the mass of the load is not known, but it is said that it is 40% of the camel's mass. Therefore, in the first step we find 40% of 700 kg, and then add the resulting number to 700 kg.

The solution to the problem with explanations is written down in a notebook:

1) 700: 100 40 = 280 (kg) - mass of the load.

2) 700 + 280 = 980 (kg)

Answer: the mass of a loaded camel is 980 kg.

8. Lesson summary.

What did you learn? What did they repeat?

What did you like? What was difficult?

9. Homework: No. 5, 12 (a), 16

Appendix 2

Training

Topic: “Solving equations”

Includes 5 tasks, as a result of which the entire algorithm of actions for solving equations is built.

In the first task, students, restoring the meaning of the operations of addition and subtraction, determine which component expresses the part and which the whole.

In the second task, having determined what the unknown is, children choose a rule to solve the equation.

In the third task, students are offered three options for solving the same equation, and the error lies in one case during the solution, and in the other in the calculation.

In the fourth task, from three equations you need to choose those that use the same action to solve. To do this, the student must “go through” the entire algorithm for solving equations three times.

In the last task you need to choose X an unusual situation that the children have not yet encountered. Thus, the depth of assimilation is tested here new topic and the child’s ability to apply the learned algorithm of actions in new conditions.

Epigraph of the lesson : “Everything secret becomes clear.” Here are some of the children's statements when summing up the results in the resource circle:

In this lesson, I remembered that the whole is found by addition, and the parts are found by subtraction.

Everything that is unknown can be found if you follow the right steps.

I realized that there are rules that need to be followed.

We realized that there is no need to hide anything.

We learn to be smart so that the unknown becomes known.

Expert review

Appendix 3

Oral exercises

The purpose of this lesson is to introduce children to the concept of a number line. In the proposed oral exercises, not only work is done on the development of mental operations, attention, memory, constructive skills, not only are counting skills practiced and advanced preparation for studying following topics course, but also offers an option for creating a problem situation, which can help the teacher organize the stage of setting an educational task when studying this topic.

Topic: “Number segment”

Main target :

1) Introduce the concept of a number line, teach

one unit.

2) Strengthen counting skills within 4.

(For this and subsequent lessons, children should have a ruler 20 cm long.) - Today in the lesson we will test your knowledge and ingenuity.

- “Lost” numbers. Find them. What can be said about the location of each missing number? (For example, 2 is 1 more than 1, but 1 less than 3.)

1… 3… 5… 7… 9

Establish a pattern in writing numbers. Continue right one number and left one number:

Restore order. What can you say about the number 3?

1 2 3 4 5 6 7 8 9 10

Divide the squares into parts by color:

+=+=

-=-=

How are all the figures labeled? How are the parts labeled? Why?

Fill in the missing letters and numbers in the boxes. Explain your decision.

What do the equalities 3 + C = K and K - 3 = C mean? What numerical equalities correspond to them?

Name the whole and parts in numerical equations.

How to find the whole? How to find a part?

How many green squares? How many blue ones?

Which squares are larger - green or blue - and by how many? Which squares are smaller and by how many? (The answer can be explained in the figure by making pairs.)

On what other basis can these squares be divided into parts? (By size - large and small.)

What parts will the number 4 be broken into then? (2 and 2.)

Make two triangles from 6 sticks.

Now make two triangles from 5 sticks.

Remove 1 stick to form a quadrangle.

Name the meanings of numerical expressions:

3 + 1 = 2-1 = 2 + 2 =

1 + 1 = 2 + 1 = 1 + 2 + 1 =

Which expression is “superfluous”? Why? (“Expression 2-1 may be superfluous, since this is a difference, and the rest are sums; in the expression 1 + 2 + 1 there are three terms, and in the rest there are two.)

Compare the expressions in the first column.

In case of difficulty, you can ask guiding questions:

What do these numerical expressions have in common? ( Same sign action, the second term is less than the first and equal to 1.)

What is the difference? ( Various firsts terms; in the second expression, both terms are equal, and in the first, one term is 2 more than the other.)

