Methods of teaching mathematics to junior schoolchildren as a pedagogical science and as a field of practical activity. Lecture on the topic: "Methods of teaching mathematics
Teaching mathematics in elementary school is very importance. It is this subject, when successfully studied, that will create the prerequisites for the mental activity of a student in the middle and senior levels.
Mathematics as a subject forms a stable cognitive interest and logical thinking skills. Mathematical tasks contribute to the development of a child's thinking, attention, observation, a strict sequence of reasoning and creative imagination.
Today's world is undergoing significant changes that place new demands on a person. If a student in the future wants to actively participate in all spheres of society, then he needs to be creative, continuously improve himself and develop his individual abilities. And this is exactly what the school should teach the child.
Unfortunately, the teaching of younger students 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 the model, that is, most mathematical tasks are training exercises that do not require the initiative and creativity of children. The priority trend is the student memorizing educational material, memorizing calculation methods and solving problems using a ready-made algorithm.
It must be said that already now many teachers are developing technologies for teaching mathematics to schoolchildren, which provide for the solution of non-standard tasks by children, that is, those that form independent thinking and cognitive activity. The main goal of schooling at this stage is the development of search, research thinking of children.
Accordingly, the tasks of modern education today have changed a lot. 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 the realization of two main goals: educational and upbringing.
Educational includes the formation of basic mathematical skills, abilities and knowledge.
The developing function of education is aimed at the development of the student, and the educational function is aimed at the formation of moral values in him.
What is the peculiarity of mathematical education? At the very beginning of his studies, the child thinks in specific categories. At the end of elementary 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 the training, this general is concretized, supplemented with facts and examples, and, therefore, turns into a truly scientific concept.
Teaching methods and techniques should fully develop the child's mental activity. This is possible only when the child finds attractive sides in the process of learning. That is, the technology of teaching younger students 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. Let's review some of them.
At the heart of the methodology according to L. V. Zankov, training is based on the mental functions of the child, which have not yet matured. The methodology involves three lines of development of the psyche of the student - the mind, feelings and will.
The idea of L. V. Zankov was embodied in the curriculum for the study of mathematics, the author of which is I. I. Arginskaya. The educational material here implies a significant independent activity of the student in acquiring and assimilating 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 activities of the students themselves in the lesson. Moreover, students do not just solve and discuss tasks, but compare, classify, generalize, and find patterns. Namely, such activity strains the mind, awakens intellectual feelings, and, therefore, gives children pleasure from the work done. In such lessons, it becomes possible to achieve the moment when students learn not for grades, but to gain new knowledge.
A feature of the methodology of I. I. Arginskaya 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's planning. In addition, it is planned to actively include weak schoolchildren in productive activities, providing them with dosed assistance.
The methodological concept of N. B. Istomina is also based on the principles of developmental education. The course is based on systematic work on the formation in schoolchildren of such techniques for studying mathematics as analysis and comparison, synthesis and classification, and generalization.
The methodology of N. B. Istomina is aimed not only at developing the necessary knowledge, skills and abilities, but also at improving logical thinking. A feature of the program is the use of special methodological techniques for developing general methods of mathematical operations, which will take into account the individual abilities of an individual student.
The use of this educational and methodological complex allows you to create a favorable atmosphere in the classroom in which children freely express their opinions, participate in the discussion and receive, if necessary, the teacher's help. 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 hunch.
In the methodology of N. B. Istomina, work is systematically and purposefully carried out to develop the mental activity of the student.
One of the traditional methods is a course in mathematics for 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, the development of the necessary skills and abilities. The methodology is based on the assertion that for the successful development of mathematics, it is necessary to create a solid foundation for learning even in the primary grades.
The traditional method forms in students conscious, sometimes brought to automatism, skills of computational actions. Much attention in the program is paid to the systematic use of comparison, comparison, generalization of educational material.
A feature of the course of M. I. Moro is that the concepts, relationships, patterns studied are applied in solving specific problems. After all, solving text problems is a powerful tool for developing imagination, speech, and logical thinking in children.
Many experts emphasize the advantage of this technique - it is the prevention of students' mistakes by performing numerous training exercises with the same techniques.
But much is said about its shortcomings - the program does not fully ensure the activation of the thinking of schoolchildren in the classroom.
Teaching mathematics to younger students assumes that each teacher has the right to choose independently the program according to which he will work. And, nevertheless, it must be taken into account that today's education requires strengthening the active thinking of students. And, after all, not every task causes the need for thinking. If the student has mastered the way of solving, then there is enough memory and perception to cope with the proposed task. Another thing is if a student is given a non-standard task that requires a creative approach, when the accumulated knowledge must be applied in new conditions. Here, then, mental activity will be fully carried out.
Thus, one of the important factors that ensure mental activity is the use of non-standard, entertaining tasks.
Another way that awakens the child's thought is the use of interactive learning in mathematics lessons. Dialogue teaches the student to defend his opinion, pose questions to a teacher or a classmate, review the answers of peers, explain incomprehensible points to weaker students, and find several different ways to solve a cognitive problem.
A very important condition for the activation of thought and the development of cognitive interest is the creation of a problem situation in a mathematics lesson. It helps to attract the student to the educational material, to put him in front of some difficulty, which can be overcome, while activating mental activity.
The activation of the mental work of students 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 the differences between objects than to determine the commonality between them. This is due to their predominantly visual-figurative thinking. In order to compare and find common ground 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. And this means that it will be possible to classify, which is carried out according to some criterion.
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 learning tasks, using a problem situation and using the “teacher-student-student” dialogue. Based on this, we can single out the main task of teaching mathematics - to teach children to think, reason, and identify patterns. At the lesson, an atmosphere of search should be created in which every student can become a pioneer.
Homework plays a very important role in the mathematical development of children. Many educators are of the opinion that the number of homework assignments should be reduced to a minimum or eliminated altogether. Thus, the workload of the student, which negatively affects health, is reduced.
On the other hand, deep research and creativity require slow reflection, which should be carried out outside the classroom. And if the student's homework involves not only learning functions, but also developing ones, then the quality of assimilation of the material will increase significantly. Thus, the teacher should think over homework so that students can join in creative and research activities both at school and at home.
Parents play an important role in the process of doing homework by a student. Therefore, the main advice to parents: the child must do his homework in mathematics himself. But, this does not mean that he should not be helped at all. If the student cannot cope with the solution of the task, then you can help him find the rule by which the example is solved, give a similar task, give him the opportunity to independently find the error and correct it. In no case should you do the task for the child. The main educational goal of both the teacher and the parent is the same - to teach the child to acquire knowledge himself, and not to receive ready-made ones.
Parents need to remember that the book “Ready-made homework” that is being purchased should not be in the hands of a student. The purpose of this book is to help parents check the correctness of homework, and not to enable the student, using it, to rewrite ready-made solutions. In such cases, you can generally forget about the child's good academic performance in the subject.
The formation of general educational skills is also facilitated by the correct organization of the work of the student at home. The role of parents is to create conditions for the work of their child. The student must do his homework in a room where the TV does not work and there are no other distractions. You need to help him plan his time correctly, for example, specifically choose an hour for doing homework and never put off this work until the very last moment. Helping a 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 in the successful education of the student. In no case should they reduce the child's independence in learning, but at the same time, they should skillfully come to his aid if necessary.
The problem of the formation and development of mathematical abilities of younger students is relevant at the present time, but at the same time it is given insufficient attention among the problems of pedagogy. Mathematical abilities refer to special abilities that are manifested only in a separate type of human activity.
Often teachers try to understand why children studying in the same school, with the same teachers, in the same class, achieve different success 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 the relevant activity, 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 there is a rapid development of intelligence. Therefore, during the period of primary school age, the possibilities for developing abilities are the highest. The development of the mathematical abilities of a junior schoolchild is understood as a purposeful, didactically and methodically organized formation and development of a set of interrelated properties and qualities of the child's mathematical style of thinking and his abilities for mathematical knowledge of reality.
The first place among academic subjects, which represent a particular difficulty in teaching, 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, it is necessary 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 into reality, because it does not exist in nature.
From the foregoing, 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", Leontiev A.N. "Ability Chapter", Zak Z.A. "Development of intellectual abilities in children" and others.
To date, the problem of developing the mathematical abilities of younger students 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 issue and identification of direct and indirect factors affecting the development of mathematical abilities.
When writing this paper, the following tasks:
1. The study of psychological and pedagogical literature in order to clarify the essence of the concept of ability in the broad sense of the word, and the concept of mathematical ability in the narrow sense.
2. Analysis of psychological and pedagogical literature, materials of periodicals devoted to the problem of studying mathematical abilities in historical development and at the present stage.
ChapterI. The essence of the concept of ability.
1.1 General concept of abilities.
The problem of abilities is one of the most complex and least developed in psychology. Considering it, first of all, it should be taken into account that the real subject of psychological research is the activity and behavior of a person. 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 the productivity of their activities. The variety of human activities and the quantitative and qualitative difference in productivity makes 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 work. The judgment about abilities is always comparative in nature, that is, it is based on a comparison of productivity, the ability of one person with the ability of others. The criterion of ability is the level (result) of activity, which one manages to achieve, while others do not. The history of social and individual development teaches that any skillful skill is achieved as a result of more or less hard 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, and others are 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 most complex problems of psychology. However, the scientific development of the question of abilities is still insufficient. Therefore, in psychology there is no single definition of abilities.
V.G. Belinsky understood the potential natural forces of the individual, or his capabilities, as abilities.
According to B.M. Teplov, abilities are individual psychological characteristics that distinguish one person from another.
S.L. Rubinstein understands abilities as suitability for a certain activity.
The psychological dictionary defines ability as a quality, opportunity, skill, experience, skill, talent. Abilities allow you to perform certain actions at a given time.
Ability is the readiness of an individual to perform some action; suitability - the available potential to perform any activity or the ability to achieve a certain level of ability development.
Based on the foregoing, we can give a general definition of abilities:
Ability is an expression of the correspondence between the requirements of activity and a complex of neuropsychological properties of a person, which ensures high qualitative and quantitative productivity and the growth of his activity, which is manifested in a high and rapidly growing (compared to the average person) ability to master this activity and own 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 the definition of the subject of study. The only thing that all researchers agree on is the opinion that one should distinguish between ordinary, “school” abilities for mastering 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 noted two aspects of mathematical abilities, reproductive (associated with the function of memory) and productive (associated with the function of thinking). In accordance with this, the author built a well-known system of mathematical tests.
The well-known psychologist Reves in his book Talent and Genius, published in 1952, considers two main forms of mathematical abilities - applicative (as the ability to quickly detect mathematical relationships without preliminary tests and apply relevant knowledge in similar cases) and productive (as the ability to discover relationships that do not directly follow from existing knowledge).
Foreign researchers show great unity of views on the question of innate or acquired mathematical abilities. If here we distinguish two different aspects of these abilities - "school" and creative abilities, then with respect to the second there is complete unity - the creative abilities of a scientist - mathematician are an innate education, a favorable environment is necessary only for their manifestation and development. Such, for example, is 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, "because probably everyone can understand someone else's thought." The thesis about the innate and hereditary nature of mathematical talent was vigorously promoted by Reves.
With regard to "school" (educational) abilities, foreign psychologists are not so unanimous. Here, perhaps, the theory of the parallel action of two factors - the biological potential and the environment - dominates. Until recently, ideas of innateness also dominated school mathematical abilities.
Back in 1909-1910. Stone and independently Curtis, studying achievements in arithmetic and ability in this subject, came to the conclusion that one can hardly speak of mathematical ability as a whole, even in relation to arithmetic. Stone pointed out that children who are good at calculations often lag behind in 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, a similar study was conducted by Davis 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 Verdelin in his book Mathematical Ability. The main intention of the author was to analyze the structure of the mathematical abilities of schoolchildren, based on the multifactorial theory of intelligence, to identify the relative role of each of the factors in this structure. Werdelin accepts as a starting point the following definition of mathematical ability: “Mathematical ability is the ability to understand the essence of mathematical (and similar) systems, symbols, methods and proofs, 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 analyzes the question of the comparative value and objectivity of measuring mathematical abilities by teachers' educational marks and special tests and notes that school marks are unreliable, subjective and far from the real measurement of abilities.
The well-known American psychologist Thorndike made a great contribution to the study of mathematical abilities. In The Psychology of Algebra, he gives a host 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 he believes characterize mathematical thinking, in particular:
1. classification;
2. ability to understand and use symbols;
3. deduction;
4. manipulation with ideas and concepts in an abstract form, without relying on the concrete.
Brown and Johnson in the article "Ways to identify and educate students with potentialities in the sciences" indicate that practicing teachers have identified those features that characterize students with potentialities 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 the review of the works of foreign psychologists, it should be noted that they do not give a more or less clear and precise idea of the structure of mathematical abilities. In addition, it must also be borne in mind that in some works the data were obtained by a slightly objective introspective method, while others are characterized by a purely quantitative approach while 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 to abstract, the ability to reason logically, a good memory, the ability to 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 each step of his thought “the sudden appearance in the mind of a ready-made solution to a problem that we could not solve for a long time,” writes the author, “we explain by unconscious thinking, which ... continued to deal with the problem ... and the result pops up beyond the threshold of consciousness.”
