Lecture on the topic: "Methods of teaching mathematics. The general concept of abilities

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

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

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

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

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

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

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

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

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

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

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

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

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

Ministry of Education, Science and Youth Policy of the Republic of Dagestan

GBOUSPO "Republican Pedagogical College" them. Z.N. Batyrmurzaev.

Course work

on TONKM with teaching methods

on the topic: " Active methods of teaching mathematics in elementary school"

Completed: St-ka 3 "in" course

Ezerkhanova Zalina

Supervisor:

Adilkhanova S.A.

Khasavyurt 2014

Introduction

Chapter I

Chapter II

Conclusion

Literature

Introduction

"A mathematician enjoys knowledge that he has already mastered, and always strives for new knowledge."

The effectiveness of teaching mathematics to schoolchildren largely depends on the choice of forms of organization of the educational process. In my work, I prefer active learning methods. Active learning methods are a set of ways to organize and manage the educational and cognitive activities of students, which have the following main features:

forced learning activity;

independent development of solutions by trainees;

a high degree of involvement of students in the educational process;

constant processing by communication between students and teachers, and control by independent work of learning.

The main meaning of the development of federal state educational standards, the solution of the strategic task of the development of Russian education - improving the quality of education, achieving new educational results. In other words, the Federal State Educational Standard is not intended to fix the state of education achieved at previous stages of its development, but orients education towards achieving a new quality that is adequate to the modern (and even predictable) needs of the individual, society and the state.

The methodological basis of the standards of primary general education of the new generation is a system-activity approach.

The system-activity approach is aimed at the development of the individual, at the formation of civic identity. Training should be organized in such a way as to purposefully lead development. Since the main form of organizing learning is a lesson, it is necessary to know the principles of building a lesson, an approximate typology of lessons and the criteria for evaluating a lesson within the framework of a system-activity approach and active methods of work used in the lesson.

At present, the student with great difficulty sets goals and draws conclusions, synthesizes material and connects complex structures, generalizes knowledge, and even more so finds relationships in them. Teachers, noting the indifference of students to knowledge, unwillingness to learn, low level of development of cognitive interests, try to design more effective forms, models, methods, conditions of learning.

The creation of didactic and psychological conditions for the meaningfulness of teaching, the inclusion of a student in it at the level of not only intellectual, but personal and social activity is possible with the use of active teaching methods. The emergence and development of active methods is due to the fact that new tasks have arisen for teaching: not only to give students knowledge, but also to ensure the formation and development of cognitive interests and abilities, skills and abilities of independent mental work, the development of creative and communicative abilities of the individual.

Active learning methods also provide a directed activation of the mental processes of students, i.e. stimulate thinking when using specific problem situations and conducting business games, facilitate memorization when highlighting the main thing in practical classes, arouse interest in mathematics and develop a need for self-acquisition of knowledge.

A chain of failures can turn away from mathematics and capable children on the other hand, learning should go close to the ceiling of the student's abilities: the feeling of success is created by the understanding that significant difficulties have been overcome. Therefore, for each lesson, you need to carefully select and prepare individual knowledge, cards, based on an adequate assessment of the student's capabilities at the moment, taking into account his individual abilities.

active method of teaching mathematics

For the organization of active cognitive activity of students in the classroom, the optimal combination of active learning methods is of decisive importance. It is very important for me to assess the work and the psychological climate in my lessons. Therefore, you need to try so that children not only actively study, but also feel confident and comfortable.

The problem of personality activity in learning is one of the most urgent in educational practice.

With this in mind, I have chosen the topic of the study: "Active methods of teaching mathematics in elementary school."

The purpose of the study: to identify, theoretically substantiate the effectiveness of the use of active methods of teaching younger students with learning difficulties in mathematics lessons.

Research problem: what methods contribute to the activation of cognitive activity in students in the learning process.

Object of study: the process of teaching mathematics to younger students.

Subject of study: the study of active methods of teaching mathematics in elementary school.

Research hypothesis: the process of teaching mathematics to younger students will be more successful under the following conditions if:

active teaching methods for younger students will be used in mathematics lessons.

Research objectives:

)study the literature on the problem of using active methods of teaching mathematics in elementary school;

2)To identify and reveal the features of active methods of teaching mathematics in elementary school;

)Consider active methods of teaching mathematics in elementary school.

Research methods:

analysis of psychological and pedagogical literature on the problem of studying active methods of teaching mathematics in elementary school;

supervision of younger students.

The structure of the work: the work consists of an introduction, 2 chapters, a conclusion, a list of references.

Chapter I

1.1 Introduction to active learning methods

Method (from the Greek methodos - the path of research) - a way to achieve.

Active teaching methods are a system of methods that ensure the activity and variety of mental and practical activities of students in the process of mastering educational material.

Active methods provide a solution to educational problems in different aspects:

The teaching method is an ordered set of didactic methods and means by which the goals of training and education are realized. Teaching methods include interrelated, sequentially alternating ways of purposeful activity of the teacher and students.

Any teaching method presupposes a goal, a system of actions, means of training and an intended result. The object and subject of the teaching method is the student.

Any one teaching method is used in its pure form only for specially planned teaching or research purposes. Usually the teacher combines different teaching methods.

Today there are different approaches to the modern theory of teaching methods.

Active teaching methods are methods that encourage students to actively think and practice in the process of mastering educational material. Active learning involves the use of such a system of methods, which is mainly aimed not at the presentation of ready-made knowledge by the teacher, their memorization and reproduction, but at the independent mastery of knowledge and skills by students in the process of active mental and practical activity. The use of active methods in mathematics lessons helps to form not just knowledge-reproductions, but the skills and needs to apply this knowledge to analyze, assess the situation and make the right decision.

Active methods ensure the interaction of participants in the educational process. When they are applied, the distribution of "duties" is carried out when receiving, processing and applying information between the teacher and the student, between the students themselves. It is clear that the active learning process on the part of the student bears a large developmental load.

When choosing active learning methods, one should be guided by a number of criteria, namely:

· compliance with the goals and objectives, the principles of training;

· compliance with the content of the topic being studied;

· according to the abilities of the trainees: age, psychological development, level of education and upbringing, etc.

· compliance with the conditions and time allotted for training;

· compliance with the capabilities of the teacher: his experience, desires, level of professional skills, personal qualities.

· Student activity can be ensured if the teacher purposefully and maximally uses assignments in the lesson: formulate a concept, prove, explain, develop an alternative point of view, etc. In addition, the teacher can use the techniques of correcting "intentionally made" mistakes, formulating and developing assignments for comrades.

· An important role is played by the formation of the skill of posing a question. Analytical and problematic questions like "Why? What follows? What does it depend on? require constant updating in work and special training in their formulation. The methods of this training are varied: from tasks for posing a question to the text in the lesson to the game "Who will ask more questions on a certain topic in a minute.

· Active methods provide a solution to educational problems in various aspects:

· formation of positive educational motivation;

· increasing the cognitive activity of students;

· active involvement of students in the educational process;

· stimulation of independent activity;

· development cognitive processes- speech, memory, thinking;

· effective assimilation of a large amount of educational information;

· development of creative abilities and non-standard thinking;

· development of the communicative-emotional sphere of the student's personality;

· revealing the personal and individual capabilities of each student and determining the conditions for their manifestation and development;

· development of skills of independent mental work;

· development of universal skills.

Let's talk about the effectiveness of teaching methods and talk in more detail.

Active teaching methods put the student in a new position. Previously, the student was completely subordinate to the teacher, now active actions, thoughts, ideas and doubts are expected from him.

The quality of education and upbringing is directly related to the interaction of thinking processes and the formation of conscious knowledge, solid skills, and active teaching methods in the student.

The activity of trainees is possible only if there are incentives. Therefore, among the principles of activation special place acquires motivation for educational and cognitive activity. Rewards are an important motivating factor. Primary school children have unstable learning motives, especially cognitive ones, so positive emotions accompany the formation of cognitive activity.

1.2 Application of active teaching methods in primary school

One of the problems that worries teachers is the question of how to develop a child's steady interest in learning, in knowledge and the need for their independent search, in other words, how to activate cognitive activity in the learning process.

If a game is a habitual and desirable form of activity for a child, then it is necessary to use this form of organizing activities for learning, combining the game and the educational process, more precisely, using a game form of organizing students' activities to achieve educational goals. Thus, the motivational potential of the game will be aimed at more effective mastering of the educational program by schoolchildren. And the role of motivation in successful learning cannot be overestimated. Conducted studies of students' motivation have revealed interesting patterns. It turned out that the value of motivation for successful study is higher than the value of the student's intellect. High positive motivation can play the role of a compensating factor in case of insufficiently high abilities of the student, however, in reverse direction this principle does not work - no abilities can compensate for the lack of a learning motive or its low severity and ensure significant academic success.

Goals school education that put the state, society and family before the school, in addition to acquiring a certain set of knowledge and skills, are the disclosure and development of the child's potential, the creation of favorable conditions for the realization of his natural abilities. A natural play environment, in which there is no coercion and there is an opportunity for each child to find their place, show initiative and independence, freely realize their abilities and educational needs, is optimal for achieving these goals.

To create such an environment in the classroom, I use active learning methods.

The use of active teaching methods in the classroom allows you to:

provide positive motivation for learning;

conduct a lesson at a high aesthetic and emotional level;

provide a high degree differentiation of learning;

increase the volume of work performed in the lesson by 1.5 - 2 times;

improve knowledge control;

rationally organize the educational process, increase the effectiveness of the lesson.

Active learning methods can be used at various stages of the educational process:

stage - the primary acquisition of knowledge. It can be a problematic lecture, a heuristic conversation, an educational discussion, etc.

stage - knowledge control (reinforcement). Methods such as collective thought activity, testing, etc. can be used.

stage - the formation of skills and abilities based on knowledge and the development of creative abilities; it is possible to use simulated learning, game and non-game methods.

In addition to the intensification of the development of educational information, active teaching methods make it possible to carry out the educational process just as effectively in the process of the lesson and in extracurricular activities. Teamwork, joint project and research activities, upholding one's position and tolerant attitude to other people's opinion, taking responsibility for oneself and the team form the personality traits, moral attitudes and value orientations of the student that meet the modern needs of society. But this is not all the possibilities of active learning methods. In parallel with training and education, the use of active teaching methods in the educational process ensures the formation and development of so-called soft or universal skills in students. These usually include the ability to make decisions and the ability to solve problems, communication skills and qualities, the ability to clearly formulate messages and clearly set goals, the ability to listen and take into account the different points of view and opinions of other people, leadership skills and qualities, the ability to work in a team, etc. And today, many already understand that, despite their softness , these skills in modern life play a key role both in achieving success in professional and social activities, and in ensuring harmony in personal life.

Innovation - important feature modern education. Education is changing in content, forms, methods, responds to changes in society, takes into account global trends.

Educational innovation- the result of the creative search of teachers and scientists: new ideas, technologies, approaches, teaching methods, as well as individual elements of the educational process.

The wisdom of the desert dwellers says: "You can lead a camel to water, but you cannot make him drink." This proverb reflects the basic principle of learning - you can create all the necessary conditions for learning, but knowledge itself will occur only when the student wants to know. How to make the student feel needed at every stage of the lesson, to be a full-fledged member of a single class team? Another wisdom teaches: "Tell me - I'll forget. Show me - I'll remember. Let me act on my own - and I'll learn" According to this principle, learning is based on one's own active activity. And therefore, one of the ways to improve performance in the study school subjects is the introduction of active forms of work at different stages of the lesson.

Based on the degree of activity of students in the educational process, teaching methods are conditionally divided into two classes: traditional and active. The fundamental difference between these methods lies in the fact that when they are applied, students create conditions under which they cannot remain passive and have the opportunity for an active mutual exchange of knowledge and work experience.

The purpose of using active teaching methods in elementary school is the formation of curiosity.Therefore, for students, you can create a journey into the world of knowledge with fairy-tale characters.

In the course of his research, the outstanding Swiss psychologist Jean Piaget expressed the opinion that logic is not innate, but develops gradually with the development of the child. Therefore, in lessons in grades 2-4, you need to use more logical tasks related to mathematics, language, knowledge of the world, etc. Tasks require the performance of specific operations: intuitive thinking based on detailed ideas about objects, simple operations (classification, generalization, one-to-one correspondence).

Let us consider several examples of the use of active methods in the educational process.

A conversation is a dialogical method of presenting educational material (from the Greek dialogos - a conversation between two or more persons), which in itself speaks of the essential specifics of this method. The essence of the conversation lies in the fact that the teacher, through skillfully posed questions, encourages students to reason, to analyze the studied facts and phenomena in a certain logical sequence and independently formulate the corresponding theoretical conclusions and generalizations.

