The purpose of the textbook is to form the future teacher's methodological knowledge, skills and experience of creative activity for the implementation IN practice of the ideas of developmental teaching of mathematics to junior schoolchildren. The manual will also be useful for teachers working in primary school.
The meaning of addition and subtraction.
The elementary school mathematics course reflects the 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, thus allowing to take into account the psychological characteristics of younger students.
However, the methodological interpretation of this approach may be different. For example, in the M1M textbook, simple word problems are used as the main means of shaping children's ideas about the meaning of addition and subtraction.
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- Mathematics, Grade 1, My academic achievements, Istomina N.B., Shmyreva G.G.
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- Education in the 4th grade according to the textbook "Mathematics", program, methodological recommendations, thematic planning, tests, Bashmakov M.I., Nefyodova M.G., 2012
- Education in the 1st grade according to the textbook "Mathematics" Bashmakova M.I., Nefyodova M.G., program, thematic planning, methodological recommendations, Bashmakov M.I., Nefyodova M.G., 2013
Developmental learning
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 methodology primary education.
1NISEYSKOV Pedagogical School*1 Smolensk "Association XXI Century"
Istomina N. B.
I89 Methods of teaching mathematics in elementary school:
Developmental training. - Smolensk: Publishing house "Association XXI century", 2005. - 2 7 2 p.
The purpose of the textbook is to form the methodological knowledge, skills and experience of creative activity in the future teacher for the implementation in practice of the ideas of developmental teaching of mathematics to junior schoolchildren.
The manual will also be useful for teachers working in primary grades.
ISBN 5-89308-193-5 © Istomina N.V., 2005 ISBN 5-89308-193-5 © XXI Century Association, 2005
INTRODUCTION
In accordance with the state standard of primary general education, the study of mathematics at the primary level is aimed at achieving the following goals:
The development of figurative and logical thinking, imagination, the formation of ~edmet skills and abilities necessary for the successful solution of educational and ~Actual tasks, continuing education;
Mastering the basics of mathematical knowledge, the formation of initial ~ ideas about mathematics;
Raising interest in mathematics, the desire 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 in itself: ~ social (mathematical), psychological, pedagogical and methodological knowledge, skills and abilities.
This manual is intended for full-time students of the primary school faculty 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 should equally be prepared to solve the problems that they will have 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 primary mathematical education (§2), about the methodological activity of the teacher in the process of teaching mathematics to younger students (§3).
The second chapter gives a methodical interpretation of the main components of the concept of "learning activity" and ways of organizing it.
Possible approaches to the development of thinking of younger 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 mastering knowledge, skills and abilities perform various functions. They can be considered:
1) as ways of organizing the educational activities of schoolchildren;
2) as ways of cognition that become the property of the child, characterizing his intellectual potential and ability to acquire knowledge, skills and abilities;
"Federal component of the state standard 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 intellectual activity of the child enters into various relationships with other aspects of his personality, primarily with direction, motivation, interests, level of claims, i.e. characterized by increasing activity of the individual.
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 the subject content of methodological activity, i.e. learn to answer the questions:
What mathematical concepts, laws, properties and methods of action are reflected in primary course 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 initial course of mathematics and the features of their assimilation by younger students." Its content includes theoretical information about various concepts of the elementary course of mathematics; types of educational tasks in the process of performing which children not only acquire knowledge, skills and abilities, but also advance in their development; guidelines to 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 master mathematical concepts. It allows you to take into account the individual characteristics of the child, his life experience, objective-effective and visual-figurative thinking and gradually introduce it 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 for organizing the computational activities of younger students in the developing course of elementary mathematics.
Chapter 6 gives a brief description of various methodological approaches to teaching younger students to solve text problems and reveals in detail the methodology for the formation of generalized problem-solving skills, which is based on various methodological techniques: choosing a scheme, expressions, conditions, reformulating the question of the problem, setting questions for a given condition and etc.
Chapter 7 gives a description of various approaches to the construction of a mathematics lesson in elementary grades and recommendations for planning and analyzing developmental lessons.
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, justify their actions and critically evaluate them;
To improve the quality of assimilation of mathematical knowledge, skills and abilities;
To ensure continuity between the primary and secondary levels of education, preparing primary school students for active mental activity;
To develop the creative methodological potential of the primary school teacher, stimulating him to independently compose educational tasks, choose the 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 Federal List textbooks and are labeled “Recommended by the Ministry of General and vocational education Russian Federation".
For the creation of an educational and methodological set in mathematics for four years elementary school Doctor of Pedagogy, Professor Istomina Natalia Orisovna in 1999 was awarded the Prize of the Government of the Russian Federation.
METHODOLOGY OF TEACHING MATHEMATICS
IN PRIMARY SCHOOL AS PEDAGOGICAL SCIENCE
AND AS A SUBJECT
§ 1. THE SCIENCE OF TEACHING MATHEMATICS
Learning is a purposeful, specially organized and managed by the teacher activity of students, during which they acquire knowledge, develop and educate.In learning, as in any process, certain patterns are manifested that express the existing connections between pedagogical phenomena, while a change in some phenomena entails a change in others. For example, learning objectives, reflecting the needs of society, influence the content and ways of organizing student activities aimed at mastering 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 independent thinking 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, apart from the mathematical essence, they cannot directly serve as the theoretical foundations of the methodology, since it remains unclear how, based 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 defined, the necessity and effectiveness of their application in the educational process has been substantiated, a number of ways to organize and manage students' independent activities have been revealed, and the most important didactic conditions for the implementation of this type of learning have been identified. However, the solution to the question of the possibility of creating problem situations in teaching mathematics to younger students remains with the methodology. And until it is presented at the methodological level, the theory of problem-based learning, which has been developed in didactics, will not become the property of the practice of primary school teachers.
The task of the mathematics teaching methodology 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 tasks. They represent a problem for the student, a way to solve which he must find on his own, creatively applying his knowledge. But at the same time, this kind of problem situations may be inaccessible to the majority of younger students, since their solution requires a high level of abstraction and generalization.
Given 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 practical actions.
So, starting to study the topic “Length of objects” (grade 1), 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, a way to solve which he was 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 practical actions. This problematic situation can be complicated by asking the question: “Can the lengths of these strips be compared using a third?” The answer to it is connected with finding a new mode of action, which underlies the measurement of quantities.
Similarly, other provisions of didactics can be illustrated, which become the theoretical foundations of the methodology for teaching mathematics only after they have been processed in connection with the specific content of the studied mathematical material.