The author notes the specific nature of mathematical talent and mathematical thinking. He argues that the ability to do mathematics is not always inherent even in brilliant people, that there is a difference between a mathematical and non-mathematical mind.
Of great interest is Mordukhai-Boltovsky's attempt to isolate the components of mathematical abilities. These components include, in particular:
1. “strong memory”, it was stipulated that “mathematical memory” is meant, memory for “an object of the type that mathematics deals with”;
2. “wit”, which is understood as the ability to “embrace in one judgment” concepts from two loosely connected areas of thought, to find in the already known something similar to the given;
3. speed of thought (speed of thought is explained by the work done by unconscious thinking in favor of the conscious).
D. Mordukhai-Boltovsky also expresses his views 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 interrelations, cannot express like a geometer." He also expressed valuable thoughts about the peculiarities of the memory of "geometers" and "algebraists".
The theory of abilities was created for a long time by the joint work of the most prominent psychologists of that time: B.M. Teplov, L.S. Vygotsky, A.N. Leontiev, S.L. Rubinstein, B.G. Anafiev and others.
In addition to general theoretical studies of the problem of abilities, B.M. Teplov, with his monograph “Psychology of Musical Abilities,” laid the foundation for an experimental analysis of the structure of abilities for specific types of activity. The significance of this work goes beyond the narrow question of the essence and structure of musical abilities, it found a solution to the main, fundamental questions of studying the problem of abilities for specific types of activity.
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. Kuzmin and F.N. Gonobolin, structural 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 guidance of A.N. Leontiev. 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 is different depending on the stage at which the main task is solved, auxiliary exercises are presented, i.e., the role of suggestive exercises was shown.
Directly related to the problem of abilities is a series of studies by L.N. Landes. In one of the first works of this series - "On some shortcomings in the study of students' thinking" - he raises the question of the need to reveal the psychological nature, the internal mechanism of "the ability to think." Cultivate abilities, according to L.N. Landa means “to teach the technique of thinking”, to form the skills and abilities of analytical and synthetic activity. In his other work - "Some Data on the Development of Mental Abilities" - L. N. Landa found significant individual differences in the assimilation of a new method of reasoning by schoolchildren when solving geometric problems for proof - differences in the number of exercises necessary to master this method, differences in the pace of work, differences in the formation of the ability to differentiate the application of operations depending on the nature of the problem conditions and differences in the assimilation of operations.
Great importance for the theory of mental abilities in general and mathematical abilities in particular, studies by D.B. Elkonin and V.V. Davydova, L.V. Zankova, A.V. Skripchenko.
It is usually believed that the thinking of children aged 7-10 has a figurative character, is distinguished by a low ability to distract and abstract. Experiential learning led by D.B. Elkonin and V.V. Davydov, showed that already in the first grade, with a special teaching methodology, it is possible to give students in alphabetical symbolism, that is, in a general form, a system of knowledge about the relationships of quantities, dependencies between them, to introduce them into the field of formally symbolic operations. A.V. Skripchenko showed that students of the third - fourth grades, under appropriate conditions, can form the ability to solve arithmetic problems by compiling an equation with one unknown.
1.3 Mathematical ability and personality
First of all, it should be noted that characterizing capable mathematicians and necessary for successful activity in the field of mathematics is the “unity of inclinations and abilities in vocation”, expressed in a selective positive attitude towards mathematics, the presence of deep and effective interests in the relevant field, the desire and need to engage in it, passionate dedication to the matter.Without an aptitude for mathematics, there can be no genuine aptitude for it. If the student does not feel any inclination towards mathematics, then even good abilities are unlikely to ensure a completely successful mastery of mathematics. The role that inclination and interest play here boils down to the fact that a person interested in mathematics is intensively engaged in it, and, consequently, vigorously exercises and develops his abilities.
Numerous studies and characteristics of gifted children in the field of mathematics indicate that abilities develop only in the presence of inclinations or even a peculiar need for mathematical activity. The problem is that often students are capable of mathematics, but have little interest in it, and therefore do not have much success in mastering this subject. But if the teacher can awaken their interest in mathematics and the desire to do it, then such a student can achieve great success.Such cases are not uncommon at school: a student capable of mathematics has little interest in it, and does not show much success in mastering this subject. But if the teacher can awaken his interest in mathematics and the inclination to do it, then such a student, "captured" by mathematics, can quickly achieve great success.
From this follows the first rule of teaching mathematics: the ability to interest in science, to push for the independent development of abilities. Emotions experienced by a person are also an important factor in the development of abilities in any activity, not excluding mathematical activity. The joy of creativity, the feeling of satisfaction from intense mental work, mobilize his strength, make him overcome difficulties. All children who are capable of mathematics are distinguished by a deep emotional attitude to mathematical activity, they experience real joy caused by each new achievement. Awakening a creative streak in a student, teaching him to love mathematics is the second rule of a mathematics teacher.Many teachers point out that the ability to quickly and deeply generalize can manifest itself in any one subject without characterizing the student's learning 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 that are general in nature, in some cases act as specific abilities. Many teachers tend to apply an objective assessment, that is, if a student is weak in reading, then in principle he cannot reach heights in the field of mathematics. This opinion is typical for primary school teachers who lead 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 students.
The problem of ability is the problem of individual differences. With the best organization of teaching methods, the student will advance more successfully and faster in one area than in another.
Naturally, success in learning is determined not only by the abilities of the student. In this sense, the content and methods of teaching, as well as the attitude of the student to the subject, are of primary importance. Therefore, success and failure in learning do not always give grounds for judgments about the nature of the student's abilities.
The presence of weak abilities in students does not relieve the teacher of 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 shows them.
It is necessary to educate and select capable ones, while not forgetting about all schoolchildren, to raise their general level of training in every possible way. In this regard, in their work, various collective and individual methods of work are needed in order to activate the activity of students in this way.
The learning process should be comprehensive both in terms of organizing the learning process itself, and in terms of developing students' deep interest in mathematics, skills and abilities in solving problems, understanding the system of mathematical knowledge, solving a special system of non-standard tasks with students, which should be offered not only in lessons, but also in tests. Thus, a special organization of the presentation of educational material, a well-thought-out system of tasks, contribute to an increase in the role of meaningful motives for studying mathematics. The number of results-oriented students is decreasing.
In the lesson, not just solving problems, 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 the course of solving the problem, but on the beauty and rationality of the method.
Teachers successfully use the technique of "setting tasks" to determine the direction of motivation. Each task is evaluated according to the system of the following indicators: the nature of the task, its correctness and relation to the original text. The same method is sometimes used in the wine version: after solving the problem, the students were asked to compose any problems somehow related to the original problem.
To create psycho-pedagogical conditions for increasing the effectiveness of the organization of the learning process system, the principle of organizing the learning process in the form of subject communication using cooperative forms of work of students is used. This is a group problem solving and collective discussion of grading, pair and team work.
Chapter II. The development of mathematical abilities in younger schoolchildren as a methodological problem.
2.1 General features of capable and talented children
The problem of developing children's mathematical abilities is one of the least developed methodological problems of teaching mathematics in primary school today.
The extreme heterogeneity of views on the very concept of mathematical ability leads to the absence of any conceptually sound methods, which in turn creates difficulties in the work of teachers. Perhaps that is why not only among parents, but also among teachers there is a widespread opinion: mathematical abilities are either given or not given. And there's nothing you can do about it.
Undoubtedly, abilities for one or another type of activity are due to 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 tissues directly affect the manifestation or absence of certain abilities.
Moreover, purposeful 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, including the ability to apply existing knowledge in mental activity, are required.
Mathematics is one of those subjects where the individual characteristics of the psyche (attention, perception, memory, thinking, imagination) of the child are crucial for its assimilation. Behind the important characteristics of behavior, behind the success (or failure) of educational activity, those natural dynamic features that were mentioned above are often hidden. Often they give rise to differences in knowledge - their depth, strength, generalization. According to these qualities of knowledge, related (along with value orientations, beliefs, skills) to the content side of a person's mental life, they usually judge the giftedness of children.
Individuality and giftedness are interrelated concepts. Researchers dealing with the problem of mathematical abilities, the problem of the formation and development of mathematical thinking, with all the differences of opinion, note first of all the specific features of the psyche of a mathematically capable child (as well as a professional mathematician), in particular, the flexibility of thinking, i.e. unconventionality, originality, the ability to vary the ways of solving a cognitive problem, the ease of transition from one solution to another, the ability to go beyond the usual way of activity and find new ways to solve a problem under changed conditions. Obviously, these features of thinking directly depend on the special organization of memory (free and connected associations), imagination and perception.
Researchers distinguish such a concept as the 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 the purposefulness of thinking, combined with breadth, i.e. the ability to form generalized methods of action, the ability to cover the problem as a whole, without missing details. Psychological analysis of these categories shows that they should be based on a specially formed or natural inclination to a structural approach to the problem and extremely high stability, concentration and a large amount of attention.
Thus, the individual typological features of the personality of each student individually, which are understood as temperament, and character, and inclinations, and the somatic organization of the personality as a whole, etc., have a significant (and perhaps even decisive!) Influence on the formation and development of the mathematical style of thinking of the child, which, of course, is a necessary condition for preserving the natural potential (inclinations) of the child in mathematics and his further development into pronounced mathematical abilities.
Experienced subject teachers know that mathematical abilities are “piece goods”, and if such a child is not dealt with individually (individually, and not as part of a circle or elective), then abilities may not develop further.
That is why we often observe how a first-grader with outstanding abilities “levels out” by the third grade, and in the fifth grade he completely ceases to differ from other children. What is this? Psychological research shows that there can be different types of age-related mental development:
. "Early rise" (at preschool or primary school age) - due to the presence of bright natural abilities and inclinations of the appropriate type. In the future, consolidation and enrichment of mental merits may occur, which will serve as a start for the formation of outstanding mental abilities.
At the same time, the facts show that almost all scientists who proved themselves before the age of 20 were mathematicians.
But “alignment” with peers can also occur. We believe that such “leveling” is largely due to the lack of a competent and methodically active individual approach to the child in the early period.
"Slow and extended rise", i.e. gradual accumulation of intelligence. The absence of early achievement in this case does not mean that the prerequisites for great or outstanding ability will not emerge later. Such a possible "rise" is the age of 16-17 years, when the factor of the "intellectual explosion" is the social reorientation of the individual, directing his activity in this direction. However, such a "rise" can occur in more mature years.
For a primary school teacher, the most urgent problem is the "early rise", which falls on 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 not letting any of the children open their mouths in the lesson. And as a result, instead of stimulating and developing the little “wunderkind” as much as possible, the teacher is forced to teach him to be 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 in 3-4 years the child “levels out” with his peers. And since mathematical abilities belong to the group of “early abilities”, then, perhaps, it is the mathematically capable children that we lose in the process of this “slowing down” and “leveling out”.
Psychological studies have shown that although the development of learning abilities and creative gifts 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 features of the nervous system of children, but on some general features of capable and talented children, which are noted by most researchers of this problem.
Different authors single out a different "set" of common features 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 the teacher to rely on certain purely procedural characteristics of the activities of capable children, which, as a comparison of a number of special psychological and pedagogical studies on this topic shows, turn out to be the same for children with different types of abilities and giftedness. Researchers note that most capable children are characterized by:
Increased propensity for mental action and a positive emotional response to any new mental challenge. These kids don't know what boredom is - they always have something to do. Some psychologists generally interpret this trait as an age factor of giftedness.
The constant need to renew and complicate the mental load, which entails a constant increase in the level of achievements. If this child is not loaded, then he finds a load for himself and can master chess, a musical instrument, radio work, etc., study encyclopedias and reference books, read special literature, etc.
The desire for independent choice of affairs and planning of their activities. This child has his own opinion about everything, stubbornly defends the unlimited initiative of his activity, has a high (almost always adequate at the same time) self-esteem and is very persistent in self-assertion in the chosen area.
Perfect self-regulation. This child is capable of full mobilization of forces to achieve the goal; is able to repeatedly resume mental efforts, striving to achieve the goal; has, as it were, an “original” attitude to overcome any difficulties, and his failures only make him strive to overcome them with enviable persistence.
Increased performance. Prolonged intellectual loads do not tire this child, on the contrary, he feels good precisely in the situation of a problem that needs to be solved. Purely instinctively, he knows how to use all the reserves of his psyche and his brain, mobilizing and switching them at the right time.
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 the human nervous system. Therefore, pedagogically and methodically, the general tactics and strategy of an individual approach to a capable child, obviously, should be based on such psychological and didactic principles that ensure that the above procedural characteristics of the activities of these children are taken into account.
