Developmental training

Recommended by the UMO in the specialties of pedagogical education as a teaching aid for students of higher educational institutions studying in the specialty 031200 (050708) - pedagogy and methods of primary education.

1 NISEYSKOV Pedagogical schools * 1 Smolensk "Association XXI century"

Istomina N.B.

I89 Methods of teaching mathematics in primary school:

Developmental training. - Smolensk: Publishing house "Association XXI century", 2005. - 2 7 2 p.

The purpose of the textbook is the formation of the future teacher's methodological knowledge, skills and experience of creative activity for the implementation in practice of the ideas of developing mathematics teaching for junior schoolchildren.

The manual will also be useful for primary school teachers.

ISBN 5-89308-193-5 © Istomina N.V., 2005 ISBN 5-89308-193-5 © Association XXI century, 2005

INTRODUCTION

In accordance with the state standard of primary general education, the study of mathematics at the initial stage is aimed at achieving the following goals:

The development of figurative and logical thinking, imagination, the formation of ~ subject skills and abilities necessary for the successful solution of educational and ~ actual problems, continuation of education;

Mastering the basics of mathematical knowledge, the formation of initial ~ ideas about mathematics;

Fostering interest in mathematics, striving to use mathematical knowledge in everyday life 1.

The task of the practical implementation of these goals is assigned to the teacher and in many ways depends on his methodological training, which should integrate: social (mathematical), psychological, pedagogical and methodological knowledge, skills and abilities.

This manual is intended for students of the daytime department of the faculty of primary classes and for students of pedagogical schools and colleges, since, "starting to study the course" Methods of teaching mathematics ", they are in equal conditions in terms of experience in methodological activities and equally should be ready to the solution of those tasks that arise in the process of practical work.

The first chapter is intended to form the future teacher's ideas about the methodology of teaching mathematics as a pedagogical science (§1), about the development of elementary mathematics education (§2), about the methodological activity of the teacher in the process of teaching mathematics to junior schoolchildren (§3).

The second chapter provides a methodological interpretation of the main components of the concept of "learning activity" and the ways of its organization.

Possible approaches to the development of thinking in junior schoolchildren are reflected in Chapter 3. It gives a brief description of such methods of mental activity as analysis and synthesis, comparison, classification, analogy, generalization ^).

These techniques in the process of assimilating knowledge, skills and abilities perform various functions. They can be considered:

1) as ways of organizing the educational activities of schoolchildren;

2) as methods of cognition that become the property of the child, characterizing his intellectual potential and ability to assimilate knowledge, skills and abilities;

"Federal component of the state standard of general education. - M., 2004 - S.

3) as ways of including various mental functions in the process of cognition:

emotions, will, feelings, attention, memory. As a result, the child's intellectual activity enters into various relationships with other aspects of his personality, primarily with focus, motivation, interests, level of aspirations, i.e. characterized by increasing personality activity.

The same chapter describes various ways of substantiating the truth of judgments by younger students (inductive and deductive reasoning, experiment, calculations, measurements (§2), as well as the relationship between logical and algorithmic thinking (§3).

In the process of studying the methodological course, the future teacher needs to master the ability to navigate in the subject content of methodological activity, that is, to learn how to answer the questions:

What mathematical concepts, laws, properties and methods of action are reflected in the initial course in mathematics?

In what form are they offered to younger students?

In what order are they studied?

In what order can they be studied?

The formation of this skill is carried out in the process of studying Chapter 4 "Basic concepts of the elementary course in mathematics and the peculiarities of their assimilation by younger students." Its content includes theoretical information about various concepts of the elementary mathematics course; types of educational tasks, in the process of completing which children not only acquire knowledge, abilities and skills, but also advance in their development; methodological recommendations for the organization of educational activities of students.

Establishing a correspondence between subject, verbal, schematic and symbolic models is considered as the main way for students to assimilate mathematical concepts. It allows you to take into account the individual characteristics of the child, his life experience, objective-effective and visual thinking and gradually introduce him into the world of mathematical concepts, terms, symbols, i.e. into the world of mathematical knowledge, thereby contributing to the development of both empirical and theoretical thinking.

Chapter 5 is devoted to the methodology of organizing the computational activity of primary schoolchildren in the developmental course of elementary mathematics.

Chapter 6 gives a brief description of various methodological approaches to teaching primary schoolchildren to solve word problems and reveals in detail the methodology for the formation of generalized problem-solving skills, which is based on various methodological techniques: the choice of a scheme, expressions, conditions, reformulation of the problem question, posing questions to a given condition and etc.

Chapter 7 describes the different approaches to building a math lesson in primary grades and provides guidance for planning and analyzing developmental lessons.

to include a small schoolchild in active cognitive activity aimed at mastering the system of mathematical concepts and general methods of action;

Create methodological conditions for the formation of educational activities, for the development of empirical and theoretical thinking, emotions and feelings of the child;

To form the ability to communicate in the process of discussing ways to solve personal problems, to justify their actions and critically evaluate them;

To improve the quality of assimilation of mathematical knowledge, abilities and skills;

Ensure continuity between primary and secondary education by preparing primary school students for active mental activity;

To develop the creative methodological potential of the primary school teacher, stimulating him to independently compile educational tasks, the choice of means and forms of organizing the activities of schoolchildren.

The elementary school works according to the textbooks of N.B. Istomina since 1993. They are included in the Federal List of textbooks and are labeled "Recommended by the Ministry of General and Professional Education of the Russian Federation."

In 1999, Doctor of Pedagogical Sciences, Professor Istomina Natalia Orisovna was awarded the Prize of the Government of the Russian Federation for the creation of an educational and methodological set in mathematics for four-year elementary school.

METHODS OF TEACHING MATHEMATICS

IN ELEMENTARY CLASSES AS A PEDAGOGICAL SCIENCE

AND AS A LEARNING SUBJECT

§ 1. THE SCIENCE OF TEACHING MATHEMATICS

Teaching is a purposeful, specially organized and teacher-controlled activity of students, during which they acquire knowledge, develop and educate.

In teaching, as in any process, certain patterns appear that express the existing connections between pedagogical phenomena, while a change in some phenomena entails a change in others. For example, learning objectives that reflect the needs of society have an impact on the content and the ways in which students' activities are organized, aimed at assimilating it. Learning outcomes depend on the nature of the activity in which the student is involved at a particular stage of development. If priority is given, for example, to reproductive activity, then the personal potential of schoolchildren, their creative attitude to learning, and independence of thought remain unclaimed.

It has been experimentally proven that the creativity of children is directly dependent on the creativity of teachers who involve students in the process of jointly solving various educational problems.

The teaching strategy is determined by didactic principles. But they are of a general nature and do not take into account the specifics of the problems that arise in teaching mathematics. Taken in an abstract form, in isolation from the mathematical essence, they cannot directly serve as the theoretical foundations of the methodology, since it remains unclear how, relying on them, to build training in specific content.

For example, in didactics, a theory of problem-based learning has been developed: the essence of its basic concepts has been determined, the necessity and effectiveness of their application in the educational process has been substantiated, a number of methods for organizing and managing students' independent activities have been revealed, and the most important didactic conditions for the implementation of this type of education have been identified. However, the solution to the question of the possibility of creating problem situations when teaching mathematics to primary schoolchildren remains with the methodology. And until it is presented at the methodological level, the theory of problem learning, which has been developed in didactics, will not become the property of the practice of primary school teachers.

The task of the methodology of teaching mathematics is not only the development of problem situations, but also general approaches to their use, which would take into account the specifics of the mathematical content and the peculiarities of its assimilation by students. So, for example, one of the means of creating problem situations at a certain stage of teaching mathematics is non-standard problems. They present a problem for the student, the solution to which he must find on his own, creatively applying his knowledge. But at the same time, such problematic situations may be inaccessible to the majority of primary schoolchildren, since their solution requires a high level of abstraction and generalization.

Considering this fact, in the initial course of mathematics, to create problem situations, it is advisable to use practical tasks, in solving which children can rely on their life experience and on practical actions.

So, starting to study the topic "Length of objects" (1st grade), the teacher offers the class two strips (red and blue) and asks: "How can you determine which one is longer?" For a younger student, this is a problematic situation, the way of solving which he is asked to find on his own.

Accessibility in this case is ensured by the fact that when finding a way to compare the lengths of the strips, he can rely only on his life experience and on practical actions. This problematic situation can be complicated by asking the question: "Is it possible to compare the lengths of these strips using the third one?" The answer to it is associated with finding a new way of action, which underlies the measurement of quantities.

Similarly, it is possible to illustrate other provisions of didactics, which become the theoretical foundations of the methods of teaching mathematics only after their revision in connection with the specific content of the studied mathematical material.