For example, the principle of accessibility of education 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 one? 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, and also taking into account the psychological characteristics of the perception and thinking of younger students, the method of primary mathematics teaching formulates general provisions that a teacher can guide when developing children's written division skills. For example: students' acquaintance with the written division algorithm should be preceded by exercises that will prepare them for the perception and understanding of the operations included in this algorithm. This includes 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 position ensures the availability of a new method of action and gives scope for greater independence of students in its assimilation.
When studying the written division algorithm, one should keep in mind the following situation: when recording a written division, it is necessary to comment on the operations performed in detail (expanded), as 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 in a timely manner students' activities on the use of the algorithm.
The above methodological recommendation takes into account one of the psychological patterns, which consists in the fact 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, reasoning is wrong. Thus, the recommendation to use the commenting technique is generalized (in this case, in relation to the study of a particular issue), theoretically substantiated (psychological position), and can be applied when studying other issues of content. Its expediency is confirmed by the practice of teaching.
One cannot but take into account 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 enter into a relationship with psychological patterns, which, like didactic ones, are usually expressed in a generalized way, in isolation from 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 provisions that reflect the characteristics of teaching a particular subject.
The development of a theory of learning, taking into account the specifics of the content, is a necessary condition for the successful development of a certain section of the methodology of teaching a particular academic discipline.
What requirements should the theoretical foundations of the methodology of teaching mathematics meet? They should: a) be based on a certain theory (psychological, pedagogical, mathematical), using it in relation to the specific content of education; b) be generalized provisions that reflect not a single case, but general approaches to the process of teaching mathematics (in particular, in primary school), to solving a certain set of issues in it; c) reflect the stable features of the process of teaching mathematics, i.e., 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 of 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 school 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 study.
The whole variety of problems of particular methods, including the methods of teaching mathematics in primary school, can be formulated in the form of questions:
Why teach? What is the purpose of teaching mathematics to children?
What to teach? That is, what should be the content of mathematical education in accordance with the goals set?
How to teach? That is:
a) in what sequence 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, means and forms of learning) should be used for this;
c) how to teach children taking into account their psychological characteristics(how, in the process of learning mathematics, to use the patterns of z most fully and correctly: perception, memory, thinking, attention of younger students)?
These problems allow us to define the methodology of teaching mathematics as a science, which, on the one hand, is addressed to a specific content, rebounds to streamline it in accordance with the goals of learning, on the other hand, to human activity (teacher and student), to the process of assimilation of this holding, management which is carried out by the teacher.
The object of study of the methodology of teaching mathematics is the process of teaching mathematics, in which four main components can be distinguished: the goal, the content, the activities of the teacher and the activities of students. Listed Components
2 WALK in interconnection and interdependence, i.e. 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 the relationships and relationships that exist between them.
Methodical problems are solved with the help of pedagogical research methods, which include: observation, conversation, questioning, summarizing the best practices of teachers, laboratory and natural experiments.
various tests and psychological techniques provide an opportunity to identify the impact of these methods of learning on the assimilation of knowledge, skills and abilities, on the overall development of children. All this makes it possible to establish certain regularities in the process of teaching mathematics.
Task 1. What concepts of teaching younger students are you familiar with? Expand the content of these concepts.
§ 2. GENERAL CHARACTERISTICS OF THE DEVELOPMENT OF THE INITIAL
MATHEMATICAL EDUCATION
At each stage in the development of primary education, methodological science gave different answers to the questions: “Why teach?”, “What to teach?”, “How to teach?”Prior to 1949, practical goals were the priority in primary education. This was due to the fact that before the introduction of a general compulsory 7-year education, elementary school was a closed stage. The main content of the initial course of mathematics was the study of four arithmetic operations, solving problems arithmetic way and acquaintance with geometric material, which was subject to the solution of practical problems (mark out rectangular land plots, measure their length, width, calculate the area and perimeter of a rectangle using formulas, etc.).
The course content was based on the concentric principle (5-6 concentres). At the end of the fourth year of study, it was supposed to generalize the studied material and familiarize with individual elements of the theory (connections between actions, components and results of actions, some properties of actions).
Teaching methods took into account those features of this age that were noted by psychological science: imagery, the predominance of "mechanical" memory over semantic, ease and strength of assimilation by younger students of numerous facts.
Based on the "mechanical" memory, the 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 taking into account the real cognitive abilities of younger students as the need to adapt the content and teaching methods to the characteristics 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 XX century, the fallacy of this position was noted, even in relation to children who were lagging behind in mental development. He noted that learning, which focuses on already completed cycles of development, does not lead the process of development, but itself trails behind it; only that training is good which gets ahead of development.
It should be noted that the 1930s and 1940s are marked by joint research by psychologists and methodologists on the methods of teaching individual subjects. Regarding the directions of these studies, psychologist N. A. Menchinskaya wrote:
“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 activities, and to explore various forms of this activity as a natural response to pedagogical influences”1.
In line with this direction, the ways of assimilation by children of the concept of number and arithmetic operations, the features 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 methodology for teaching mathematics, A.S. Pchelko (author of a textbook on arithmetic for elementary grades) complained that the main attention of methodologists is focused on the teacher, on the methods and techniques that he teaches children, and the questions of whether how students perceive the teacher's explanations, what difficulties they have in mastering one or another section of arithmetic, what is the reason for these difficulties and how they can be prevented.
In the 1940s and 1950s, methodical works appeared based on research and 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 the primary grades.
However, the changes made to the program of the arithmetic course, which was introduced in 1960, did not affect its essence. They amounted to minor amendments, aimed mainly at further simplification of the course. New trends, brought to life 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 junior schoolchildren the general methods of working on a problem, the importance of forming correct generalizations in children and organizing various tasks for independent work.
In 1965, the book by M. I. Moreau and N. A. Menchinskaya “Issues 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 younger students. Here are some of them1.
“In order for a younger student to be active in the learning process, it is necessary: firstly, to provide him with a wide opportunity for independence in academic 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.
“A well-known old saying goes: “Repetition is the mother of learning.”
Now, sometimes it is contrasted with another: "Application is the mother of learning." The second formulation is more in line 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 at the same time, repetition is not monotonous or monotonous, but one that involves change as the knowledge itself and the conditions for its use.
“The ability to solve problems, although it is of a general nature, is amenable to development, like all others, but this requires a special system of exercises aimed at instilling in schoolchildren the need for creative thinking, interest in independent problem solving, and consequently, to the search for the most rational methods of solving them.
“Full awareness of assimilation can be achieved by the student only on the condition that he does not passively perceive the communicated new material, but actively operates with it.
“It is necessary to avoid not only extremely difficult, but also extremely easy material for the student to master, 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 classes. - 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 the same lesson, in comparison with each other, and sequential, when one of the compared concepts is studied first, and the second is introduced based on the opposition to the first, only when the first has already been mastered.
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 (the UDE method).