From a pedagogical standpoint, a capable child is most in need of an instructive style of relations with the teacher, which requires greater information content and validity of the requirements put forward by the teacher. The instructive style, as opposed to the imperative style that prevails in elementary school, involves appealing to the personality of the student, taking into account his individual characteristics and focusing on them. This style of relationship contributes to the development of independence, initiative and creativity, which is noted by many research educators. It is equally obvious that from a didactic point of view, capable children need, at a minimum, to ensure the optimal pace of progress in the content and the optimal amount of teaching load. Moreover, it is optimal for oneself, for one's abilities, i.e. higher than for normal children. If we take into account the need for a constant complication of the mental load, the persistent craving for self-regulation of their activities and the increased efficiency of these children, it can be stated with sufficient confidence that these children are by no means “prosperous” students at school, since their educational activity constantly takes place not in the zone of proximal development (!), but 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 the zone of his proximal development.
Working with talented children in primary school today is no less a "sore" problem than working with underachieving ones.
Its lesser "popularity" in special pedagogical and methodical publications is explained by its lesser "conspicuousness", since the loser is an eternal source of trouble for the teacher, and the fact that Petya's five does not even half reflect his capabilities is known only to the teacher (and even then not always), and Petya's parents (if they deal with this issue on purpose). At the same time, the constant “underload” of a capable child (and the norm for everyone is 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 paragraphs above), but also to the possible extinction of these abilities as unclaimed in educational activities (leading to 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 develop enough ability to overcome difficulties, does not develop immunity to failures, which largely explains the massive "collapse" of the progress of such children when moving from primary to secondary.
In order for a teacher of a mass school to be able to successfully cope with work with a capable child in mathematics, it is not enough to indicate the pedagogical and methodological aspects of the problem. As the thirty-year practice of implementing the system of developmental education has shown, in order for this problem to be solved in the conditions of education in a mass primary school, a specific and fundamentally new methodological solution is needed, which is 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 of the Mathematical Box type. To work with capable and gifted children, tasks that are not entertaining are needed, this is too poor food for their minds! We need a special system and special "parallel" to the existing teaching aids. The lack of methodological support for individual work with a capable child in mathematics leads to the fact that elementary school teachers do not do this work at all (it cannot be considered individual circle or optional work, where a group of children solve entertaining tasks with a teacher, as a rule, not systematically selected). One can understand the problems of a young teacher who does not have enough time or knowledge to select and organize the relevant materials. But a teacher with experience is not always ready to solve such a problem. Another (and, perhaps, the main!) constraint here is the presence of a single textbook for the entire class. Work according to a single textbook for all children, according to a single calendar plan, simply does not allow the teacher to realize the requirement of individualization of the pace of learning for a capable child, and the content of the textbook, which is the same for all children, does not allow realizing the requirement of individualization of the volume of the teaching load (not to mention the requirement of self-regulation and independent planning of activities).
We believe that the creation of special teaching materials in mathematics for working with gifted children is the only possible way to implement the principle of individualization of education in relation to these children in the conditions of teaching a whole class.
2.2 Methodology for long-term assignments
The methodology for using the system of long-term tasks was considered by E.S. Rabunsky when organizing work with high school students in the process of teaching German at school.
In a number of pedagogical studies, the possibility of creating systems of such tasks in various subjects for high school students was considered, both in terms of mastering new material and eliminating gaps in knowledge. 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, essays, etc.), turned out to be the most preferable for the majority of the students surveyed. It turned out that such “delayed work” satisfies the student more than individual lessons and assignments, since the main criterion for student satisfaction at any age is success in work. The absence of a sharp time limit (as happens in the classroom) and the possibility of free multiple return to the content of the work 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 all of the above 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 experience of Beloshistaya A.V. and teachers who took part in the experimental verification 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 their 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 of acquaintance and constant control at the stage of consolidation.
During the experimental work, a large number of printed sheets were developed, combined into blocks covering the whole topic. Each block contains 12-20 sheets. The sheet 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 come to an understanding of the essence and method of performing a new computational technique, and then consolidate the new method of activity. A sheet (or sheet system, i.e. a thematic block) is a “long-term task”, the deadlines for which are individualized in accordance with the desire and capabilities of the student working on this system. Such a sheet can be offered at the lesson or instead of homework in the form of a task “with a delayed deadline” for execution, which the teacher either sets individually or allows the student (this way is more productive) to set the deadline for its completion for himself (this is the way to form self-discipline, since independent planning of activities in connection with independently defined goals and deadlines is the basis of a person’s self-education).
The teacher determines the tactics of working with sheets for the student individually. At first, they can be offered to the student as homework (instead of the usual assignment), individually agreeing on the timing of its implementation (2-4 days). As you master this system, you can switch to a preliminary or parallel way of working, i.e. give the student a sheet before getting to know the topic (on the eve of the lesson) or at the lesson itself for self-learning of the material. Attentive and friendly observation of the student in the process of activity, the “contractual style” of relations (let the child decide 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 (one question can always be answered immediately, passing by the child in the lesson) - all this will help the teacher to fully make the learning process of a capable child individualized without spending a lot of time.
Children should not be forced to rewrite tasks from a sheet. The student works with a pencil on a sheet, writing down answers or adding actions. Such an organization of education causes positive emotions in the child - he likes to work on a printed basis. Saved from the need for tedious rewriting, the child works with greater productivity. Practice shows that although the sheets contain up to fifty tasks (the usual homework norm is 6-10 examples), the student works with them with pleasure. Many children ask for a new leaf every day! In other words, they exceed the working norm of the lesson and homework several times, while experiencing positive emotions and working on their own.
During the experiment, such sheets were developed on the topics: "Oral and written computational techniques", "Numbering", "Values", "Fractions", "Equations".
Methodological principles for constructing the proposed system:
1. The principle of compliance with the program in mathematics for elementary grades. Content sheets are tied to a stable (standard) program in mathematics for elementary grades. Thus, we believe that it is possible to implement the concept of individualization of teaching mathematics to a capable child in accordance with the procedural features of his educational activity when working with any textbook that corresponds to a standard program.
2. Methodically, each sheet implements the principle of dosage, i.e. in one sheet, only one technique, or one concept, is introduced, or one connection, but essential for this concept, is revealed. This, on the one hand, helps the child to clearly understand the purpose of the work, and on the other hand, it helps the teacher to easily monitor the quality of assimilation of this technique or concept.
3. Structurally, the sheet is a detailed methodological solution to the problem of introducing or getting to know and fixing 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 also 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 at the first steps is fully disclosed in smaller actions that are the basis of this technique. As you move along the sheet, these small actions are gradually assembled into larger blocks. This allows the student to master the technique as a whole, which is the logical conclusion of the entire methodological "construction". Such a structure of the sheet allows you to fully implement the principle of a gradual increase in the level of complexity at all stages.
4. Such a sheet structure also makes it possible to implement the principle of accessibility, and to a much deeper extent than it is possible to do today when working only with a textbook, since the systematic use of sheets allows you to assimilate the material at an individual pace 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 (delayed) preparation require long-term planning. The ability to organize one's work, planning it for a certain period of time, is the most important learning skill.
6. The system of sheets on the topic also makes it possible to implement the principle of individualization of testing and assessing students' knowledge, and not on the basis of differentiation of the level of complexity 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 do not need to make higher demands. Sheets at a certain stage allow such children to use more intellectually rich material, which in a propaedeutic plan will introduce them to the following mathematical concepts of a higher level of complexity.
Conclusion
An analysis of the psychological and pedagogical literature on the problem of the formation and development of mathematical abilities shows that all researchers without exception (both domestic and foreign) associate 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 possible only if there are significant natural data for this, i.e. most often in the practice of teaching it is believed that it is necessary to develop abilities only in those children who already have them. But the experimental studies of Beloshistaya A.V. showed that work on the development of mathematical abilities is necessary for every child, regardless of his natural giftedness. It's just that the results of this work will be expressed in varying degrees of development of these abilities: for some children this will be a significant advance in the level of development of mathematical abilities, for others it will be a correction of natural insufficiency in their development.
A great difficulty for the teacher in 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 elementary 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, possibly, the source of his giftedness. And the individualization of the education of such a child 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 schoolchildren's mathematical abilities as a methodological problem [Text] / A.V. White // Primary school. - 2003. - No. 1. - pp. 45 - 53
2. Vygotsky, L.S. Collection of works in 6 volumes (volume 3) [Text] / L.S. Vygotsky. - M, 1983. - S. 368
3. Dorofeev, G.V. Mathematics and intellectual development of schoolchildren [Text] / G.V. Dorofeev // The world of education in the world. - 2008. - No. 1. - pp. 68 - 78
4. Zaitseva, S.A. Activation of the mathematical activity of younger schoolchildren [Text] / S.A. Zaitseva // Primary education. - 2009. - No. 1. - S. 12 - 19
5. Zak, A.Z. Development of intellectual abilities in children 8 - 9 years old [Text] / A.Z. Zach. - M.: New School, 1996. - S. 278
6. Krutetsky, V.A. Fundamentals of pedagogical psychology [Text] / V.A. Krutetsky - M., 1972. - S. 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. - S. 560
9. Nemov, R.S. Psychology: in 3 books (vol. 1) [Text] / R.S. Nemov. - M.: VLADOS, 2006. - S. 688
10. Ozhegov, S.I. Explanatory dictionary of the Russian language [Text] / S.I. Ozhegov. - Onyx, 2008. - S. 736
11. Reverse, J.. Talent and Genius [Text] / J. Reverse. - M., 1982. - S. 512
12. Teplov, B.M. The problem of individual abilities [Text] / B.M. Teplov. - M.: APN RSFSR, 1961. - S. 535
13. Thorndike, E.L. Principles of teaching based on psychology [electronic resource]. - Access mode. - http://metodolog.ru/vigotskiy40.html
14. Psychology [Text] / ed. A.A. Krylova. - M.: Nauka, 2008. - P. 752
15. Shadrikov V.D. Development of abilities [Text] / V.D. Shadrikov // Primary school. - 2004. - No. 5. - from 18-25
16. Volkov, I.P. Are there many talents in the school? [Text] / I.P. Volkov. - M.: Knowledge, 1989. - P.78
17. Dorofeev, G.V. Does teaching mathematics help to increase the level of intellectual development of schoolchildren? [Text] /G.V. Dorofeev // Mathematics at school. - 2007. - No. 4. - S. 24 - 29
18. Istomina, N.V. Methods of teaching mathematics in elementary grades [Text] / N.V. Istomin. - M.: Academy, 2002. - S. 288
19. Savenkov, A.I. A gifted child in a mass school [Text] / ed. M.A. Ushakov. - M.: September, 2001. - S. 201
20. Elkonin, D.B. Questions of psychology of educational activity of junior schoolchildren [Text] / Ed. V. V. Davydova, V. P. Zinchenko. - M.: Enlightenment, 2001. - S. 574
Consider the purpose of studying the course "Methods of teaching mathematics in elementary school" in the process of preparing a future elementary school teacher.
Discussion at a lecture with students
2. Methods of teaching mathematics to younger students 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, to outline the range of problems that it is designed to solve, to determine its object, subject and features.
In the system of sciences, methodological sciences are considered in the block didactics. As you know, didactics is divided into theory education Andtheory learning. In turn, in the theory of learning, general didactics (general issues: methods, forms, means) and particular didactics (subject) are distinguished. Private didactics are also called differently - teaching methods or, as is customary in recent years, educational technologies.
Thus, methodological disciplines belong to the pedagogical cycle, but at the same time, they are purely subject areas, since the methodology for teaching literacy, of course, will be very different from the methodology for teaching mathematics, although both of them are private didactics.
The methodology of teaching mathematics to junior 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. The rock paintings of the Paleolithic era tell about learning to count. Magnitsky's Arithmetic (1703) and V.A. Lai "Guide to the initial teaching of arithmetic, based on the results of didactic experiments" (1910) ... In 1935, SI. Shokhor-Trotsky wrote the first textbook "Methods of Teaching Mathematics". But only in 1955, the first book “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 assimilation of 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 education and the psychology of learning and development. IN Lately an important role in the formation of teaching methods begins to play the psychophysiology of the development of the child's brain. At the intersection of these areas, answers to three “eternal” questions of the methodology of teaching subject content are born today:
Why teach? What is the purpose of teaching a young child math? Is it necessary? And if necessary, why?
What to teach? What content should be taught? What should be the list of mathematical concepts intended for learning with a child? Are there any criteria for selecting this content, the hierarchy of its construction (sequence) and how are they justified?
How to teach? What methods of organizing the child's activity (methods, techniques, means, forms of education) should be selected and applied so that the child can usefully assimilate the selected content? What is meant by “benefit”: the amount of knowledge and skills of the child or something else? How to take into account the psychological characteristics of age and individual differences of children when organizing training, but at the same time “fit” into the allotted time (curriculum, program, daily routine), and also take into account the real content of the class in connection with the system of collective education adopted in our country (class-lesson system)?