For example, the principle of accessibility of teaching in didactics is understood as a requirement to present students with material of such complexity that they could overcome on their own or with the help of a teacher. But how to do this, for example, when studying the division of a multi-digit number by a single-digit number? The answer can only be given by the methodology of teaching mathematics. Guided by the algorithm of written division and the principle of constructing a decimal number system, as well as taking into account the psychological characteristics of the perception and thinking of younger students, the methodology of primary teaching in mathematics formulates general provisions that a teacher can be guided by when developing children's writing division skills. For example: acquaintance of students with the algorithm of written division should be preceded by exercises that will prepare them for the perception and understanding of the operations included in this algorithm. This is determining the number of tens, hundreds, thousands in a multi-digit number, and performing division with a remainder, and checking division by multiplication, etc. The guidance of this methodological provision ensures the availability of a new method of action and gives scope for greater independence of students in its assimilation.

When studying the algorithm of written division, the following should be borne in mind: when recording a written division, it is necessary to comment in detail (expanded) on the operations performed, since this will allow the teacher not only to control the correctness of the final result, but also the process of its calculation, and thereby correct it in a timely manner. student activity on the use of the algorithm.

The given methodical recommendation takes into account one of the psychological laws, which is that external activity does not always coincide with internal activity. This means that outwardly, children can perform the right actions, but in their minds at this time, reason is wrong. Thus, the recommendation to use the commenting technique is generalized (in this case, in relation to the study of a specific issue), theoretically justified (psychological position), and can be applied when studying other content issues. Its expediency is confirmed by the practice of teaching.

It should be borne in mind that the peculiarity of using the theoretical provisions of didactics in teaching a specific subject lies in the fact that they become effective only when they interact with psychological laws, which, like didactic ones, are usually expressed in a generalized way, in isolation from the specific content.

So, the process of assimilation by children of various content, obeying general laws, has its own specifics, which should be expressed in theoretical propositions that reflect the characteristics of teaching a specific subject.

The development of a teaching theory, taking into account the specifics of the content, is a necessary condition for the successful development of a certain section of teaching methods for a specific academic discipline.

What requirements should the theoretical foundations of the methodology for teaching mathematics meet? They should: a) rely on a certain theory (psychological, pedagogical, mathematical), using it in relation to the specific content of education; b) be generalized provisions reflecting not a separate case, but general approaches to the process of teaching mathematics (in particular, in primary grades), to solving a certain set of issues in it; c) reflect the stable features of the process of teaching mathematics, that is, the patterns of this process or important facts about it; d) be confirmed in practice by experiments or the experience of teachers.

Consequently, the theoretical foundations of the methodology for teaching mathematics is a system of provisions that underlie the construction of the process of teaching mathematics, which are theoretically substantiated and characterize the general methodological approaches to its organization.

Considering the methodology of teaching mathematics in primary grades as a science, we will single out the range of problems that it is designed to solve, and define the object and subject of its research.

The whole variety of problems of private methods, including the methods of teaching mathematics in primary grades, can be formulated in the form of questions:

Why teach? That is, what is the purpose of teaching children mathematics?

What to teach? That is, what should be the content of mathematical education in accordance with the set goals?

How to teach? That is:

a) in what order to arrange the content questions so that students can consciously assimilate them, effectively moving forward in their development;

b) what methods of organizing the activities of students (methods, techniques, environments and forms of education) should be used for this;

c) how to teach children, taking into account their psychological characteristics (how in the process of learning mathematics the most complete and correct use of the laws of z: perception, memory, thinking, attention of younger students)?

The named problems make it possible to determine the methodology of teaching mathematics as a science, which, on the one hand, is addressed to a specific content, refusals to streamline it in accordance with the set learning goals, on the other, to human activity (teacher and student), to the process of mastering this holding, management which is carried out by the teacher.

The object of the study of the methods of teaching mathematics is the process of teaching mathematics, in which four main components can be distinguished: purpose, content, teacher's activity and student's activity. Listed components

2 ARE in interconnection and interdependence, that is, they form a system in which a change in one of the components causes changes in others.

The subject of research can be each of the components of this system, as well as those interconnections and relationships that exist between them.

Methodological problems are solved using methods of pedagogical research, which include: observation, conversation, questionnaires, generalization of the best practices of teachers, laboratory and natural experiments.

Various tests and psychological methods make it possible to reveal the influence of eazy ways of teaching on the assimilation of knowledge, skills and abilities, on the general development of children. All this makes it possible to establish certain patterns of the process of teaching mathematics.

Assignment 1. What concepts of teaching younger students are you familiar with? Expand the content of these concepts.

§ 2. GENERAL CHARACTERISTIC OF THE DEVELOPMENT OF THE INITIAL

MATHEMATICAL EDUCATION

At each stage of the development of primary education, methodological science answered the questions in different ways: "Why teach?", "What to teach?", "How to teach?"

Prior to 1949, the priority in primary education was practical goals. This was due to the fact that before the introduction of the general compulsory 7-year education, primary school was a closed stage. The main content of the initial course in mathematics was the study of four arithmetic operations, solving problems by the arithmetic method and acquaintance with geometric material, which was subordinated to the solution of practical problems (marking out rectangular land plots, measuring their length, width, calculating the area and perimeter of a rectangle, etc. ).

The content of the course was based on the concentric principle (5-6 concentrates). At the end of the fourth year of study, it was supposed to generalize the material studied and familiarize oneself with individual elements of the theory (connections between actions, components and results of actions, some properties of actions).

The teaching methods took into account those features of a given age that were noted by psychological science: imagery, the predominance of “mechanical” memory over semantic memory, the ease and strength of the assimilation of numerous facts by younger schoolchildren.

Based on "mechanical" memory, children were instructed to memorize 4 tables (2 multiplication tables and 2 division tables, each of which included 100 examples). This approach to teaching mathematics in the primary grades was substantiated by the data of developmental psychology, which interpreted the consideration of the real cognitive capabilities of younger schoolchildren as the need to adapt the content and teaching methods to the peculiarities of the mental development of children of a given age.

However, in the works of L. S. Vygotsky, the most prominent Russian psychologist, back in the early 30s of the 20th century, the erroneousness of this position was noted, even in relation to children who were lagging behind in mental development. He noted that learning, which focuses on the already completed development cycles, does not lead the development process, but itself lags behind him; only that learning is good that gets ahead of development.

It should be noted that the 30-40s are marked by joint research of psychologists and methodologists on the methods of teaching individual subjects. Psychologist N.A.Menchinskaya wrote about the directions of these studies:

“In order for psychology to be able to directly respond to the demands of teaching practice, it is necessary to study specific types of educational activity, and to explore various forms of this activity as a natural response to pedagogical influences” 1.

In the mainstream of this direction, the ways of assimilation by children of the concept of number and arithmetic operations, the peculiarities of mastering the process of counting and the formation of computational skills, the ability to solve textual arithmetic problems were studied.

At the same time, much attention was paid to the study of the role of analysis and synthesis, concretization, abstraction and generalizations. The results of these studies have played a certain role in the development of methodological science.

Speaking about the shortcomings of the methods of teaching mathematics, A.S. Pchelko (the author of the arithmetic textbook for primary grades) lamented that the main attention of the methodologists is focused on the teacher, on the methods and techniques he teaches children, and questions about how how students perceive the teacher's explanations, what difficulties they have in mastering a particular section of arithmetic, what is the reason for these difficulties and how they can be prevented.

In the 40-50s, methodical works appeared, built on research, experimental material (N.N. Nikitin, G.B. Polyak, M.N. Skatkin,

Menchinskaya N.A.Psychology of teaching arithmetic. - M., 1947.

A.S. Pchelko) and there is a need to revise the content of education in primary grades.

However, the changes made to the curriculum of the arithmetic course, which was introduced in 1960, did not affect its essence. They boiled down to minor amendments aimed mainly at further simplifying the course. New trends caused by research in the field of methodology and psychology were reflected only in the explanatory note of the program. It emphasized the need to teach primary school students general methods of working on a problem, the importance of forming correct generalizations in children and organizing various types of independent work.

In 1965, the book by MI Moro and NA Menchinskaya "Questions of methodology and psychology of teaching arithmetic ..." was published. A number of provisions formulated in this book remain relevant today, being the basis for the development of new methodological approaches to the assimilation of mathematical content by primary schoolchildren. Here are some of them: 1.

“In order for a junior schoolchild to be active in the learning process, it is necessary: ​​first, to provide him with ample opportunities for self-reliance in educational work; secondly, to teach him the techniques and methods of independent work; thirdly, to awaken in him the desire for independence, creating in him the appropriate motivation, that is, to make his independent creative approach to solving educational problems vital for him. "

"There is a well-known old saying:" Repetition is the mother of learning. "

Now sometimes it is opposed by another: "Application is the mother of learning." The second formulation is more consistent with the modern tasks facing our school, but it must be borne in mind that the application of knowledge does not exclude repetition, but includes it, but this means repetition is not monotonous or monotonous, but one that presupposes change as the knowledge itself, and the conditions for its use. "

“The ability to solve problems, although it is of a general nature, lends itself to development, like all others, but this requires a special system of exercises aimed at forming in schoolchildren the need for creative thinking, an interest in solving problems-problems independently, and consequently, and to the search for the most rational methods of their solution ”.