Education, built in accordance with this idea, is effective in improving the quality of students' knowledge with significant savings in the time spent on studying the mathematics course.
a) simultaneous study of similar concepts; b) simultaneous study of mutually inverse actions; c) transformation of mathematical exercises; d) drawing up tasks by schoolchildren; e) deformed examples.
Among the studies that played an invaluable role in the development of the methodology of primary education, two should be mentioned: one under the direction of L. V. Zankov (1957), the other - under the direction of D. B. Elkonin and V. V. Davydov (1959). .).
And although the object of experimental research by L. V. Zankov was not individual academic subjects, and the didactic system, covering all primary education, nevertheless didactic principles developed in the laboratory (training on high level difficulties, the study of 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 ones) 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 called versatility, collisions, processuality. L. V. Zankov considered the development of a methodological system to be especially relevant.
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 construction of the learning process. As such new formations were named: educational activity, theoretical thinking and arbitrary control of behavior (reflection).
In parallel with the 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.
A huge contribution to the preparation of the reform of mathematical education at this stage was made by methodologists M. I. Moro, A. S. Pchelko, M. A. Bantova, G. V. Beltyukova, N. V. Melentsova, E. M. Semenov, P. M. Erdniev, I. K. Andronov, Yu. Psychologists (N. A. Menchinskaya, A. A. Lyublinskaya) actively participated 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 initial 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.
Modernization of the subject content of elementary mathematical education was accompanied by instructions: "One of the important educational tasks associated with the study of mathematics is the development of students' cognitive abilities"; "Mathematics lessons should contribute to the education of children's independence, initiative, creativity, work culture"; “Learning and development in the study of mathematical material should be carried out in close connection with each other”1.
However, the implementation of these instructions in school practice turned out to be an even more difficult task than the introduction of the new content of the unified natural course of mathematics. “Teachers received new programs and started their implementation, having no idea about the new methodology,” writes Sh. A. Amonashvili.
The task of developing a child in the learning process remained unresolved in a stable mathematics course (M. I. Moro and others). and its reinforcement. The educational tasks were monotonous, and the tasks that required the activation of the mental activity of schoolchildren were classified as material of "increased difficulty" and were "acquired" only by years 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 to organize the educational activities of younger students continued both in theory and in practice.
In the 70-80s, thousands of schoolchildren worked according to the system of L. V. Zankov, the experiment on the system of D. B. Elkonin, V. V. Davydov continued, the UDE system was actively introduced into school practice, the experiment of A. M. Pyshkalo and K. I. Neshkov, which tested the possibility of constructing an initial course in mathematics on a set-theoretic basis.
Actual problems of methods of teaching mathematics in elementary grades / Ed. M. I. Moro, A. M. Pyshkalo. - M., 1977.
Amonashvili Sh. A. in Sat. 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 into school practice, new teaching technologies, variable authoring programs and textbooks.
On the wave of this innovative movement, “Russian primary education acquires a developing 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 performance of various mental operations (analysis, synthesis, comparison, classification, generalization), with organization of control over their actions and their evaluation.
Understanding these areas at the methodological level is an urgent task of modern methodological science.
§ 3. OBJECTIVES OF THE METHOD OF TEACHING MATHEMATICS
AS A SUBJECT
The main objective of the course "Methods of Teaching Mathematics in Primary School" in college and 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 in communication and cooperation in the process of assimilation of 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 a methodological course, students learn to apply this knowledge to solve methodological problems. Consequently, the methodological activity of the teacher 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 training tasks, includes:
Regularities of the processes of education and upbringing;
Psychological features of the development of the child and the assimilation of knowledge, skills and abilities.
How better 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 primary teaching of mathematics and analyze it from the point of view of the concept of "methodological task".
Imagine that you 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 that the student gave the wrong answer, i.e. completed entry 68. What will you do? Contact another student or try to figure out the reasons for the mistake? In other words, how will you solve this methodological problem?
"Davydov V.V. The concept of humanization of Russian primary education. - Sat. "Primary education in Russia." - M., 1994.
The choice of methodological actions in this case can be determined by a whole range of psychological and pedagogical factors: the personality of the student, the level of his mathematical training, the purpose for which this task was offered, etc. understand the causes of the error. But = to do it?
If the student reads it as “six is less than eight”, then the reason for the error is “: and that the mathematical symbol has not been mastered. Children simultaneously get acquainted with know and, therefore they may confuse their meanings.
Having established the cause in this way, you can continue to work. But at the same time
It is necessary to take into account the peculiarities of the perception of the younger student. Since it has
Visually-shaped character, then the teacher uses the method of comparing the sign with a conet (for a child) image, for example, with a beak that is open to a larger number and closed to a smaller one (5 8, 8 5). Such a comparison will help the child remember mathematical symbolism.
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 knowing such mathematical concepts as "quantitative number", "establishment of one-to-one correspondence" and the set-theoretic approach to determining the relation "more" ("less"). This will allow him to choose the right way to organize the activities of students associated with the implementation given task. Considering the visual-effective nature of the thinking of younger students, the teacher invites one student to lay out 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 suggest a course of action or find it with the help of a teacher, i.e., establish a one-to-one correspondence between the elements of these subject sets.
§ §§!§ till id Now imagine that the student successfully completes the task of comparing numbers. In this case, it is important to establish how conscious his actions are, that is, whether he can justify them, while expressing the necessary reasoning that is related to 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 number series”, since they are the basis of the rationale that the student can give: “The number that is called earlier when counting 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 suggest underlining the numbers 6 and 8 (1, 2, 3, 4, 5, 6, 7, 8, 9) in it or designate these numbers on the number line.
Thus, the process of a student performing a rather simple task required the teacher to solve four methodological problems and apply mathematical, psychological, and methodological knowledge.
Consider another situation related to written division by a single digit. For example, 8463:7. Each of you, of course, can easily cope with this task.
But suppose that the student received in the answer not 1209, but 129, i.e. he missed a private zero (this typical mistake). The reason for such an error may be either his inattention or lack of necessary knowledge and skills.
How to find out? Probably, by analogy with the first situation, you will already be able to answer this question: “It is necessary for the student to say the actions that he performed.” In the methodology, this technique is called "commenting".
The use of this technique allows the teacher to control the correctness of not only the final result, but also the process of obtaining it, and thereby correct the activity of students in using the algorithm.
But in order to teach children to consciously comment on the sequence of operations that are included in the written division algorithm, the teacher must himself 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 zero in the quotient is usually commented as follows: "6 is not divisible by 7 - we put zero." This formal explanation can be more justified if we rely on the concept of division with a remainder.