These questions actually determine the range of problems of any methodological science. The methodology of teaching mathematics to junior schoolchildren as a science, on the one hand, is addressed to the specific content, its selection and ordering in accordance with the set learning goals, on the other hand, to the pedagogical methodological activity of the teacher and the educational (cognitive) activity of the child in the lesson, to the process of assimilation of the selected content, which is managed by the teacher.
Object of study This science is the process of mathematical development and the process of forming mathematical knowledge and ideas of a child of primary school age, in which the following components can be distinguished: the purpose of learning (Why teach?), content (What to teach?) and the activities of the teacher and the activity of the child (How to teach?). These components form methodological systemmu, in which a change in one of the components will cause a change in the other. Above, the modifications of this system were considered, which entailed a change in the purpose of primary education in connection with a change in the educational paradigm in the last decade. Later we will consider the modifications of this system, which entail the psychological-pedagogical and physiological research of the last half century, the theoretical results of which gradually penetrate into methodological science. It can also be noted that an important factor in changing approaches to the construction of a methodological system is the change in the views of mathematicians on the definition of a system of basic postulates for constructing a school mathematics course. For example, in 1950-1970. the prevailing belief was that the set-theoretic approach should be the basis for constructing a school mathematics course, which was reflected in the methodological concepts of school mathematics textbooks, and therefore required an appropriate orientation of initial mathematical training. In recent decades, mathematicians have been talking more and more about the need to develop functional and spatial thinking in schoolchildren, which is reflected in the content of textbooks published in the 90s. In accordance with this, the requirements for the initial mathematical preparation of the child are gradually changing.
Thus, the process of development of methodological sciences is closely connected with the process of development of other pedagogical, psychological and natural sciences.
Let us consider the relationship between the methodology of teaching mathematics in elementary school and other sciences.
1. The method of mathematical development of the child uses OSnew ideas, theoretical provisions and results of researchny other sciences.
For example, philosophical and pedagogical ideas play a fundamental and guiding role in the development of methodological theory. In addition, borrowing the ideas of other sciences can serve as the basis for the development of specific methodological technologies. Thus, the ideas of psychology and the results of its experimental studies are widely used by the methodology to substantiate the content of education and the sequence of its study, to develop methodological techniques and systems of exercises that organize the assimilation of various mathematical knowledge, concepts and methods of action by children. The ideas of physiology about conditioned reflex activity, two signal systems, feedback, and age stages of maturation of the subcortical zones of the brain help to understand the mechanisms for acquiring skills, habits, and skills in the learning process. Of particular importance for the development of methods of teaching mathematics in recent decades are the results of psychological and pedagogical research and theoretical research in the field of constructing the theory of developmental education (L.S. Vygotsky, J. Piaget, L.V. Zankov, V.V. Davydov, D.B. Elkonin, P.Ya. Galperin, N.N. Poddyakov, L.A. Venger, etc.). This theory is based on the position of L.S. Vygotsky that learning is based not only on completed cycles of a child's development, but primarily on those mental functions that have not yet matured ("zones of proximal development"). Such training contributes to the effective development of the child.
2. The methodology creatively borrows research methods, withchanged in other sciences.
In fact, any method of theoretical or empirical research can find application in methodology, since in the context of the integration of sciences, research methods very quickly become general scientific. Thus, the method of literature analysis familiar to students (compiling bibliographies, taking notes, summarizing, compiling abstracts, plans, writing out citations, etc.) is universal and is used in any science. The method of analyzing programs and textbooks is commonly used in all didactic and methodological sciences. From pedagogy and psychology, the methodology borrows the method of observation, questioning, conversation; from mathematics - methods of statistical analysis, etc.
3. The methodology uses specific research resultspsychology, physiology of higher nervous activity, mathematicski and other sciences.
For example, the specific results of J. Piaget's research on the process of perception by young children of the conservation of quantity gave rise to a whole series of specific mathematical tasks in various programs for younger schoolchildren: using specially designed exercises, a child is taught to understand that a change in the shape of an object does not entail a change in its quantity (for example, when water is poured from a wide jar into a narrow bottle, its visually perceived level increases, but this does not mean that there is more water in the bottle than there was in the jar).
4. The technique is involved in complex developmental studieschild in the course of his education and upbringing.
For example, in 1980-2002. a number of scientific studies of the process of personal development of a child of primary school age appeared in the course of teaching him mathematics.
Summarizing the question of the relationship between the methodology of mathematical development and the formation of mathematical representations in preschoolers, the following can be noted:
It is impossible to deduce from any one science a system of methodological knowledge and methodological technologies;
Data from other sciences are necessary for the development of methodological theory and practical methodological recommendations;
The methodology, like any science, will develop if it is replenished with more and more new facts;
The same facts or data can be interpreted and used in different (and even opposite) ways, depending on what goals are realized in the educational process and what system of theoretical principles (methodology) is adopted in the concept;
The methodology does not just borrow and use data from other sciences, but processes them in such a way as to develop ways for the optimal organization of the learning process;
Methodology, determines the corresponding concept of the mathematical development of the child; Thus, concept - this is not something abstract, far from life and real educational practice, but a theoretical base that determines the construction of the totality of all components of the methodological system: goals, content, methods, forms and means of teaching.
Let's consider the ratio of modern scientific and "everyday" ideas about teaching mathematics to younger students.
At the heart of any science lies the experience of people. For example, physics is based on the knowledge we acquire in everyday life about the movement and fall of bodies, about light, sound, heat, and much more. Mathematics also proceeds from ideas about the forms of objects of the surrounding world, their location in space, quantitative characteristics and ratios of parts of real sets and individual objects. The first coherent mathematical theory - the geometry of Euclid (4th century BC) was born from practical surveying.
The situation is quite different with regard to methodology. Each of us has a life experience of teaching someone something. However, it is possible to engage in the mathematical development of a child only with special methodological knowledge. With what different special (scientific) methodical knowledgeand skills from life Tey ideas that it is enough to have some understanding of counting, calculations and solving simple arithmetic problems to teach mathematics to a younger student?
1. Everyday methodological knowledge and skills are specific; they are dedicated to specific people and specific tasks. For example, a mother, knowing the peculiarities of the perception of her child, through repeated repetitions, teaches the child to name numerals in the correct order and recognize specific geometric shapes. With sufficient perseverance of the mother, the child learns to fluently name numerals, recognizes a fairly large number of geometric shapes, recognizes and even writes numbers, etc. Many believe that this is what the child should be taught before school. Does this training guarantee the development of mathematical abilities in a child? Or at least the continued success of this child in mathematics? Experience shows that it does not guarantee. Can this mother teach the same to another child who is not like her child? Unknown. Will this mother be able to help her child learn other mathematical material? Most likely - no. Most often, one can observe a picture when the mother herself knows, for example, how to add or subtract numbers, solve this or that problem, but she cannot even explain to her child so that he learns the way to solve it. Thus, everyday methodological knowledge is characterized by the specificity, limitation of the task, situations and persons to which they apply,
Scientific methodological knowledge (knowledge of educational technology) tends to to generalization. They use scientific concepts and generalized psychological and pedagogical patterns. Scientific methodological knowledge (educational technologies), consisting of clearly defined concepts, reflects their most significant interrelations, which makes it possible to formulate methodological patterns. For example, an experienced highly professional teacher can often determine by the nature of a child's mistake which methodological patterns in the formation of a given concept were violated when teaching this child.
2. Everyday methodological knowledge is intuitiveter. This is due to the way they are obtained: they are acquired through practical trials and "adjustment". A sensitive, attentive mother goes this way, experimenting and vigilantly noticing the slightest positive results (which is not difficult to do when spending a lot of time with a child. Often the subject of “mathematics” itself leaves specific imprints on the perception of parents. You can often hear: “I myself suffered from mathematics at school, he has the same problems. that a person either has mathematical abilities or not, and nothing can be done about it.The idea that mathematical abilities (as well as musical, visual, sports and others) can be developed and improved by most people is perceived with skepticism. Such a position is very convenient for justifying doing nothing, but from the point of view of general methodological scientific knowledge about the nature, character and genesis of the mathematical development of a child, it is, of course, inadequate.
It can be said that, unlike intuitive methodological knowledge, scientific methodological knowledge rational And conscious. A professional methodologist will never point to heredity, "planid", lack of materials, poor quality of teaching aids and insufficient attention of parents to the educational problems of the child. He has a fairly large arsenal of effective methodological techniques, you just need to select from it those that are most suitable for this child.
Scientific methodological knowledge can be transferred to anotherto a person. The accumulation and transfer of scientific methodological knowledge is possible due to the fact that this knowledge is crystallized in concepts, regularities, methodological theories and fixed in the scientific literature, educational and methodological manuals that future teachers read, which allows them to come even to the first practice in their life with a fairly large baggage of generalized methodological knowledge.
Everyday knowledge about the methods and techniques of teaching is receivedusually through observation and reflection. In scientific activity, these methods are supplemented methodical experiment. The essence of the experimental method is that the teacher does not wait for a confluence of circumstances, as a result of which a phenomenon of interest arises, but causes the phenomenon himself, creating the appropriate conditions. Then he purposefully varies these conditions in order to reveal the patterns that this phenomenon obeys. This is how any new methodological concept or methodological regularity is born. We can say that when creating a new methodological concept, each lesson becomes such a methodological experiment.
5. Scientific methodological knowledge is much broader, more diverse,than worldly; it has unique factual material, inaccessible in its scope to any carrier of worldly methodological knowledge. This material is accumulated and comprehended in separate sections of the methodology, for example: a methodology for teaching problem solving, a method for forming the concept of a natural number, a method for forming ideas about fractions, a method for forming ideas about quantities, etc., as well as in certain branches of methodological science, for example: teaching mathematics in groups for correcting mental retardation, teaching mathematics in compensation groups (visually impaired, hearing impaired, etc.), teaching mathematics to children with mental retardation, teaching schoolchildren capable of mathematics and etc.
The development of special branches of methodology for teaching mathematics to young children is in itself the most effective method of general didactics for teaching mathematics. L.S. Vygotsky began working with mentally retarded children, and as a result, the theory of "zones of proximal development" was formed, which formed the basis of the theory of developmental education for all children, including for teaching mathematics.
One should not think, however, that worldly methodological knowledge is an unnecessary or harmful thing. The "golden mean" is to see in small facts a reflection of general principles, and how to move from general principles to real life problems is not written in any book. Only constant attention to these transitions, constant exercise in them can form in the teacher what is called "methodological intuition." Experience shows that the more worldly methodological knowledge a teacher has, the more likely this intuition is to form, especially if this rich worldly methodological experience is constantly accompanied by scientific analysis and comprehension.
The methodology for teaching mathematics to younger students is applied field of knowledge(applied Science). As a science, it was created to improve the practical activities of teachers working with children of primary school age. It has already been noted above that the methodology of mathematical development as a science is actually making its first steps, although the methodology of teaching mathematics has a thousand-year history. Today there is not a single program of primary (and preschool) education that does without mathematics. But until recently, it was only about teaching young children the elements of arithmetic, algebra and geometry. And only in the last twenty years of the XX century. began to talk about a new methodological direction - theory and practice mathematical development child.
This direction became possible in connection with the formation of the theory of developmental education of a young child. This direction in the traditional methodology of teaching mathematics is still debatable. Not all teachers today stand on the positions of the need to implement developmental education. in progress teaching mathematics, the purpose of which is not so much the formation of a certain list of knowledge, skills and abilities of a subject nature in the child, but the development of higher mental functions, his abilities and the disclosure of the internal potential of the child.
For a progressively thinking teacher, it is obvious that practicallysome results from the development of this methodological direction should become incommensurably more significant than the results of just a methodology for teaching elementary mathematical knowledge and skills to children of primary school age, in addition, they should be qualitatively different. After all, to know something means to master this “something”, to learn it. manage.
Learning to control the process of mathematical development (ie, the development of a mathematical style of thinking) is, of course, a grandiose task that cannot be solved overnight. The methodology has already accumulated a lot of facts today, showing that the new knowledge of the teacher about the essence and meaning of the learning process makes it significantly different: it changes his attitude both to the child and to the content of education, and to the methodology. Learning the essence of the process of mathematical development, the teacher changes his attitude to the educational process (changes himself!), to the interaction of the subjects of this process, to its meaning and goals. It can be said that technique is a scienceconstructing teacher as a subject of educational interaction. In real practical activity today, this has been expressed in modifications of the forms of work with children: teachers are paying more and more attention to individual work, since it is obvious that the effectiveness of the learning process is determined by the individual differences of children. More and more attention is paid by teachers to productive methods of working with children: search and partial search, children's experimentation, heuristic conversation, organization of problem situations in the classroom. Further development of this direction can lead to significant meaningful modifications of the programs of mathematical education of younger students, since many psychologists and mathematicians in recent decades have expressed doubts about the correctness of the traditional filling of primary school mathematics programs with mainly arithmetic material.
