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

Teaching mathematics in primary school is very important. 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 sustainable 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 memorization by the student 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, tasks 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 working out common methods mathematical operations that 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.

Pupils primary school it is easier to find differences between objects than to determine what is common 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 the 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 homework, and not to give the student the opportunity, 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.

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" in the allotted time ( syllabus, 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, selection and ordering of it in accordance with the goals of education, 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 managed by the teacher.

Object of study of this science - the process of mathematical development and the process of formation of mathematical knowledge and ideas of a younger child school age, in which the following components can be distinguished: the purpose of learning (Why teach?), content (What to teach?) and the activity of the teacher and the child's activity (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 building a school course in mathematics, which was reflected in the methodological concepts school textbooks mathematics, and therefore required an appropriate orientation of the 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. Wenger and others). 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 into the process of perception by young children of quantity conservation gave rise to a whole series of specific mathematical tasks in various programs for younger students: using specially constructed 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 pouring water 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. appeared whole line scientific research of the process of personal development of a child of primary school age 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 call numerals in the correct order and recognize specific geometric figures. 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 educational technology) strive 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 “mathematics” itself leaves specific imprints on the perception of parents. You can often hear: “I myself suffered with mathematics at school , he has the same problems. This is hereditary with us. " Or vice versa: "I had no problems with mathematics at school, I don’t understand who he was born into!" It is widely believed that a person either has mathematical abilities, or no, 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 skeptically. scientific knowledge about the nature, character and genesis of the mathematical development of the 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, patterns, methodological theories and fixed in 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 Math in Delay Correction Groups mental development, teaching mathematics in compensation groups (visually impaired, hearing impaired, etc.), teaching mathematics to children with mental retardation, teaching schoolchildren capable of mathematics, 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, but how to move from general principles to real life problems 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 the 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 human mathematical thinking shows that many of its most important components actually coincide with the components of such a category as the general intellectual abilities of a person - this is logic, breadth and flexibility of thinking, spatial mobility, conciseness 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.

The modern requirements of society for the development of the individual dictate the need to more fully implement the idea of individualization of education, taking into account the readiness of children for school, their state of health, individual typological characteristics of students. Building an educational process taking into account the individual development of the student is important for all levels of education, but of particular the implementation of this principle is at the initial stage, when the foundation for successful learning in general is laid. Omissions at the initial stage of education are manifested by gaps in the knowledge of children, the lack of formation of general educational skills and abilities, a negative attitude towards school, which can be difficult to correct and compensate. Observations of unsuccessful schoolchildren showed that among them there are children who have learning difficulties due to mental retardation.

Difficulties in learning are characterized by cognitive passivity, increased fatigue during intellectual activity, a slow pace of the formation of knowledge, skills, poverty of the dictionary and an insufficient level of development of oral coherent speech.

Insufficiency of cognitive activity during learning is manifested in the fact that these students do not strive to effectively use the time allotted for the task, make few presumptive judgments before solving problems, need special work aimed at developing cognitive interest, stimulating cognitive activity, and activating cognitive activity. .

That's why great importance acquires a deep disclosure of the essence of the principle of activity in learning, taking into account the individual, psychophysiological characteristics of younger students with learning difficulties and determining ways to implement it in school education.

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Explanatory note

Therefore, a deep disclosure of the essence of the principle of activity in learning, taking into account the individual, psychophysiological characteristics of younger students with learning difficulties and determining ways to implement it in school education, is of great importance.

Pedagogical science has accumulated quite a lot of experience on the problem of activating learning.

In the 60s of the last century in our country, independence and activity were proclaimed the leading didactic principle. Work on the intensification of learning has led to the need to find ways to intensify the educational and cognitive activity of students, as well as methods to stimulate their learning. In the School Law of 1958, the development of cognitive activity and independence of students was considered as the main task of restructuring secondary school.

The study of cognitive activity was carried out by scientists-teachers Z.A. Abasov, B.I. Korotyaev, N.A. Tomin and others, who revealed the content and structure of this concept.

B.P. Esipov, O.A. Nilson investigated issues related to the problem of activating learning, considering independent work as one of the most effective means of activating cognitive activity.

The development of ways to activate and develop the cognitive activity of students was carried out by modern scientists and methodologists: V.V. Davydov, A.V. Zankov, D.B. Elkonin and others.