Recall the definition that you considered in the course of mathematics: “To divide with a remainder a non-negative integer a by a natural number b means to find non-negative integers q and r such that a = bq + g\lo r b.”
Understanding that this definition is the basis of students' actions when performing division with a remainder will allow the teacher to methodically correctly organize their activities to master these methods. For example, when performing division for the case 29:4, students first find the largest number up to 29, which is divisible by 4 without a remainder (this operation requires a solid mastery of 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 transfer the same reasoning to the case of 6:7. The largest number up to 6 that is divisible by 7 is 0. 0:7=0. Find the remainder by subtracting 6-0=6. End result: 6:7=0 (rest. 6). So knowing mathematical concepts helps the teacher to find reasonable ways to explain to students the actions that they perform.
Mathematical knowledge is necessary for the teacher in order to properly organize the acquaintance of younger students with new concepts. For example, some teachers try to explain cases of multiplying by 1 like this: "The number was repeated once, so it remained." When studying the case of division by 1, they turn to specific example: “Imagine that the boy has 5 apples. He kept them all for himself, that is, he divided them by 1, which is why he got 5 apples. It would seem that the methodological actions of the teacher 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, there is no mathematical basis in his actions, without which correct mathematical representations and concepts cannot be formed.
It is clear that the teacher's methodical actions in teaching mathematics to younger students largely depend on the level of his mathematical training. In addition, mathematical training has a positive effect on the clarity of the teacher's eyes, on the correct use of terminology and the validity of the selection of methodological techniques related to 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 educating and developing 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. The basis of these skills is his psychological and pedagogical knowledge. So, the lack of a teacher psychological knowledge about the peculiarities of the attention of younger schoolchildren leads to the fact that, organizing their attention, he usually uses only the method of setting, that is, he says: "be careful." If this installation does not work, he resorts to various measures of punishment. But it is enough to understand the psychological essence of his actions in order to understand their fallacy. Namely: the “be careful” setting is designed mainly for the arbitrary attention of children. This kind of attention requires strong-willed efforts and quickly tires them. Therefore, the effectiveness of this installation is very short-lived. In an attempt to reinforce it, some teachers, when asking a question to the whole class, ask exactly the student who is in this moment got distracted. Naturally, he cannot answer. The teacher begins to shame him, lecture him, punish him. But this only increases the mental load and causes negative emotions in the child:
feeling of fear, insecurity, anxiety. How to avoid it? Knowledge of psychological patterns will help the teacher to find the right solution.
In psychology, for example, the following pattern has been established: the attention of students is activated if: a) mental activity is accompanied by motor activity; b) the objects that the student operates with are perceived visually.
In addition to regularities, psychological science the conditions under the influence of which attention is maintained are highlighted. These include: a) intensity, YENISEI!
P "Duchnlyash"
Novelty, the unexpectedness of the 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 associated with specific mathematical content, the use of subject visualization, observation techniques, comparisons, 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 goal, but without volitional efforts. This plays an important role in the construction of education, as it opens up the prospect of purposeful control of children's attention for the teacher.
But it is quite possible that there may be situations where 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: “Now Kolya will offer you tasks for oral counting, which are written out on cards. He will control the correctness of their decision.” As a result, Kolya is included in the work, experiencing positive emotions caused by the trust that the teacher has placed in him.
In the above examples, 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 connected with the solution of design problems, which he thinks through in preparation for the lesson, choosing the way of setting the learning task, selecting the learning task for its solution.
As you can see, the methodological activity of the teacher is associated with the solution of various methodological problems. The formation of the ability to identify, set and solve them is one of the important tasks of the methodological course.
Task 3. Give examples of methodological tasks, the solution of which you observed in pedagogical practice.
Can you, using your psychological, pedagogical and mathematical knowledge, suggest other options for action in the lesson?
LEARNING ACTIVITY OF THE JUNIOR STUDENT
IN THE PROCESS OF TEACHING MATH
§ 1. THE CONCEPT OF LEARNING ACTIVITY AND ITS STRUCTURE
Activity is a form of an active attitude of a person to the surrounding reality. It is primarily characterized by the presence of a goal and is caused by various needs and interests (motives).Educational activity is aimed directly at the assimilation of knowledge, skills and abilities, its content is scientific concepts and general methods for solving practical problems. Being leading 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 personality structure and activity, those mental and social changes that first occur at a given stage and in the most important and fundamental way determine the child’s consciousness, his attitude to the environment, his inner and outer life, the whole course of his development during this period.
The structure of learning activity includes the following components: motives, learning objectives, methods of action, as well as self-control and self-assessment. The relationship of these components ensures the integrity of learning activities.
The motive is the motivating force of the activity, for the sake of which it is carried out. The motives of learning activity are dynamic and change depending on the social attitudes of the individual. Initially, they are formed under the influence of external factors in relation to educational activity, not related to its content.
With the help of thinking, the student evaluates different motives, compares them, correlates them with his beliefs and aspirations, and after an emotional assessment of these motives, he proceeds to learning activities, realizing their necessity. Therefore, the learning process should be structured in such a way that the tasks that are set for 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 a cognitive motivation that is closely related to the content and methods of learning.
Motivation (i.e., the student's focus on learning activities) most often arises when a learning task is set. But in some cases, it can also appear in the process of the activity itself, its control and self-assessment. This is usually facilitated by the student's successful completion of those learning tasks that the teacher offers both in the process of solving a learning problem and at the stage of self-control.
"Vygotsky L. S. Pedagogical psychology. - M., 1991.
§ 2. LEARNING TASK AND ITS TYPES A learning task is a key component of learning activity.
On the one hand, it clarifies the general goals of learning, specifies 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, tasks). For example, mastering the written multiplication algorithm is a learning task that is solved in the process of performing a certain system of learning tasks (exercises). It is obvious that several, often many mathematical tasks (exercises) can be used to solve one educational problem. At the same time, in the process of performing one mathematical task (exercises), several educational tasks can be solved.
For example:
Numbers are given: 18, 81, 881, 42, 442, 818. On what basis can these numbers be divided into two groups?
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The purpose of this course is the formation of mathematical ZUN and the general development of students. The concept of the course is the purposeful development of the thinking of all students in the process of mastering the program content. The course is built on the thematic principle and is focused on mastering the system of concepts and general methods of action. At the same time, the repetition of previously studied issues is organically included in all stages of assimilation of new content.
The organization of such a productive repetition ensures continuity between topics and creates conditions for the active use of mental activity techniques in the process of assimilation of mathematical content. Thus, at the methodological level, the psychological and pedagogical ideas of developmental education are implemented.
In Istomina's program, the sequence of studying some issues of the program has been changed in comparison with the Moro program. The geometric line has been significantly strengthened and the use of calculators is envisaged when performing a number of tasks.