"Full consciousness of assimilation can be achieved by a student only if he does not passively perceive the new material being communicated, but actively operates with it."

"It is necessary to avoid not only extremely difficult, but also extremely easy material for the student to assimilate, when in the process of assimilation for him there are no problems or tasks that require mental effort."

Menchinskaya N.A., Moro M.I. Questions of methodology and psychology of teaching arithmetic in primary grades. - M., 1965.

The book not only notes the role of comparisons and contrasts as concepts mixed by children, but also suggests the main ways of their application in the process of teaching mathematics. This is a simultaneous opposition, when both concepts or rules are introduced in one lesson, in comparison with each other, and sequential, when one of the compared concepts is first studied, and the second is introduced on the basis of opposition to the first, only when the first has already been mastered.

The works of P.M. Erdniev made a great contribution to the development of methods of teaching mathematics. Under his leadership, an experimental study was carried out in order to substantiate the idea of ​​enlarging didactic units in the process of teaching children mathematics (UDE method).

Learning built in accordance with this idea is effective in improving the quality of students' knowledge while significantly saving time spent on studying a math course.

a) the simultaneous study of similar concepts; b) the simultaneous study of mutually opposite actions; c) transformation of math exercises; d) drawing up tasks by schoolchildren; e) deformed examples.

Among the studies that played an invaluable role in the development of the method of primary education, two should be named: one under the leadership of L.V. Zankov (1957), the other under the leadership of D. B. Elkonin and V. V. Davydov (1959). .).

And although the object of L.V. Zankov's experimental research was not individual academic subjects, but a didactic system covering all primary education, nevertheless the didactic principles developed in the laboratory (teaching at a high level of difficulty, studying program material at a fast pace; the leading role of theoretical knowledge ; students' awareness of the learning process; purposeful and systematic work on the development of all students in the class, including the weakest) could serve as an effective basis for improving the methods of teaching mathematics.

A large-scale experiment conducted under the leadership of L. V. Zankov led to a theoretical understanding of the typical properties of the methodological system of primary education. As such properties, the scientist named versatility, collisions, processuality. L. V. Zankov considered the development of a methodological system to be especially urgent.

In a study led by D. B. Elkonin and V. V. Davydov, those neoplasms were identified, the formation of which in primary school students turned out to be possible with a certain structure of the learning process. As such new formations were named: educational activity, theoretical thinking and voluntary control of behavior (reflection).

In parallel with psychological and pedagogical studies, methodological studies were carried out aimed at preparing the reform of primary education. Variants of programs were developed, experimental textbooks were created.

Methodologists M.I. Moro, A.S. Pchelko, M.A.Bantova, G.V. Beltyukova, N.V. Melentsova, E.M. P. M. Erdniev, I. K. Andronov, Yu. M. Kolyag in. Psychologists (N. A. Menchinskaya, A. A. Lyublinskaya) took an active part in the preparation of the reform of primary education.

As a result of the research, conclusions were drawn about the need to enrich the content of the elementary course in mathematics, to strengthen the role of theory in it and to include elements of algebra and geometry in the content of the course.

The modernization of the subject content of primary mathematical education was accompanied by instructions: "One of the important educational tasks associated with the study of the course of mathematics is the development of the cognitive abilities of students"; "Classes in mathematics should contribute to the upbringing of independence, initiative, creativity, work culture in children"; "Education and development in the study of mathematical material should be carried out in close connection with each other" 1.

However, the implementation of these guidelines in school practice turned out to be, perhaps, an even more difficult task than the introduction of the new content of a unified na- tional mathematics course. “The teachers received new programs and started their existence, having no idea about the new methodology,” writes Sh. A. Amonashvili.

The task of the child's development in the learning process remained unresolved in the stable mathematics course (M.I.Moro and others) - Despite its meaningful generalization in comparison with the arithmetic course and the focus on increasing the level of theoretical knowledge of primary schoolchildren, the leading method remained the display of the survey and its consolidation. The study assignments were monotonous, and the assignments requiring the activation of the mental activity of schoolchildren were classified as material of "increased difficulty" and "got" only for those who were capable of mathematics. The main task for all students was still the formation of computational skills, skills and the ability to solve certain types of problems.

Meanwhile, the search for ways of organizing the educational activities of younger schoolchildren continued both in theory and in teaching practice.

In the 70-80s, thousands of schoolchildren worked according to the system of L.V. Zankov, the experiment continued according to the system of D. B. Elkonin, V. V. Davydov, the UDE system was actively introduced into school practice, the experiment was carried out by A. M. Pyshkalo and KI Neshkov, in which the possibility of constructing an initial course in mathematics on a set-theoretical basis was tested.

Actual problems of methods of teaching mathematics in primary grades / Ed. M.I. Moro, A.M. Pyshkalo. - M., 1977.

Amonashvili Sh. A. in collection. articles "New time - new didactics": Pedagogical ideas of L. V. Zankov and school practice. - Moscow - Samara, 2000.

The beginning of the 90s is marked by the introduction of various innovations, new teaching technologies, variable copyright programs and textbooks into school practice.

In the wake of this innovative movement, “Russian primary education is acquiring a developmental character” 1.

The tasks of developing a child's interest in learning, the formation of educational independence and the skills necessary for it, associated with the awareness of the educational task, with the search for its solution, with the implementation of various mental operations (analysis, synthesis, comparison, classification, generalization), are put forward. organization of control over their actions and their assessment.

Comprehension of these directions at the methodological level is an urgent task of modern methodological science.

§ 3. TASKS OF THE METHODS OF TEACHING MATHEMATICS

AS A LEARNING SUBJECT

The main objective of the course "Methods of teaching mathematics in primary grades" in college and at the university is to prepare students for professional methodological activities aimed at educating the child's personality, developing his thinking, developing his ability and desire to learn, and gaining experience of communication and cooperation in the process of assimilating mathematical content.

A certain contribution to the solution of this problem is made by courses in mathematics, psychology, developmental psychology, didactics, etc. In the process of studying the methodological course, students learn to apply this knowledge to solve methodological problems. Consequently, the teacher's methodological activity is integrative in nature.

The complex mechanism of such integration is due to the fact that methodological knowledge, presented in the form of ideas, provisions, descriptions of recommendations, techniques, types of educational tasks, include:

Laws of teaching and upbringing processes;

Psychological characteristics of a child's development and assimilation of knowledge, skills and abilities.

The better the teacher realizes this connection, the higher the level of his methodological training, the wider his possibilities in the implementation of creative methodological activity.

Let's consider a typical situation from the practice of elementary teaching of mathematics and analyze it from the point of view of the concept of "methodological problem".

Imagine that you have offered the children a task: "Compare the numbers 6 and 8" or "Put a sign between the numbers 6 and 8, = so that you get the correct record." Suppose the student gave the wrong answer, that is, completed entry 68. What will you do? Contact another student or try to figure out the reasons for the mistake? In other words, how do you solve this methodological problem?

"Davydov V. V. The concept of humanization of Russian primary education. - Collection of articles" Primary education in Russia. "- M., 1994.

The choice of methodological actions in this case may be due to a number of psychological and pedagogical factors: the personality of the student, the level of his mathematical training, the purpose for which the given task was proposed, etc. understand the reasons for the error. But = to do it?

If the student reads it as "six less than eight", then the reason for the error is ": and, that the mathematical symbol has not been learned. Children simultaneously get acquainted with the knowledge and, therefore, they may confuse their meanings.

Having established the cause in this way, you can continue to work. But at the same time

Take into account the peculiarities of the perception of a younger student. Since it has

A visual-shaped character, the teacher uses the technique of comparing a sign with a cognitive (for a child) image, for example, with a beak, which is open to a greater number and closed to a smaller one (5 8, 8 5). Such a comparison will help the child remember mathematical symbols.

But if the student read this entry "6 8" as "six more than eight", then the error is due to another reason. How to proceed in this case?

Here the teacher cannot do without knowledge of such mathematical concepts as "quantitative number", "establishing a one-to-one correspondence" and a set-theoretic approach to determining the relationship "more" ("less"). This will allow him to choose the right way to organize the activities of students associated with the implementation of this task. Taking into account the visual-effective nature of the thinking of younger students, the teacher suggests that one student put 6 objects on the desk, and the other - 8 and think about how to arrange them in order to find out who has more objects and who has less.

Based on his life experience, the child can independently propose a method of action or find it with the help of a teacher, that is, establish a one-to-one correspondence between the elements of these subject sets.

§ §§! § till id Now imagine that a student is successfully completing a number comparison task. In this case, it is important to establish how conscious his actions are, that is, can he justify them, while expressing the necessary reasoning that is associated with the answer to the question: "Why is 6 less than 8?"