Relevance The identified problem determined the choice of the topic: "Active methods of teaching mathematics as a means of stimulating the cognitive activity of younger students with learning difficulties."

Target - identify, theoretically substantiate and experimentally test the effectiveness of the use of active methods of teaching younger students with learning difficulties in mathematics lessons.

An object research - the process of teaching younger students with learning difficulties in elementary school.

Item research - active teaching methods as a means of stimulating the cognitive activity of younger students with learning difficulties.

Hypothesis research: the process of teaching younger students with learning difficulties will be more successful if:

in the lessons of mathematics, active methods of teaching a younger student with learning difficulties will be used;

active teaching methods will act as a means of stimulating the cognitive activity of younger students with learning difficulties.

Tasks :

To identify active teaching methods in mathematics lessons that stimulate the cognitive activity of younger students with learning difficulties.

Use a variety of forms and methods of work to stimulate the cognitive activity of younger students with learning difficulties.

Determine, substantiate and test the effectiveness of the use of active teaching methods for younger students with learning difficulties in mathematics lessons.

The practical significance of the work lies in the definition of active teaching methods that stimulate the cognitive activity of younger students with learning difficulties in mathematics lessons.

Cognitive activity is a qualitative characteristic of the effectiveness of teaching younger students.

Cognitive activity is a socially significant personality trait and is formed in schoolchildren in learning activities. The problem of developing the cognitive activity of younger schoolchildren, as studies show, has been in the center of attention of teachers since ancient times. Pedagogical reality proves every day that the learning process is more effective if the student is cognitively active. This phenomenon It is fixed in pedagogical theory as the principle of "activism and independence of students in learning". The means of implementing the leading pedagogical principle are determined depending on the content of the concept of "cognitive activity". In the content of the concept of "cognitive activity", a number of scientists consider cognitive activity as a natural desire of schoolchildren for knowledge.

Cognitive activity reflects a certain interest of younger students in acquiring new knowledge, skills, internal purposefulness and a constant need to use different methods of action to fill knowledge, expand knowledge, and broaden their horizons.

Cognitive interest is a form of manifestation of needs, expressed in the desire to learn.

Interest depends on:

The level and quality of acquired knowledge, skills, the formation of ways of mental activity;

Student-teacher relationship.

The most important components of teaching as an activity are its content and form.

Features of the formation of mathematical knowledge, abilities, skills in younger students with learning difficulties

One of the most important conditions for the effectiveness of the educational process is the prevention and overcoming of the difficulties that younger students experience in their studies.

Among the students of general education schools there are a significant number of children with insufficient mathematical training. Already by the time they enter school, students have different levels of school maturity due to individual characteristics of psychophysical development. Insufficient formation of readiness of some children for schooling often aggravated by health and other adverse factors.

Difficulties in teaching mathematics cannot but be affected by such characteristics of students as reduced cognitive activity, fluctuations in attention and working capacity, insufficient development of basic mental operations (analysis, synthesis, comparison, generalization, abstraction), and some underdevelopment of speech. Reduced activity of perception is expressed in the fact that children do not always recognize familiar geometric shapes if they are presented in an unusual perspective, upside down. For the same reason, some students cannot find numerical data in the text of the problem if they are written in words, highlight the question of the problem if it is not at the end, but in the middle or at the beginning. The imperfection of visual perception and motor skills of younger students causes increased difficulties in teaching them to write numbers: children master this skill for much longer, often mix numbers, write them in a mirror image, and poorly orient themselves in the cells of a notebook. Flaws speech development children, in particular poverty vocabulary, affect when solving problems: students do not always adequately understand some words and expressions contained in the text, which leads to an incorrect decision. When independently compiling tasks, they come up with template texts containing the same type of situations and life actions, repeating the same questions and numerical data.

All these features of children with some developmental delay, together with the insufficiency of their initial mathematical knowledge and ideas, create increased difficulties in mastering them. school knowledge mathematics. It is possible to achieve successful mastery of program material by students provided that special corrective techniques are used in teaching, differentiated approach to children, taking into account the peculiarities of their mental development.

Methods and means of stimulating the cognitive activity of younger students

Teaching methods - a system of consistent, interrelated actions of the teacher and students, ensuring the assimilation of the content of education, the development of mental strength and abilities of students, their mastery of the means of self-education and self-learning. Teaching methods indicate the purpose of learning, the method of assimilation and the nature of the interaction of learning subjects.