There is no doubt that the fact that child learning process ka mathematics is constructive for the development of it personalities . The process of learning any subject content leaves its mark on the development of the cognitive sphere of the child. However, the specificity of mathematics as an academic subject is such that its study can largely influence the overall personal development of the child. Even 200 years ago, this idea was expressed by M.V. Lomonosov: "Mathematics is good because it puts the mind in order." The formation of a systematic thought processes is only one side of the development of the mathematical style of thinking. Deepening the knowledge of psychologists and methodologists about the various aspects and properties of a person’s mathematical thinking shows that many of its most important components actually coincide with the components of such a category as a person’s general intellectual abilities - these are logic, breadth and flexibility of thinking, spatial mobility, laconism and consistency, etc. And such character traits as purposefulness, perseverance in achieving a goal, the ability to organize oneself, “intellectual endurance”, which are formed during active mathematics, are already personal characteristics of a person .
To date, there are a number of psychological studies showing that a systematic and specially organized system of doing mathematics actively influences the formation and development of an internal plan of action, lowers the child's level of anxiety, developing a sense of confidence and control of the situation; increases the level of development of creativity (creative activity) and the overall level of mental development of the child. All of these studies support the idea that mathematical content is the most powerful means of development intelligence and a means of personal development of the child.
Thus, theoretical research in the field of methods of mathematical development of a child of primary school age, refracted through a set of methodological techniques and the theory of developmental education, are implemented when teaching a specific mathematical content in the teacher's practical activities in the classroom.
Lecture 3Traditional and Alternative Systems for Teaching Mathematics to Primary School Students
Brief review of learning systems.
Peculiarities of assimilation of mathematical knowledge, skills and abilities by students with severe speech disorders.
LECTURE 1.
Methods of elementary teaching of mathematics as a subject.
Primary Mathematics Teaching Methodology Answers Questions
· For what? -
· What? -
The methodology of primary teaching of mathematics as a subject is associated with
Essay "Methods of teaching mathematics science, art or craft?"
Objectives of elementary education in mathematics.
1. Educational goals.
2. Development goals.
3. Educational goals.
Features of the construction of the initial course of mathematics.
1. The main content of the course is arithmetic material.
2. The elements of algebra and geometry do not constitute special sections of the course. They are organically associated with arithmetic material.
The elementary course of mathematics is structured in such a way that elements of algebra and geometry are included simultaneously with the study of arithmetic material. Consequently, in one lesson, besides the arithmetic material, algebraic and geometrical material is very often considered. The inclusion of material from different sections of the course, of course, affects the construction of a mathematics lesson and the methodology for conducting it.
4. Relationship between practical and theoretical issues. Therefore, in each lesson of mathematics, work on the assimilation of knowledge goes simultaneously with the development of skills and abilities.
5. Many questions of the theory are introduced inductively.
6. Mathematical concepts, their properties and patterns are revealed in their relationship. Each concept gets its own development.
7. Convergence in time of studying some of the questions of the course, for example, addition and subtraction are introduced at the same time.
1. Arithmetic stuff.
The concept of a natural number, the formation of a natural number.
A visual representation of fractions
The concept of the number system.
The concept of arithmetic operations.
2. Algebra elements.
3.Geometric material.
4. The concept of magnitude and the idea of measuring magnitudes.
5. Tasks. (As the goal and means of teaching mathematics).
Messages.
Analysis of various programs in mathematics
1. Elkonin-Davydov
2. Zankov (Arginskaya)
3. Peterson L.G.
4. Istomina N.B.
5. Checkin
Methods and techniques for teaching mathematics to younger students.
1. Define the concepts of "teaching method", "learning method".
The problem of teaching methods is formulated briefly with the question how to teach?
To solve the problem of how to teach something to students, it is necessary,
Speaking about the methods of teaching mathematics, it is natural, first of all, to clarify this concept.
The method is
The description of each teaching method should include:
1) description of the teaching activity of the teacher;
2) a description of the educational (cognitive) activity of the student and
3) the connection between them, or the way in which the teaching activity of the teacher controls the cognitive activity of students.
The subject of didactics, however, is only general teaching methods, i.e., methods that generalize a certain set of systems of sequential actions of a teacher and a student in the interaction of teaching and learning, which do not take into account the specifics of individual academic subjects.
In addition to specifying and modifying general teaching methods, taking into account the specifics of mathematics, the subject of the methodology is also the addition of these methods with private (special) teaching methods that reflect the main methods of cognition used in mathematics itself.
Thus, the system of teaching methods in mathematics consists of general teaching methods developed by didactics, adapted to teaching mathematics, and of particular (special) methods of teaching mathematics, reflecting the main methods of cognition used in mathematics.
1. EMPIRICAL METHODS: OBSERVATION, EXPERIENCE, MEASUREMENTS.
Observation, experience, measurements are the empirical methods used in the experimental natural sciences.
Observation, experience and measurements should be aimed at creating special situations in the learning process and providing students with the opportunity to extract from them obvious patterns, geometric facts, ideas of proof, etc. Most often, the results of observation, experience and measurements serve as premises of inductive conclusions, with the help of which new truths are discovered. Therefore, observation, experience and measurement are also referred to as heuristic methods of learning, i.e., to methods that contribute to discoveries.
observation.
2. COMPARISON AND ANALOGY - logical methods of thinking used both in scientific research and in education.
By using comparisons the similarity and difference of the objects compared are revealed, i.e., the presence of common and non-common (different) properties in them.
The comparison produces the correct output if the following conditions are met:
1) the compared concepts are homogeneous and
2) the comparison is carried out on such grounds that are essential.
By using analogy the similarity of objects revealed as a result of their comparison extends to a new property (or new properties).
Reasoning by analogy has the following general outline:
A has properties a, b, c, d;
B has properties a, b, c;
Probably (possibly) B also has property d.
The conclusion by analogy is only probable (plausible), but not reliable.
3. GENERALIZATION AND ABSTRAGING - two logical techniques that are almost always used together in the process of cognition.
Generalization- this is a mental selection, fixation of some common essential properties that belong only to a given class of objects or relations.
abstraction- this is a mental abstraction, the separation of general, essential properties, highlighted as a result of generalization, from other non-essential or non-general properties of the objects or relations under consideration and the rejection (within the framework of our study) of the latter.
Under oh bobbling they also understand the transition from the singular to the general, from the less general to the more general.
Under specification understand the reverse transition - from the more general to the less general, from the general to the singular.
If generalization is used in the formation of concepts, then concretization is used in the description of specific situations with the help of previously formed concepts.
4. SPECIFICATION is based on the well-known inference rule
called the specification rule.
5. INDUCTION.
The transition from the particular to the general, from individual facts established with the help of observation and experience, to generalizations is the law of knowledge. An integral logical form of such a transition is induction, which is a method of reasoning from the particular to the general, the conclusion of a conclusion from particular premises (from Latin inductio - guidance).
Usually, when they say "inductive teaching methods", they mean the use of incomplete induction in teaching. Further, when we say "induction", we mean incomplete induction.
At certain stages of education, in particular in elementary school, mathematics is taught mainly by inductive methods. Here the inductive conclusions are psychologically convincing enough and for the most part remain so far (at this stage of learning) unproven. One can only find isolated "deductive islands" consisting in the application of simple deductive reasoning as proofs of individual propositions.
6. DEDUCTION (from Latin deductio - inference) in a broad sense is a form of thinking, consisting in the fact that a new sentence (or rather, the thought expressed in it) is derived in a purely logical way, i.e., according to certain rules of logical inference (following) from some well-known sentences (thoughts).
Taking into account the needs of mathematics, it received special development in the form of proof theory in mathematical logic.
By teaching proof, we mean teaching the thought processes of finding and constructing evidence, rather than reproducing and memorizing ready-made proofs. To teach to prove means first of all to teach to reason, and this is one of the main tasks of teaching in general.
7. ANALYSIS - a logical technique, a method of research, consisting in the fact that the object under study is mentally (or practically) divided into constituent elements (features, properties, relationships), each of which is studied separately as part of a divided whole.
SYNTHESIS is a logical technique by which individual elements are combined into a whole.
In mathematics, most often, analysis is understood as reasoning in the "reverse direction", i.e. from the unknown, from what needs to be found, to the known, to what has already been found or given, from what needs to be proved, to what has already been proven or accepted as true.
In this understanding, which is the most important for learning, analysis is a means of finding a solution, a proof, although in most cases a solution in itself is not yet a proof.
Synthesis, based on the data obtained during the analysis, gives a solution to a problem or a proof of a theorem.
Ministry of Education, Science and Youth Policy of the Republic of Dagestan
GBOUSPO "Republican Pedagogical College" them. Z.N. Batyrmurzaeva.
Course work
on TONKM with teaching methods
on the topic of: " Active methods of teaching mathematics in elementary school"
Completed: St-ka 3 "in" course
Ezerkhanova Zalina
Scientific adviser:
Adilkhanova S.A.
Khasavyurt 2014
Introduction
Chapter I
Chapter II
Conclusion
Literature
Introduction
"A mathematician enjoys knowledge that he has already mastered, and always strives for new knowledge."
The effectiveness of teaching mathematics to schoolchildren largely depends on the choice of forms of organization of the educational process. In my work, I prefer active learning methods. Active learning methods are a set of ways to organize and manage the educational and cognitive activities of students, which have the following main features:
forced learning activity;
independent development of solutions by trainees;
a high degree of involvement of students in the educational process;
constant processing by communication between students and teachers, and control by independent work of learning.
The main meaning of the development of federal state educational standards, the solution of the strategic task of the development of Russian education - improving the quality of education, achieving new educational results. In other words, the Federal State Educational Standard is not intended to fix the state of education achieved at previous stages of its development, but orients education towards achieving a new quality that is adequate to the modern (and even predictable) needs of the individual, society and the state.
The methodological basis of the standards of primary general education of the new generation is a system-activity approach.
The system-activity approach is aimed at the development of the individual, at the formation of civic identity. Training should be organized in such a way as to purposefully lead development. Since the main form of organizing learning is a lesson, it is necessary to know the principles of building a lesson, an approximate typology of lessons and the criteria for evaluating a lesson in the framework of a system-activity approach and active methods of work used in the lesson.
At present, the student with great difficulty sets goals and draws conclusions, synthesizes material and connects complex structures, generalizes knowledge, and even more so finds relationships in them. Teachers, noting the indifference of students to knowledge, unwillingness to learn, low level of development of cognitive interests, try to design more effective forms, models, methods, conditions of learning.
The creation of didactic and psychological conditions for the meaningfulness of teaching, the inclusion of a student in it at the level of not only intellectual, but personal and social activity is possible with the use of active teaching methods. The emergence and development of active methods is due to the fact that new tasks have arisen for teaching: not only to give students knowledge, but also to ensure the formation and development of cognitive interests and abilities, skills and abilities of independent mental work, the development of creative and communicative abilities of the individual.
Active learning methods also provide a directed activation of the mental processes of students, i.e. stimulate thinking when using specific problem situations and conducting business games, facilitate memorization when highlighting the main thing in practical classes, arouse interest in mathematics and develop a need for self-acquisition of knowledge.
A chain of failures can turn away from mathematics and capable children, on the other hand, learning should go close to the ceiling of the student's abilities: the feeling of success is created by the understanding that significant difficulties have been overcome. Therefore, for each lesson, you need to carefully select and prepare individual knowledge, cards, based on an adequate assessment of the student's capabilities at the moment, taking into account his individual abilities.
active method of teaching mathematics
For the organization of active cognitive activity of students in the classroom, the optimal combination of active learning methods is of decisive importance. It is very important for me to assess the work and the psychological climate in my lessons. Therefore, you need to try so that children not only actively study, but also feel confident and comfortable.
The problem of personality activity in learning is one of the most urgent in educational practice.
With this in mind, I have chosen the topic of the study: "Active methods of teaching mathematics in elementary school."
The purpose of the study: to identify, theoretically substantiate the effectiveness of the use of active methods of teaching younger students with learning difficulties in mathematics lessons.
Research problem: what methods contribute to the activation of cognitive activity in students in the learning process.
Object of study: the process of teaching mathematics to younger students.
Subject of study: the study of active methods of teaching mathematics in elementary school.
Research hypothesis: the process of teaching mathematics to younger students will be more successful under the following conditions if:
active teaching methods for younger students will be used in mathematics lessons.
Research objectives:
)study the literature on the problem of using active methods of teaching mathematics in elementary school;
2)To identify and reveal the features of active methods of teaching mathematics in elementary school;
)Consider active methods of teaching mathematics in elementary school.
Research methods:
analysis of psychological and pedagogical literature on the problem of studying active methods of teaching mathematics in elementary school;
supervision of younger students.
The structure of the work: the work consists of an introduction, 2 chapters, a conclusion, a list of references.