The conversation is not a communication, but a question-answer method of educational work to comprehend new material. The main point of the conversation is to encourage students, with the help of questions, to reason, analyze the material and generalize, to independently "discover" new conclusions, ideas, laws, etc. for them. Therefore, when conducting a conversation to comprehend new material, it is necessary to pose questions in such a way that they require not monosyllabic affirmative or negative answers, but detailed reasoning, certain arguments and comparisons, as a result of which students isolate the essential features and properties of the objects and phenomena being studied and in this way acquire new knowledge. It is equally important that the questions have a clear sequence and focus, allowing students to deeply comprehend the internal logic of the acquired knowledge.

These specific features of the conversation make it a very active method of learning. However, the use of this method has its limitations, because not every material can be presented through conversation. This method is most often used when the topic being studied is relatively simple and when students have a certain stock of ideas or life observations on it, allowing them to comprehend and assimilate knowledge in a heuristic (from Greek heurisko - I find) way.

Active methods provide for conducting classes through the organization of students' gaming activities. The pedagogy of the game collects ideas that facilitate communication in the group, the exchange of thoughts and feelings, the understanding of specific problems and the search for ways to solve them. It has an auxiliary function in the entire learning process. The task of the pedagogy of the game is to provide methods that help the work of the group and create an atmosphere that makes the participants feel safe and well.

The pedagogy of the game helps the facilitator to realize the various needs of the participants: the need for movement, experiences, overcoming fear, the desire to be with other people. It also helps to overcome shyness, shyness, as well as existing social stereotypes.

For active teaching methods, a special place is occupied by the forms of organization of the educational process - non-standard lessons: a lesson - a fairy tale, a game, a journey, a scenario, a quiz, lessons - reviews of knowledge.

At such lessons, the activity of children increases, they are happy to help Kolobok escape from the fox, save ships from pirate attacks, store food for the squirrel for the winter. At such lessons, the children are in for a surprise, so they try to work fruitfully and complete various tasks as much as possible. The very beginning of such lessons captivates children from the first minutes: “We will go to the forest today for science” or “A floorboard creaks about something ...” Books from the series “I am going to a lesson in elementary school” and, of course, the work of teachers. They help the teacher prepare for lessons in less time, make them more meaningful, modern, and interesting.

In my work, feedback means have acquired particular importance, which make it possible to quickly obtain information about the movement of each student’s thoughts, about the correctness of his actions at any moment of the lesson. Means of feedback using to control the quality of assimilation of knowledge, skills. Each student has means of feedback (we make them ourselves at labor lessons or purchase them in stores), they are an essential logical component of his cognitive activity. These are signal circles, cards, numerical and alphabetic fans, traffic lights. The use of feedback tools makes it possible to make the work of the class more rhythmic, forcing each student to study. It is important that such work be carried out systematically.

One of the new means of checking the quality of education are tests. This is a qualitative way to test learning outcomes, characterized by such parameters as reliability and objectivity. Tests test theoretical knowledge and practical skills. With the advent of the computer in the school, new methods of activating learning activities open up for the teacher.

Modern teaching methods are mainly focused on teaching not ready-made knowledge, but activities for the independent acquisition of new knowledge, i.e. cognitive activity.

In the practice of many teachers, independent work of students is widely used. It is carried out in almost every lesson within 7-15 minutes. The first independent works on the topic are mainly educational and corrective in nature. With their help, operational feedback in learning is carried out: the teacher sees all the shortcomings in the knowledge of students and eliminates them in a timely manner. You can refrain from entering grades "2" and "3" in the class journal for the time being (putting them in a student's notebook or diary). Such an assessment system is quite humane, mobilizes students well, helps them to better comprehend their difficulties and overcome them, and improves the quality of knowledge. Students are better prepared for the test, their fear of such work disappears, the fear of getting a deuce. The number of unsatisfactory ratings, as a rule, is sharply reduced. Students develop a positive attitude towards business, rhythmic work, rational use of lesson time.

Do not forget about the restorative power of relaxation in the classroom. After all, sometimes a few minutes are enough to shake things up, have fun and actively relax, and restore energy. Active methods - "physical minutes" "Earth, air, fire and water", "Bunnies" and many others will allow you to do this without leaving the classroom.

If the teacher himself takes part in this exercise, in addition to benefiting himself, he will also help insecure and shy students to participate more actively in the exercise.

1.3 Features of active methods of teaching mathematics in elementary school

· use of an activity approach to learning;

· the practical orientation of the activities of the participants in the educational process;

· playful and creative nature of learning;

· interactivity of the educational process;

· inclusion in the work of various communications, dialogue and polylogue;

· use of knowledge and experience of students;

· reflection of the learning process by its participants

Another essential quality of a mathematician is an interest in regularities. Regularity is the most stable characteristic of an ever-changing world. Today cannot be like yesterday. You cannot see the same face twice from the same angle. Patterns are found at the very beginning of arithmetic. There are many elementary examples of regularities in the multiplication table. Here is one of them. Usually children like to multiply by 2 and by 5, because the last digits of the answer are easy to remember: when multiplied by 2, even numbers are always obtained, and when multiplied by 5, even easier, it is always 0 or 5. But even multiplying by 7 has its own patterns . If we look at the last digits of the products 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, i.e. by 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, we will see that the difference between the next and previous digits is: - 3; +7; - 3; - 3; +7; - 3; - 3, - 3. A very definite rhythm is felt in this row.

If we read the final numbers of the answers when multiplying by 7 in reverse order, then we get the final numbers from multiplying by 3. Even in elementary school, one can develop the skill of observing mathematical patterns.

During the period of adaptation of first-graders, one should try to be attentive to the little personality, support it, worry about it, try to get interested in learning, help so that further education for the child is successful and brings mutual joy to the teacher and student. The quality of education and upbringing is directly related to the interaction of thinking processes and the formation of conscious knowledge, solid skills, and active teaching methods in the student.

The key to the quality of education is love for children and a constant search.

The direct involvement of students in educational and cognitive activities during the educational process is associated with the use of appropriate methods, which have received the generalized name of active learning methods. For active learning, the principle of individuality is important - the organization of educational and cognitive activities, taking into account individual abilities and capabilities. This includes pedagogical techniques, and special forms of classes. Active methods help to make the learning process easy and accessible to every child. The activity of trainees is possible only if there are incentives. Therefore, among the principles of activation, a special place is occupied by the motivation of educational and cognitive activity. Rewards are an important motivating factor. Primary school children have unstable learning motives, especially cognitive ones, so positive emotions accompany the formation of cognitive activity.

The age and psychological characteristics of younger students indicate the need to use incentives to achieve the activation of the educational process. Encouragement not only evaluates the positive results visible at the moment, but in itself it encourages further fruitful work. Encouragement is the factor of recognition and assessment of the achievements of the child, if necessary - the correction of knowledge, a statement of success, stimulating further achievements. Encouragement contributes to the development of memory, thinking, forms cognitive interest.

The success of learning also depends on the means of visualization. These are tables, reference diagrams, didactic and handouts, individual teaching aids that help make the lesson interesting, joyful, and provide a deep assimilation of the program material.

Individual teaching aids (mathematical pencil cases, cash registers of letters, abacuses) ensure the involvement of children in the active learning process, they become active participants in the educational process, activate the attention and thinking of children.

1The use of information technology in the lesson of mathematics in elementary school .

In elementary school it is impossible to conduct a lesson without the involvement of visual aids, problems often arise. Where can I find the material I need and how best to demonstrate it? The computer came to the rescue.

1.2The most effective means of including a child in the creative process in the classroom are:

· gaming activity;

· creating positive emotional situations;

work in pairs;

· problem learning.

Over the past 10 years, there has been a radical change in the role and place of personal computers and information technology in society. Knowledge of information technology is put in the modern world on a par with such qualities as the ability to read and write. A person who skillfully and effectively masters technologies and information has another, new style thinking, a fundamentally different approach to the assessment of the problem that has arisen, to the organization of their activities. As practice shows, it is already impossible to imagine a modern school without new information technologies. Obviously, in the coming decades, the role of personal computers will increase and, in accordance with this, the requirements for computer literacy of primary school students will increase. The use of ICT in primary school classes helps students navigate the information flows of the world around them, master practical ways of working with information, and develop skills that allow them to exchange information using modern technical means. In the process of studying, diverse application and use of ICT tools, a person is formed who is able to act not only according to the model, but also independently, receiving the necessary information from the largest possible number of sources; able to analyze it, put forward hypotheses, build models, experiment and draw conclusions, make decisions in difficult situations. In the process of using ICT, the student develops, prepares students for a free and comfortable life in the information society, including:

development of visual-figurative, visual-effective, theoretical, intuitive, creative types of thinking; - aesthetic education through the use of computer graphics, multimedia technology;

development of communication skills;

the formation of skills to make the best decision or offer solutions in a difficult situation (use of situational computer games, focused on optimizing decision-making activities);

formation of information culture, skills to process information.

ICT leads to the intensification of all levels of the educational process, providing:

improving the efficiency and quality of the learning process through the implementation of ICT tools;

providing motivational motives (stimuli) that cause the activation of cognitive activity;

deepening interdisciplinary connections through the use of modern means information processing, including audiovisual, in solving problems from various subject areas.

The use of information technology in the classroom in elementary schoolis one of the most modern means of developing the personality of a younger student, the formation of his information culture.

Teachers are increasingly using computer capabilities in preparing and conducting lessons in elementary school.Modern computer programs make it possible to demonstrate vivid visualization, offer various interesting dynamic types of work, and reveal the level of knowledge and skills of students.

The role of the teacher in culture is also changing - he must become the coordinator of the information flow.

Today, when information becomes a strategic resource for the development of society, and knowledge is a relative and unreliable subject, as it quickly becomes obsolete and requires constant updating in the information society, it becomes obvious that modern education is a continuous process.

The rapid development of new information technologies and their introduction in our country have left their mark on the development of the personality of a modern child. Today, a new link is being introduced into the traditional scheme "teacher - student - textbook" - a computer, and school consciousness - computer training. One of the main parts of informatization of education is the use of information technologies in educational disciplines.

For an elementary school, this means a change in priorities in setting the goals of education: one of the results of education and upbringing in a first-stage school should be the readiness of children to master modern computer technologies and the ability to update the information obtained with their help for further self-education. To achieve these goals, there is a need to apply in the practice of the work of a primary school teacher different strategies for teaching younger students, and, first of all, the use of information and communication technologies in the educational process.

Lessons using computer technology make them more interesting, thoughtful, mobile. Almost any material is used, there is no need to prepare a lot of encyclopedias, reproductions, audio accompaniment for the lesson - all this is already prepared in advance and is contained on a small CD or flash card Lessons using ICT are especially relevant in elementary school. Pupils in grades 1-4 have visual-figurative thinking Therefore, it is very important to build their education, using as much high-quality illustrative material as possible, involving not only vision, but also hearing, emotions, and imagination in the process of perceiving the new. Here, by the way, we have the brightness and entertainment of computer slides, animations.

The organization of the educational process in elementary school, first of all, should contribute to the activation of the cognitive sphere of students, the successful assimilation of educational material and contribute to the mental development of the child. Therefore, ICT should perform a certain educational function, help the child understand the flow of information, perceive it, remember it, and, in no case, undermine health. ICT should act as an auxiliary element of the educational process, and not the main one. Given the psychological characteristics of a younger student, work using ICT should be clearly thought out and dosed. Thus, the use of ITC in the classroom should be sparing. When planning a lesson (work) in elementary school, the teacher must carefully consider the purpose, place and method of using ICT. Therefore, the teacher needs to master modern methods and new educational technologies in order to communicate in the same language with the child.

Chapter II

2.1 Classification of active methods of teaching mathematics in primary school on various grounds

According to the nature of cognitive activity:

explanatory and illustrative (story, lecture, conversation, demonstration, etc.);

reproductive (problem solving, repetition of experiments, etc.);

problematic (problematic tasks, cognitive tasks etc.);

partial search - heuristic;

research.

By activity components:

organizational and effective - methods of organization and implementation of educational and cognitive activities;

stimulating - methods of stimulation and motivation of educational and cognitive activity;

control and evaluation - methods of control and self-control of the effectiveness of educational and cognitive activity.

For didactic purposes:

methods of studying new knowledge;

methods of consolidating knowledge;

control methods.

By way of presentation of educational material:

monologic - information-reporting (story, lecture, explanation);

dialogic (problematic presentation, conversation, dispute).

According to the sources of knowledge transfer:

verbal (story, lecture, conversation, briefing, discussion);

visual (demonstration, illustration, diagram, display of material, graph);

practical (exercise, laboratory work, workshop).

According to the personality structure:

consciousness (story, conversation, instruction, illustration, etc.);

behavior (exercise, training, etc.);

feelings - stimulation (approval, praise, censure, control, etc.).