Facilities - material objects and objects of spiritual culture, intended for the organization and implementation of the pedagogical process and performing the functions of the development of students; substantive support of the pedagogical process, as well as a variety of activities in which pupils are included: work, play, teaching, communication, knowledge.

Teaching aids (TUT)- devices and devices that serve to improve the pedagogical process, increase the efficiency and quality of education by demonstrating audiovisual means.

The effectiveness of mastering any type of activity largely depends on the child's motivation for this type of activity. The activity proceeds more efficiently and gives better results if the student has strong, vivid and deep motives that cause a desire to act actively, overcome inevitable difficulties, persistently moving towards the intended goal.

Learning activity is more successful if students have a positive attitude towards learning, have a cognitive interest and need for cognitive activity, and also if they have a sense of responsibility and obligation.

Incentive methods.

Creating Success Situations in Learningis the creation of a chain of situations in which the student achieves in learning good results, which leads to a feeling of self-confidence and ease of learning process.This method is one of the most effective means of stimulating interest in learning.

It is known that without experiencing the joy of success, it is impossible to truly count on further progress in overcoming learning difficulties. One way to create a situation of success is toselection for students of not one, but a small number of tasksincreasing complexity. The first task is chosen to be easy so that students who need stimulation can solve it and feel knowledgeable and experienced. What follows are large and complex exercises. For example, you can use special dual tasks: the first one is available to the student and prepares the basis for solving the next, more complex task.

Another technique that contributes to creating a situation of success isdifferentiated assistance to schoolchildren in the performance of educational tasks of the same complexity.So, low-performing schoolchildren can receive consultation cards, analogue examples, plans for the upcoming answer and other materials that allow them to cope with the task presented. Next, you can invite the student to perform an exercise similar to the first, but on their own.

Encouragement and reprimand in education.Experienced teachers often achieve success as a result of the widespread use of this particular method. To praise a child in time at the moment of success and emotional upsurge, to find words for a short reprimand when he goes beyond the boundaries of what is acceptable is a real art that allows you to manage the student's emotional state.

The circle of rewards is very diverse. In the educational process, this can be the praise of the child, a positive assessment of some of his individual qualities, the encouragement of his chosen direction of activity or the way he performs the task, setting a higher mark, etc.

The use of censures and other types of punishment is an exception in the formation of the motives of the teaching and, as a rule, is used only in forced situations.

The use of games and game forms of organizing educational activities.A valuable method of stimulating interest in learning is the method of using various games and game forms of organizing cognitive activity. It can be used ready-made, for example, Board games with cognitive content or game shells of finished educational material. Game shells can be created for one lesson, a separate discipline, or the entire educational activity over a long period of time. In total, there are three groups of games suitable for use in educational institutions.

Short games. By the word "game" we most often mean the games of this particular group. These include subject, plot-role-playing and other games used to develop interest in learning activities and solve individual specific problems. Examples of such tasks are the assimilation of a particular rule, the development of a skill, etc. So, for practicing mental counting skills in mathematics lessons, chain games are suitable, built (like the well-known game “to the cities”) on the principle of transferring the right to answer along the chain.

Game shells. These games (more likely not even games, but game forms of organizing educational activities) are longer in time. Most often they are limited to the scope of the lesson, but they can last a little longer. For example, in elementary school, such a game can cover the entire school day.

Long educational games.Games of this type are designed for various time periods and can last from several days or weeks to several years. They are oriented, according to A.S. Makarenko, to the far promising line, i.e. to a distant ideal goal, and are aimed at the formation of slowly formed mental and personal qualities of the child. A feature of this group of games are seriousness and efficiency. The games of this group are no longer like games, as we imagine them to be - with jokes and laughter, but like a responsible job. Actually, they teach responsibility - these are educational games. To form the cognitive interest of students, we used tasks in the form of "Tasks-jokes".

1. Who has a piglet, but you can’t buy anything with it? (At the piglet).

2. When a heron stands on one leg, it weighs 3 kg. How much will a heron weigh if it stands on two legs? (Weight will not change).

There were 3 glasses of cherries on the table. Kostya ate cherries from one glass. How many glasses are left? (Three).