The essence of this concept is connected with certain answers to 3 main questions of methodological science:
1. why teach?
2. what to teach?
3.how to teach?
The answer to the first question “why teach?” was reflected in the orientation of the course in elementary mathematics on the formation of mental activity techniques in schoolchildren (analysis, synthesis, generalization, classification, etc.), which perform various functions in the process of teaching mathematics and can be considered:
1.how to organize the educational activities of students
2. as ways of knowing that become the property of the child, characterizing his intellectual potential and ability to assimilate knowledge
3. as ways to include in the knowledge of various mental processes: emotions, will, feelings and attention.
As a result, the child's intellectual activity enters into various relationships with other aspects of his personality, primarily with its orientation, motivation, interests, level of claims, i.e. characterized by increasing activity of the individual in various areas of its activity.
The question "How to teach?" is the core concept of the course. The answer to it requires, first of all, the adoption of a certain position in relation to the process of assimilation of knowledge by children, the formation of skills and abilities. Depending on the answer to this question, 2 positions can be distinguished:
In one case, knowledge and methods of action are offered to students in the form of a model known to the teacher, which children must remember and reproduce. Then by training exercises"work them out".
In another case, the student is first involved in the activity, he has a need to learn new knowledge, the ion himself obtains them under the guidance of the teacher.
The second position, according to psychologists, is more effective for the development of thinking, but it requires significant changes in the organization of educational activities of schoolchildren. It was these changes that necessitated the creation of textbooks, which reflected:
1. A new logic for constructing the content of the course, which is based on the thematic principle, which allows you to orient the course towards the assimilation of a system of concepts and general methods of action.
2. new methodological approaches to the assimilation of mathematical concepts by schoolchildren, which are based on established correspondences between subject verbal, graphic, schematic and symbolic models, as well as the formation of their general ideas about changing rules and dependence, which is the basis not only for studying mathematics, but for the regularity and dependence of the surrounding world.
3. A new system of educational tasks, which is adequate to the concept of the course of the logic of building its content and is aimed at understanding the learning tasks by schoolchildren, at mastering the methods for solving them and at forming the ability to control and evaluate their actions.
4. A new methodological approach to teaching problem solving, which is focused on the formation of generalized changes: read the problem, highlight the condition and the question, establish the relationship between them and, using mathematical concepts, transfer the verbal model to the symbolic one.
5. Active use of mental activity techniques in the formation of geometric representations, focus on the development of spatial thinking of schoolchildren and the ability to establish correspondences between models of geometric shapes, their image and scan. Along with this, students master the skill of working with a ruler, compass and square.
6. The method of using a calculator, which is considered as a means of teaching mathematics to younger students, with certain methodological capabilities.
7.Organization of differentiated learning.
8. Dialogues of Masha and Misha, which help to teach younger students to analyze the proposed information, condemn it, express and justify their point of view.
Educational literature 1. Istomina NB Methods of teaching mathematics in elementary grades: Textbook for students of higher and secondary ped. textbook establishments. – 4th ed. , erased - M. : Academy Publishing Center, 2001. - 288 p. 2. Bantova M. A., Beltyukova G. V. Methods of teaching mathematics in primary grades: A textbook for school students. dept. ped. Uchsch - 3rd ed. , corr. - M. : Enlightenment, 1984. - 335 p. 3. Kalinchenko A. V., Shikova R. N., Leonovich E. N. Methods of teaching the initial course of mathematics: textbook. allowance for students. medium institutions. prof. education - 2nd ed. , erased - M.: Publishing Center "Academy", 2014. - 208 p. 4. Tikhonenko A. V., Rusinova M. M., Nalesnaya S. L., Trofimenko Yu. V. Theoretical and methodological foundations of the study of mathematics in elementary school - Rostov n / D: Phoenix, 2008. -349 p.
Questions of methodology What to teach? How to teach? Content of training 1. Requirements of the Federal State Standard of Primary General Education of the Second Generation (FSES IEO) 2. Programs for teaching mathematics in primary school learning 4. Means of learning Way of learning 5. Forms of learning
The content of teaching mathematics in elementary school 1) the use of basic mathematical knowledge to describe and explain the surrounding objects, 12. Subject results of mastering the main processes, phenomena, as well as assessing their quantitative and spatial relationships; educational program of primary general education 2) mastering the basics of logical and algorithmic thinking, spatial imagination and mathematical speech, measurement, recalculation, estimation and evaluation, visual presentation of data and taking into account the specifics of the content of subject areas, processes, recording and execution of algorithms; 3) the acquisition of initial experience in the application of mathematical knowledge to solve educational and cognitive tasks that include specific academic subjects, should be educational and practical tasks; 4) the ability to perform oral and written arithmetic operations with numbers and numerical expressions, solve textual reflections: tasks, the ability to act in accordance with an algorithm and build simple algorithms, explore, recognize and 12. 2. Mathematics and computer science: depict geometric shapes, work with tables, charts, graphs and diagrams, chains, collections, present, analyze and interpret data; 5) acquisition of initial ideas about computer literacy.
The program of teaching mathematics in elementary grades "School of Russia" Moro M.I., Volkova S.I., Stepanova S.V. and others. Mathematics. Work programs. Subject line of textbooks "School of Russia". Grades 1-4 1. Moro M. I., Volkova S. I., Stepanova S. V. Mathematics. 1 class. In 2 parts. - M. : Education, 2011 2. Moro M. I., Bantova M. A., Beltyukova G. V. Mathematics. Grade 2 In 2 parts. - M. : Education, 2011 3. Moro M. I., Volkova S. I., Bantova M. A. Mathematics. Grade 3 In 2 parts. - M. : Education, 2012 4. Moro M. I., Volkova S. I., Bantova M. A. Mathematics. 4th grade. In 2 parts. - M. : Enlightenment, 2014
The program of teaching mathematics in primary school "Harmony" Istomina NB Mathematics. Textbook for grades 1-4 educational institutions. In two parts. - Programs of educational institutions Smolensk: Association of the XXI century, 2014. Mathematics: program 1-4 grades. Lesson-thematic planning: grades 1–4 / N. B. Istomina. - Smolensk: Association XXI century, 2013. - 160 p.