The choice of teaching methods is a creative matter, but it is based on knowledge of learning theory. Teaching methods cannot be divided, universalized or considered in isolation. In addition, the same teaching method may or may not be effective depending on the conditions of its application. The new content of education gives rise to new methods in teaching mathematics. An integrated approach is needed in the application of teaching methods, their flexibility and dynamism.

The main methods of mathematical research are: observation and experience; comparison; analysis and synthesis; generalization and specialization; abstraction and specification.

Modern methods of teaching mathematics: problematic (promising), laboratory, programmed learning, heuristic, building mathematical models, axiomatic, etc.

Consider the classification of teaching methods:

Information-developing methods are divided into two classes:

Transfer of information in finished form (lecture, explanation, demonstration of educational films and videos, listening to tape recordings, etc.);

Independent acquisition of knowledge (independent work with a book, with a training program, with information databases - the use of information technology).

Problem-search methods: problematic presentation of educational material (heuristic conversation), educational discussion, laboratory search work (preceding the study of the material), organization of collective mental activity in work in small groups, organizational and activity game, research.

Reproductive methods: retelling of educational material, performing exercises according to the model, laboratory work according to instructions, exercises on simulators.

Creative-reproductive methods: composition, variational exercises, analysis of production situations, business games and other types of imitation professional activity.

An integral part of teaching methods are the methods of educational activity of the teacher and students. Methodological techniques - actions, methods of work aimed at solving a specific problem. Behind the methods of educational work are hidden methods of mental activity (analysis and synthesis, comparison and generalization, proof, abstraction, concretization, identification of the essential, formulation of conclusions, concepts, methods of imagination and memorization).

2.2 Heuristic method of teaching mathematics

One of the main methods that allows students to be creative in the process of teaching mathematics is the heuristic method. Roughly speaking, this method consists in the fact that the teacher poses a certain educational problem to the class, and then, through successively set tasks, "leads" students to independently discover this or that mathematical fact. Students gradually, step by step, overcome difficulties in solving the problem and "discover" its solution themselves.

It is known that in the process of studying mathematics, students often face various difficulties. However, in heuristically designed learning, these difficulties often become a kind of incentive for learning. So, for example, if schoolchildren have an insufficient stock of knowledge to solve a problem or prove a theorem, then they themselves seek to fill this gap by independently “discovering” this or that property and thereby immediately discovering the usefulness of studying it. In this case, the role of the teacher is reduced to organizing and directing the work of the student, so that the difficulties that the student overcomes are within his power. Often the heuristic method appears in the practice of teaching in the form of the so-called heuristic conversation. The experience of many teachers who widely use the heuristic method has shown that it affects the attitude of students to learning activities. Having acquired a "taste" for heuristics, students begin to regard work on "ready-made instructions" as uninteresting and boring work. The most significant moments of their educational activity in the classroom and at home are independent "discoveries" of one or another way of solving a problem. There is a clear increase in students' interest in those types of work in which heuristic methods and techniques are used.

Modern experimental studies conducted in Soviet and foreign schools testify to the usefulness of the widespread use heuristic method in the study of mathematics by secondary school students, starting from the primary school age. Naturally, in this case, only those learning problems can be presented to students that can be understood and resolved by students at this stage of learning.

Unfortunately, the frequent use of the heuristic method in the process of teaching the posed educational problems requires much more study time than the study of the same issue by the method of communication by the teacher. ready solution(evidence, result). Therefore, the teacher cannot use the heuristic method of teaching in every lesson. In addition, long-term use of only one (even a very effective method) is contraindicated in training. However, it should be noted that "the time spent on fundamental questions worked out with the personal participation of students is not wasted time: new knowledge is acquired almost effortlessly thanks to the deep thinking experience previously gained." Heuristic activity or heuristic processes, although they include mental operations as an important component, at the same time have some specifics. That is why heuristic activity should be considered as a kind of human thinking that creates a new system of actions or reveals previously unknown patterns of objects surrounding a person (or objects of the science being studied).

The beginning of the application of the heuristic method as a teaching method - mathematics can be found in the book of the famous French teacher - mathematician Lezan "Development of mathematical initiative". In this book, the heuristic method does not yet have a modern name and appears in the form of advice to the teacher. Here are some of them:

The basic principle of teaching is "keep the appearance of the game, respect the freedom of the child, maintaining the illusion (if any) of his own discovery of the truth"; "to avoid in the initial upbringing of the child the dangerous temptation of abusing the exercises of memory," for this kills his innate qualities; teach based on interest in what is being studied.

Well-known methodologist-mathematician V.M. Bradis defines the heuristic method as follows: "A heuristic method is called such a teaching method when the leader does not inform students of ready-made information to be learned, but leads students to independently rediscover the relevant proposals and rules"

But the essence of these definitions is the same - an independent, planned only in common features ah search for a solution to the problem.

The role of heuristic activity in science and in the practice of teaching mathematics is covered in detail in the books of the American mathematician D. Poya. The purpose of heuristics is to investigate the rules and methods that lead to discoveries and inventions. Interestingly, the main method by which one can study the structure of the creative thought process is, in his opinion, the study of personal experience in solving problems and observing how others solve problems. The author is trying to derive some rules, following which one can come to discoveries, without analyzing the mental activity in relation to which these rules are proposed. "The first rule is to have the ability, and along with them good luck. The second rule is to hold firm and not retreat until a happy idea appears." The problem solving scheme given at the end of the book is interesting. The diagram indicates the sequence in which actions must be performed in order to succeed. It includes four stages:

Understanding the problem statement.

Drawing up a solution plan.

Implementation of the plan.

Looking back (studying the solution obtained).

During these steps, the problem solver must answer the next questions: What is unknown? What is given? What is the condition? Have I encountered this problem before, at least in a slightly different form? Is there any related task to this? Can't you use it?

From the point of view of applying the heuristic method at school, the book of the American teacher W. Sawyer "Prelude to Mathematics" is very interesting.

“For all mathematicians,” writes Sawyer, “the audacity of the mind is characteristic. The mathematician does not like to be told about something, he himself wants to get to everything”

This "impudence of the mind", according to Sawyer, is especially pronounced in children.

2.3 Special methods of teaching mathematics

These are the basic methods of cognition adapted for teaching, used in mathematics itself, methods of studying reality characteristic of mathematics.

PROBLEM LEARNING Problem-based learning is a didactic system based on the laws of creative assimilation of knowledge and methods of activity, including a combination of teaching and learning techniques and methods, which are characterized by the main features of scientific research.

The problematic method of learning is learning that proceeds in the form of the removal (permission) of successively created in educational purposes problematic situations.

A problem situation is a conscious difficulty generated by a discrepancy between the available knowledge and the knowledge that is necessary to solve the proposed problem.

A task that creates a problem situation is called a problem, or a problem task.

The problem must be understandable students, and its wording - to arouse the interest and desire of students to solve it.

It is necessary to distinguish between a problematic task and a problem. The problem is broader, it breaks down into a sequential or branched set of problematic tasks. A problem task can be considered as the simplest, particular case of a problem consisting of one task. Problem-based learning is focused on the formation and development of students' ability to creative activity and the need for it. It is advisable to start problem-based learning with problematic tasks, thereby preparing the ground for setting learning objectives.

PROGRAMMED LEARNING

Programmed learning is such learning when the solution of a problem is presented in the form of a strict sequence of elementary operations; in training programs, the material being studied is presented in the form of a strict sequence of frames. In the era of computerization, programmed learning is carried out with the help of training programs that determine not only the content, but also the learning process. There are two different systems for programming educational material - linear and branched.

As the advantages of programmed learning, one can note: the dosage of educational material that is assimilated accurately, which leads to high learning outcomes; individual assimilation; constant monitoring of assimilation; the possibility of using technical automated learning devices.

Significant disadvantages of using this method: not every educational material lends itself to programmed processing; the method limits the mental development of students to reproductive operations; when using it, there is a lack of communication between the teacher and students; there is no emotional-sensory component of learning.

2.4 Interactive methods of teaching mathematics and their benefits

The learning process is inextricably linked with such a concept as teaching methods. Methodology is not what books we use, but how our training is organized. In other words, teaching methodology is a form of interaction between students and teachers in the learning process. Within the framework of the current conditions of learning, the learning process is seen as a process of interaction between the teacher and students, the purpose of which is to familiarize the latter with certain knowledge, skills, abilities and values. Generally speaking, from the first days of the existence of education, as such, to the present day, only three forms of interaction between the teacher and students have developed, established and become widespread. Methodological approaches to learning can be divided into three groups:

.passive methods.

2.active methods.

.interactive methods.

A passive methodological approach is a form of interaction between students and a teacher, in which the teacher is the main active figure in the lesson, and students act as passive listeners. Feedback in passive lessons is carried out through surveys, self-study, tests, tests, etc. The passive method is considered the most inefficient in terms of student learning of educational material, but its advantages are relatively easy lesson preparation and the ability to present a relatively large amount of educational material in a limited time frame. Given these advantages, many teachers prefer it to other methods. Indeed, in some cases this approach works well in the hands of a skilled and experienced teacher, especially if the students already have clear goals for a thorough study of the subject.

An active methodological approach is a form of interaction between students and the teacher, in which the teacher and students interact with each other during the lesson and the students are no longer passive listeners, but active participants in the lesson. If in a passive lesson the teacher was the main acting figure, then here the teacher and students are on an equal footing. If passive lessons suggested an authoritarian style of learning, then active lessons suggest a democratic style. Active and Interactive methodological approaches have a lot in common. In general, the interactive method can be seen as the most modern form of active methods. Just unlike active methods, interactive ones are focused on a wider interaction of students not only with the teacher, but also with each other and on the dominance of student activity in the learning process.

Interactive ("Inter" is mutual, "act" is to act) - means to interact or is in the mode of conversation, dialogue with someone. In other words, interactive teaching methods are a special form of organizing cognitive and communicative activities in which students are involved in the process of cognition, have the opportunity to hire and reflect on what they know and think. The place of the teacher in interactive lessons is often reduced to the direction of students' activities to achieve the goals of the lesson. He also develops a lesson plan (as a rule, this is a set of interactive exercises and tasks in the course of which the student studies the material).

Thus, the main components of interactive lessons are interactive exercises and tasks that are performed by students.

The fundamental difference between interactive exercises and tasks is that in the course of their implementation, not only and not so much the already studied material is consolidated, but new material is studied. And then the interactive exercises and tasks are designed for the so-called interactive approaches. In modern pedagogy, a rich arsenal of interactive approaches has been accumulated, among which the following can be distinguished:

Creative tasks;

Work in small groups;

Educational games (role-playing games, simulations, business games and educational games);

Use of public resources (invitation of a specialist, excursions);

Social projects, classroom teaching methods (social projects, competitions, radio and newspapers, films, performances, exhibitions, performances, songs and fairy tales);

Warm-ups;

Learning and consolidating new material (interactive lecture, working with visual video and audio materials, "student as a teacher", everyone teaches everyone, mosaic (openwork saw), use of questions, Socratic dialogue);

Discussion of complex and debatable issues and problems ("Take a position", "opinion scale", POPS - formula, projective techniques, "One - together - all together", "Change position", "Carousel", "Discussion in the style of television talk - show", debate);

Problem solving ("Decision Tree", "Brainstorming", "Case Analysis")

Creative tasks should be understood as such study tasks, which require students not to simply reproduce information, but to be creative, since tasks contain a greater or lesser element of uncertainty and, as a rule, have several approaches.

The creative task is the content, the basis of any interactive method. An atmosphere of openness and search is created around him. A creative task, especially a practical one, gives meaning to learning, motivates students. The choice of a creative task in itself is a creative task for the teacher, since it is required to find a task that would meet the following criteria: does not have an unambiguous and monosyllabic answer or solution; is practical and useful for students; connected with the life of students; arouses interest among students; serve the purposes of education to the maximum. If students are not accustomed to working creatively, then you should gradually introduce simple exercises first, and then more and more difficult tasks.

Small group work - this is one of the most popular strategies, as it gives all students (including shy ones) the opportunity to participate in the work, practice the skills of cooperation, interpersonal communication (in particular, the ability to listen, develop a common opinion, resolve differences that arise). All this is often impossible in a large team. Work in small group an integral part of many interactive methods, such as mosaics, debates, public hearings, almost all types of imitations, etc.

At the same time, working in small groups requires a lot of time, this strategy should not be abused. Group work should be used when it is necessary to solve a problem that students cannot solve on their own. Group work should be started slowly. You can organize couples first. Pay special attention to students who have difficulty adjusting to work in a small group. When students learn to work in pairs, move on to work in a group, which consists of three students. As soon as we are convinced that this group is able to function independently, we gradually add new students.

Students spend more time presenting their point of view, are able to discuss an issue in more detail, and learn to look at an issue from different angles. In such groups, more constructive relationships are built between the participants.