When evaluating, for each correctly solved problem, the team received two tokens.. In didactics, the following classification of forms of educational activity is adopted, which is based on the quantitative characteristics of the group of students interacting with the teacher in this moment lesson:

general or frontal (work with the whole class);

individual (with a specific student);

group (link, brigade, pair, etc.).

The first involves the joint actions of all students in the class under the guidance of a teacher, the second - the independent work of each student individually; group - students work in groups of three to six people or in pairs. Tasks for groups can be the same or different.basic active learning methods

Problem learning- such a form in which the process of cognition of students approaches search, research activities. The success of problem-based learning is ensured by the joint efforts of the teacher and students. The main task of the teacher is not so much to convey information as to introduce students to the objective contradictions in the development of scientific knowledge and ways to resolve them. In cooperation with the teacher, students "discover" new knowledge for themselves, comprehend the theoretical features of a particular science.

The main didactic method of "turning on" the thinking of students in problem-based learning is the creation of a problem situation that has the form of a cognitive task, fixing some contradiction in its conditions and ending with a question (questions) that objectifies this contradiction. The unknown is the answer to the question that resolves the contradiction.

Case Study- one of the most effective and widespread methods of organizing active cognitive activity of students. The method of analysis of specific situations develops the ability to analyze unrefined life and production tasks. Faced with a specific situation, the student must determine whether there is a problem in it, what it consists of, determine their attitude to the situation.

role playing- game method of active learning, characterized by the following main features:

O the presence of tasks and problems and the distribution of roles between the participants in their solution. For example, using the role-playing method, a production meeting can be simulated;

"Round table" - is an active learning method organizational forms cognitive activity of students, which allows to consolidate the previously acquired knowledge, fill in the missing information, form the ability to solve problems, strengthen positions, teach the culture of discussion. characteristic feature "round table"is a combination of a thematic discussion with a group consultation. Along with an active exchange of knowledge, students develop professional skills to express thoughts, argue their views, justify proposed solutions and defend their beliefs. At the same time, information is consolidated and independent work with additional material and identifying issues and issues for discussion.

An important condition for organizing a "round table" is that it must be really round, i.e. the process of communication, communication, took place "eye to eye". The principle of the "round table" (it is no coincidence that it was adopted at the negotiations), i.e. the location of the participants facing each other, and not in the back of the head, as in a normal lesson, generally leads to an increase in activity, an increase in the number of statements, the possibility of personal inclusion of each student in the discussion, increases the motivation of students, includes non-verbal means communication, such as facial expressions, gestures, emotional manifestations.

The teacher is also located in general circle, as an equal member of the group, which creates a less formal environment compared to the generally accepted one, where he sits separately from the students, they face him. IN classic version the participants in the discussion address their statements mainly to him, and not to each other. And if the teacher sits among the children, the group members' addresses to each other become more frequent and less constrained, this also contributes to the formation of a favorable environment for discussion and the development of mutual understanding between teachers and students. The main part of the "round table" on any topic is the discussion. Discussion (from Latin discussio - research, consideration) is a comprehensive discussion controversial issue in a public meeting, in a private conversation, dispute. In other words, the discussion consists in a collective discussion of any issue, problem or comparison of information, ideas, opinions, proposals. The goals of the discussion can be very diverse: education, training, diagnostics, transformation, changing attitudes, stimulating creativity, etc.

One of the effective ways to activate the educational activities of younger students areunconventional lessons.

In my work I often use:

Lesson - a fairy tale
Lesson-KVN
Lesson Journey
quiz lesson
Relay lesson
Competition lesson

The use of multimedia technologies in mathematics lessons

In my pedagogical practice, along with traditional ones, I use information technologies of education in order to create conditions for the choice of an individual educational trajectory for each student, I strive to inspire students to satisfy their cognitive interest, therefore, I consider it my main task to create conditions for the formation of motivation among students, the development of their abilities , improve learning efficiency.

When conducting mathematics lessons, I use multimedia presentations. At such lessons, the principles of accessibility and visibility are more clearly implemented. Lessons are effective in their aesthetic appeal. Presentation lessons provide a large amount of information and tasks in a short period. You can always return to the previous slide (normal blackboard cannot accommodate the volume that can be put on the slide).