The program of teaching mathematics in elementary grades "Perspektiva" Peterson LG Mathematics. Work programs. Subject line of textbooks of the system "PERSPECTIVE" grades 1-4. Handbook for teachers of educational institutions. - 2nd ed. - M. : Enlightenment, 2011 Peterson L. G. Mathematics "Learning to learn." 1-4 class. In 3 parts. Textbook set "Textbook + workbooks". - M. : Yuventa, 2013
The program of teaching mathematics in primary grades "School 2100" Demidova T. E., Kozlova S. A., Tonkikh A. P. Mathematics. Textbook for grades 1-4 in 3 parts. - M. : Balass, 2012 Educational system "School 2100". Federal state educational standard. Approximate basic educational program. In 2 books. Book 1. Book 2. Elementary school. Preschool education/ Under scientific. ed. D. I. Feldstein. -M. : Balass, 2011. - 192 p. (Educational system "School 2100"). PROGRAM "MATHEMATICS" for a four-year elementary school / T. E. Demidova, S. A. Kozlova, A. G. Rubin, A. P. Tonkikh
The program of teaching mathematics in primary grades "Planet of Knowledge" Programs of educational institutions. Primary School. 1-4 classes. - M. : Astrel, 2012 Bashmakov M. I., Nefedova M. G. Mathematics. 1-4 class. In 2 parts. Textbook. - M. : Astrel, 2011
What to teach in mathematics lessons in elementary school? 1. Numbering 2. Arithmetic operations (addition, subtraction, multiplication and division), their properties, oral and written algorithms 3. Values and their measurement 4. Arithmetic operations with numbers obtained during measurement 5. Algebraic material 6. Shares, ordinary fractions , finding a number by its part and part of a number 7. Geometric material
ANO secondary school "Dimitrievskaya",
MO elementary school teachers
Abstract on the topic of self-education
Features of the organization of students' activities in mathematics lessons when studying the topic "Problem Solving" according to the textbook by N.B. Istomina
Made 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
- 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 Good to study! And one of the powerful levers for the emergence of the desire and ability to study well is the creation of conditions that ensure the child's success in work, a sense of joy on the path of progress from ignorance to knowledge, from inability to skill, i.e. awareness of the meaning and result of their efforts. “Vain, fruitless work even for an adult becomes hateful, stupefying, meaningless, and yet we are dealing with children,” wrote Z.A. Sukhomlinsky.
If all children cope with the task set before them, if they work with passion and pleasure, helping each other, if they go home satisfied 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 teacher's work. “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 ... ”(Z.A. 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 methodology of teaching mathematics in the primary grades are contained.
To the features of the concept underlying the construction of the initial course of mathematics N.B. Istomina, include the following:
- a new logic for constructing the content of the course, which is based on the thematic principle, which makes it possible 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 repeating 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 of their general ideas about change, rule (regularity) and dependence, which is a reliable basis not only for further studying mathematics, but also for understanding the patterns and dependencies of the world around them in their various interpretations;
- a new system of educational tasks, the process of performing which is productive, compiled taking into account the psychological characteristics of younger students, is determined by maintaining a balance between logic and intuition, word and visual image, conscious and subconscious, conjecture and reasoning;
- a technique for the formation of geometric representations, which is based on the active use of mental activity techniques, a focus on the development of spatial thinking of schoolchildren and the ability to establish correspondences between models of geometric bodies, their image and scan;
- the possibility of using a calculator in the process of teaching mathematics to younger students, while the calculator is considered not only and so much as a computing device, but as a means of organizing the cognitive activity of students.
And finally
- a new methodical approach to teaching problem solving, which is focused on the formation of generalized skills: to read a problem, highlight a condition and a question, establish a relationship between them, consciously use mathematical concepts to answer a question of a problem.
In our work, we will consider the features of the organization of students' activities in mathematics lessons in the formation of skills to solve problems according to the textbook by N.B. Istomina.
1. Features of the methodological approach to teaching problem solving in the course of N.B. Istomina.
In the primary school mathematics course, text 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 the theory and practice of teaching, contribute to the development of students' thinking.
A special place in the course of primary school mathematics has always been given 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 technique focuses students on memorizing and recognizing the types of simple tasks, on consolidating the skills of solving problems of this type. But it forms a formal approach to problem solving.
Traditionally, younger students start solving text problems quite early. True, at first these are simple tasks, for the solution of which it is necessary to perform one arithmetic operation (addition or subtraction). But already at this stage, students are introduced to the structure of the problem (condition, question), with such concepts as known, unknown, data sought, with a brief record 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, identify known and unknown values 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 introduce children to the structure of the word problem and its solution later, when they learn to read?
But certain traditions have already developed in the teaching of mathematics. So they taught to solve problems in the course "Arithmetic", focusing on the types of simple problems and considering it as the main means of forming in younger students ideas about the specific meaning of arithmetic operations. The same methodology was reflected in mathematics textbooks (author M.I. Moro and others), which primary school teachers have been using since 1969. Later, additions were made to them related to the names of the structural components of the problem. The same methodological approach, in which a simple problem is the main means of forming mathematical concepts in younger students, remained in the 2002 edition of mathematics textbooks for grades 1–4, although it should be noted that the authors increased the preparatory period to familiarize students with the problem .
Representing a certain cognitive value, this approach has one significant drawback: when solving simple problems using 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 the count of objects. In this regard, writing down the solution of the 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. "It's 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 received, since the student can count the remaining carrots on the board. But now we need to write down the solution to the problem. “There are fewer carrots than there were, so 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 meaningless. 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 the actions from which he will choose. But it turns out that these ideas are only formed in younger students in the process of solving simple problems. And for the choice of arithmetic operations, everyday representations of children are used, which are oriented in most cases to words-actions in the text of the task: gave - took, was - left, came - left, flew away - arrived - or on the child's ability to imagine the situation that is described in the task . But not all children cope with this, because they were not taught this.
Therefore, the second question arises: maybe it is advisable to first explain to the children the meaning of the operations of addition and subtraction, and then proceed to solving simple problems?
Note that the progressive Russian Methodist F.A. Ern, who believed that the student must first form the concepts of arithmetic operations, and only after that - the ability to choose one or another action to solve this simple problem.
As you know, the process of solving a problem is associated with the selection of premises and the construction of conclusions. 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 when analyzing the text of the problem.
From the above reflections it follows that the solution of text problems should be preceded by a lot of preparatory work, the purpose of which is to form in younger students: a) reading skills; b) methods of mental activity (analysis and synthesis, comparison, generalization); c) ideas about the meaning of arithmetic operations, on which they can rely when searching for a solution to the problem.
Considering a text task as a verbal model of a situation (phenomenon, event, process), and its solution as a translation of the verbal model into a symbolic (mathematical) one - an expression, equality, equation, etc., it is advisable to create conditions for students to acquire experience in interpreting a given situation on various models. The means of creating these conditions can be a technique for forming students' ideas about the meaning of arithmetic operations, which is based on establishing a correspondence between verbal (verbal), subject, graphic (schematic) and symbolic models. Having mastered these skills before solving text 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 methodical approach to teaching younger students to solve text problems is the answer to the question of how to teach younger students to solve text problems.