- Problems in verse(the solution to the problems is justified):

Anya has two goals, Tanya has two goals. (We are looking for the whole. To find

Two balls and two, baby, the whole, the parts must be added:

How many are there, can you imagine? 2 + 2 = 4.)

Four magpies came to class. (We are looking for a part. To find

One of the forty did not know the lesson. part must be subtracted from the whole

How diligently did forty work? other part: 4 -1 = 3.)

Today we are waiting for a meeting with our favorite heroes: Boa Constrictor, Monkey, Baby Elephant and Parrot. The boa constrictor really wanted to measure its length. All attempts by Monkey and Baby Elephant to help him were in vain. Their trouble was that they did not know how to count, they did not know how to add and subtract numbers. And so the smart Parrot advised me to measure the length of the boa constrictor with my own steps. He took the first step, and everyone shouted in unison... (One!)

The teacher lays out a red segment on the flannelgraph and puts the number 1 at the end of it. Students draw a red segment 3 cells long in their notebooks and write down the number 1. The blue, yellow and green segments are completed in the same way, each with 3 cells. A colored drawing appears on the board and in students’ notebooks - a numerical segment:

Did the Parrot take the same steps? (Yes, all steps are equal.)

- What does each number show? (How many steps taken.)

How do numbers change when moving left and right? (When moving 1 step to the right, they increase by 1, and when moving 1 step to the left, they decrease by 1.)

The material of oral exercises should not be used formally - “everything in a row”, but should be correlated with specific working conditions - the level of preparation of children, their number in the class, the technical equipment of the office, the level of pedagogical excellence teachers, etc. To use this material correctly, you must be guided by the following in your work principles.

1. The atmosphere in the lesson should be calm and friendly. You shouldn’t allow “races,” overloading children - it’s better to deal with them one task fully and efficiently than seven, but superficially and chaotically.

2. Forms of work need to be diversified. They should change every 3-5 minutes - collective dialogue, work with subject models, cards or a cash register of numbers, mathematical dictation, work in pairs, independent answer at the board, etc. Thoughtful organization of the lesson allows significantly increase the volume of material, which can be considered with children without overload.

3. The introduction of new material should begin no later than 10-12 minutes into the lesson. Exercises prior to learning something new should be aimed primarily at updating the knowledge that is necessary for its full assimilation.

Lecture session Topic: Methods of teaching mathematics to junior schoolchildren as an academic subject.

Purpose of the lesson:

1).Didactic:

To achieve students' understanding of the methods of teaching mathematics to junior schoolchildren as an academic subject.

2). Developmental:

Expand the concepts of methods of teaching mathematics to primary schoolchildren. Develop students' logical thinking.

3). Educating:

Teach students to realize the importance of studying this topic for their future profession.

6.Form of training: frontal.

7. Teaching methods:

Verbal: explanation, conversation, questioning.

Practical: independent work.

Visual: Handout, tutorials.

Lesson plan:

1. Methods of teaching mathematics to junior schoolchildren as a pedagogical science and as a field of practical activity.
2. Methods of teaching mathematics as an academic subject. Principles of designing a mathematics course in elementary school.
3. Methods of teaching mathematics.

Basic concepts:

Methods of teaching mathematics- is the science of mathematics as a scientific subject and the laws of teaching mathematics to students of various age groups, in its research this science is based on various psychological, pedagogical, mathematical foundations and generalizations of the practical experience of mathematics teachers.

1. Methods of teaching mathematics to junior schoolchildren as a pedagogical science and as a field of practical activity.

Considering the methodology of teaching mathematics to junior schoolchildren as a science, it is necessary, first of all, to determine its place in the system of sciences, outline the range of problems that it is designed to solve, determine its object, subject and features.

In the system of sciences, methodological sciences are considered in the block didactics. As is known, didactics is divided into education theory And theory training. In turn, in the theory of learning, general didactics (general issues: methods, forms, means) and particular didactics (subject-specific) are distinguished. Private didactics are called differently - teaching methods or, as has become common in recent years - educational technologies.