To solve this problem, the teacher will need knowledge of such mathematical concepts as "counting" and "natural series of numbers", because they are the basis of the justification that the student can give: "The number that is called when counting earlier is always less any number following it. "

To make this rationale clear to all children, it is useful to turn to a segment of the natural series and propose to emphasize the numbers 6 and 8 (1, 2, 3, 4, 5, 6, 7, 8, 9) in it, or to designate these numbers on the number ray.

Thus, the process of a student completing a fairly simple task required the teacher to solve four methodological problems and apply mathematical, psychological and methodological knowledge.

Consider another situation involving written single-digit division. For example, 8463: 7. Each of you, of course, will easily cope with this task.

But suppose that the student received in the answer not 1209, but 129, that is, he missed the quotient zero (this is a typical error). The reason for such an error can be either his inattention, or the lack of the necessary knowledge and skills.

How do you find out? Probably, by analogy with the first situation, you can already answer this question: "It is necessary for the student to speak the actions that he performed." In the methodology, this technique is called "commenting".

The use of such a technique allows the teacher to control the correctness of not only the final result, but also the process of obtaining it, and thereby adjust the activity of schoolchildren in using the algorithm.

But in order to teach children to consciously comment on the sequence of operations that are included in the algorithm of written division, the teacher himself must own the necessary mathematical concepts. Under this condition, he will be able to clearly explain the mathematical essence of the operations performed. For example, for the case 8463: 7, the appearance of a zero in the quotient is usually commented as follows: "6 by 7 is not divisible - we put zero." This formal explanation can be more reasonable if we rely on the concept of division with remainder.

Remember the definition that you considered in the course of mathematics: "To divide with the remainder a non-negative integer a by a natural number b means to find such non-negative integers q and r so that a = bq + r \ n0 r b".

Understanding that this definition is the basis of students' actions when performing division with a remainder will allow the teacher to methodically organize correctly their activities to master these methods. For example, performing division for the case of 29: 4, the students first find the largest number up to 29, which is evenly divisible by 4 (this operation requires a firm grasp of the tabular division cases): 28: 4 = 7. The remainder is found by subtracting 29-28 = 1. End result: 29: 4 = 7 (rest 1).

Let us now carry over the same reasoning to case 6: 7. The largest number up to 6 that is even divisible by 7 is 0. 0: 7 = 0. Find the remainder by subtracting 6-0 = 6. End result: 6: 7 = 0 (rest 6). So knowledge of mathematical concepts helps the teacher find reasonable ways to explain to students the actions they perform.

Mathematical knowledge is necessary for a teacher in order to properly organize the acquaintance of younger students with new concepts. For example, some teachers try to explain the cases of multiplication by 1 as follows: "The number was repeated once, so it remains." When studying the case of division by 1, they turn to a specific example: “Imagine that a boy has 5 apples. He kept them all to himself, that is, he divided them by 1, therefore he received 5 apples. " It would seem that the teacher's methodological actions take into account the psychological characteristics of children, and he seeks to ensure that the introduction of a new concept is accessible to them. Nevertheless, his actions lack that mathematical basis, without which the correct mathematical concepts and concepts cannot be formed.

It is clear that the teacher's methodological actions in teaching mathematics to primary schoolchildren largely depend on the level of his mathematical training. In addition, mathematical training has a positive effect on the clarity of the teacher's words, on the correct use of terminology and the validity of the selection of methodological techniques associated with the study of mathematical concepts.

Task 2. Think about what mathematical knowledge the teacher should rely on when introducing students to the cases of multiplication and division by 1.

Activities aimed at the upbringing and development of a younger student in the process of teaching mathematics require the teacher to master not only private, but also general methodological skills. They can be called didactic, since they can be used by the teacher not only in teaching mathematics, but also in other academic subjects (Russian, reading, natural history, etc.).

For example, the ability to purposefully apply various ways of organizing children's attention is also a component of the teacher's methodological activity. These skills are based on his psychological and pedagogical knowledge. So, the teacher's lack of psychological knowledge about the peculiarities of the attention of younger students leads to the fact that, when organizing their attention, he uses, as a rule, only the method of setting, that is, he says: "be careful." If this attitude does not work, he resorts to various punishments. But it is enough to understand the psychological essence of his actions to understand their erroneousness. Namely: the attitude "be careful" is designed mainly for the voluntary attention of children. This kind of attention requires volitional efforts and quickly tires them. Therefore, the effectiveness of this installation is very short-lived. In an attempt to reinforce it, some teachers ask a question to the whole class, asking exactly the student who is currently distracted. Naturally, he cannot answer. The teacher begins to shame him, lecture him, punish him. But this only increases the mental stress and causes negative emotions in the child:

a feeling of fear, insecurity, anxiety. How can you avoid this? Knowledge of psychological patterns will help the teacher find the right solution.

In psychology, for example, the following pattern has been established: pupils' attention is activated if: a) mental activity is accompanied by motor activity; b) the objects that the student operates are perceived visually.

In addition to regularities, in psychological science, conditions are identified under the influence of which attention is maintained. These include: a) intensity, YENISEI!

P "duchnlyash"

Novelty, unexpected appearance of stimuli and the contrast between them; b) waiting for a specific event; c) positive emotions. Here, the teacher will be helped by various methodological techniques that implement these patterns: didactic games related to specific mathematical content, the use of subject visualization, methods of observation, comparison, appeal to the child's experience, the possibility of choice.

The use of various methodological techniques makes it possible to organize the activities of students on the basis of post-voluntary attention, that is, in accordance with the set goal, but without volitional efforts. This plays an important role in the construction of learning, as it opens up the perspective of the teacher to purposefully control the attention of children.

But it is quite possible that there may be situations when even proven methodological techniques are insufficient. In this case, measures of pedagogical influence are necessary. For example, you can turn to an inattentive student with the following sentence: “And now Kolya will offer you the tasks for verbal counting, which are written out on the cards. He will also control the correctness of their decisions. " As a result, Kolya gets involved in the work, experiencing positive emotions caused by the trust placed in him by the teacher.

In the examples given, the teacher solves operational methodological problems, that is, he must quickly respond to the circumstances that arise during the lesson.

In addition, the methodological activity of the teacher is associated with the solution of design problems, which he thinks over in preparation for the lesson, choosing the way of setting the educational problem, selecting the educational task for its solution.

As you can see, the teacher's methodological activity is associated with the solution of various methodological problems. The formation of the ability to identify, pose and solve them is one of the important tasks of the methodological course.

Task 3. Give examples of methodological problems, the solution of which you observed in teaching practice.

Can you, using your psychological, pedagogical and mathematical knowledge, suggest other options for actions in the lesson?

LEARNING ACTIVITY OF THE JUNIOR SCHOOL STUDENT

IN THE PROCESS OF LEARNING MATHEMATICS

§ 1. CONCEPT OF LEARNING ACTIVITY AND ITS STRUCTURE

Activity is a form of an active relationship of a person to the surrounding reality. It is primarily characterized by the presence of a goal and is caused by different needs and interests (motives).

Educational activity is aimed directly at the assimilation of knowledge, abilities and skills, its content is scientific concepts and general ways of solving practical problems. Being the leading one for primary school students, it stimulates the emergence of central mental neoplasms of a given age, the development of the psyche and personality of the student. Age-related neoplasms are understood as “that new type of structure of the personality and its activity, those mental and social changes that first arise at this stage and in the most important and basic way determine the consciousness of the child, his attitude to the environment, his inner and outer life, his entire course development in this period "1.

The structure of learning activity includes the following components: motives, learning objectives, methods of action, as well as self-control and self-esteem. The interrelation of these components ensures the integrity of the learning activity.

Motive is the driving force of activity, that for the sake of which it is carried out. The motives of educational activity are dynamic and change depending on the social attitudes of the individual. Initially, they are formed under the influence of factors external to educational activity that are not related to its content.

With the help of thinking, the student evaluates different motives, compares them, correlates with his existing beliefs and aspirations, and after an emotional assessment of these motives, begins learning actions, realizing their necessity. Therefore, the learning process should be structured so that the tasks that are posed to the student are not only understandable, but also internally accepted by him, so that they acquire significance for him. In other words, it is necessary to form cognitive motivation closely related to the content and methods of teaching.

Motivation (that is, the orientation of the student towards learning actions) most often arises when setting a learning task. But in some cases, it can appear in the process of the activity itself, its control and self-esteem. This is usually facilitated by the student's successful completion of those educational tasks that the teacher offers both in the process of solving the educational problem and at the stage of self-control.

"Vygotsky LS Educational psychology. - M., 1991.

§ 2. THE LEARNING PROBLEM AND ITS KINDS An educational task is a key component of educational activity.

On the one hand, it clarifies the general goals of learning, concretizes cognitive motives, on the other hand, it helps to make meaningful the very process of actions aimed at solving it.