Chapter I
1.1 Introduction to active learning methods
Method (from the Greek methodos - the path of research) - a way to achieve.
Active teaching methods are a system of methods that ensure the activity and variety of mental and practical activities of students in the process of mastering educational material.
Active methods provide a solution to educational problems in various aspects:
The teaching method is an ordered set of didactic methods and means by which the goals of training and education are realized. Teaching methods include interrelated, sequentially alternating ways of purposeful activity of the teacher and students.
Any teaching method presupposes a goal, a system of actions, means of training and an intended result. The object and subject of the teaching method is the student.
Any one teaching method is used in its pure form only for specially planned teaching or research purposes. Usually the teacher combines different teaching methods.
Today there are different approaches to the modern theory of teaching methods.
Active teaching methods are methods that encourage students to actively think and practice in the process of mastering educational material. Active learning involves the use of such a system of methods, which is mainly aimed not at the presentation of ready-made knowledge by the teacher, their memorization and reproduction, but at the independent mastery of knowledge and skills by students in the process of active mental and practical activity. The use of active methods in mathematics lessons helps to form not just knowledge-reproductions, but the skills and needs to apply this knowledge to analyze, assess the situation and make the right decision.
Active methods ensure the interaction of participants in the educational process. When they are applied, the distribution of "duties when receiving, processing and applying information between the teacher and the student, between the students themselves. It is clear that the active learning process on the part of the student bears a large developmental load.
When choosing active learning methods, one should be guided by a number of criteria, namely:
· compliance with the goals and objectives, the principles of training;
· compliance with the content of the topic being studied;
· compliance with the capabilities of the trainees: age, psychological development, level of education and upbringing, etc.
· compliance with the conditions and time allotted for training;
· compliance with the capabilities of the teacher: his experience, desires, level of professional skills, personal qualities.
· Student activity can be ensured if the teacher purposefully and maximally uses assignments in the lesson: formulate a concept, prove, explain, develop an alternative point of view, etc. In addition, the teacher can use the techniques of correcting "intentionally made" mistakes, formulating and developing assignments for comrades.
· An important role is played by the formation of the skill of posing a question. Analytical and problematic questions like "Why? What follows? What does it depend on? require constant updating in work and special training in their formulation. The methods of this training are varied: from tasks for posing a question to the text in the lesson to the game "Who will ask more questions on a certain topic in a minute.
· Active methods provide a solution to educational problems in various aspects:
· formation of positive educational motivation;
· increasing the cognitive activity of students;
· active involvement of students in the educational process;
· stimulation of independent activity;
· development of cognitive processes - speech, memory, thinking;
· effective assimilation of a large amount of educational information;
· development of creative abilities and non-standard thinking;
· development of the communicative-emotional sphere of the student's personality;
· revealing the personal and individual capabilities of each student and determining the conditions for their manifestation and development;
· development of skills of independent mental work;
· development of universal skills.
Let's talk about the effectiveness of teaching methods and talk in more detail.
Active teaching methods put the student in a new position. Previously, the student was completely subordinate to the teacher, now active actions, thoughts, ideas and doubts are expected from him.
The quality of education and upbringing is directly related to the interaction of thinking processes and the formation of conscious knowledge, strong skills, and active teaching methods in the student.
The direct involvement of students in educational and cognitive activities during the educational process is associated with the use of appropriate methods, which have received the generalized name of active learning methods. For active learning, the principle of individuality is important - the organization of educational and cognitive activities, taking into account individual abilities and capabilities. This includes pedagogical techniques, and special forms of classes. Active methods help to make the learning process easy and accessible to every child.
The activity of trainees is possible only if there are incentives. Therefore, among the principles of activation, a special place is occupied by the motivation of educational and cognitive activity. Rewards are an important motivating factor. Primary school children have unstable learning motives, especially cognitive ones, so positive emotions accompany the formation of cognitive activity.
1.2 Application of active teaching methods in primary school
One of the problems that worries teachers is the question of how to develop a child's steady interest in learning, in knowledge and the need for their independent search, in other words, how to activate cognitive activity in the learning process.
If a game is a habitual and desirable form of activity for a child, then it is necessary to use this form of organizing activities for learning, combining the game and the educational process, more precisely, using a game form of organizing students' activities to achieve educational goals. Thus, the motivational potential of the game will be aimed at more effective mastering of the educational program by schoolchildren. And the role of motivation in successful learning cannot be overestimated. Conducted studies of students' motivation have revealed interesting patterns. It turned out that the value of motivation for successful study is higher than the value of the student's intellect. High positive motivation can play the role of a compensating factor in case of insufficiently high student abilities, but this principle does not work in the opposite direction - no abilities can compensate for the absence of a learning motive or its low severity and ensure significant academic success.
The goals of school education, which the state, society and family set before the school, in addition to acquiring a certain set of knowledge and skills, are the disclosure and development of the child's potential, the creation favorable conditions for the realization of his natural abilities. A natural play environment, in which there is no coercion and there is an opportunity for each child to find his place, show initiative and independence, freely realize his abilities and educational needs, is optimal for achieving these goals.
To create such an environment in the classroom, I use active learning methods.
The use of active teaching methods in the classroom allows you to:
provide positive motivation for learning;
conduct a lesson at a high aesthetic and emotional level;
ensure a high degree of differentiation of training;
increase the volume of work performed in the lesson by 1.5 - 2 times;
improve knowledge control;
rationally organize the educational process, increase the effectiveness of the lesson.
Active learning methods can be used at various stages of the educational process:
stage - the primary acquisition of knowledge. It can be a problematic lecture, a heuristic conversation, an educational discussion, etc.
stage - knowledge control (reinforcement). Methods such as collective thought activity, testing, etc. can be used.
stage - the formation of skills and abilities based on knowledge and the development of creative abilities; it is possible to use simulated learning, game and non-game methods.
In addition to the intensification of the development of educational information, active teaching methods make it possible to carry out the educational process just as effectively in the process of the lesson and in extracurricular activities. Teamwork, joint project and research activities, upholding one's position and a tolerant attitude towards other people's opinions, taking responsibility for oneself and the team form personality traits, moral attitudes and value orientations of a student that meet the modern needs of society. But this is not all the possibilities of active learning methods. In parallel with training and education, the use of active teaching methods in the educational process ensures the formation and development of so-called soft or universal skills in students. These usually include the ability to make decisions and the ability to solve problems, communication skills and qualities, the ability to clearly formulate messages and clearly set goals, the ability to listen and take into account the different points of view and opinions of other people, leadership skills and qualities, the ability to work in a team, etc. And today, many already understand that, despite their softness, these skills in modern life play a key role both for achieving success in professional and social activities, and for ensuring harmony in personal life.
Innovation is an important feature of modern education. Education is changing in content, forms, methods, responds to changes in society, takes into account global trends.
Educational innovations are the result of the creative search of teachers and scientists: new ideas, technologies, approaches, teaching methods, as well as individual elements of the educational process.
The wisdom of the desert dwellers says: "You can lead a camel to water, but you cannot make him drink." This proverb reflects the basic principle of learning - you can create all the necessary conditions for learning, but knowledge itself will occur only when the student wants to know. How to make the student feel needed at every stage of the lesson, to be a full-fledged member of a single class team? Another wisdom teaches: "Tell me - I'll forget. Show me - I'll remember. Let me do it myself - and I'll learn" According to this principle, learning is based on one's own activity. And therefore, one of the ways to increase the effectiveness in the study of school subjects is the introduction of active forms of work at different stages of the lesson.
Based on the degree of activity of students in the educational process, teaching methods are conditionally divided into two classes: traditional and active. The fundamental difference between these methods lies in the fact that when they are applied, students create conditions under which they cannot remain passive and have the opportunity for an active mutual exchange of knowledge and work experience.
The purpose of using active teaching methods in elementary school is the formation of curiosity.Therefore, for students, you can create a journey into the world of knowledge with fairy-tale characters.
In the course of his research, the outstanding Swiss psychologist Jean Piaget expressed the opinion that logic is not innate, but develops gradually with the development of the child. Therefore, in lessons in grades 2-4, more logical tasks related to mathematics, language, knowledge of the world, etc. should be used. Tasks require the performance of specific operations: intuitive thinking based on detailed ideas about objects, simple operations (classification, generalization, one-to-one correspondence).
Let us consider several examples of the use of active methods in the educational process.
A conversation is a dialogical method of presenting educational material (from the Greek dialogos - a conversation between two or more persons), which in itself speaks of the essential specifics of this method. The essence of the conversation lies in the fact that the teacher, through skillfully posed questions, encourages students to reason, to analyze the studied facts and phenomena in a certain logical sequence and independently formulate the corresponding theoretical conclusions and generalizations.
The conversation is not a communication, but a question-answer method of educational work to comprehend new material. The main point of the conversation is to encourage students, with the help of questions, to reason, analyze the material and generalize, to independently "discover" new conclusions, ideas, laws, etc. for them. Therefore, when conducting a conversation to comprehend new material, it is necessary to pose questions in such a way that they require not monosyllabic affirmative or negative answers, but detailed reasoning, certain arguments and comparisons, as a result of which students isolate the essential features and properties of the objects and phenomena being studied and in this way acquire new knowledge. It is equally important that the questions have a clear sequence and focus, allowing students to deeply comprehend the internal logic of the acquired knowledge.
These specific features of the conversation make it a very active method of learning. However, the use of this method has its limitations, because not every material can be presented through conversation. This method is most often used when the topic being studied is relatively simple and when students have a certain stock of ideas or life observations on it, allowing them to comprehend and assimilate knowledge in a heuristic (from Greek heurisko - I find) way.
Active methods provide for conducting classes through the organization of students' gaming activities. The pedagogy of the game collects ideas that facilitate communication in the group, the exchange of thoughts and feelings, the understanding of specific problems and the search for ways to solve them. It has an auxiliary function in the entire learning process. The task of the pedagogy of the game is to provide methods that help the work of the group and create an atmosphere that makes the participants feel safe and well.
The pedagogy of the game helps the facilitator to realize the various needs of the participants: the need for movement, experiences, overcoming fear, the desire to be with other people. It also helps to overcome shyness, shyness, as well as existing social stereotypes.
For active teaching methods, a special place is occupied by the forms of organization of the educational process - non-standard lessons: a lesson - a fairy tale, a game, a journey, a script, a quiz, lessons - reviews of knowledge.
At such lessons, the activity of children increases, they are happy to help Kolobok escape from the fox, save ships from pirate attacks, store food for the squirrel for the winter. At such lessons, the children are in for a surprise, so they try to work fruitfully and complete various tasks as much as possible. The very beginning of such lessons captivates children from the first minutes: “We will go to the forest today for science” or “A floorboard creaks about something ...” Books from the series “I am going to a lesson in elementary school” and, of course, the work of the teacher himself help to lead such lessons. They help the teacher prepare for lessons in less time, make them more meaningful, modern, and interesting.
In my work, feedback means have acquired particular importance, which make it possible to quickly obtain information about the movement of each student’s thoughts, about the correctness of his actions at any moment of the lesson. Means of feedback using to control the quality of assimilation of knowledge, skills. Each student has means of feedback (we make them ourselves at labor lessons or purchase them in stores), they are an essential logical component of his cognitive activity. These are signal circles, cards, numerical and alphabetic fans, traffic lights. The use of feedback tools makes it possible to make the work of the class more rhythmic, forcing each student to study. It is important that such work be carried out systematically.
One of the new means of checking the quality of education are tests. This is a qualitative way to test learning outcomes, characterized by such parameters as reliability and objectivity. Tests test theoretical knowledge and practical skills. With the advent of the computer in the school, new methods of activating learning activities open up for the teacher.
Modern teaching methods are mainly focused on teaching not ready-made knowledge, but activities for the independent acquisition of new knowledge, i.e. cognitive activity.
In the practice of many teachers, independent work of students is widely used. It is carried out in almost every lesson within 7-15 minutes. The first independent works on the topic are mainly educational and corrective in nature. With their help, operational feedback in learning is carried out: the teacher sees all the shortcomings in the knowledge of students and eliminates them in a timely manner. You can refrain from entering grades "2" and "3" in the class journal for the time being (putting them in a student's notebook or diary). Such an assessment system is quite humane, mobilizes students well, helps them to better comprehend their difficulties and overcome them, and improves the quality of knowledge. Students are better prepared for the test, their fear of such work disappears, the fear of getting a deuce. The number of unsatisfactory ratings, as a rule, is sharply reduced. Students develop a positive attitude towards business, rhythmic work, rational use of lesson time.
Do not forget about the restorative power of relaxation in the classroom. After all, sometimes a few minutes are enough to shake things up, have fun and actively relax, and restore energy. Active methods - "physical minutes" "Earth, air, fire and water", "Bunnies" and many others will allow you to do this without leaving the classroom.
If the teacher himself takes part in this exercise, in addition to benefiting himself, he will also help insecure and shy students to participate more actively in the exercise.