Thus, methodological disciplines belong to the pedagogical cycle, but at the same time, they represent purely subject areas, since the methods of teaching literacy will certainly be very different from the methods of teaching mathematics, although both of them are private didactics.

The methodology of teaching mathematics to primary schoolchildren is a very ancient and very young science. Learning to count and calculate was a necessary part of education in ancient Sumerian and ancient Egyptian schools. Rock paintings from the Paleolithic era tell stories about learning to count. To the first textbooks for teaching children mathematics, we can include “Arithmetic” by Magnitsky (1703) and the book by V.A. Laya "Guide to initial training arithmetic based on results didactic experiences"(1910). In 1935 S.I. Shokhor-Trotsky wrote the first textbook “Methods of teaching mathematics”. But only in 1955, the first book “The Psychology of Teaching Arithmetic” appeared, the author of which was N.A. Menchinskaya turned not so much to the characteristics of the mathematical specifics of the subject, but to the patterns of mastering arithmetic content by a child of primary school age. Thus, the emergence of this science in its modern form was preceded not only by the development of mathematics as a science, but also by the development of two large areas of knowledge: general didactics of learning and the psychology of learning and development.

The teaching technology is based on a methodological system of meaning that includes the following 5 components:

2) learning goals.

3) means

Didactic principles are divided into general and basic.

When considering didactic principles, the main provisions determine the content of the organizational forms and methods of educational work of the school. In accordance with the goals of education and the laws of the learning process.

Didactic principles express what is common to any academic subject and are a guideline for planning the organization and analysis of a practical task.

In the methodological literature there is no single approach to identifying systems of principle:

A. Stolyar identifies the following principles:

1) scientific character

3) visibility

4) activity

5) strength

6) individual approach

Yu.K. Babansky identifies 5 groups of principles:

2) to select the learning task

3) to select the form of training

4) choice of teaching methods

5) analysis of results

The development of modern education is based on the principle of lifelong learning.

The principles of learning are not established once and for all; they deepen and change.

The scientific principle, as a didactic principle, was formulated by N.N. Skatkin in 1950.

Feature of the principle:

Displays, but does not reproduce the accuracy of the scientific system, preserving, as far as possible, the general features of their inherent logic, stages and system of knowledge.

Reliance for subsequent knowledge on previous ones.

Systematic pattern of arrangement of material by year of study in accordance with age characteristics and the age of the students, as well as further development teaching.

Disclosure of internal connections between the concepts of patterns and connections with other sciences.

The redesigned programs emphasized the principles of clarity.

The principle of visibility ensures the transition from living contemplation to real thinking. Visualization makes it more accessible, concrete and interesting, develops observation and thinking, provides a connection between the concrete and the abstract, and promotes the development of abstract thinking.

Excessive use of visualization can lead to undesirable results.

Types of visibility:

natural (models, handouts)

visual clarity (drawings, photos, etc.)

symbolic clarity (schemes, tables, drawings, diagrams)

2.Methods of teaching mathematics as an academic subject. Principles of designing a mathematics course in elementary school.

Methods of teaching mathematics (MTM) is a science whose subject is teaching mathematics, and in a broad sense: teaching mathematics at all levels, from preschool institutions to higher education.

MPM develops on the basis of a certain psychological theory of learning, i.e. MPM is a “technology” for applying psychological and pedagogical theories to primary mathematics teaching. In addition, the MPM should reflect the specifics of the subject of study - mathematics.

The goals of primary mathematics education: general education (mastery of a certain amount of mathematical knowledge by students in accordance with the program), educational (formation of a worldview, the most important moral qualities, readiness to work), developmental (development of logical structures and mathematical style of thinking), practical (formation of the ability to apply mathematical knowledge in specific situations, when solving practical problems).

The relationship between teacher and student occurs in the form of information transfer in two opposite directions: from teacher to student (direct), from teaching to teacher (reverse).

Principles of constructing mathematics in elementary school (L.V. Zankov): 1) teaching at a high level of difficulty; 2) learning at a fast pace; 3) the leading role of theory; 4) awareness of the learning process; 5) purposeful and systematic work.