Interactive learning helps the child not only learn, but also live. Thus, interactive learning is undoubtedly an interesting, creative, and promising area of our pedagogy.

Conclusion

Lessons using active learning methods are interesting not only for students, but also for teachers. But their unsystematic, ill-conceived use does not give good results. Therefore, it is very important to actively develop and implement your own game methods in the lesson in accordance with the individual characteristics of your class.

It is not necessary to apply these techniques all in one lesson.

In the classroom, quite acceptable working noise is created when discussing problems: sometimes, due to their psychological age characteristics, elementary school children cannot cope with their emotions. Therefore, it is better to introduce these methods gradually, cultivating a culture of discussion and cooperation among students.

The use of active methods strengthens the motivation for learning and develops the best sides student. At the same time, one should not use these methods without looking for an answer to the question: why do we use them and what consequences can there be as a result of this (both for the teacher and for the students).

Without well-designed teaching methods, it is difficult to organize the assimilation of program material. That is why it is necessary to improve those teaching methods and means that help to involve students in a cognitive search, in the labor of learning: they help teach students to actively, independently acquire knowledge, excite their thoughts and develop interest in the subject. A lot in mathematics various forms st. In order for students to be able to freely operate with them when solving problems and exercises, they must know the most common of them, often encountered in practice, by heart. Thus, the task of the teacher is to create conditions for the practical application of abilities for each student, to choose such teaching methods that would allow each student to show their activity, and also to activate the student's cognitive activity in the process of teaching mathematics. The correct selection of types of educational activities, various forms and methods of work, the search for various resources to increase the motivation of students to study mathematics, the orientation of students to acquire the competencies necessary for life and

activities in a multicultural world will allow you to get the required

learning outcome.

The use of active teaching methods not only increases the effectiveness of the lesson, but also harmonizes the development of the individual, which is possible only in vigorous activity.

Thus, active teaching methods are ways to enhance the educational and cognitive activity of students, which encourage them to active mental and practical activities in the process of mastering the material, when not only the teacher is active, but the students are also active.

Summing up, I will note that each student is interesting for his uniqueness, and my task is to preserve this uniqueness, grow a self-valuable personality, develop inclinations and talents, expand the capabilities of each Self.

Literature

1.Pedagogical technologies: Textbook for students of pedagogical specialties / under the general editorship of V.S. Kukushina.

2.Series "Pedagogical education". - M.: ICC "Mart"; Rostov n / a: Publishing Center "Mart", 2004. - 336s.

.Pometun O.I., Pirozhenko L.V. Modern lesson. Interactive technologies. - K.: A.S.K., 2004. - 196 p.

.Lukyanova M.I., Kalinina N.V. Educational activity of schoolchildren: the essence and possibilities of formation.

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.Kharlamov I.F. Pedagogy. - M.: Gardariki, 1999. - 520 p.

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.Modern ways of activating learning: a textbook for students. Higher textbook institutions / ed. T.S. Panina. - 4th ed., erased. - M.: Publishing center "Academy", 2008. - 176 p.

."Active teaching methods". Electronic course.

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13. Educational portal "My university",

Anatolyeva E. In "The use of information and communication technologies in the classroom in elementary school" edu/cap/ru

Efimov V.F. The use of information and communication technologies in the primary education of schoolchildren. "Elementary School". №2 2009

Molokova A.V. Information technology in traditional elementary school. Primary Education No. 1 2003.

Sidorenko E.V. Methods of mathematical processing: OO "Rech" 2001 pp. 113-142.

Bespalko V.P. Programmed learning. - M.: graduate School. Large encyclopedic Dictionary.

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Methods of teaching mathematics to younger students as a subject

Lecture 2

1. Methods of teaching mathematics to younger students as a subject

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

Consider the purpose of studying the course "Methods of teaching mathematics in elementary school" in the process of preparing a future elementary school teacher.

Discussion at a lecture with students

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

In the system of sciences, methodological sciences are considered in the block didactics. As you know, didactics is divided into theory of education and theory learning. In turn, in the theory of learning, general didactics is distinguished ( general issues: methods, forms, means) and private didactics (subject). Private didactics are also called differently - teaching methods or, as is customary in recent years, educational technologies.

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

The methodology of teaching mathematics to junior schoolchildren is a very ancient and very young science. Learning to count and calculate was a necessary part of education in ancient Sumerian and ancient Egyptian schools. The rock paintings of the Paleolithic era tell about learning to count. Magnitsky's Arithmetic (1703) and V.A. Lai "Guide to the initial teaching of arithmetic, based on the results of didactic experiments" (1910) ... In 1935, SI. Shokhor-Trotsky wrote the first textbook "Methods of Teaching Mathematics". But only in 1955, the first book “Psychology of teaching arithmetic” appeared, the author of which was N.A. Menchinskaya turned not so much to the characteristics of the mathematical specifics of the subject, but to the patterns of assimilation of arithmetic content by a child of primary school age. Thus, the emergence of this science in its modern form was preceded not only by the development of mathematics as a science, but also by the development of two large areas knowledge: general didactics of learning and psychology of learning and development. AT recent times an important role in the formation of teaching methods begins to play the psychophysiology of the development of the child's brain. At the intersection of these areas, answers to three “eternal” questions of the methodology of teaching subject content are born today:

1. Why teach? What is the purpose of teaching a young child math? Is it necessary? And if necessary, why?

2. What to teach? What content should be taught? What should be the list mathematical concepts designed to be explored with a child? Are there any criteria for selecting this content, the hierarchy of its construction (sequence) and how are they justified?

3. How to teach? What are the ways of organizing the activities of the child
(methods, techniques, means, forms of education) should be selected and applied so that the child can usefully assimilate the selected content? What is meant by “benefit”: the amount of knowledge and skills of the child or something else? How to take into account, when organizing training, the psychological characteristics of age and the individual differences of children, but at the same time "fit" in the allotted time (curriculum,
gram, daily routine), and also take into account the real content of the class in connection with the system of collective education adopted in our country (class-lesson system)?

These questions actually determine the range of problems of any methodological science. The methodology of teaching mathematics to junior schoolchildren as a science, on the one hand, is addressed to the specific content, selection and ordering of it in accordance with the goals of education, on the other hand, to the pedagogical methodological activity of the teacher and the educational (cognitive) activity of the child in the lesson, to the process of assimilation of the selected content managed by the teacher.

Object of study of this science is the process of mathematical development and the process of forming mathematical knowledge and ideas of a child of primary school age, in which the following components can be distinguished: the purpose of learning (Why teach?), content (What to teach?) and the activities of the teacher and the activities of the child (How to teach?) . These components form methodical system, in which a change in one of the components will cause a change in the other. Above, the modifications of this system were considered, which entailed a change in the purpose of primary education in connection with a change in the educational paradigm in the last decade. Later we will consider the modifications of this system, which entail the psychological-pedagogical and physiological research of the last half century, the theoretical results of which gradually penetrate into methodological science. It can also be noted that an important factor in changing approaches to the construction of a methodological system is the change in the views of mathematicians on the definition of a system of basic postulates for constructing a school mathematics course. For example, in 1950-1970. the prevailing belief was that the set-theoretic approach should be the basis for constructing a school mathematics course, which was reflected in the methodological concepts of school mathematics textbooks, and therefore required an appropriate orientation of initial mathematical training. In recent decades, mathematicians have been talking more and more about the need to develop functional and spatial thinking in schoolchildren, which is reflected in the content of textbooks published in the 90s. In accordance with this, the requirements for the initial mathematical preparation of the child are gradually changing.

Thus, the process of development of methodological sciences is closely connected with the process of development of other pedagogical, psychological and natural sciences.

Let us consider the relationship between the methodology of teaching mathematics in elementary school and other sciences.

1. The method of mathematical development of the child uses the main ideas, theoretical positions and results of research in other sciences.

For example, philosophical and pedagogical ideas play a fundamental and guiding role in the development of methodological theory. In addition, borrowing the ideas of other sciences can serve as the basis for the development of specific methodological technologies. Thus, the ideas of psychology and the results of its experimental studies are widely used by the methodology to substantiate the content of education and the sequence of its study, to develop methodological techniques and systems of exercises that organize the assimilation of various mathematical knowledge, concepts and methods of action by children. Ideas of physiology about conditioned reflex activity, two signaling systems, feedback and age stages of maturation of the subcortical areas of the brain help to understand the mechanisms of acquiring skills, habits and skills in the learning process. Special meaning for the development of methods of teaching mathematics in recent decades have the results of psychological and pedagogical research and theoretical research in the field of constructing the theory of developmental education (L.S. Vygotsky, J. Piaget, L.V. Zankov, V.V. Davydov, D.B. Elkonin, P. Ya. Galperin, N. N. Poddyakov, L. A. Wenger and others). This theory is based on the position of L.S. Vygotsky that learning is based not only on completed cycles of a child's development, but primarily on those mental functions that have not yet matured ("zones of proximal development"). Such training contributes to the effective development of the child.

2. The methodology creatively borrows research methods used in other sciences.

In fact, any method of theoretical or empirical research can find application in methodology, since in the context of the integration of sciences, research methods very quickly become general scientific. Thus, the method of literature analysis familiar to students (compiling bibliographies, taking notes, summarizing, compiling abstracts, plans, writing out quotations, etc.) is universal and is used in any science. The method of analyzing programs and textbooks is commonly used in all didactic and methodological sciences. From pedagogy and psychology, the methodology borrows the method of observation, questioning, conversation; from mathematics - methods statistical analysis etc.

3. The technique uses concrete results of researches of psychology, physiology of the higher nervous activity, mathematics and other sciences.

For example, the specific results of J. Piaget's research on the process of perception by young children of the conservation of quantity gave rise to a whole series of specific math assignments in various programs for younger students: using specially constructed exercises, a child is taught to understand that changing the shape of an object does not entail a change in its quantity (for example, when pouring water from a wide jar into a narrow bottle, its visually perceived level increases, but this does not mean that there was more water in the bottle than there was in the jar).

4. The technique is involved in integrated research development of the child in the process of his education and upbringing.

For example, in 1980-2002. there have been a number of scientific studies of the process personal development a child of primary school age in the course of teaching him mathematics.

Summarizing the question of the relationship between the methodology of mathematical development and the formation of mathematical representations in preschoolers, the following can be noted:

It is impossible to deduce from any one science a system of methodological knowledge and methodological technologies;

Data from other sciences are necessary for the development of methodological theory and practical methodological recommendations;

The methodology, like any science, will develop if it is replenished with more and more new facts;

The same facts or data can be interpreted and used in different (and even opposite) ways, depending on what goals are realized in the educational process and what system of theoretical principles (methodology) is adopted in the concept;

The methodology does not just borrow and use data from other sciences, but processes them in such a way as to develop ways for the optimal organization of the learning process;

Methodology, determines the corresponding concept of the mathematical development of the child; thus, concept - this is not something abstract, far from life and real educational practice, but a theoretical base that determines the construction of the totality of all components of the methodological system: goals, content, methods, forms and means of teaching.

Let's consider the ratio of modern scientific and "everyday" ideas about teaching mathematics to younger students.

At the heart of any science lies the experience of people. For example, physics is based on the knowledge we acquire in everyday life about the movement and fall of bodies, about light, sound, heat, and much more. Mathematics also proceeds from ideas about the forms of objects of the surrounding world, their location in space, quantitative characteristics and ratios of parts of real sets and individual objects. The first coherent mathematical theory - the geometry of Euclid (4th century BC) was born from practical surveying.

The situation is quite different with regard to methodology. Each of us has a life experience of teaching someone something. However, it is possible to engage in the mathematical development of a child only with special methodological knowledge. With what special (scientific) methodological knowledge and skills from life Tey ideas that it is enough to have some understanding of counting, calculations and solving simple arithmetic problems to teach mathematics to a younger student?

1. Everyday methodological knowledge and skills are specific; they are dedicated to specific people and specific tasks. For example, a mother, knowing the peculiarities of the perception of her child, through repeated repetitions, teaches the child to name numerals in the correct order and recognize specific geometric shapes. With sufficient perseverance of the mother, the child learns to fluently name numerals, recognizes a fairly large number of geometric shapes, recognizes and even writes numbers, etc. Many believe that this is what the child should be taught before school. Does this training guarantee the development of mathematical abilities in a child? Or at least the continued success of this child in mathematics? Experience shows that it does not guarantee. Can this mother teach the same to another child who is not like her child? Unknown. Will this mother be able to help her child learn other mathematical material? Most likely - no. Most often, one can observe a picture when the mother herself knows, for example, how to add or subtract numbers, solve this or that problem, but she cannot even explain to her child so that he learns the way to solve it. Thus, everyday methodological knowledge is characterized by the specificity, limitation of the task, situations and persons to which they apply,

Scientific methodological knowledge (knowledge of educational technology) tends to to generalization. They use scientific concepts and generalized psychological and pedagogical patterns. Scientific methodological knowledge (educational technologies), consisting of clearly defined concepts, reflects their most significant relationships, which makes it possible to formulate methodological patterns. For example, an experienced highly professional teacher can often determine by the nature of a child's mistake what methodological patterns of formation of a given concept were violated when teaching this child.