1.3 Features of active methods of teaching mathematics in elementary school
· use of an activity approach to learning;
· the practical orientation of the activities of the participants in the educational process;
· playful and creative nature of learning;
· interactivity of the educational process;
· inclusion in the work of various communications, dialogue and polylogue;
· use of knowledge and experience of students;
· reflection of the learning process by its participants
Another essential quality of a mathematician is an interest in regularities. Regularity is the most stable characteristic of an ever-changing world. Today cannot be like yesterday. You cannot see the same face twice from the same angle. Patterns are found at the very beginning of arithmetic. There are many elementary examples of regularities in the multiplication table. Here is one of them. Usually, children like to multiply by 2 and by 5, because the last digits of the answer are easy to remember: when multiplied by 2, even numbers are always obtained, and when multiplied by 5, even easier, it is always 0 or 5. But even multiplication by 7 has its own patterns. If we look at the last digits of the products 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, i.e. by 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, we will see that the difference between the next and previous digits is: - 3; +7; - 3; - 3; +7; - 3; - 3, - 3. A very definite rhythm is felt in this row.
If you read the final numbers of the answers when multiplying by 7 in reverse order, then we get the final numbers from multiplying by 3. Even in elementary school, you can develop the skill of observing mathematical patterns.
During the period of adaptation of first-graders, one should try to be attentive to the little personality, support her, worry about her, try to interest her in learning, help so that further education for the child is successful and brings mutual joy to the teacher and student. The quality of education and upbringing is directly related to the interaction of thinking processes and the formation of conscious knowledge, strong skills, and active teaching methods in the student.
The key to the quality of education is love for children and a constant search.
The direct involvement of students in educational and cognitive activities during the educational process is associated with the use of appropriate methods, which have received the generalized name of active learning methods. For active learning, the principle of individuality is important - the organization of educational and cognitive activities, taking into account individual abilities and capabilities. This includes pedagogical techniques, and special forms of classes. Active methods help to make the learning process easy and accessible to every child. The activity of trainees is possible only if there are incentives. Therefore, among the principles of activation, a special place is occupied by the motivation of educational and cognitive activity. Rewards are an important motivating factor. Primary school children have unstable learning motives, especially cognitive ones, so positive emotions accompany the formation of cognitive activity.
The age and psychological characteristics of younger students indicate the need to use incentives to achieve the activation of the educational process. Encouragement not only evaluates the positive results visible at the moment, but in itself it encourages further fruitful work. Encouragement is the factor of recognition and assessment of the achievements of the child, if necessary - the correction of knowledge, a statement of success, stimulating further achievements. Encouragement contributes to the development of memory, thinking, forms cognitive interest.
The success of learning also depends on the means of visualization. These are tables, reference diagrams, didactic and handouts, individual teaching aids that help make the lesson interesting, joyful, and provide deep assimilation of the program material.
Individual teaching aids (mathematical pencil cases, cash registers of letters, abacuses) ensure the involvement of children in the active learning process, they become active participants in the educational process, activate the attention and thinking of children.
1The use of information technology in the lesson of mathematics in elementary school .
In elementary school it is impossible to conduct a lesson without the involvement of visual aids, problems often arise. Where can I find the material I need and how best to demonstrate it? The computer came to the rescue.
1.2The most effective means of including a child in the creative process in the classroom are:
· gaming activity;
· creating positive emotional situations;
work in pairs;
· problem learning.
Over the past 10 years, there has been a radical change in the role and place of personal computers and information technology in society. Knowledge of information technology is put in the modern world on a par with such qualities as the ability to read and write. A person who skillfully and effectively masters technologies and information has a different, new style of thinking, a fundamentally different approach to assessing the problem that has arisen, to organizing his activities. As practice shows, it is already impossible to imagine a modern school without new information technologies. Obviously, in the coming decades, the role of personal computers will increase and, in accordance with this, the requirements for computer literacy of primary school students will increase. The use of ICT in primary school classes helps students navigate the information flows of the world around them, master practical ways of working with information, and develop skills that allow them to exchange information using modern technical means. In the process of studying, diverse application and use of ICT tools, a person is formed who is able to act not only according to the model, but also independently, receiving the necessary information from the largest possible number of sources; able to analyze it, put forward hypotheses, build models, experiment and draw conclusions, make decisions in difficult situations. In the process of using ICT, the student develops, prepares students for a free and comfortable life in the information society, including:
development of visual-figurative, visual-effective, theoretical, intuitive, creative types of thinking; - aesthetic education through the use of computer graphics, multimedia technology;
development of communication skills;
the formation of skills to make the best decision or offer solutions in a difficult situation (the use of situational computer games focused on optimizing decision-making activities);
formation of information culture, skills to process information.
ICT leads to the intensification of all levels of the educational process, providing:
improving the efficiency and quality of the learning process through the implementation of ICT tools;
providing motivational motives (stimuli) that cause the activation of cognitive activity;
deepening interdisciplinary connections through the use of modern means of processing information, including audiovisual, in solving problems from various subject areas.
The use of information technology in the classroom in elementary schoolis one of the most modern means of developing the personality of a younger student, the formation of his information culture.
Teachers are increasingly using computer capabilities in preparing and conducting lessons in elementary school.Modern computer programs make it possible to demonstrate vivid visualization, offer various interesting dynamic types of work, and reveal the level of knowledge and skills of students.
The role of the teacher in culture is also changing - he must become the coordinator of the information flow.
Today, when information becomes a strategic resource for the development of society, and knowledge is a relative and unreliable subject, as it quickly becomes obsolete and requires constant updating in the information society, it becomes obvious that modern education is a continuous process.
The rapid development of new information technologies and their introduction in our country have left their mark on the development of the personality of a modern child. Today, a new link is being introduced into the traditional scheme "teacher - student - textbook" - a computer, and computer training is being introduced into school consciousness. One of the main parts of informatization of education is the use of information technologies in educational disciplines.
For an elementary school, this means a change in priorities in setting the goals of education: one of the results of education and upbringing at the first stage school should be the readiness of children to master modern computer technologies and the ability to update the information obtained with their help for further self-education. To achieve these goals, it becomes necessary to apply in the practice of the work of a primary school teacher different strategies for teaching younger students, and, first of all, the use of information and communication technologies in the educational process.
Lessons using computer technology make them more interesting, thoughtful, mobile. Almost any material is used, there is no need to prepare a lot of encyclopedias, reproductions, audio accompaniment for the lesson - all this is already prepared in advance and is contained on a small CD or flash card Lessons using ICT are especially relevant in elementary school. Pupils in grades 1-4 have visual-figurative thinking, so it is very important to build their education, using as much high-quality illustrative material as possible, involving not only vision, but also hearing, emotions, and imagination in the process of perceiving the new. Here, by the way, we have the brightness and entertainment of computer slides, animations.
The organization of the educational process in elementary school, first of all, should contribute to the activation of the cognitive sphere of students, the successful assimilation of educational material and contribute to the mental development of the child. Therefore, ICT should perform a certain educational function, help the child understand the flow of information, perceive it, remember it, and, in no case, undermine health. ICT should act as an auxiliary element of the educational process, and not the main one. Given the psychological characteristics of a younger student, work using ICT should be clearly thought out and dosed. Thus, the use of ITC in the classroom should be sparing. When planning a lesson (work) in elementary school, the teacher must carefully consider the purpose, place and method of using ICT. Therefore, the teacher needs to master modern methods and new educational technologies in order to communicate in the same language with the child.
Chapter II
2.1 Classification of active methods of teaching mathematics in primary school on various grounds
According to the nature of cognitive activity:
explanatory and illustrative (story, lecture, conversation, demonstration, etc.);
reproductive (problem solving, repetition of experiments, etc.);
problematic (problematic tasks, cognitive tasks, etc.);
partial search - heuristic;
research.
By activity components:
organizational and effective - methods of organization and implementation of educational and cognitive activities;
stimulating - methods of stimulation and motivation of educational and cognitive activity;
control and evaluation - methods of control and self-control of the effectiveness of educational and cognitive activity.
For didactic purposes:
methods of studying new knowledge;
methods of consolidating knowledge;
control methods.
By way of presentation of educational material:
monologic - information-reporting (story, lecture, explanation);
dialogic (problematic presentation, conversation, dispute).
According to the sources of knowledge transfer:
verbal (story, lecture, conversation, briefing, discussion);
visual (demonstration, illustration, diagram, display of material, graph);
practical (exercise, laboratory work, workshop).
According to the personality structure:
consciousness (story, conversation, instruction, illustration, etc.);
behavior (exercise, training, etc.);
feelings - stimulation (approval, praise, censure, control, etc.).
The choice of teaching methods is a creative matter, but it is based on knowledge of learning theory. Teaching methods cannot be divided, universalized or considered in isolation. In addition, the same teaching method may or may not be effective depending on the conditions of its application. The new content of education gives rise to new methods in teaching mathematics. An integrated approach is needed in the application of teaching methods, their flexibility and dynamism.
The main methods of mathematical research are: observation and experience; comparison; analysis and synthesis; generalization and specialization; abstraction and specification.
Modern methods of teaching mathematics: problematic (promising), laboratory, programmed learning, heuristic, building mathematical models, axiomatic, etc.
Consider the classification of teaching methods:
Information-developing methods are divided into two classes:
Transfer of information in finished form (lecture, explanation, demonstration of educational films and videos, listening to tape recordings, etc.);
Independent acquisition of knowledge (independent work with a book, with a training program, with information databases - the use of information technology).
Problem-search methods: problematic presentation of educational material (heuristic conversation), educational discussion, laboratory search work (preceding the study of the material), organization of collective mental activity in work in small groups, organizational and activity game, research work.
Reproductive methods: retelling of educational material, performing exercises according to the model, laboratory work according to instructions, exercises on simulators.
Creative and reproductive methods: composition, variational exercises, analysis of production situations, business games and other types of imitation of professional activities.
An integral part of teaching methods are the methods of educational activity of the teacher and students. Methodological techniques - actions, methods of work aimed at solving a specific problem. Behind the methods of educational work are hidden methods of mental activity (analysis and synthesis, comparison and generalization, proof, abstraction, concretization, identification of the essential, formulation of conclusions, concepts, methods of imagination and memorization).
2.2 Heuristic method of teaching mathematics
One of the main methods that allows students to be creative in the process of teaching mathematics is the heuristic method. Roughly speaking, this method consists in the fact that the teacher poses a certain educational problem to the class, and then, through successively set tasks, "leads" students to independently discover this or that mathematical fact. Students gradually, step by step, overcome difficulties in solving the problem and "discover" its solution themselves.
It is known that in the process of studying mathematics, students often face various difficulties. However, in heuristically designed learning, these difficulties often become a kind of incentive for learning. So, for example, if schoolchildren reveal an insufficient stock of knowledge to solve a problem or prove a theorem, then they themselves seek to fill this gap by independently “discovering” this or that property and thereby immediately discovering the usefulness of studying it. In this case, the role of the teacher is reduced to organizing and directing the work of the student, so that the difficulties that the student overcomes are within his power. Often the heuristic method appears in the practice of teaching in the form of the so-called heuristic conversation. The experience of many teachers who widely use the heuristic method has shown that it affects the attitude of students to learning activities. Having acquired a "taste" for heuristics, students begin to regard work on "ready-made instructions" as uninteresting and boring work. The most significant moments of their educational activity in the classroom and at home are independent "discoveries" of one or another way of solving a problem. There is a clear increase in students' interest in those types of work in which heuristic methods and techniques are used.
Modern experimental studies carried out in Soviet and foreign schools testify to the usefulness of the wide use of the heuristic method in the study of mathematics by secondary school students, starting from the primary school age. Naturally, in this case, only those learning problems can be presented to students that can be understood and resolved by students at this stage of learning.
Unfortunately, the frequent use of the heuristic method in the process of teaching the posed educational problems requires much more study time than the study of the same issue by the method of giving the teacher a ready solution (proof, result). Therefore, the teacher cannot use the heuristic method of teaching in every lesson. In addition, long-term use of only one (even a very effective method) is contraindicated in training. However, it should be noted that "the time spent on fundamental issues worked out with the personal participation of students is not wasted time: new knowledge is acquired almost effortlessly thanks to the deep thinking experience previously gained." Heuristic activity or heuristic processes, although they include mental operations as an important component, at the same time have some specifics. That is why heuristic activity should be considered as a kind of human thinking that creates a new system of actions or reveals previously unknown patterns of objects surrounding a person (or objects of the science being studied).
The beginning of the application of the heuristic method as a method of teaching - mathematics can be found in the book of the famous French teacher - mathematician Lezan "Development of mathematical initiative". In this book, the heuristic method does not yet have a modern name and appears in the form of advice to the teacher. Here are some of them:
The basic principle of teaching is "keep the appearance of the game, respect the freedom of the child, maintaining the illusion (if any) of his own discovery of the truth"; "to avoid in the initial upbringing of the child the dangerous temptation of abusing the exercises of memory," for this kills his innate qualities; teach based on interest in what is being studied.