In most cases, the means of solving educational problems in mathematics are mathematical tasks (exercises, problems). For example, mastering the algorithm of written multiplication is an educational problem, which is solved in the process of performing a certain system of educational tasks (exercises). Obviously, to solve one educational problem, several, often many, mathematical tasks (exercises) can be used. At the same time, in the process of completing one mathematical task (exercise), several educational tasks can be solved.

For example:

Given the numbers: 18, 81, 881, 42, 442, 818. On what basis can these numbers be divided into two groups?

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Notebook with a printed basis “Learning to solve problems. Grade 1 "contains additional material to the textbook" Mathematics. Grade 1 "for a four-year elementary school (by NB Istomina). It presents tasks in the process of completing which students master the skills of reading and various types of educational activities necessary for independent and conscious solution of arithmetic problems. The tasks are aimed at the formation of universal educational actions that meet the requirements of the Federal State Educational Standard of Primary General Education.

Excerpt from the book:
For each child, paint the balloon in the right hand in green and in the left hand in red.
Katya (K), Misha (M), Lena (L) and Tanya (T) are sitting at the table. Katya is to the right of Misha, and Lena is to the left of Misha.

Download and read Visual geometry, Notebook on mathematics, grade 1, Istomina N.B., Redko Z.B., 2016

10. Draw a line around a couple of shapes that have:
1) the same shape;
2) different shape.

Cards with math assignments were compiled in addition to the textbook “Mathematics. Grade 2 "(author - Professor NB Istomina), but can be used when working with other textbooks. The manual includes assignments on the main topics of the mathematics course studied in the second grade: “Two-digit numbers. Addition and Subtraction "; "Multiplication". Sections dedicated to testing computational skills include punch cards. For reusable use, it is advisable to stick them on thick paper, and then cut out the marked rectangles. By placing a card on a checkered sheet of paper, the student will write down only the necessary numbers or signs in the "windows", which is very convenient for testing knowledge.

Download and read Didactic flashcards in mathematics, grade 2, Istomina N.B., Shmyreva G.G., 2002

The notebook with a printed base contains additional material to the textbooks “Mathematics. Grade 1 "and" Mathematics. Grade 2 "(by professor NB Istomina). Completing the tasks proposed in the notebook contributes to the formation of the methods of mental activity in students (analysis, synthesis, comparison), develops such qualities of thinking as flexibility and criticality, expands the understanding of younger students about the methods of modeling when solving word problems.
The notebook can be used when working with children and other mathematics textbooks for primary grades, as well as in gymnasiums and in preparing children for school.

ANO secondary school "Dimitrievskaya",

MOE elementary school teachers

An essay on the topic of self-education

The peculiarities of organizing the activities of students in mathematics lessons when studying the topic "Problem solving" according to the textbook N. B. Istomina

Completed by primary school teacher

Kobeleva Nadezhda

Konstantinovna

MOSCOW, 2013

Plan:

I. Introduction

II. Main part:

1) Features of the methodological approach to teaching problem solving in the course of N.B. Istomina

1. Organization of students' activities in mathematics lessons in the formation of skills to solve problems according to the textbook by N.B. Istomina

III. Conclusion

IV. Bibliography

Introduction. General characteristics of the course "Mathematics" N. B. Istomina.

Everyone knows the truth - children love to learn, but often one word is omitted here - children love OK study! And one of the powerful levers of the emergence of the desire and ability to learn well is the creation of conditions that ensure the child's success in work, a sense of joy on the way from ignorance to knowledge, from inability to skill, i.e. awareness of the meaning and result of their efforts. "Waste, fruitless work for an adult becomes hateful, stupefying, meaningless, and we are dealing with children," wrote Z.A. Sukhomlinsky.

If all children cope with the task assigned to them, if they work with enthusiasm and pleasure, helping each other, if they go home happy with the school day and look forward to tomorrow, the desire to learn grows stronger. And this is one of the results, indicators and success of teaching. “There is success - there is a desire to learn. This is especially important at the first stage of education - primary school, where the child does not know how to overcome difficulties, where failure brings real grief ... ”(ZA Sukhomlinsky. Ibid.)

Namely, the course of N.B. Istomina.

Significant changes within the proposed concept are related to the answer to the question "How to teach?" This is where the main differences from the traditional methods of teaching mathematics in primary grades are contained.

To the peculiarities of the concept underlying the construction of an initial course in mathematics by N.B. Istomina, include the following:

• a new logic for constructing the content of the course, which is based on a thematic principle that allows you to orient the course towards the assimilation of a system of concepts and general methods of action. In line with this logic, the course is structured in such a way that each next topic is organically linked to the previous one, and thus conditions are created for repetition of previously studied issues at a higher level;
• new methodological approaches to the assimilation of mathematical concepts by schoolchildren, which are based on the establishment of correspondence between subject, verbal, schematic and symbolic models, as well as the formation in them of general ideas about change, rule (pattern) and dependence, which is a reliable basis not only for further studying mathematics, but also for understanding the patterns and dependencies of the surrounding world in their various interpretations;
• a new system of educational tasks, the implementation process of which is productive in nature, compiled taking into account the psychological characteristics of primary schoolchildren, is determined by maintaining a balance between logic and intuition, word and visual image, conscious and subconscious, guess and reasoning;
• methodology for the formation of geometric representations, which is based on the active use of methods of mental activity, focus on the development of spatial thinking of schoolchildren and the ability to establish correspondences between models of geometric bodies, their image and development;
• the possibility of using the calculator in the process of teaching mathematics to junior schoolchildren, while the calculator is considered not only and so much as a calculating device, but as a means of organizing the cognitive activity of students.

And finally

• a new methodological approach to teaching problem solving, which is focused on the formation of generalized skills: reading a problem, highlighting a condition and a question, establishing a relationship between them, deliberately using mathematical concepts to answer a problem question.

In our work, we will consider the features of the organization of students' activities in mathematics lessons in the formation of the ability to solve problems according to the textbook of N.B. Istomina.

1. Features of the methodological approach to teaching problem solving in the course of NB. Istomina.

In the course of mathematics in elementary grades, word problems act, on the one hand, as an object of study, assimilation, and the formation of certain skills. On the other hand, word problems are one of the means of forming mathematical concepts (arithmetic operations, their properties, etc.). Tasks serve as a link between theory and practice of teaching, contribute to the development of students' thinking.

A special place in the primary school mathematics course has always been assigned to simple problems. It is in the primary grades that students must master the ability to confidently solve simple problems for all 4 arithmetic operations. Work on simple tasks is carried out throughout all 4 years of study. The methodology focuses students on memorizing and recognizing the types of simple tasks, on consolidating the skills of solving problems of this type. But this forms a formal approach to problem solving.

Traditionally, junior schoolchildren start solving word problems quite early. True, at first these are simple tasks, for the solution of which you need to perform one arithmetic operation (addition or subtraction). But already at this stage, students are familiarized with the structure of the problem (condition, question), with such concepts as known, unknown, data sought, with a short note of the problem and with the design of its solution and answer.

Obviously, most first-graders are not only unable at this stage to analyze the text of the problem, establish the relationship between the condition and the question, highlight known and unknown quantities and choose an arithmetic operation to solve the problem, but they cannot even read the problem.

Naturally, the question arises: maybe it is more expedient to acquaint children with the structure of a word problem and with its solution later, when they learn to read?

But in the teaching of mathematics, certain traditions have already developed. This is how they taught to solve problems in the course "Arithmetic", focusing on the types of simple problems and considering them as the main means of forming in younger schoolchildren ideas about the specific meaning of arithmetic operations. The same technique is reflected in mathematics textbooks (authored by MI Moro and others), according to which primary school teachers have been working since 1969. Later, they were supplemented with the names of the structural components of the problem. The same methodological approach, in which a simple task is the main means of forming mathematical concepts in junior schoolchildren, remained in the mathematics textbooks of 2002 edition for grades 1-4, although it should be noted that the authors increased the time of the preparatory period to familiarize students with the problem ...

While presenting a certain cognitive value, this approach has one significant drawback: when solving simple problems with the help of subject models, the student does not realize the need to choose an arithmetic operation to answer the question of the problem, since he can answer it using object counting. In this regard, writing down the solution to a problem turns out to be a formal operation for him, an additional burden. For example, solving the problem: "The bunny had 9 carrots, he ate 3 carrots. How many carrots did the bunny have left?", The student puts 9 carrots on the typesetting canvas. "This is known in the problem," he says. Then he removes 3 carrots: "This is also known, the bunny ate these carrots." In fact, the answer to the question of the problem has been obtained, since the student can count the carrots remaining on the board. But now we need to write down the solution to the problem. "There are fewer carrots than there were, which means you need to subtract," the child says and writes down the solution to the problem.