2. Everyday methodological knowledge is intuitive. This is due to the way they are obtained: they are acquired through practical trials and "adjustment". So way goes a sensitive, attentive mother, experimenting and vigilantly noticing the slightest positive results (which is not difficult to do when spending a lot of time with a child. Often the subject “mathematics” itself leaves specific imprints on the perception of parents. You can often hear: “I myself suffered with mathematics at school, he the same problems. This is hereditary." Or vice versa: "I had no problems with mathematics at school, I don’t understand who he was born into!" It is widely believed that a person either has mathematical abilities or not, and nothing can be done about it.The idea that mathematical abilities (as well as musical, visual, sports and others) can be developed and improved by most people is perceived with skepticism. Such a position is very convenient for justifying doing nothing, but from the point of view of general methodological scientific knowledge about nature, character and genesis of the mathematical development of the child, it is, of course, inadequate.

It can be said that, unlike intuitive methodological knowledge, scientific methodological knowledge rational and conscious. A professional methodologist will never point to heredity, "planid", lack of materials, poor quality of teaching aids and insufficient attention of parents to the educational problems of the child. He has a fairly large arsenal of effective methodological techniques, you just need to select from it those that are most suitable for this child.

3. Scientific methodological knowledge can be transferred to another
to a person. Accumulation and transfer of scientific methodological knowledge
are possible due to the fact that this knowledge is crystallized in concepts, regularities, methodological theories and fixed in the scientific literature, educational and methodological manuals that future teachers read, which allows them to come even to their first practice in their life with a fairly large baggage of generalized methodological knowledge.

4. Everyday knowledge about the methods and techniques of teaching is received
usually through observation and reflection. In scientific activity, these methods are supplemented methodical experiment. essence experimental method consists in the fact that the teacher does not wait for a confluence of circumstances, as a result of which the phenomenon of interest to him arises, but causes the phenomenon himself, creating the appropriate conditions. Then he purposefully varies these conditions in order to reveal the patterns by which this phenomenon
obeys. This is how any new methodological concept or methodological regularity is born. We can say that when creating a new methodological concept, each lesson becomes such a methodological experiment.

5. Scientific methodological knowledge is much broader, more diverse than everyday knowledge; it has unique factual material, inaccessible in its scope to any carrier of worldly methodological knowledge. This material is accumulated and comprehended in separate sections of the methodology, for example: a methodology for teaching problem solving, a method for forming the concept of a natural number, a method for forming ideas about fractions, a method for forming ideas about quantities, etc., as well as in certain branches of methodological science, for example : teaching mathematics in groups for the correction of mental retardation, teaching mathematics in compensation groups (visually impaired, hearing impaired, etc.), teaching mathematics to children with mental retardation, teaching schoolchildren capable of mathematics, etc.

The development of special branches of methodology for teaching mathematics to young children is in itself the most effective method of general didactics for teaching mathematics. L.S. Vygotsky began working with mentally retarded children, and as a result, the theory of “zones of proximal development” was formed, which formed the basis of the theory of developmental education for all children, including for teaching mathematics.

One should not think, however, that worldly methodological knowledge is an unnecessary or harmful thing. The "golden mean" is to see in small facts the reflection of general principles, and how to move from general principles to real life problems is not written in any book. Only constant attention to these transitions, constant exercise in them can form in the teacher what is called "methodological intuition." Experience shows that the more worldly methodological knowledge a teacher has, the more likely this intuition is to form, especially if this rich worldly methodological experience is constantly accompanied by scientific analysis and comprehension.

The methodology for teaching mathematics to younger students is applied field of knowledge(applied Science). As a science, it was created to improve the practical activities of teachers working with children of primary school age. It has already been noted above that the methodology of mathematical development as a science is actually making its first steps, although the methodology of teaching mathematics has a thousand-year history. Today there is not a single program of primary (and preschool) education that does without mathematics. But until recently, it was only about teaching young children the elements of arithmetic, algebra and geometry. And only in the last twenty years of the XX century. began to talk about a new methodological direction - theory and practice mathematical development child.

This direction became possible in connection with the formation of the theory of developmental education of a young child. This direction in the traditional methodology of teaching mathematics is still debatable. Not all teachers today stand on the positions of the need to implement developmental education. during teaching mathematics, the purpose of which is not so much the formation of a certain list of knowledge, skills and abilities of a subject nature in the child, but the development of higher mental functions, his abilities and the disclosure of the internal potential of the child.

For a progressively thinking teacher, it is obvious that practical results from the development of this methodological direction should become incommensurably more significant than the results of just a methodology for teaching elementary mathematical knowledge and skills to children of primary school age, in addition, they should be qualitatively different. After all, to know something means to master this “something”, to learn it. manage.

Learning to control the process of mathematical development (ie, the development of a mathematical style of thinking) is, of course, a grandiose task that cannot be solved overnight. The methodology has already accumulated a lot of facts today, showing that the new knowledge of the teacher about the essence and meaning of the learning process makes it significantly different: it changes his attitude both to the child and to the content of education, and to the methodology. Learning the essence of the process of mathematical development, the teacher changes his attitude to the educational process (changes himself!), to the interaction of the subjects of this process, to its meaning and goals. It can be said that methodology is a science that constructs a teacher as a subject of educational interaction. In real practical activity today, this has been expressed in modifications of the forms of work with children: teachers are paying more and more attention to individual work, since it is obvious that the effectiveness of the learning process is determined by the individual differences of children. More and more attention is paid by teachers to productive methods of working with children: search and partial search, children's experimentation, heuristic conversation, organization of problem situations in the classroom. Further development of this direction can lead to significant meaningful modifications of the programs of mathematical education of younger students, since many psychologists and mathematicians in recent decades have expressed doubts about the correctness of the traditional filling of elementary school mathematics programs mainly with arithmetic material.

There is no doubt that the fact that the process of teaching a child mathematics is constructive for the development of his personality . The process of learning any subject content leaves its mark on the development of the cognitive sphere of the child. However, the specificity of mathematics as an academic subject is such that its study can largely influence the overall personal development of the child. Even 200 years ago, this idea was expressed by M.V. Lomonosov: "Mathematics is good because it puts the mind in order." The formation of a systematic thought processes is only one side of the development of the mathematical style of thinking. Deepening the knowledge of psychologists and methodologists about the various aspects and properties of human mathematical thinking shows that many of its most important components actually coincide with the components of such a category as the general intellectual abilities of a person - this is logic, breadth and flexibility of thinking, spatial mobility, conciseness and consistency, etc. And such character traits as purposefulness, perseverance in achieving a goal, the ability to organize oneself, “intellectual endurance”, which are formed during active mathematics, are already personal characteristics person.

To date, there are a number of psychological studies showing that a systematic and specially organized system of doing mathematics actively influences the formation and development of an internal plan of action, lowers the child's level of anxiety, developing a sense of confidence and control of the situation; increases the level of development of creativity (creative activity) and the general level mental development child. All of these studies support the idea that mathematical content is the most powerful means of development intelligence and a means of personal development of the child.

Thus, theoretical research in the field of methods of mathematical development of a child of primary school age, refracted through a set of methodological techniques and the theory of developmental education, are implemented when teaching a specific mathematical content in the teacher's practical activities in the classroom.

Lecture topic Topic: Methods of teaching mathematics to junior schoolchildren as a subject.

Purpose of the lesson:

1).Didactic:

Achieve the assimilation by students of representations of the methodology of teaching mathematics to younger students as an academic subject.

2). Developing:

To expand the concepts of the methodology of teaching mathematics to younger students. Develop students' logical thinking.

3). Nurturing:

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

6. Form of training: frontal.

7. Teaching methods:

Verbal: explanation, conversation, survey.

Practical: independent work.

Visual: handouts, teaching aids.

Lesson plan:

Methods of teaching mathematics to younger schoolchildren as a pedagogical science and as a sphere of practical activity.
Methods of teaching mathematics as a subject. Principles of building a course of mathematics in elementary school.
Methods of teaching mathematics.

Basic concepts:

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

Methods of teaching mathematics to younger schoolchildren as a pedagogical science and as a sphere of practical activity.

In the system of sciences, methodological sciences are considered in the block didactics. As you know, didactics is divided into theory of education and theory learning. In turn, in the theory of learning, there are general didactics (general issues: methods, forms, means) and private didactics (subject). Private didactics are also called differently - teaching methods or, as is customary in recent years, educational technologies.

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

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

2) learning objectives.

3) funds

Didactic principles are divided into general and basic.

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

Didactic principles express the general that is inherent in any academic subject and are a guideline for organization planning and analysis of a practical task.

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

A. Stolyar highlights the following principles:

1) scientific

3) visibility

4) activity

5) strength

6) individual approach

Yu.K. Babansky identifies 5 groups of principles:

2) on the selection of learning tasks

3) for the selection of the form of education

4) choice of teaching methods

5) analysis of results

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

The principles of education are not fixed once and for all, they are deepened and changed.

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

Feature of the principle:

Displays, but does not reproduce the accuracy of the system of science, preserving, if possible, the common features of their inherent logic, phasing and system of knowledge.

Reliance on subsequent knowledge on previous ones.

The system regularity of the arrangement of material by year of study in accordance with age characteristics and the age of the trainees, as well as the further development of the trainees.

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

In the revised programs, the principles of visualization were emphasized.

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

Excessive use of visualization can lead to undesirable results.

Types of visibility:

natural (models, handouts)

visual clarity (drawings, photos, etc.)

symbolic clarity (diagrams, tables, drawings, diagrams)

2.Methods of teaching mathematics as a subject. Principles of building a course of mathematics in elementary school.

Mathematics Teaching Methods (MTM) is a science whose subject is teaching mathematics, and in a broad sense: teaching mathematics at all levels, starting from preschool institutions and ending with high school.

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

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

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

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

The learning objective is the key. On the one hand, it reflects common goals learning, concretizes cognitive motives. On the other hand, it makes the process of performing educational actions meaningful.

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

Methods of enlargement of didactic units (PM Erdniev): 1) simultaneous study of similar concepts; 2) simultaneous study of reciprocal actions; 3) transformation of mathematical exercises; 4) drawing up tasks by students; 5) deformed examples.

3.Methods of teaching mathematics.

Question about methods of primary teaching of mathematics and their classification has always served as a subject of attention on the part of Methodists. In most modern methodological manuals, special chapters are devoted to this problem, in which the main features of individual methods are revealed and the conditions for their practical application in the learning process are shown.

Elementary Mathematics Course consists of several sections, different in their content. This includes: problem solving; the study of arithmetic operations and the formation of computational skills; study of measures and formation of measuring skills; the study of geometric material and the development of spatial representations. Each of these sections, having its own special content, at the same time has its own, private, methodology, its own methods, which are in accordance with the specifics of the content and form of training sessions.

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

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

The study of arithmetic operations is based on the use of methods and techniques that are unique to this section and different from the methods used in other sections of mathematics.

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

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

Teaching methods are ways of joint activity of the teacher and students aimed at solving learning problems.

The teaching method is a system of purposeful actions of the teacher, organizing the cognitive and practical activities of the student, ensuring the assimilation of the content of education.

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

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

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

The most common classification is teaching methods

- according to the sources of obtaining knowledge;

- for didactic purposes;

- by the level of activity of students;

- by the nature of the cognitive activity of students.

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

Let us consider in more detail each classification and its inherent goals.

In the classification of teaching methods for didactic purpose allocate :

Methods for acquiring new knowledge;

Methods for the formation of skills and abilities;

Methods for consolidating and testing knowledge, skills and abilities.

Often used to introduce students to new knowledge storytelling method.

In the methodology of mathematics, this method is usually called - knowledge presentation method.

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

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

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

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

To passive include methods in which students only listen and look (story, explanation, excursion, demonstration, observation).

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

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

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

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

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

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

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

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

Verbal methods include storytelling, explanation, conversation, discussion.

The second group in this classification is visual teaching methods.

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

Practical Methods learning is based on the practical activities of students. The main purpose of this group of methods is the formation of practical skills and abilities.

Practices include exercises, practical and laboratory work.

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

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

There are the following methods:

Explanatory and illustrative;

Problem presentation methods;

Partial search (heuristic);

Research.