The learning task is the key. On the one hand, it reflects the general goals of learning and specifies cognitive motives. On the other hand, it allows you to make the process of performing educational actions meaningful.

Stages of the theory of the gradual formation of mental actions (P.Ya. Galperin): 1) preliminary familiarization with the purpose of the action; 2) drawing up an indicative basis for action; 3) performing an action in material form; 4) speaking the action; 5) automation of action; 6) performing an action mentally.

Techniques for consolidating didactic units (P.M. Erdniev): 1) simultaneous study of similar concepts; 2) simultaneous study of reciprocal actions; 3) transformation of mathematical exercises; 4) preparation of tasks by students; 5) deformed examples.

3.Methods of teaching mathematics.

Question about methods of primary mathematics teaching and their classification has always been the subject of attention from methodologists. In most modern methodological manuals, special chapters are devoted to this problem, which reveal the main features of individual methods and show the conditions for them. practical application in the learning process.

Beginning mathematics course consists of several sections, different in content. This includes: problem solving; studying arithmetic operations and developing computational skills; studying measures and developing measurement skills; study of geometric material and development of spatial concepts. Each of these sections, having its own special content, at the same time has its own, private, methodology, its own methods, which are in accordance with the specifics of the content and form of training sessions.

Thus, in the methodology of teaching children to solve problems, the logical analysis of the problem conditions using analysis, synthesis, comparison, abstraction, generalization, etc. comes to the fore as a methodological technique.

But when studying measures and geometric material, another method comes to the fore - laboratory, which is characterized by a combination of mental work and physical work. It combines observations and comparisons with measurements, drawing, cutting, modeling, etc.

The study of arithmetic operations occurs on the basis of the use of methods and techniques that are unique to this section and differ from the methods used in other branches of mathematics.

Therefore, developing mathematics teaching methods, it is necessary to take into account psychological and didactic patterns of a general nature, which are manifested in general methods and principles related to the course as a whole.

The most important task of the school at the present stage of its development is to improve the quality of education. This problem is complex and multifaceted. During today's lesson, our attention will be focused on teaching methods, as one of the most important links in improving the learning process.

Teaching methods are ways joint activities teachers and students aimed at solving learning problems.

The teaching method is a system of purposeful actions of the teacher that organizes the cognitive and practical activities of the student, ensuring that he masters the content of education.

Ilyina: “Method is the way in which the teacher directs the teacher’s cognitive activity” (there is no student as an object of activity or educational process)

The teaching method is a way of transferring knowledge and organizing cognitive practical activities of students in which students master knowledge of knowledge, while developing their abilities and forming their scientific worldview.

Currently, intensive attempts are being made to classify teaching methods. It is of great importance for bringing all known methods into a certain system and order, identifying their common features and features.

The most common classification is teaching methods

- by sources of knowledge;

- for didactic purposes;

- according to the level of activity of students;

- by the nature of students’ cognitive activity.

The choice of teaching methods is determined by a number of factors: the objectives of the school at the current stage of development, the academic subject, the content of the material being studied, the age and level of development of students, as well as their level of readiness to master the educational material.

Let's take a closer look at each classification and its inherent purposes.

In the classification of teaching methods for didactic purposes allocate :

Methods of acquiring new knowledge;

Methods of developing skills and abilities;

Methods of consolidating and testing knowledge, abilities, skills.

Often used to introduce students to new knowledge story method.

In mathematics, this method is usually called - method of presenting knowledge.

Along with this method, the most widely used conversation method. During the conversation, the teacher poses questions to the students, the answers to which involve the use of existing knowledge. Based on existing knowledge, observations, and past experience, the teacher gradually leads students to new knowledge.

At the next stage, the stage of formation of skills and abilities, practical teaching methods. These include exercises, practical and laboratory methods, and work with a book.

Contributes to the consolidation of new knowledge, the formation of skills and abilities, and their improvement independent work method. Often, using this method, the teacher organizes the students’ activities in such a way that the students acquire new theoretical knowledge on their own and can apply them in a similar situation.