As you can see, the logic of the actions performed by the student is devoid of any sense. First, he answered the question of the problem, then he concluded that "it turned out less", and therefore chose subtraction.

If we turned to the student with the question "What action will you choose to solve the problem?", Then he should already have certain ideas about those actions from which he will make a choice. But it turns out that these ideas are only being formed in younger schoolchildren in the process of solving simple problems. And to select arithmetic actions, everyday representations of children are used, which in most cases are focused on words-actions in the text of the problem: presented - took, was - left, came - left, flew away - arrived - or on the child's ability to imagine a situation that is described in the problem ... But not all children cope with this, since they were not taught this.

Therefore, a second question arises: maybe it is advisable to first explain to children the meaning of the actions of addition and subtraction, and then proceed to solving simple problems?

Note that the proponent of this point of view was the progressive Russian methodist F.A. Ern, who believed that the student must first have the concept of arithmetic operations, and only after that - the ability to choose one or another action to solve a given simple problem.

As you know, the process of solving a problem is associated with the allocation of premises and the construction of inferences. Therefore, before starting to solve problems, it is necessary to carry out some work on the formation of the basic methods of mental activity in schoolchildren (analysis and synthesis, comparison, generalization), the use of which is necessary in the analysis of the text of the problem.

From the above reflections, it follows that the solution of word problems should be preceded by a lot of preparatory work, the purpose of which is to form in younger students: a) reading skills; b) techniques of mental activity (analysis and synthesis, comparison, generalization); c) ideas about the meaning of arithmetic operations, on which they can rely, while searching for a solution to the problem.

Considering a word problem as a verbal model of a situation (phenomenon, event, process), and its solution - as a translation of a verbal model into a symbolic (mathematical) one - expression, equality, equation, etc., it is advisable to create conditions for students to acquire experience in interpreting a particular situation on various models. A means of creating these conditions can be a method of forming students' ideas about the meaning of arithmetic operations, which is based on the establishment of a correspondence between verbal (verbal), objective, graphic (schematic) and symbolic models. Having mastered these skills before solving word problems, students will be able to use modeling techniques as a general way of activity, and not as a private technique for solving a particular problem.

This methodological approach to teaching younger students to solve word problems is the answer to the question of how to teach younger students to solve word problems.

The following features of the course can be distinguished in the formation of the ability to solve problems:

1. there is no division of tasks into simple and complex ones.
2. the short entry is completely excluded. Six-year-olds and seven-year-olds do not yet possess stable skills for reading and comprehending text at the same time. Consequently, the task from the verbal must be transferred to some other form so that the child understands what is being reported, what is asked in the task. The subject model also cannot always help in understanding the meaning of the problem. For example: “There are 2 apples on the plate, 3 apples on the other. How many apples are there? " There is no visibility of the unknown here. For children to understand this problem, you need to show a diagram on which they will see 5 apples. Thus, the schematic representation gives the most complete picture of the content of the problem.
3. The work is not going on solving problems of different types, but on various tasks for the formation of the ability to solve problems.
4. There are 2 stages in the formation of the ability to solve problems: preparatory and basic. The main period begins only in the 2nd grade, when the children have already formed the reading skill at the proper level, and with special exercises in the 1st and the beginning of the 2nd grade they are already prepared for the formation of the ability to solve problems and draw up a solution in a notebook.

When solving problems in the course, special attention is paid not to the connection of these numbers by any action, but to the conscious choice of this very action. This is achieved by a specially built system of tasks.

2 . Organization of students' activities in mathematics lessons in the formation of skills to solve problems according to the textbook by N.B. Istomina.

The methodological approach to teaching problem solving, laid down in the course of N.B. Istomina, includes 2 stages: preparatory and main.

Preparatory stage.

A prerequisite for the implementation of this approach in teaching practice is a specially thought-out preparatory work for learning to solve problems. The preparatory stage begins in grade 1 and includes:

1. developing students' reading skills. Without this skill, it is impossible to read the problem and, therefore, understand and solve it;
2. assimilation by children of the specific meaning of addition and subtraction, the relationship "more by", "less by", differential comparison. For this purpose, not the solution of simple typical problems is used, but the method of correlating different models:

a) subject (work with specific objects or drawings)

b) verbal (frontal conversation with the text, which helps students to correctly establish the relationship between these values)

c) symbolic model (equality and inequality)

d) graphic (numerical ray);

1. formation of methods of mental activity;
2. the ability to add and subtract segments and interpret various situations with their help.

As mentioned above, to clarify the meaning of arithmetic operations, a method of correlating various models is used: subject, verbal, graphic and symbolic. We will show how you can organize such an activity for students in a specific lesson on the topic "Addition".

The first version of the lesson

Teacher. Read the word at the top of the page.

Children. Addition.

W. Maybe someone knows what this word means?

D. This is a plus, this is to add. The bunny has one carrot, and the squirrel has 3. They have 4 carrots in total. This is addition.

In addition to these answers, there were others, but they were less related to the content of this concept.

W. Today in the lesson we will try to figure out what addition is. Who can read the assignment? (No. 152). Tell us what Misha and Masha are doing?

D. Misha and Masha put the fish into one aquarium, they plant the fish together. Masha puts three fish into the aquarium, and Misha two; fish will swim together, etc.

Pay attention to how many important and necessary words that characterize the meaning of the action "addition" were uttered by the children. At the same time, mind you, they were not given any sample. Each of them worked at his own level and used only those words that he understood.

W. I will try to depict on the blackboard what is drawn in the picture.

The teacher lays out three fish on the flannelgraph.

- Did I do everything right?

D. You showed only Masha's fish, you also need to add Misha's fish. He has two fish.

The teacher puts two more fish on the flannelgraph.

Similar work is carried out with the upper right picture, which is given in the textbook. Misha puts four tulips in a vase, and Masha puts five cornflowers. They combine flowers together in one vase.

W. You told very well what was drawn in the pictures. Now let's try what you told in words, write it down using mathematical symbols. Look, there are some entries in frames under the pictures. Maybe some of you can read them, but as they are called, you probably do not know.

Some children try to guess the titles of the recordings. Some say - examples, others - inequalities, still others - a multiplication table.

W. No, no one guessed right. These records are called "mathematical expressions".

D. And here it is written.

W. That's right, read to all the guys what is written in the textbook. (The actions of Misha and Masha can be written in mathematical expressions.)

– Now consider these expressions carefully. Maybe someone will guess which expressions refer to the top left picture.

Focusing on numbers, the children name the expressions 3 + 2 and 2 + 3 and explain what each number in the expression means: 3 is the number of fish that Masha puts into the aquarium, 2 is the number of fish that Misha puts into the aquarium.

W. That's right, the expressions 3 + 2 and 2 + 3 mean that the fish are combined together.

Now match the expressions to the top right picture.

Children easily cope with the task and explain what the numbers 4 and 5 represent in the picture.

W. Now try to match expressions to other pictures yourself. Each of you has a leaflet that is divided into four parts. You should write down expressions that fit the bottom left picture and the bottom right picture.

Children independently complete the task. The teacher observes their work, walks around the classroom, and helps some of the children. Then he writes mathematical expressions on the board, which is divided into four parts.

On the desk:

3 + 2 2 + 3

- Look at the blackboard. I wrote down two expressions that I saw in one student in a notebook. Does everyone agree with him?

D. This should be written down to the top picture.

- This is not true. Here you need to write 3 + 1 and 1 + 3, because Masha has 3 candies, and Misha has one. They put them in one vase.

W. Well, if I write the expression 2 + 2 to the lower left picture, will it be correct?

There are students who agree with this, since 2 + 2 is 4. But others object. This is not true, because Masha puts three candies in a vase, and Misha puts one.

W. Now, guess what picture the entry 4 + 5 = 9 is suitable for?

Look, a new sign has appeared here, which is called "equal", and the notation 4 + 5 = 9 is called "equal".

Equality can be true or false. What does “true equalities” mean?

Each of the equalities proposed in the textbook is written on the board and checked on subject models (these can be any subjects).

4 + 5 = 9

Children count or count objects to test for equality.

W. Let's now read in the textbook how Misha proposes to check the equalities.

(The drawing of the number ray, which the teacher brings to the blackboard, is discussed..)

Component names can be entered in the second lesson on the topic. The second lesson also includes exercises in which children choose a drawing on the number line corresponding to the picture, or choose an expression corresponding to the picture on the number line, or choose a picture corresponding to the picture on the number line.

Thus, to explain the action of addition, previously studied material (counting, counting, number ray) is actively involved. A simple task is replaced by a method of correlating various models: subject (pictures), verbal (description of pictures), graphic (drawing on a number ray), symbolic (writing an expression, equality).

Second version of the lesson

There is a number ray on the board. The teacher calls two students to the blackboard. The children turn their backs to the class and the teacher gives each of them some items.

The teacher comments:

W. I give the mushrooms to Lena and Vera. They will count them and tell me the number in my ear. And I will show you on the beam how many mushrooms each of them has.