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

The teacher communicates information using the spoken word (story, conversation, explanation, lecture), printed word (textbook, additional aids), visual aids (tables, diagrams, pictures, films and filmstrips), practical demonstration of methods of activity (showing experience, work on the machine, the method of solving the problem, etc.).

reproductive method assumes that the teacher communicates, explains the knowledge in a finished form, and the students learn them and can reproduce, repeat the method of activity on the instructions of the teacher. The criterion for assimilation is the correct reproduction (reproduction) of knowledge.

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

A higher level of cognitive activity brings partially search (heuristic) method.

The method is called partly exploratory because students independently solve a complex educational problem not from beginning to end, but only partially. The teacher guides the students through the individual search steps. Part of the knowledge is communicated by the teacher, and part of the knowledge is obtained by the students on their own, answering the questions posed or solving problematic tasks. Educational activity develops according to the scheme: teacher - students - teacher - students, etc.

Thus, the essence of the partially search method of teaching is that:

Not all knowledge is offered to students in finished form, they partially need to be obtained independently;

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

One modification of this method is heuristic conversation.

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

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

The teacher together with the students formulates the problem;

Students independently resolve it;

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

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

Homework:

Prepare for the practical session

Belarusian State Pedagogical University named after Maxim Tank

Faculty of Pedagogy and Methods of Primary Education

Department of Mathematics and Methods of its Teaching

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

Graduate work

INTRODUCTION… 3

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

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

2.2. The essence of educational technology… 9

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

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

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

2.1. The use of the activity method in teaching mathematics to junior schoolchildren ... 20

2.1.1. Statement of the learning task… 21

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

2.1.3. Primary fastening… 22

2.1.4. Independent work with checking in the classroom ... 22

2.1.5. Training exercises… 23

2.1.6. Delayed control of knowledge… 23

2.2. Training Lesson… 25

2.2.1. Structure of training lessons… 25

2.2.2. Training Lesson Model… 28

2.3. Oral exercises in mathematics lessons ... 28

2.4. Knowledge control… 29

Chapter 3. Analysis of the experiment… 36

3.1. Ascertaining experiment… 36

3.2. Teaching experiment… 37

3.3. Control experiment… 40

Conclusion… 43

Literature… 46

Appendix 1… 48

Appendix 2… 69

2.2. Essence of educational technology

Before giving a definition of educational technology, it is necessary to reveal the etymology of the word “technology” (the science of craftsmanship, art, because from the Greek. - techne craftsmanship, art and logos- the science). The concept of technology in modern meaning It is used primarily in production (industrial, agricultural), various types of scientific and production activities of a person and involves a body of knowledge about the methods (a set of methods, operations, actions) for the implementation of production processes that guarantee a certain result.

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

A set (combination, connection) of any components.

· Logic, sequence of components.

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

· Guaranteed result.

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

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

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

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

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

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

In a specific educational process, the object of internalization is the target group. The relations of degree in the target group correspond to the internalization of the corresponding components by the subject of the teaching: the primary elements must be assigned, the secondary elements must be mastered. Pedagogical target groups interpreted in the described way will be called targets. For example, a target group with primary elements “facts and methods of activity” and a secondary element “values” set the target for knowledge, skills and norms. The assignment of primary goals occurs explicitly as a result of specially organized and managed educational activities (education), while the assimilation of secondary goals occurs implicitly, as a result of unmanaged educational activities and a by-product of education.

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

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

An empirically obtained or theoretically designed system of rules of educational activity, not related to a specific content, is educational technology .

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

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

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

· Informational.

· Information and value.

· Activity.

· Activity-valuable.

· Valuable.

· Value-information.

· Value-activity.

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

The final classification of educational technologies by technological targets (objects of appropriation) looks like this:

· Information-perceptual.

· Information and activity.

· Information and value.

· Activity.

· Activity-informational.

· Activity-valuable.

· Valuable.

· Value-information.

· Value-activity.

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

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

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

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

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

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

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

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

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

In accordance with this principle, the main task of teaching mathematics is not the study of the foundations of mathematical science as such, but general intellectual development - the formation in students in the process of studying mathematics of the qualities of thinking necessary for the full functioning of a person in modern society, for the dynamic adaptation of a person to this society.

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

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

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

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

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

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

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

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

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

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

Formation of activity skills and development of students' moral and ethical qualities of a person, adequate to full-fledged mathematical activity;

Realization of the possibilities of mathematics in the formation of the scientific worldview of students, in their mastering the scientific picture of the world;

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

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

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

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

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

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

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

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

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

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

It should be emphasized that the orientation towards developmental education does not does not mean a rejection of the formation of knowledge, skills, without which self-determination of the personality, its self-realization is impossible.

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

· Didactic principles - visibility, accessibility, scientific character, systematic, conscientiousness of assimilation of educational material.

· Teaching method - explanatory and illustrative.

· Form of study - classroom class.

However, it is obvious to everyone that the existing didactic system, having not exhausted its significance, at the same time does not allow the developmental function of education to be effectively carried out. In recent years, in the works of L.V. Zankova, V.V. Davydova, P.Ya. Galperin and many other teachers, scientists and practitioners, new didactic requirements have been formed that solve modern educational problems, taking into account the demands of the future. The main ones are:

1. Operation principle

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

Thus, the main mechanism for implementing the goals and objectives of developmental education is inclusion of the child in educational and cognitive activities. AT this is what operating principle, Training that implements operating principle is called the activity approach.

2. The principle of a holistic view of the world

More Ya.A. Comenius noted that phenomena should be studied in mutual connection, and not separately (not as a “heap of firewood”). In our time, this thesis acquires even greater significance. It means that the child should form a generalized, holistic view of the world (nature - society - himself), about the role and place of each science in the system of sciences. Naturally, in this case, the knowledge formed by students should reflect the language and structure of scientific knowledge.

The principle of a unified picture of the world in the activity approach is closely related to the didactic principle of scientific character in the traditional system, but much deeper than it. Here we are talking not just about the formation of a scientific picture of the world, but also about the personal attitude of students to the knowledge gained, as well as about ability to apply them in their practice. For example, if it is about ecological knowledge, then the student must not just to know that it is not good to pluck certain flowers, leave rubbish in the forest, etc., but make your own decision don't do that.

3. The principle of continuity

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

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

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

4. Minimax principle

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

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

The way out is simple: select only two levels - maximum, determined by the zone of proximal development of children, and the necessary minimum. The minimax principle is as follows: the school must offer the student the content of education at the maximum level, and the student is obliged to learn this content at the minimum level(see annex 1) .

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

The work is carried out at a high level of difficulty, but only the obligatory result, and success, is evaluated. This will allow students to form an attitude to achieve success, and not to avoid the "deuce", which is much more important for the development of the motivational sphere.

5. The principle of psychological comfort

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

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

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

6. The principle of variability

Modern life requires a person to be able to to make a choice from choosing goods and services to choosing friends and choosing a life path. The principle of variability involves the development of students' variative thinking, that is, understanding the possibility of various options for solving the problem and the ability to carry out a systematic enumeration of options.

Education, in which the principle of variability is implemented, relieves students of the fear of making a mistake, teaches them to perceive failure not as a tragedy, but as a signal for its correction. Such an approach to solving problems, especially in difficult situations, is also necessary in life: in case of failure, do not become discouraged, but seek and find a constructive way.

On the other hand, the principle of variability ensures the teacher's right to independence in choosing educational literature, forms and methods of work, the degree of their adaptation in the educational process. However, this right gives rise to a great responsibility of the teacher for the final result of his activity - the quality of education.

7. The principle of creativity (creativity)

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

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

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

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

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

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

The minimax system effectively contributes to the development of personal qualities, forms a motivational sphere. It also solves the problem of multi-level teaching, which allows you to advance in the development of all children, both strong and weak (L.V. Zankov).

The requirements of psychological comfort are ensured by taking into account the psychophysiological state of the child, contribute to the development of cognitive interests and the preservation of children's health (L.V. Zankov, A.A. Leontiev, Sh.A. Amonashvili).

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

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

Thus, the listed didactic principles of the educational technology "School 2100" to a certain extent necessary and sufficient for the implementation of modern goals of education and already today can be carried out in a comprehensive school.

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

CHAPTER 2. Features of work on educational technology "School 2100" in mathematics lessons

2.1. Using the activity method in teaching mathematics to younger students

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

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

· setting a learning task;

· learning activities;

· actions of self-control and self-evaluation.

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

· motivation;

Creation of an indicative framework for action (OOA):

· material or materialized action;

· external speech;

· inner speech;

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

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

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

2.1.1. Statement of the learning task

Any process of cognition begins with an impulse that prompts action. Surprise is necessary, coming from the impossibility of momentary provision of this or that phenomenon. Delight is needed emotional outburst coming from participation in this phenomenon. In a word, motivation is needed that encourages the student to join the activity.

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

2.1.2. “Discovery” of new knowledge by children

The next stage of work on the concept is the solution of the problem, which is carried out by the students themselves in the course of discussion, discussion on the basis of substantive actions with material or materialized objects. The teacher organizes an introductory or inciting dialogue. In conclusion, he sums up, introducing the generally accepted terminology.

This stage includes students in active work, in which there are no disinterested, because the dialogue of the teacher with the class is the dialogue of the teacher with each student, focusing on the degree and speed of assimilation of the desired concept and adjusting the number and quality of tasks that will help to solve the problem. The dialogical form of the search for truth - the most important aspect activity method.

2.1.3. Primary fastening

Primary consolidation is carried out through commenting on each desired situation, pronouncing in a loud speech the established action algorithms (what I do and why, what follows what, what should happen).

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

2.1.4. Independent work with class check

The task of the fourth stage is self-control and self-esteem. Self-control encourages students to be responsible for the work performed, teaches them to adequately evaluate the results of their actions.

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

The four stages of work on the concept listed above are best done in one lesson, without breaking them in time. Usually it takes about 20-25 minutes of the lesson. The remaining time is devoted, on the one hand, to consolidating the knowledge, skills and abilities accumulated earlier and integrating them with new material, and on the other hand, to advanced preparation for the following topics. Here, on an individual basis, errors on a new topic that could have arisen at the stage of self-control are finalized: positive self-esteem is important for every student, so every effort should be made to correct the situation in the same lesson.

Attention should also be paid to organizational issues, setting common goals and objectives at the beginning of the lesson and summing up the activities at the end of the lesson.

Thus, lessons of introduction of new knowledge in the activity approach have the following structure:

1) Organizational moment, general lesson plan.

2) Statement of the learning task.

3) “Discovery” of new knowledge by children.

4) Primary fastening.

5) Independent work with checking in the class.

6) Repetition and consolidation of previously studied material.

7) The result of the lesson.

(See Appendix 2.)

The principle of creativity determines the nature of fixing new material in homework assignments. Not reproductive, but productive activity is the key to lasting assimilation. Therefore, as often as possible, homework should be offered tasks in which it is required to correlate the particular and the general, to isolate stable connections and patterns. Only in this case, knowledge becomes thinking, acquires consistency and dynamics.

2.1.5. Training exercises

In subsequent lessons, the studied material is worked out and consolidated, it is brought to the level of automated mental action. Knowledge undergoes a qualitative change: there is a turn in the process of cognition.

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

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

2.1.6. Delayed knowledge control

The final control work should be offered to students on the basis of the minimax principle (readiness according to the upper level of knowledge, control - according to the lower one). Under this condition, the negative reaction of schoolchildren to grades, the emotional pressure of the expected result in the form of a mark, will be minimized. The task of the teacher is to evaluate the assimilation of educational material according to the bar necessary for further advancement.

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

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

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

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

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

On the other hand, the activity method ensures the passage of all the necessary stages of assimilation of concepts, which can significantly increase the strength of knowledge. Indeed, the formulation of a learning task provides the motivation for the concept and the construction of an orienting basis for action (OOF). The “discovery” of new knowledge by children is carried out by means of their performance of objective actions with material or materialized objects. Primary consolidation ensures the passage of the stage of external speech - children speak out loud and at the same time perform the established action algorithms in writing. In teaching independent work, the action is no longer accompanied by speech; students pronounce the action algorithms “to themselves”, inner speech (see Appendix 3). And, finally, in the process of performing the final training exercises, the action passes into the internal plan and is automated (mental action).

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

New education goals need updating content education and search forms training, which will enable their optimal implementation. The whole set of information should be subordinated to the orientation to life, to the ability to act in any situations, to get out of crisis, conflict situations, which include situations of knowledge search. A student at school learns not only to solve mathematical problems, but through them also life tasks, not only the rules of spelling, but also the rules of social coexistence, not only the perception of culture, but also its creation.

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

2.2. Lesson-training

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

Trainings can be used when studying new material, consolidating what has been learned. However, the special expediency of their use in the generalization and systematization of students' knowledge.

Conducting training is no easy task. A special skill is required from the teacher. In such a lesson, the teacher is the conductor, whose task is to skillfully switch and concentrate the attention of students.