Well-known methodologist-mathematician V.M. Bradis defines the heuristic method as follows: "A heuristic method is called such a teaching method when the leader does not inform students of ready-made information to be learned, but leads students to independently rediscover the relevant proposals and rules"
But the essence of these definitions is the same - an independent, planned only in general terms, search for a solution to the problem posed.
The role of heuristic activity in science and in the practice of teaching mathematics is covered in detail in the books of the American mathematician D. Poya. The purpose of heuristics is to investigate the rules and methods that lead to discoveries and inventions. Interestingly, the main method by which one can study the structure of the creative thought process is, in his opinion, the study of personal experience in solving problems and observing how others solve problems. The author is trying to derive some rules, following which one can come to discoveries, without analyzing the mental activity in relation to which these rules are proposed. "The first rule is to have the ability, and along with them good luck. The second rule is to hold fast and not retreat until a happy idea appears." The problem solving scheme given at the end of the book is interesting. The diagram indicates the sequence in which actions must be performed in order to succeed. It includes four stages:
Understanding the problem statement.
Drawing up a solution plan.
Implementation of the plan.
Looking back (studying the solution obtained).
During these steps, the problem solver must answer the following questions: What is unknown? What is given? What is the condition? Have I encountered this problem before, at least in a slightly different form? Is there any related task to this? Can't you use it?
From the point of view of applying the heuristic method in school, the book of the American teacher W. Sawyer "Prelude to Mathematics" is very interesting.
“For all mathematicians,” writes Sawyer, “the audacity of the mind is characteristic. The mathematician does not like to be told about something, he himself wants to get to everything”
This "impudence of the mind", according to Sawyer, is especially pronounced in children.
2.3 Special methods of teaching mathematics
These are the basic methods of cognition adapted for teaching, used in mathematics itself, methods of studying reality that are characteristic of mathematics.
PROBLEM LEARNING Problem-based learning is a didactic system based on the laws of creative assimilation of knowledge and methods of activity, including a combination of teaching and learning techniques and methods, which are characterized by the main features of scientific research.
The problematic method of teaching is learning that proceeds in the form of removal (resolution) of problem situations consistently created for educational purposes.
A problematic situation is a conscious difficulty generated by a discrepancy between the available knowledge and the knowledge that is necessary to solve the proposed problem.
A task that creates a problem situation is called a problem, or a problem task.
The problem should be accessible to the understanding of students, and its formulation should arouse the interest and desire of students to solve it.
It is necessary to distinguish between a problem task and a problem. The problem is broader, it breaks down into a sequential or branched set of problematic tasks. A problem task can be considered as the simplest, particular case of a problem consisting of one task. Problem-based learning is focused on the formation and development of students' ability to creative activity and the need for it. It is advisable to start problem-based learning with problematic tasks, thereby preparing the ground for setting learning objectives.
PROGRAMMED LEARNING
Programmed learning is such learning when the solution of a problem is presented in the form of a strict sequence of elementary operations; in training programs, the material being studied is presented in the form of a strict sequence of frames. In the era of computerization, programmed learning is carried out with the help of training programs that determine not only the content, but also the learning process. There are two different systems for programming educational material - linear and branched.
The advantages of programmed learning include: the dosage of educational material, which is assimilated accurately, which leads to high learning outcomes; individual assimilation; constant monitoring of assimilation; the possibility of using technical automated learning devices.
Significant disadvantages of using this method: not every educational material lends itself to programmed processing; the method limits the mental development of students to reproductive operations; when using it, there is a lack of communication between the teacher and students; there is no emotional-sensory component of learning.
2.4 Interactive methods of teaching mathematics and their benefits
The learning process is inextricably linked with such a concept as teaching methods. Methodology is not what books we use, but how our training is organized. In other words, teaching methodology is a form of interaction between students and teachers in the learning process. Within the framework of the current conditions of learning, the learning process is seen as a process of interaction between the teacher and students, the purpose of which is to familiarize the latter with certain knowledge, skills, abilities and values. Generally speaking, from the first days of the existence of education, as such, to the present day, only three forms of interaction between the teacher and students have developed, established and become widespread. Methodological approaches to learning can be divided into three groups:
.passive methods.
2.active methods.
.interactive methods.
A passive methodological approach is a form of interaction between students and a teacher, in which the teacher is the main active figure in the lesson, and students act as passive listeners. Feedback in passive lessons is carried out through surveys, self-study, tests, tests, etc. The passive method is considered the most inefficient in terms of students learning the educational material, but its advantages are the relatively labor-intensive preparation of the lesson and the ability to present a relatively large amount of educational material in a limited time frame. Given these advantages, many teachers prefer it to other methods. Indeed, in some cases this approach works well in the hands of a skilled and experienced teacher, especially if the students already have clear goals for a thorough study of the subject.
An active methodological approach is a form of interaction between students and the teacher, in which the teacher and students interact with each other during the lesson and the students are no longer passive listeners, but active participants in the lesson. If in a passive lesson the teacher was the main acting figure, then here the teacher and students are on an equal footing. If passive lessons suggested an authoritarian style of learning, then active lessons suggest a democratic style. Active and interactive methodological approaches have much in common. In general, the interactive method can be seen as the most modern form of active methods. Just unlike active methods, interactive ones are focused on a wider interaction of students not only with the teacher, but also with each other and on the dominance of student activity in the learning process.
Interactive ("Inter" is mutual, "act" is to act) - means to interact or is in the mode of conversation, dialogue with someone. In other words, interactive teaching methods are a special form of organizing cognitive and communicative activities in which students are involved in the process of cognition, have the opportunity to hire and reflect on what they know and think. The place of the teacher in interactive lessons is often reduced to the direction of students' activities to achieve the goals of the lesson. He also develops a lesson plan (as a rule, this is a set of interactive exercises and tasks in the course of which the student studies the material).
Thus, the main components of interactive lessons are interactive exercises and tasks that are performed by students.
The fundamental difference between interactive exercises and tasks is that in the course of their implementation, not only and not so much the already studied material is consolidated, but new material is studied. And then the interactive exercises and tasks are designed for the so-called interactive approaches. In modern pedagogy, a rich arsenal of interactive approaches has been accumulated, among which the following can be distinguished:
Creative tasks;
Work in small groups;
Educational games (role-playing games, simulations, business games and educational games);
Use of public resources (invitation of a specialist, excursions);
Social projects, classroom teaching methods (social projects, competitions, radio and newspapers, films, performances, exhibitions, performances, songs and fairy tales);
Warm-ups;
Studying and consolidating new material (interactive lecture, working with visual video and audio materials, "student as a teacher", everyone teaches everyone, mosaic (openwork saw), use of questions, Socratic dialogue);
Discussion of complex and debatable issues and problems ("Take a position", "opinion scale", POPS - formula, projective techniques, "One - together - all together", "Change position", "Carousel", "Discussion in the style of a television talk show", debates);
Problem solving ("Decision Tree", "Brainstorming", "Case Analysis")
Creative tasks should be understood as such educational tasks that require students not to simply reproduce information, but to be creative, since tasks contain a greater or lesser element of uncertainty and, as a rule, have several approaches.
The creative task is the content, the basis of any interactive method. An atmosphere of openness and search is created around him. A creative task, especially a practical one, gives meaning to learning, motivates students. The choice of a creative task in itself is a creative task for the teacher, since it is required to find a task that would meet the following criteria: does not have an unambiguous and monosyllabic answer or solution; is practical and useful for students; connected with the life of students; arouses interest among students; serve the purposes of education to the maximum. If students are not accustomed to working creatively, then you should gradually introduce simple exercises first, and then more and more complex tasks.
Small group work - this is one of the most popular strategies, as it gives all students (including shy ones) the opportunity to participate in the work, practice the skills of cooperation, interpersonal communication (in particular, the ability to listen, develop a common opinion, resolve differences that arise). All this is often impossible in a large team. Small group work is an integral part of many interactive methods, such as mosaics, debates, public hearings, almost all types of simulations, etc.
At the same time, working in small groups requires a lot of time, this strategy should not be abused. Group work should be used when it is necessary to solve a problem that students cannot solve on their own. Group work should be started slowly. You can organize couples first. Pay special attention to students who have difficulty adjusting to work in a small group. When students learn to work in pairs, move on to work in a group, which consists of three students. As soon as we are convinced that this group is able to function independently, we gradually add new students.
Students spend more time presenting their point of view, are able to discuss an issue in more detail, and learn to look at an issue from different angles. In such groups, more constructive relationships are built between the participants.
Interactive learning helps the child not only learn, but also live. Thus, interactive learning is undoubtedly an interesting, creative, and promising area of our pedagogy.
Conclusion
Lessons using active learning methods are interesting not only for students, but also for teachers. But their unsystematic, ill-conceived use does not give good results. Therefore, it is very important to actively develop and implement your own game methods in the lesson in accordance with the individual characteristics of your class.
It is not necessary to apply these techniques all in one lesson.
In the classroom, quite acceptable working noise is created when discussing problems: sometimes, due to their psychological age characteristics, elementary school children cannot cope with their emotions. Therefore, it is better to introduce these methods gradually, cultivating a culture of discussion and cooperation among students.
The use of active methods strengthens the motivation for learning and develops the best sides of the student. At the same time, one should not use these methods without looking for an answer to the question: why do we use them and what consequences can there be as a result of this (both for the teacher and for the students).
Without well-designed teaching methods, it is difficult to organize the assimilation of program material. That is why it is necessary to improve those teaching methods and means that help to involve students in a cognitive search, in the labor of learning: they help teach students to actively, independently acquire knowledge, excite their thoughts and develop interest in the subject. There are many different formulas in the course of mathematics. In order for students to be able to freely operate with them when solving problems and exercises, they must know the most common of them, often encountered in practice, by heart. Thus, the task of the teacher is to create conditions for the practical application of abilities for each student, to choose such teaching methods that would allow each student to show their activity, and also to activate the student's cognitive activity in the process of teaching mathematics. The correct selection of types of educational activities, various forms and methods of work, the search for various resources to increase the motivation of students to study mathematics, the orientation of students to acquire the competencies necessary for life and
activities in a multicultural world will allow you to get the required
learning outcome.
The use of active teaching methods not only increases the effectiveness of the lesson, but also harmonizes the development of the individual, which is possible only in vigorous activity.
Thus, active teaching methods are ways to enhance the educational and cognitive activity of students, which encourage them to active mental and practical activities in the process of mastering the material, when not only the teacher is active, but the students are also active.
Summing up, I will note that each student is interesting for his uniqueness, and my task is to preserve this uniqueness, grow a self-valuable personality, develop inclinations and talents, expand the capabilities of each Self.
Literature
1.Pedagogical technologies: Textbook for students of pedagogical specialties / under the general editorship of V.S. Kukushina.
2.Series "Pedagogical education". - M.: ICC "Mart"; Rostov n / a: Publishing Center "Mart", 2004. - 336s.
.Pometun O.I., Pirozhenko L.V. Modern lesson. Interactive technologies. - K.: A.S.K., 2004. - 196 p.
.Lukyanova M.I., Kalinina N.V. Educational activity of schoolchildren: the essence and possibilities of formation.
.Innovative pedagogical technologies: Active learning: textbook. allowance for students. higher textbook institutions / A.P. Panfilov. - M.: Publishing center "Academy", 2009. - 192 p.
.Kharlamov I.F. Pedagogy. - M.: Gardariki, 1999. - 520 p.
.Modern ways of activating learning: a textbook for students. Higher textbook institutions / T.S. Panina, L.N. Vavilovva;
.Modern ways of activating learning: a textbook for students. Higher textbook institutions / ed. T.S. Panina. - 4th ed., erased. - M.: Publishing center "Academy", 2008. - 176 p.
."Active teaching methods". Electronic course.
.International Development Institute "EcoPro".
13. Educational portal "My university",
Anatolyeva E. In "The use of information and communication technologies in the classroom in elementary school" edu/cap/ru
Efimov V.F. The use of information and communication technologies in the primary education of schoolchildren. "Elementary School". №2 2009
Molokova A.V. Information technology in traditional elementary school. Primary Education No. 1 2003.
Sidorenko E.V. Methods of mathematical processing: OO "Rech" 2001 pp. 113-142.
Bespalko V.P. Programmed learning. - M.: Higher school. Big encyclopedic dictionary.
Zankov L.V. Assimilation of knowledge and development of younger schoolchildren / Zankov L.V. - 1965
Babansky Yu.K. Methods of teaching in a modern comprehensive school. M: Enlightenment, 1985.
Dzhurinsky A.N. The development of education in the modern world: textbook. allowance. M.: Enlightenment, 1987.
Tutoring
Need help learning a topic?
Our experts will advise or provide tutoring services on topics of interest to you.
Submit an application indicating the topic right now to find out about the possibility of obtaining a consultation.