The teacher performs a drawing on the blackboard:

The teacher comments on his actions:

Lena has so many mushrooms (draws the first arc), and Vera has so many mushrooms (draws the second arc).
Who guessed how many mushrooms Lena has? How many mushrooms does Vera have? How many mushrooms do Lena and Vera have?

W. Let's check if you answered my questions correctly. Girls spread mushrooms on a flannelgraph (4 large and 4 small).
And now I will combine large and small mushrooms (draws a curved closed line, inside which there are large and small mushrooms). Who can write down in the language of mathematics what I have done?

The children write down 4 + 4 and explain what each number in this expression means.

As you can see, in the second lesson, the teacher first used the graphic model to explain the meaning of addition, then moved on to the subject model, then to the verbal one (the children described what they see in the picture) and then introduced them to the symbolic model (expression, equality).

Likewise, by focusing on the textbook page, you can build a lesson when introducing children to subtraction.

Thus, the solution of simple problems is replaced by various exercises (educational tasks), in the process of performing which children learn the specific meaning of the addition and subtraction actions. Here are the following exercises: (notebook with a printed basis No. 1) No. 63, 64–67, 68, 70, 79.

To clarify the concept of "difference comparison" - "How much more? How much less? " - the choice of the subject model is of particular importance. The fact is that if a drawing is used as a subject model, in which objects are located one under the other, then it is rather difficult for children to realize that the answer to the question "How much more (less)?" is associated with performing a subtraction action. If the child is not aware of this connection, but only remembers the rule: "To find out how much one number is greater than another, you need to subtract the smaller one from the larger number," then when solving problems he will be guided only by an external sign, namely the word " how much ".

An example is the following problem: “At the bus stop, 3 girls and 7 boys got off the bus. How many people are on the bus less? " (Up to 50% of children solve the problem by subtraction.)

Not understanding the meaning of difference comparison, many children, answering the question "How much less?", Choose subtraction. And to answer the question "How much more?" choose addition.

Here are examples of tasks in the course of which children learn the meaning of the difference comparison: No. 261, 267 (textbook for the 1st grade), No. 18, 19, 24 (notebook with a printed basis No. 2, 1st grade).

For the formation of children’s ability to imagine a situation described in words, tasks are proposed for correlating verbal and subject models: No. 393, 402 (textbook for the 1st grade).

In the first quarter of the 2nd grade, students get acquainted with the diagram: No. 41, 42, 49, 58 (textbook for the 2nd grade).

The main stage.

The main period of learning to solve problems begins with an acquaintance with the problem, its structure. This material is well described in the textbook of the 2nd grade in the form of a dialogue between the heroes of the textbook Masha and Misha (pp. 49-51: №129). From this dialogue, students learn which text can be called a task, that a task consists of a condition and a question that are interconnected.

1) Comparison of problem texts, identification of their similarities and differences: № 131, 132, 138, 149 (textbook for the 2nd grade).

2) Compilation of tasks according to these conditions and the question: № 35 (a), 36 (a) (notebook "Learning to solve problems", 1–2 grades).

3) Translation of the verbal model of the problem or its condition into a schematic model: № 41 (a), 43 (a) (notebook "Learning to solve problems", 1–2 grades).

4) Choice of scheme No. 44 (a) (exercise book "Learning to solve problems", grades 1–2).

5) Completion of the begun scheme, corresponding to the given task: № 49 (a), 59 (a), (b) (notebook "Learning to solve problems", 1–2 nd grades).

6) Explanation of expressions compiled according to the condition of the problem: № 179 (textbook for the 2nd grade).

7) Selection of questions that meet this condition: No. 191; which can be answered using this condition: No. 222 (textbook for the 2nd grade).

8) The choice of conditions corresponding to this question: No. 230 (textbook for the 2nd grade).

9) Completion of the text of the problem in accordance with the given decision: No. 65 (notebook "Learning to solve problems").

10) Supplementing the text of the problem in accordance with this scheme: No. 42 (a), (b), No. 72 (a), (b).

11) Selection of the problem corresponding to the given scheme: No. 77.

12) The choice of the solution to this problem: № 37 (notebook).

13) Setting various questions to this condition and recording the expression corresponding to each question: No. 34 (notebook).

14) Designation on the diagram of known and unknown quantities in the problem: № 51 (a), (b), 69 (a), (b) (notebook).

To check the formation of the ability to solve problems, the teacher invites the children to write down the solution to various problems on their own. If children have difficulties, then the teacher can use any combination of methodological techniques, depending on the content of the problem.

Math lesson

2nd grade

Theme. "Problem solving"

Target. Formation of skills to analyze the text of the problem and interpret it on a schematic model (translation of a verbal model into a schematic).

Teacher. We continue today in the lesson to learn to solve problems. Tasks from the notebook "Learning to Solve Problems" will help us with this.... Open task number 48. Read the task (s) silently, then out loud.

- Now read the task (b).

- Let's try to complete the task ourselves. This will help you conclude whether you understood the text of the problem statement or not.

Children work independently (use a simple pencil). Everyone copes with the task by choosing Scheme 4 and denoting the quantities known in the problem statement. The teacher opens on the blackboard pre-drawn, the same as in a notebook with a printed basis, diagrams.

Teacher. Who wants to draw a diagram on a chalkboard?

There are many who wish. Two students come to the blackboard and quickly "animate" diagram 4:

Teacher. We read the task c). Before answering the questions, let's mark them on the selected diagram.

Children complete the task on their own in a notebook, the teacher observes their work and calls those who have difficulties to the blackboard. Three children come out to the board in turn. Each designates one question on the diagram.

The diagram on the board takes the following form:

W. Now you can independently answer each question by writing down the arithmetic operations.

All children quickly cope with the first question: 7 + 2 = 9 (l.). The second question is also straightforward. Everyone has a note in their notebooks: 9 + 3 = 12 (l.). Children carefully study the scheme, checking it against the actions already performed. The teacher fixes the children's answer options on the blackboard and invites them to discuss them:

Children. 12 - 9 = 3 - this is not true. It was already known that Lena was 3 years older than Vera.

– The question asks how many years Lena is older than Masha; Lena is 12 years old, and Masha is 7. So, you need to subtract 7 from 12.

W. And who can tell me how much Masha is younger than Lena?

D. You do not need to do this here; how much Lena is older than Masha, and how much Masha is younger than Lena.

W. And who answered the third question like this: 3 + 2 = 5? (Five hands are raised.) I don’t understand something, how did you reason?

D. And this can be seen in the diagram. (He comes out to the board and shows a segment equal to the sum of two segments: one denotes the number 2, and the other denotes the number 3.)

W. I think that without a diagram it would be difficult to offer such a way to answer the question.

Children agree with the teacher.

W. Well, now let's try to change the condition of the problem so that it corresponds to Scheme 1.

D. Masha is 7 years old, Vera is the same, and Lena is 3 years older than Masha. ()
– Masha and Vera are 7 years old. And Lena is 3 years older than Vera. (Goes to the board and shows the condition on the diagram.)

W. But is such a condition suitable? Masha is the same age as Vera. And Lena is 3 years older than Vera.

D. In general, it will do. Only not a single question can be answered.
– If you ask the question, you get a problem that lacks data.

Similar work is carried out with diagram 2. The children "animate" the diagram on the board and orally answer the same questions.

The third question changes: "How old is Lena younger than Masha?"

W. I see that you know how to work with a diagram, so let's try to draw a diagram for another task on our own. But before reading the problem, open your notebooks and draw a free line segment.

Children draw a segment, after which they open task number 159 from the textbook.

Read the assignment.

- Let's answer the question of the assignment first.

D. Here the beginning is exactly the same.

W. I don’t understand, what does the beginning mean?

D. Well, the conditions are the same ...
- I do not agree. Conditions are different. The problem on the left does not say how many chairs were in the room, while the second says: there were 84 chairs in the room.

D. The task on the left is missing data.

W. What is missing? To answer the first question?

D. No, the first question can be answered, but the second cannot.

W. Well, can you answer two questions in the second problem?

D. In the second, you can.

W. Let's label all the chairs in the room with the line you drew. Using this line segment, draw a diagram that matches the task.

Children work independently. The teacher draws a diagram on the blackboard:

The children are discussing it.

D. Well, everything is wrong here. After all, you said to mark with a segment all the chairs in the hall.

D. I drew like this. (Goes to the board, draws a segment from the hand and marks it.)

On the desk:

- Now we will take out the chairs. (Draws on the diagram and comments.)First they took out 24 chairs, then 10 more.

W. Well, let someone else pose the questions according to the scheme.

Children finish the circuit.

– Write down the solution to the problem in a notebook.

The children write down the solution themselves. The teacher helps those who are in difficulty. Those who quickly wrote down the solution to the problem are invited to complete the task number 162.
Children are happy to do it. For the rest, the chalkboard reads "No. 162", and the children already know that this is a homework assignment.