The main character in the lesson-training is the student.

2.2.1. The structure of training lessons

1. Goal setting

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

Motivation to work is reinforced in the resource circle. Children stand in a circle, hold hands. The task of the teacher is to make each child feel support, a good attitude towards him. The feeling of unity with the class, the teacher helps to create an atmosphere of trust and mutual understanding.

2. Independent work. Making your own decision

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

Before starting the tasks, the children pronounce the “rules” of the work that will help them organize a dialogue. Each class may be different. Here is one of the options: "Everyone should speak out and listen to everyone." Pronouncing these rules in a loud speech helps to create an attitude for participation in the dialogue of all the children of the group.

At the stage of independent work, the student must consider all three answers, comparing, comparing them, make a choice and prepare to explain his choice to a friend: why he thinks so and not otherwise. To do this, everyone needs to delve into the luggage of their knowledge. The knowledge gained by students in the classroom is built into a system and becomes a means for evidence-based choice. The child learns to carry out a systematic enumeration of options, to compare them, to find the best option.

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

3. Work in pairs (fours)

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

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

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

4. Listening to different opinions as a class

By providing a word for expression to various groups of students, the teacher has an excellent opportunity to track how well the concepts are formed, the knowledge is strong, how well the children have mastered the terminology, whether they include it in their speech.

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

5. Expert review

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

6. Self-esteem

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

Have you listened carefully to your friend?

Could you prove the correctness of your choice?

If not, why not?

What happened that was difficult? Why?

What needs to be done to be successful?

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

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

In total, trainings include from 4 to 7 tasks.

7. Summing up

Summing up takes place in the resource circle. Everyone has the opportunity to express (or not to express) their attitude to the epigraph, as he understood it. At this stage, the “mystery of the words” of the epigraph is revealed. This technique allows the teacher to come to the problems of morality, the relationship of educational activity with the real problems of the world around, allows students to perceive educational activity as their social experience.

Trainings should not be confused with practical lessons, where, due to the many training exercises, strong skills and abilities are formed. They also differ from testing, although they also provide for the choice of an answer. However, when testing, it is difficult for the teacher to trace how justified the choice was made by the student, the choice at random is not ruled out, since the student's reasoning remains at the level of inner speech.

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

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

Lessons-trainings are developed in accordance with the thematic planning of L.G. Peterson and are held at the expense of reserve lessons. Topics of training lessons: numbering, the meaning of arithmetic operations, methods of calculation, procedure, quantities, solving problems and equations. During the academic year, from 5 to 10 trainings are held, depending on the class.

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

November: Addition and subtraction within 9 .

December: Task .

February: Quantities .

March: Solving Equations .

April: Problem solving .

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

2.2.2. Lesson-training model

2.3. Oral exercises in mathematics lessons

Changing priorities in the goals of mathematics education has significantly affected the process of teaching mathematics. The main idea is the priority of the developmental function in learning. Oral exercises serve as one of the means in the educational and cognitive process that make it possible to realize the idea of development.

Oral exercises contain great potential for the development of thinking, enhancing the cognitive activity of students. They allow you to organize the educational process in such a way that as a result of their implementation, students form a complete picture of the phenomenon under consideration. This provides an opportunity not only to retain in memory, but also to reproduce exactly those fragments that are necessary in the process of passing the subsequent steps of cognition.

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

Oral exercises break the stereotyped thinking by constantly involving the student in the analysis background information, error prediction. The main thing when working with information is to involve students themselves in creating an indicative framework that shifts the focus of the educational process from the need to memorize to the need to be able to apply information, and thereby contributes to the transfer of students from the level of reproductive assimilation of knowledge to the level of research activity.

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

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

b) formation of heuristic techniques;

c) development of combinatorial thinking;

d) the formation of spatial representations.

2.4. Knowledge control

Modern learning technologies can significantly improve the efficiency of the learning process. At the same time, most of these technologies leave out of their attention innovations related to such important components of the educational process as knowledge control. The methods of organizing control over the level of students' preparation currently used at the school have not undergone any significant changes over a long period. Until now, many believe that teachers successfully cope with this type of activity and do not experience significant difficulties in their practical implementation. At best, the question of what is expedient to be brought under control is discussed. Issues related to the forms of control, and even more so the methods of processing and storing educational information obtained during the control, remain without due attention on the part of teachers. At the same time, an information revolution has already taken place in modern society for quite a long time, new methods of analyzing, collecting and storing data have appeared that have made this process more efficient in terms of the volume and quality of information retrieved.

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

The first level is the student, who must consciously manage his activity, directing it to achieve learning goals. If there is no management at this level or is not consistent with the goals of learning, then a situation is realized when the student is taught, but he himself does not learn. Accordingly, in order to effectively manage their activities, a student must have all the necessary information about the learning outcomes he achieves. Naturally, at the lower levels of education, the student mainly receives this information from the teacher in finished form.

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

The third level - controls public education. This level is a hierarchical system of public education management institutions. The governing bodies deal both with information that they receive independently and independently of the teacher, and with information transmitted to them by teachers.

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

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

Evaluation is secondary to measurement and maybe be obtained only after the measurement. In the modern school, these two processes are often not distinguished, since the process of measurement takes place as if in a collapsed form, and the assessment itself has the form of a number. Teachers do not think about the fact that by fixing the number of actions correctly performed by a student (or the number of mistakes made by him) in the performance of a particular work, they thereby measure the results of students' activities, and when grading a student, they correlate the identified quantitative indicators with those available in their disposal of evaluation criteria. Thus, teachers themselves, having, as a rule, the measurement results that they use to mark students, rarely inform other participants in the educational process about them. This significantly narrows the information available to students, their parents and authorities.

Knowledge assessment can be both numerical and verbal, which, in turn, gives rise to additional confusion that often exists between measurements and assessments. Measurement results can only have a numerical form, since in general terms measurement is establishing a correspondence between an object and a number. The form of the assessment is its insignificant characteristic. So, for example, a judgment like “student fully has mastered the studied material” can be equivalent to the judgment “the student knows the material Great” or “the student has a grade of 5 for the completed educational material”. The only thing that researchers and practitioners should keep in mind is that in the latter case, the assessment 5 is not a number in the mathematical sense and no arithmetic operations are allowed with it. Grade 5 serves to assign this student to a certain category, the meaning of which can be deciphered unambiguously only taking into account the accepted grading system.

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

As measuring tools in the educational process, standard control and independent work is used, common to all students. The results of these tests are evaluated by the teacher. In modern methodological literature, much attention is paid to the content of these tests, they are improved and brought into line with the set learning objectives. At the same time, the issues of processing the results of examinations, measuring the results of students' activities and their evaluation in most of the methodological literature are worked out at an insufficiently high level of detail and formalization. This leads to the fact that teachers often give different grades for the same results of work by students. Even more can be differences in the results of evaluation of the same work by different teachers. The latter is due to the fact that in the absence of strictly formalized rules defining carrying out algorithm measurement and assessment, different teachers may perceive the proposed measurement algorithms and assessment criteria in different ways, replacing them with their own.

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

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

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

Traditional control is represented by independent and control works (12 books-notebooks that make up a set of mathematics for elementary school).

When conducting independent work, the goal is primarily to identify the level of mathematical training of children and to eliminate the existing knowledge gaps in a timely manner. At the end of each independent work there is a place for work on bugs. At first, the teacher should help the children in choosing tasks that allow them to correct their mistakes in a timely manner. During the year, independent work with corrected errors is collected in a folder, which helps students to trace their path in mastering knowledge.

Control works sum up this work. Unlike independent work, the main function of control work is precisely the control of knowledge. From the very first steps, the child should be taught to be especially attentive and precise in his actions during the control of knowledge. The results of the control work, as a rule, are not corrected - you need to prepare for the knowledge control before him, not after. But this is how any competitions, exams, administrative tests are carried out - after their implementation, the result cannot be corrected, And children need to be gradually psychologically prepared for this. At the same time, preparatory work, timely correction of errors during independent work gives a certain guarantee that the test will be written successfully.

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

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

Independent work is designed, as a rule, for 7-10 minutes (sometimes up to 15). If the child does not have time to complete the task of independent work within the allotted time, after checking the work by the teacher, he finalizes these tasks at home.

The assessment for independent work is put after the work on the bugs has been carried out. It is not so much what the child managed to do during the lesson that is evaluated, but how he eventually worked on the material. Therefore, even those independent works that are not written very well in the lesson can be evaluated with a good and excellent score. In independent work, the quality of work on oneself is fundamentally important and only success is evaluated.

Testing takes 30 to 45 minutes. If one of the children on the tests does not fit in the allotted time, then at the initial stages of training, you can allocate some extra time for him to give him the opportunity to calmly finish the work. Such “finishing” of the work is excluded when carrying out independent work. But in the control work is not provided for the subsequent "refinement" - the result is evaluated. The assessment for the control work is corrected, as a rule, in the next control work.

When grading, you can focus on the following scale (tasks with an asterisk are not included in the mandatory part and are evaluated by an additional assessment):

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

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

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

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

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

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

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

The main goal of the final work is to reveal the real level of knowledge of children, their mastery of general educational skills and abilities, to enable children themselves to realize the result of their work, to emotionally experience the joy of victory.

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

Chapter 3. Analysis of the experiment

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

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

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

1. Condition.

2. Question.

4. Expression.

5. Decision.

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

Control. The students were offered tasks similar to those from the ascertaining experiment, as well as tasks of a more complex level.

3.1. Ascertaining experiment

The students were given the following tasks:

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

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

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

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

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

Decided: 25 tasks - students of gymnasium No. 5

24 tasks - students of high school No. 74

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

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

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

Reasons for low results:

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

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

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

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

d) knowledge of tabular cases of addition within 10;

e) the ability to compare numbers within 10.

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

3.2. Teaching experiment

Purpose of the experiment: to continue work on solving problems using the activity method with students from gymnasium No. 5 studying under the program “School 2100”. In order to form more solid knowledge, skills and abilities in solving problems, special attention was paid to drawing up a scheme (“dressing” a scheme) and drawing up an expression according to a scheme.

The following tasks were offered.

1. Game "Part or whole?"

The teacher at a fast pace with the movement of the pointer shows a part or whole in a segment, the students name. In order to activate the activity of students, feedback tools should be used. Taking into account the fact that in the letter it was agreed to designate the part and the whole with special signs, instead of answering “the whole”, students depict a “circle”, connecting the thumb and forefinger of the right hand, and “part” - placing the index finger of the right hand horizontally. The game allows you to complete up to 15 tasks with a specified goal in one minute.

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

¾ - whole

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

2. Game "What changed?"

Schematic for students:

It turns out what is known: a part or a whole. Then the students close their eyes, the diagram becomes 2), the students answer the same question, close their eyes again, the diagram is transformed, and so on. as many times as the teacher deems necessary.

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

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

3. Game “Dress Scheme”

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

- a- part;

- b- whole;

unknown integer;

Unknown part.

4. Game “Choose a scheme”

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

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

Modeling the text of the problem, the student must clearly imagine what needs to be found in the problem: a part or a whole. To this end, the following work can be carried out.

5. Game “What is unknown?”

The teacher reads the text of the problem, and the students give an answer to the question of what is unknown in the problem: a part or a whole. As a means of feedback, a card can be used that looks like:

on the one hand, on the other: .

for example: in one bunch 3 carrots, and in the other 5 carrots. How many carrots are in two bunches? (unknown integer).

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

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

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

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

What more part or whole?

How to find the whole?

How to find a part?

What can be found by knowing the whole and the part? How? (What action?).

What can be found by knowing the parts of the whole? How? (What action?).

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

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

Write an expression for each scheme?

The reference schemes used at this stage of work on the task can be as follows:

During the experiment, the students came up with their own tasks, illustrated them, “dressed” the schemes, commenting was used, independent work with various types checks.

3.3. Control experiment

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

Tasks were proposed:

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

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

6 birds were sitting on the birch. Several birds flew away, 4 birds remained. How many birds have flown?

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

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

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

Decided: 63 tasks - students of gymnasium No. 5

50 tasks - students of school No. 74

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

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

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

Calmly answer at the blackboard 100%

They are able to express their thoughts more clearly and clearly 100%

Don't be afraid to make a mistake 100%

Became more active and independent 86.7%

Not afraid to express their point of view 93.3%

Better justify their answers 100%

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

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

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

At the same time, children are at ease and bold in communicating with adults, easily come into contact with them;

They have excellent self-control skills, including in the field of relationships and rules of conduct.

Conclusion

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

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

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

advances in the study of the material at a pace convenient for him, gradually assimilating the material;

masters the material in the volume that is available and necessary for him (minimax principle);

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

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

2. The teacher in the lesson does not act as an informant, but as an organizer search activities of students. A specially selected system of tasks, in the course of solving which students analyze the situation, express their suggestions, listen to others and find the right answer, help the teacher in this.