When studying new topic I am giving a lesson-lecture using a multimedia presentation. This allows students to focus on the significant points of the information presented. The combination of oral lecture material with a slide show allows you to focus visual attention on especially significant moments of educational work.

Multi-slide presentations are effective in any lesson due to significant time savings, the ability to demonstrate a large amount of information, visibility and aesthetics. Such lessons arouse students' cognitive interest in the subject, which contributes to a deeper and more solid mastery of the material being studied, and increases the creative abilities of students.

I also use a presentation to systematically check that all students in the class have done their homework correctly. When checking homework, it usually takes a lot of time to reproduce the drawings on the board, explaining those fragments that caused difficulties.

I use a presentation for oral exercises. Work on the finished drawing contributes to the development of constructive abilities, the development of speech culture skills, logic and reasoning sequence, teaches the preparation of oral plans for solving problems of varying complexity. It is especially good to apply this in high school in geometry lessons. It is possible to offer students samples of the design of solutions, write down the conditions of the problem, repeat the demonstration of some fragments of constructions, organize an oral solution of tasks that are complex in content and formulation.

Work experience shows that the use of computer technology in teaching mathematics makes it possible to differentiate learning activities in the classroom, activates the cognitive interest of students, develops their creative abilities, stimulates mental activity, encourages research activities.

The use of multimedia technologies is one of the promising directions informatization of the educational process and is one of the actual problems modern techniques teaching mathematics. I consider the use of information technologies necessary and motivate this by the fact that they contribute to:

Improving practical skills and abilities;

Allows you to effectively organize independent work and individualize the learning process;

Increase interest in lessons;

Activate the cognitive activity of students;

Update the lesson.

Conclusions:

I note that the systematic use of active methods of teaching younger students with learning difficulties in mathematics lessons forms the level of cognitive activity, and this contributes to an increase in the effectiveness of the learning process in mathematics lessons.

All this allows us to confirm the correctness of the chosen path in the use of active methods in the classroom in elementary school.

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

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

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

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

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

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

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

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

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

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

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

The idea of L.S. Vygotsky that training should be carried out in the zone of proximal development of students, and its effectiveness is determined by what zone (large or small) it prepares, is well known to everyone. At the theoretical (conceptual) level, it is shared almost all over the world. The problem lies in its practical implementation: how to determine (measure) this zone and what should be the technology of education, so that the process of learning the scientific foundations and mastering (“appropriation”) of human culture takes place precisely in it, providing the maximum developmental effect?

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

between the need for a high level of mathematical development for modern man and the discrepancy to this task of the integral system of the process of teaching mathematics in elementary school;

between the discreteness of the education system and the need to create a holistic picture of the world in the mind of the child;

between the basic postulate of the theory of developmental education, which considers the essence of the child's personality as a "self-developing system" that develops in the educational process, amenable to controlled processes of formation and development, through the use of developmental education technologies and the lack of such technologies in primary school mathematical education;

between the need for teachers of mathematics to use an activity approach to teaching and their practical unpreparedness for such teaching, for thoughtful joint activity of a teacher and a student in the "zone of proximal development".

Summarizing the above, it can be argued that the problem of the mathematical development of younger schoolchildren is undoubtedly relevant and requires, for its solution, the expansion of general approaches, going beyond "pure didactics", taking into account modern achievements not only in the field of psychology and physiology, creating a general concept of formation and development of mathematical thinking of students on a broader theoretical basis than is currently accepted.

The purpose of our study was to build, on the basis of the dominant individual typological features of thinking, the concept of mathematical development, which makes it possible to ensure the continuity of mathematical education at the preschool, primary school level and in grades V-VI of the main school, its continuity and improving the quality of mathematical training of a child of primary school age. , as well as in the development and testing of its applied aspect in the form of educational technology (methods, tools, forms).

The main provisions of the concept of mathematical development of a child of primary school age are formulated by us as follows.

1. As a starting point, the concept of educational and mathematical activity is singled out, which should be characterized by a set of interrelated main components and qualities of the child's mathematical thinking and his abilities for mathematical knowledge of reality. In the process of all educational and mathematical activities at school, such mental actions as analysis, planning, reflection should be formed, which provide mastery of generalized methods for solving mathematical problems.

LECTURE 1.

Methodology primary education mathematics as academic 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 discover new truths. 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 analogies 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 scheme:

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.

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