So, the use of various methodological techniques in teaching problem solving contributes to the development of the students 'outlook, the correct understanding of the mathematical meaning of various life situations, which is very important for the implementation of the practical orientation of the mathematics course, and forms the students' ability to see various connections between data and the desired, i.e. solve the problem in different ways.

All of these techniques can be found in the course tutorials.

Conclusion

Solving problems, students acquire new mathematical knowledge, prepare for practical activities. Tasks contribute to the development of their logical thinking. The solution of problems is also of great importance in the education of the personality of students.

Acting as a concrete material for the formation of knowledge, tasks make it possible to connect theory with practice, learning with life. Problem solving forms practical skills in children that are necessary for every person in everyday life. For example, calculate the cost of a purchase, calculate what time you need to get off so as not to miss the train, etc.

Through solving problems, children get acquainted with facts that are important in cognitive and educational terms. Thus, the content of many tasks solved in the primary grades reflects the work of children and adults, the achievements of our country in the field of the national economy, technology, science, and culture.

Tasks perform a very important function in the initial course of mathematics - they are a useful means of developing logical thinking in children, the ability to analyze and synthesize, generalize, abstract and concretize, and reveal the connections that exist between the phenomena under consideration.

Problem solving - exercises that develop thinking. Moreover, problem solving contributes to the education of patience, perseverance, will, helps to awaken interest in the very process of finding a solution, makes it possible to experience deep satisfaction associated with a successful decision.

All of the above proves how important it is to teach a younger student to solve problems not automatically, but meaningfully. This is exactly what the carefully thought-out system of teaching problem solving by N. B. Istomina.

In conclusion, I would like to quote the words of L.N. Tolstoy, which, in my opinion, perfectly reflect the purpose of working on mathematics textbooks by N.B. Istomina: "Knowledge is only then knowledge when it is acquired by the effort of one's thought, and not by memory ..."

Bibliography:

1. Istomina NB Mathematics. Grade 1: Textbook for a four-year

2. Istomina NB Mathematics. Grade 2: Textbook for a four-year

Primary school. - Smolensk: Association XXI century, 2000.

3. Istomina NB Methods of teaching mathematics in primary grades. - M .:

LINKA - PRESS, 1997.

4. Istomina NB Learning to solve problems. A notebook on mathematics for the 1st and 2nd grade of a four-year elementary school. M .: M .: LINKA - PRESS, 2005.

6. Sukhomlinsky Z.A. I give my heart to children: Fav. ped. op. - M., 1979

7. Tolstoy L.N. Complete works - vol. 42, M., 1992.

The purpose of the textbook is the formation of methodological knowledge, skills and experience of creative activity in the future teacher for the implementation in practice of the ideas of developing mathematics teaching for junior schoolchildren. The manual will also be useful for primary school teachers.

The meaning of the actions of addition and subtraction.
The primary school mathematics course reflects a set-theoretic approach to the interpretation of addition and subtraction of non-negative integers (natural and zero), according to which the addition of non-negative integers is associated with the operation of combining pairwise disjoint finite sets, subtraction - with the operation of complementing a selected subset. This approach is easily interpreted at the level of objective actions, thereby allowing to take into account the psychological characteristics of primary schoolchildren.

However, the methodological interpretation of this approach can be different. For example, in the M1M textbook, simple word problems act as the main means of forming children's ideas about the meaning of addition and subtraction actions.

Free download an e-book in a convenient format, watch and read:
Download the book Methods of teaching mathematics in primary grades, Istomina N.B., 2001 - fileskachat.com, fast and free download.

• Mathematics, Grade 1, My academic achievements, Istomina N.B., Shmyreva G.G.

The following tutorials and books:

• Education in the 4th grade according to the textbook "Mathematics", program, methodological recommendations, thematic planning, tests, Bashmakov M.I., Nefedova M.G., 2012
• Teaching in the 1st grade according to the textbook "Mathematics" Bashmakova M.I., Nefedova M.G., program, thematic planning, methodological recommendations, Bashmakov M.I., Nefedova M.G., 2013

The main idea of ​​the approach to teaching problem solving when working on the teaching and learning method "Harmony" lies in the fact that the meaning of arithmetic operations is realized by students even before solving simple problems. Psychologist N.A. Menchinskaya considered the choice of an arithmetic operation as a new mental operation, the essence of which boils down to translating a specific situation described in a task into a plan of arithmetic operations. Of course, in order to perform operations on the mental plane, the student must master them on the subject level. In this regard, the acquaintance of students with the text problem is postponed to a later period, which is preceded by a lot of preparatory work.

Preparatory work shapes

Reading skill

Concepts of Mathematical Concepts and Relationships

Logical methods of thinking - analysis and synthesis, comparison, analogy, generalization

Certain experience in correlating textual, subject, schematic and symbolic models

The basis of the content line of the preparatory stage is: the meaning of arithmetic operations (addition, subtraction), relations: "increase by ...", "decrease by ...", "how much more?", "How much less?"

The mathematical basis for explaining the meaning of addition is the set-theoretic interpretation of a sum as a union of sets that do not have common elements, subtraction - as the removal of a part of a set. And the organization of student activity is based on the correlation of subject, verbal, schematic, symbolic models and the transition from one model to another. For this, tasks are used with various instructions: to correlate a picture and a mathematical record; to choose a mathematical notation corresponding to the figure; to choose a pattern corresponding to the mathematical notation.

At the preparatory stage, students also master the ability to build segments of a given length, add and subtract them.

As reading skills develop, students are offered tasks for interpreting texts that describe various situations in the form of a mathematical record or a schematic drawing.

Examples of such tasks:

1. There are 15 mushrooms in the basket. Of these, 5 are white, the rest are chanterelles. Mark all the mushrooms with circles and show how many chanterelles are in the basket.

Masha completed the task as follows:

chanterelles

Misha like this:

chanterelles

Who completed the task correctly?

2. 11 monkeys and 7 tigers performed in the circus. Mark animals with squares and show how many more monkeys than tigers.

Masha made the following drawing:

And Misha is like this:

Who is right: Masha or Misha?

At the preparatory stage, special work is also carried out to form ideas about the scheme.

An example of such a task:

1. The pencil is 2 cm longer than the pen. Guess how to show this using line segments.

Masha: I think this task cannot be completed. After all, we do not know the length of the handle.Misha : And I think it can be shown like this:

2 cm

The picture that Misha drew will be called a diagram.

The answers given in the textbook do not at all mean that after reading the assignment, the students will immediately consider the options for its implementation, which are proposed by Misha and Masha. You should resort to the statements of Misha and Masha when students cannot cope with the task. In this case, they perform the function of methodological assistance to the teacher, helping to activate students or to correct and self-control those judgments expressed by children.

Chapter 2. The main methodological stages of work on the problem

Work to clarify the text of the problem

It is to find out if all the words and phrases of the text are clear to children. When solving problems of addition and subtraction, these are terms: older - younger, more expensive - cheaper, etc.

Analysis of the problem (analysis), search for a solution

Finding a solution and drawing up a plan for solving a problem is usually called its analysis. The approach to parsing can be analytical - "from the question" and synthetic - "from the data."

In grades 1-2, it is easier for a child to master the synthetic method of parsing a problem, especially if it is accompanied by a visual interpretation or a graphical diagram, because from the point of view of psychology, at the age of 6 - 8 years, the formation of the ability to synthesize in a child is somewhat ahead of the formation of the ability to analyze.

Record of decision and response

Recording can be done in different ways:

for actions without explanation - in this case, write a full answer

on actions with explanations - in this case, write a short answer

as an expression (in a compound problem)

in the case of solving the problem using an equation, they gradually write the equation with explanations

Working on a problem after solving it

This work is as follows:

if the task was recorded by actions, then the solution is recorded in the form of an expression (in a compound task);

verifying the solution:

In primary grades, the following verification methods are used:

estimation of the answer (setting the possible boundaries of the values ​​of the sought-for)

solving the problem in another way

inverse problem solution

variation of data, conditions and question.

This is the best developmental technique at the stage of working on a problem after its solution. Varying the question in some simple problems organically leads children to get acquainted with the compound problem. Varying the data and the desired gradually leads to the ability to compose the inverse problem.

The considered stages of work on the problem are the stages of the teacher's work. These stages should not be confused with the methods of the child's independent work on the task. When working independently on a task at home or on a control child, you must be well able to:

to simulate the situation assigned to the problem, while it is important that the model is not formal, it should lead to a way to solve the problem;

compose a mathematical expression according to the meaning of the situation (choice of action);

draw up a record of the decision and response;

control the result (own methods of checking the answer to the problem).

The most difficult for the child are skills 2 and 5, however, the formation of these particular skills guarantees that the child will solve the problem not by “remembering” the learned solution, but by approaching any task as an object requiring the performance of the above actions.