The following features of the course in the formation of skills to solve problems can be distinguished:
- there is no division of tasks into simple and compound.
- abbreviation is completely excluded. Six-year-old and seven-year-old children do not yet have stable skills of simultaneous reading and understanding of the text. Consequently, the task must be translated from a verbal one into some other form, so that the child understands what is being reported, what is being asked in the task. The subject model is also not always able to 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. In order for the children to understand this task, you need to show a diagram on which they will see 5 apples. Thus, a schematic representation gives the most complete picture of the content of the problem.
- The work is not on solving problems of different types, but on various tasks for the formation of the ability to solve problems.
- It is possible to single out 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 reading skill is already formed in children at the proper level, and with special exercises in the 1st and early 2nd grades they are already prepared to develop the skills 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 some action, but to the conscious choice of this action itself. 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.
A methodical approach to teaching problem solving, laid down in the course of N.B. Istomina, includes 2 stages: preparatory and main.
Preparatory stage.
A necessary condition for the implementation of this approach in the practice of teaching is a specially thought out preparatory work for learning to solve problems. The preparatory stage begins in grade 1 and includes:
- the formation of students' reading skills. Without this skill, it is impossible to read the problem and, therefore, understand it and solve it;
- the assimilation by children of the specific meaning of addition and subtraction, the relations “more by”, “less by”, difference 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 a text that helps students to correctly establish the relationship between these values)
c) symbolic model (equalities and inequalities)
d) graphic (numerical beam);
- formation of methods of mental activity;
- the ability to add and subtract segments and interpret various situations with their help.
As mentioned above, to clarify the meaning of arithmetic operations, the method of correlating various models is used: subject, verbal, graphic and symbolic. Let's show how you can organize such activities for students in a particular lesson on the topic "Addition".
First version of the lesson
Teacher. Read the word that is written at the top of the page.
Children. Addition.
U. 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.
U. Today in the lesson we will try to figure out what addition is. Who can read the task? (No. 152). Tell me, what are Misha and Masha doing?
D. Misha and Masha put the fish in the same aquarium, they plant the fish together. Masha launches three fish into the aquarium, and Misha two; fish will swim together, etc.
Pay attention to how many important and necessary words characterizing the meaning of the “addition” action were uttered by the children. Note that they were not given any sample. Each of them worked at his own level and used only those words that he understood.
U. I will try to draw on the board 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 lays out two more fish on the flannelograph.
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.
U. You are very good at telling what is drawn in the pictures. And now let's try what you said in words, write down using mathematical signs. Look, under the pictures there are some entries in the frames. Maybe some of you can read them, but you probably don't know what they are called.
Some children try to guess the names of the records. Some say - examples, others - inequalities, others even - the multiplication table.
U. No, no one guessed. These entries are called "mathematical expressions".
D. And here it is written.
U. 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 upper left picture.
Focusing on the numbers, the children call the expressions 3 + 2 and 2 + 3 and explain what each number in the expression means: 3 is the number of fish that Masha launches into the aquarium, 2 is the number of fish that Misha launches into the aquarium.
U. 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 mean in the picture.
U. Now try to find expressions for other pictures on your own. Each of you has a piece of paper, which is divided into four parts. You must write down the expressions that match the bottom left picture and the bottom right picture.
The children complete the task on their own. The teacher watches their work, walks around the classroom, helps some children. Then he writes on the board, which is divided into four parts, mathematical expressions.
On the desk:
3 + 2 |
- Look at the desk. I wrote down two expressions that I saw from one student in a notebook. Does everyone agree with him?
D. This should be added to the top picture.
- This is not true. Here you need to write down 3 + 1 and 1 + 3, because Masha has 3 sweets, and Misha has one. They put them in one bowl.
U. Well, if I write down the expression 2 + 2 to the lower left picture, will that be true?
There are students who agree with this, since 2 + 2 is 4. But others object. This is not true, because Masha puts three sweets in a vase, and Misha puts one.
U. Now guess which picture fits the entry 4 + 5 = 9?
Look, there is a new sign here, which is called "equal", and the notation 4 + 5 = 9 is called "equality".
Equality can be true or false. What does "correct equality" mean?
Each of the equalities proposed in the textbook is written on the blackboard and tested on object models (these can be any objects).
4 + 5 = 9 |
To check equality, children count or count objects.
U. Let's now read in the textbook how Misha suggests checking the equalities.
(The drawing of the number beam, which the teacher puts on the board, is discussed..)
The names of the components can be entered in the second lesson on the topic. The second lesson also includes exercises in which children choose a picture 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, numerical beam) is actively involved. A simple task is replaced by a method of correlating different models: subject (drawings), verbal (description of pictures), graphic (drawing on a number line), symbolic (writing an expression, equality).
The second version of the lesson
There is a number line on the board. The teacher calls two students to the board. The children turn their backs to the class and the teacher gives each of them some objects.
The teacher comments:
U. I give 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 draws on the blackboard:
The teacher comments on his actions:
Lena has so many mushrooms (makes the first arc), and Vera has so many mushrooms (makes a second arc).
Who guessed how many mushrooms Lena has? How many mushrooms does Vera have? How many mushrooms do Lena and Vera have in total?
U. Let's see if you answered my questions correctly. Girls lay out mushrooms on a flannelograph (4 large and 4 small).
And now I will combine large and small mushrooms (draws a curved closed line, inside which are large and small mushrooms). Who can write down in the language of mathematics what I did?
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 graphical model to explain the meaning of addition, then moved on to the subject, then to the verbal one (the children described what they see in the picture) and then introduced them to the symbolic model (expression, equality).
Similarly, focusing on the page of the textbook, you can build a lesson when introducing children to subtraction.
Thus, the solution of simple problems is replaced by various exercises (learning tasks), in the process of performing which children learn the specific meaning of the actions of addition and subtraction. Here are the exercises: (notebook with a printed base 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 an object model, in which objects are located one below the other, then it is rather difficult for children to realize that the answer to the question “How much more (less)?” associated with the operation of subtraction. If the child is not aware of this connection, but only remembers the rule: “To find out how much one number is more than another, you need to subtract the smaller number from a larger number,” then when solving problems, he will focus only on an external sign, namely the word “ how much."
As an example, we can give the following problem: “At the bus stop, 3 girls and 7 boys got off the bus. How many fewer people were on the bus? (Up to 50% of children solve the problem by subtraction.)
Not representing the substantive meaning of the 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 process of completing which children learn the objective meaning of difference comparison: No. 261, 267 (textbook for the 1st grade), No. 18, 19, 24 (notebook with a printed base No. 2, 1st grade).