The following classification of teaching methods by student activity level- one of the early classifications. According to this classification, teaching methods are divided into passive and active, depending on the degree of student involvement in learning activities.

TO passive These include methods in which students only listen and watch (story, explanation, excursion, demonstration, observation).

TO active - methods that organize independent work students ( laboratory method, practical method, working with a book).

Consider the following classification of teaching methods by source of knowledge. This classification is most widely used due to its simplicity.

There are three sources of knowledge: word, visualization, practice. Accordingly, they allocate

- verbal methods(the source of knowledge is the spoken or printed word);

- visual methods(sources of knowledge are observed objects, phenomena, visual aids );

- practical methods(knowledge and skills are formed in the process of performing practical actions).

Let's take a closer look at each of these categories.

Verbal methods occupy a central place in the system of teaching methods.

TO verbal methods include story, explanation, conversation, discussion.

The second group according to this classification consists of visual teaching methods.

Visual teaching methods are those methods in which the assimilation of educational material is significantly dependent on the methods used. visual aids.

Practical methods training is based on the practical activities of students. The main purpose of this group of methods is the formation of practical skills.

Practical methods include exercises, practical and laboratory work.

The next classification is teaching methods by the nature of students’ cognitive activity.

The nature of cognitive activity is the level of mental activity of students.

The following methods are distinguished:

Explanatory and illustrative;

Methods of problem presentation;

Partially search (heuristic);

Research.

Explanatory and illustrative method. Its essence lies in the fact that the teacher communicates ready-made information through various means, and students perceive it, realize it and record it in memory.

The teacher communicates information using spoken word(story, conversation, explanation, lecture), printed word(textbook, additional manuals), visual aids (tables, diagrams, pictures, films and filmstrips), practical demonstration of methods of activity (showing experience, working on a machine, how to solve a problem, etc.).

Reproductive method assumes that the teacher communicates and explains knowledge in a ready-made form, and students assimilate it and can reproduce and repeat the method of activity according to the teacher’s instructions. The criterion for assimilation is the correct reproduction (reproduction) of knowledge.

Method of problem presentation is a transition from performing to creative activity. The essence of the problem presentation method is that the teacher poses a problem and solves it himself, thereby showing the train of thought in the process of cognition. At the same time, students follow the logic of presentation, mastering the stages of solving holistic problems. At the same time, they not only perceive, understand and remember ready-made knowledge and conclusions, but also follow the logic of evidence and the movement of the teacher’s thoughts.

A higher level of cognitive activity carries with it partially search (heuristic) method.

The method was called partially search because students independently solve a complex educational problem not from beginning to end, but only partially. The teacher involves students in performing individual search steps. Some of the knowledge is conveyed by the teacher, some of the knowledge is obtained by students on their own, answering questions posed or resolving problematic tasks. Educational activities develops according to the scheme: teacher - students - teacher - students, etc.

Thus, the essence of the partially search method of teaching comes down to the fact that:

Not all knowledge is offered to students in a ready-made form; some of it needs to be acquired on their own;

The teacher’s activity consists of operational management of the process of solving problematic problems.

One of the modifications this method is heuristic conversation.

The essence of a heuristic conversation is that the teacher, by asking students certain questions and joint logical reasoning with them, leads them to certain conclusions that constitute the essence of the phenomena, processes, rules under consideration, i.e. Students, through logical reasoning, in the direction of the teacher, make a “discovery.” At the same time, the teacher encourages students to reproduce and use their existing theoretical and practical knowledge, production experience, compare, contrast, and draw conclusions.

The next method in classification according to the nature of students’ cognitive activity is research method training. It provides for the creative assimilation of knowledge by students. Its essence is as follows:

The teacher, together with the students, formulates the problem;

Students resolve it independently;

The teacher provides assistance only when difficulties arise in solving the problem.

Thus, the research method is used not only to generalize knowledge, but mainly so that the student learns to acquire knowledge, investigate an object or phenomenon, draw conclusions and apply the acquired knowledge and skills in life. Its essence comes down to organizing the search and creative activities of students to solve problems that are new to them.

1. Homework:

Prepare for practical training