To develop in children the ability to imagine a situation described in words, tasks are offered for correlating verbal and object models: No. 393, 402 (textbook for the 1st grade).
In the first quarter of the 2nd grade, students get acquainted with the scheme: No. 41, 42, 49, 58 (textbook for the 2nd grade).
Main stage.
The main period of learning to solve problems begins with getting to know the problem, its structure. This material is well presented in the 2nd grade textbook in the form of a dialogue between the heroes of the textbook Masha and Misha (pp. 49-51: No. 129). From this dialogue, students will learn what text can be called a task, that the task consists of a condition and a question related to each other.
1) Comparison of texts of tasks, identification of their similarities and differences: No. 131, 132,138, 149 (textbook for the 2nd grade).
2) Drawing up tasks according to the given conditions and question: No. 35 (a), 36 (a) (notebook "Learning to solve problems", grades 1–2).
3) Translation of the verbal model of the problem or its conditions into a schematic model: No. 41 (a), 43 (a) (notebook "Learning to solve problems", grades 1–2).
4) Choice of scheme No. 44 (a) (notebook "Learning to solve problems", grades 1–2).
5) Completion of the started scheme corresponding to this task: No. 49 (a), 59 (a), (b) (notebook "Learning to solve problems", grades 1–2).
6) Explanation of expressions compiled according to the condition of the problem: No. 179 (textbook for the 2nd grade).
7) Choice of questions corresponding to 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 issue: No. 230 (textbook for the 2nd grade).
9) Supplementing the text of the problem in accordance with this decision: No. 65 (notebook "Learning to solve problems").
10) Addition of the text of the problem in accordance with this scheme: No. 42 (a), (b), No. 72 (a), (b).
11) Choice of the task corresponding to the given scheme: No. 77.
12) Choosing a solution to this problem: No. 37 (notebook).
13) Statement of various questions to this condition and recording of the expression corresponding to each question: No. 34 (notebook).
14) Designation on the diagram of known and unknown quantities in the problem: No. 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 of various problems on their own. If children have difficulties, the teacher can use any combination of methodological techniques, depending on the content of the task.
Math lesson
2nd grade
Topic. "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 one).
Teacher. We continue today in the lesson to learn how to solve problems. This will help us with tasks from the notebook "Learning to solve problems". Open task number 48. Read task (a) to yourself, then aloud.
– Now read task (b).
Let's try to do the task on our own. This will help you to conclude whether you understood the text of the problem or not.
Children work independently (use a simple pencil). Everyone copes with the task by choosing scheme 4 and denoting on it the quantities known in the condition of the problem. The teacher opens on the blackboard the schemes drawn in advance, the same as in a notebook with a printed basis.
Teacher. Who wants to draw a diagram on the board?
There are many who wish. Two students come to the board and quickly "revive" scheme 4:
Teacher. Read assignment 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 are having difficulty to the board. Three children come to the board in turn. Each represents one question on the diagram.
The diagram on the board looks like this:
U. Now you can independently answer each question by writing arithmetic operations.
All children quickly cope with the first question: 7 + 2 = 9 (l.). The second question is also not difficult. Everyone has an entry in their notebooks: 9 + 3 = 12 (l.). Children carefully study the scheme, comparing it with the actions already performed. The teacher writes the children's answers on the blackboard and invites them to discuss:
Children. 12 - 9 = 3 is wrong. 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.
U. And who will tell me how much Masha is younger than Lena?
D. No action is required here; how much Lena is older than Masha, how much Masha is younger than Lena.
U. 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 is visible on the diagram. (He goes to the blackboard and shows a segment equal to the sum of two segments: one indicates the number 2, and the other the number 3.)
U. I think without a diagram it would be difficult to suggest this way of answering the question.
The children agree with the teacher.
U. 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 age, 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.)
U. Would such a condition fit? Masha is the same age as Vera. And Lena is 3 years older than Vera.
D. In general, it will do. Just don't answer a single question.
–
If you ask a question, you get a task in which there is not enough data.
Similar work is carried out with scheme 2. Children "revive" the scheme on the board and orally answer the same questions.
The third question changes: "How many years is Lena younger than Masha?"
U. 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 an arbitrary segment.
Children draw a segment, after which they open task No. 159 from the textbook.
Read the task.
Let's answer the question first.
D. Here the beginning is exactly the same.
U. I do not understand something, what does the beginning mean?
D. Well, conditions are the same...
- I disagree. Conditions are different. The left problem does not say how many chairs were in the hall, but the second one says: there were 84 chairs in the hall.
D. There is not enough data in the left task.
U. What is missing? To answer the first question?
D. No, the first question can be answered, but the second cannot.
U. Well, in the second task, can you answer two questions?
D. In the second it is possible.
U. Let's mark all the chairs in the hall with the line segment that you drew. Using this segment, draw a diagram that matches the problem.
Children work independently. The teacher draws a diagram on the board:
The children are discussing it.
D. Well, everything here is wrong. After all, you said to mark all the chairs in the hall with a segment.
D. I drew like this. (He goes to the board, draws a segment by hand and marks it.)
On the desk:
“Now let’s take out the chairs.” (Draws on the diagram and comments.)First they took out 24 chairs, then 10 more.
U. Well, let someone else put the questions according to the scheme.
Children complete the diagram.
– Write the solution to the problem in your notebook.
The children write their own solution. The teacher helps those who are in trouble. Those who quickly wrote down the solution to the problem are invited to complete task No. 162.
Children enjoy doing it. For the rest, it is written on the board: "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 students' horizons, a 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 ability of students 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 manuals.
Conclusion
Solving problems, students acquire new mathematical knowledge, prepare for practical activities. Tasks contribute to the development of their logical thinking. Of great importance is the solution of problems in the education of the personality of students.
Acting as a specific material for the formation of knowledge, tasks provide an opportunity to connect theory with practice, learning with life. Solving problems forms in children the practical skills necessary for every person in everyday life. For example, calculate the cost of a purchase, calculate what time you need to leave 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 are performed 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, reveal the connections that exist between the phenomena under consideration.
Problem solving - exercises that develop thinking. Moreover, problem solving contributes to the development of patience, perseverance, will, contributes to the awakening of interest in the very process of finding a solution, makes it possible to experience deep satisfaction associated with a successful solution.
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 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 the work on 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 N. B. Mathematics. Grade 1: Textbook for a four-year-old
2. Istomina N. B. Mathematics. Grade 2: Textbook for a four-year-old
elementary school. - Smolensk: Association XXI century, 2000.
3. Istomina N. B. Methods of teaching mathematics in primary classes. – M.:
LINKA - PRESS, 1997.
4. Istomina N.B. We learn to solve problems. 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 - v. 42, M., 1992.