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

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

3. Developmental aspect. It is impossible not to say about special exercises aimed at developing the creative abilities of students. It is important that these tasks are given in the system, starting from the first lessons. Children come up with their own examples, tasks, equations, etc. They love this activity. It is not by accident that creative work children on their own initiative are usually brightly and colorfully decorated.

Textbooks are multilevel, allow organizing differentiated work with textbooks in the classroom. Tasks, as a rule, include both working out the standard of mathematical education and questions requiring the application of knowledge at a constructive level. The teacher builds his system of work, taking into account the characteristics of the class, the presence in it of groups of poorly prepared students and students who have achieved high rates in the study of mathematics.

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

Students from the very beginning of studying the course of mathematics are accustomed to working with algebraic expressions. Moreover, the work is carried out in two directions: the compilation and reading of expressions.

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

The early use of elements of algebra makes it possible to lay a solid foundation for the study of mathematical models and for revealing to students at the senior levels of education the role and significance of the method of mathematical modeling.

This program makes it possible through activity to lay the foundation for further study of geometry. Already in elementary school, children “discover” various geometric patterns: they derive the formula for the area of a right-angled triangle, put forward a hypothesis about the sum of the angles of a triangle.

6. The program develops interest in the subject. It is impossible to achieve good results in learning if students have a low interest in mathematics. For its development and consolidation in the course, a lot of exercises are proposed that are interesting in content and form. A large number of numerical crosswords, rebuses, tasks for ingenuity, transcripts help the teacher to make the lessons truly exciting and interesting. In the course of performing these tasks, children decipher either a new concept or a riddle ... Among the deciphered words are names literary heroes, titles of works, names of historical figures that are not always familiar to children. This stimulates to learn new things, there is a desire to work with additional sources (dictionaries, reference books, encyclopedias, etc.)

7. Textbooks have a multi-line structure, giving the ability to systematically work on the repetition of the material. It is well known that knowledge that is not included in the work for a certain time is forgotten. It is difficult for a teacher to independently conduct work on the selection of knowledge for repetition. searching for them takes a lot of time. These textbooks are of great help to the teacher in this matter.

8. Printed basis of textbooks in elementary school saves time and focuses students on solving problems, which makes the lesson more voluminous and informative. At the same time, the most important task of forming students of the skill is being solved. self-control.

The work carried out confirmed the proposed hypothesis. The use of the activity approach in teaching mathematics to junior schoolchildren has shown that cognitive activity, creativity, and emancipation of students increase, and fatigue decreases. The program "School 2100" meets the tasks of modern education and the requirements for the lesson. For several years, children did not have unsatisfactory marks at the entrance exams to the gymnasium - an indicator of the effectiveness of the "School 2100" program in schools of the Republic of Belarus.

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Appendix 1

Topic: SUBTRACTION OF TWO-DIGITAL NUMBERS WITH TRANSITION THROUGH THE DISCHARGE

Grade 2 1 hour (1 - 4)

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

2) To consolidate the learned computational techniques, the ability to independently analyze and solve complex problems.

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

During the classes:

1. Organizational moment.

2. Statement of the learning task.

2.1. Solving examples for subtraction with the transition through the discharge within 20.

The teacher asks the children to solve examples:

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

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

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

What subtraction examples do you still know how to solve? (For subtraction of two-digit numbers.)

2.2. Solving examples for subtracting two-digit numbers without crossing the digit.

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

Examples are best placed one below the other. Children should notice that in the reduced one digit is unknown; unknown tens and ones alternate; all known numbers in the minuend are odd, go in descending order: in the subtrahend, the number of tens decreases by 1, and the number of units does not change.

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

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

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

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

2.3. Formulation of the problem. Goal setting.

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

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

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

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

Why is there a difficulty here? (The minuend lacks units.)

Is the minuend less than the subtrahend? (No, lessened more.)

Where are the units hiding? (At ten.)

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

Well done! Solve an example.

Children replace in the reduced triangle-ten with a triangle on which 10 units are drawn:

11e -4e \u003d 7e, Zd-2d \u003d 1d. In total, it turned out 1 day and 7 e., or 17.

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

And how do you think of what you should always remember when using this technique, where a mistake is possible? (The number of tens decreases by 1.)

4. Physical education.

5. Primary fastening.

1) No. 1, p. 16.

Comment on the first example like this:

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

Solve the following examples with explanation.

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

2) No. 2, p. 16

Once again, the decision and commenting on the example in a column are clearly spelled out:

81 _82 _83 _84 _85 _86

29 29 29 29 29 29

I write: units under units, tens under tens.

I subtract units: from 1 unit. you can not subtract 9 units. I take 1 day and put an end to it. 11-9 = 2 units I write in units.

Subtract tens: 7-2 = 5 dec.

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

3) № 3, page 16.

Let's play the game "Guess":

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

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

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

The teacher invites the children to write off examples from the board in a column to a new computing technique

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

Children write down the necessary examples in a notebook in a cell, and then check the correctness of their notes according to the finished model:

19 18 17

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

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

Write your own example that would continue this pattern.

7. Tasks for repetition.

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

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

1 option 2 option

Perform a cross check. What did you notice? (The answers in the tasks are the same. These are reciprocal tasks.)

8. The result of the lesson.

What examples did you learn to solve?

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

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

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

9. Homework.

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

Compose your example for a new computational technique and solve it graphically and in a column.

Topic: MULTIPLICATION BY 0 AND BY 1.

Grade 2, 2 hours (1-4)

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

2) To consolidate the meaning of multiplication and the commutative property of multiplication, to develop computational skills,

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

During the classes:

1. Organizational moment.

2.1. Tasks for the development of attention.

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

(blue)

(red)

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

What is the excess number? (10 is round and the others are not; 10 is two digits and the rest are single digits; 5 is repeated twice and the rest are one at a time.)

I will close the number 10. Is there an extra among the other numbers? (3 - he doesn't have a pair under 10, but the others do.)

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

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

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

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

What action were you looking for? (Subtraction.)

2.2. Tasks for the development of memory and speech. Knowledge update.

a) -Repeat in order the words that I will name: term, term, sum, reduced, subtracted, difference. (Children try to reproduce word order.)

What action components are named? (Addition and subtraction.)

What new action did we meet? (Multiplication.)

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

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

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

Write down the definition of multiplication.

b) Review the notes. What task will you be doing?

12 + 12 + 12 + 12 + 12

33 + 33 + 33 + 33

(Replace sum by product.)

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

12 5. Similarly - 33 4, and 3)

c) Name the reverse operation. (Replace the product with the sum.)

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

d) Equations are written on the board:

21 3 = 21+22 + 23

44 + 44 + 44 + 44 = 44 + 4

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

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

The animals of the forest school were on a mission. Did they do it right?

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

e) - Compare the expressions:

8 – 5… 5 – 8 34 – 9… 31 2

5 6… 3 6 a – 3… a 2 + a

(8 5 \u003d 5 8, since the sum does not change from a rearrangement of the terms; 5 6\u003e 3 6, since there are 6 terms on the left and right, but there are more terms on the left; 34 9\u003e 31 - 2. since there are more terms on the left and themselves the terms are larger; a 3 \u003d a 2 + a, since there are 3 terms on the left and on the right, equal to a.)

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

2.3. Formulation of the problem. Goal setting.

Consider the picture. Are equalities true? Why? (True, since the sum 5 + 5 + 5 = 15. then the sum becomes one more term 5, and the sum increases by 5.)

5 3 = 15 5 5 = 25

5 4 = 20 5 6 = 30

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

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

And what does the expression 5 1 mean? fifty? (? Trouble!) Outcome discussions:

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

3. “Discovery” of new knowledge by children.

1) No. 1, p. 80.

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

Children solve examples with comments in a textbook-notebook:

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

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

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

Make a conclusion: 1 a -? (1 a \u003d a.) The teacher exposes a card: 1 a \u003d a

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

What should they be equal to in order not to violate the commutative property of multiplication? (7 1 must also equal 7, so 7 1 = 7.)

4 1 = 4; 5 1 = 5.

Make a conclusion: a 1 =? (a 1 = a.)

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

Does our conclusion coincide with what we got on the numerical ray? (Yes.)

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

a 1 = 1 a = a.

2) Similarly, the case of multiplication from 0 in No. 4, p. 80 is investigated. Conclusion - multiplying a number by 0 or 0 by a number results in zero:

a 0 = 0 a = 0.

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

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

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

4. Physical education.

5. Primary fastening.

Examples are written on the board:

23 1 = 0 925 = 364 1 =

1 89= 156 0 = 0 1 =

Children solve them in a notebook with pronunciation in loud speech of the received rules, for example:

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

2) No. 1, p. 80.

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

When multiplying 145 by unknown number it turned out 145. So, multiplied by 1 x= 1. Etc.

3) No. 6, p. 81.

a) 8 x = 0; b) x 1 \u003d 0.

Multiplying 8 by an unknown number resulted in 0. So, multiplied by 0 x = 0. And so on.

6. Independent work with checking in the class.

1) No. 2, p. 80.

1 729 = 956 1 = 1 1 =

No. 5, p. 81.

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

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

7. Tasks for repetition.

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

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

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

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

To whom are we invited? (To Fortran.)

b) - Professor Fortran is a connoisseur of computers. But the thing is, we don't have an address. Cat X - the best student of Professor Fortran - left a program for us (A poster is posted such as on page 56, M-2, part 1.) We set off on the path according to X's program, Which house did you come to?

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

c) - We are met by Professor Fortran with his students. His best student - a caterpillar - has prepared a task for you: “I conceived a number, subtracted 7 from it, added 15, then added 4 and got 45. What number did I conceive?”

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

G) Competition game.

- Asam Professor Fortran suggested that we play the game "Computing Machines".

a	1	4	7	8	9
x

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

8. The result of the lesson.

Have you done everything you planned in the lesson?

What are the new rules?

9. Homework.

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

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

Subject: PROBLEM SOLVING.

Grade 2, 4 hours (1 - 3).

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

2) Consolidate computational skills, compiling literal expressions for text tasks.

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

During the classes:

1. Organizational moment .

2. Statement of the learning task.

2.1. oral exercises.

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

Front work:

Reduce the number 244 by 2 times (122)

Find the product of 57 and 2 (114)

Decrease the number 350 by 230 (120)

How much more is 134 than 8? (126)

Decrease the number 1280 by 10 times (128)

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

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

Arrange the resulting numbers in ascending order:

114	120	121	122	124	126	128
W	BUT	Y	H	BUT	T	BUT

Individual work at the blackboard:

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

What numbers are missing? (Number 7.)

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

2.2. Statement of the educational task.

Each team receives 4 tasks of the “Blitz Tournament”, a sign and a diagram.

"Blitz Tournament"

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

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

c) There were a red rings, b white rings and pink rings. They were distributed equally among 4 rabbits. How many rings did each bunny get?

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

Team I:

Team II:

Team III:

It has become fashionable among rabbits to wear rings in their ears. Read the problems on your slips and determine which problem your scheme and your expression are suitable for?

Students discuss problems in groups and find the answer together. One person from the group “protects” the opinion of the team.

For which task did I not choose a scheme and an expression?

Which of these schemes is suitable for the fourth problem?

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

Is this decision correct? Why not? Under what condition can we consider it correct? (If the number of rings in both rabbits were equal.)

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

3. "Discovery" of new knowledge.

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

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

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

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

How to write down the expression, how many rings have become? (a-n)

This is twice the smaller or more? (Less.)

How can you find the smaller number? ((a-n): 2.)

Did we answer the question? (Not.)

What else should you know? (Higher number.)

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

Tablets with the received expressions are fixed on the board:

(a-n): 2 is the smaller number,

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

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

How to do it? (a + n)

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

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

1 way 2 way

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

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

4. Physical education.

5. Primary fastening.

Students work with a textbook. Tasks are solved with commenting, the solution is recorded on a printed basis.

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

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

- “Dress” the scheme and analyze the problem. (We know the sum is 56 people, and the difference is 2 students. First we find twice the smaller number: 56 - 2 \u003d 54 people. Then we find out how many students are in the second grade: 54: 2 \u003d 27 people. Now we find out how many students are in first class - 27 + 2 = 29 people.)

How else to find how many students are in the first grade? (56 - 27 = 29 people.)

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

How else could the problem be solved? (Find first the number of students in the first class, and subtract 2 from it.)

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

After a minute of reasoning, a representative of the team that was ready earlier speaks in the teams. Both methods of solving the problem are discussed orally. After discussing each method, a ready-made sample solution record is opened and compared with the student's answer:

I method II method

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

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

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

6. Independent work with checking in the class.

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

No. 7 (a), p. 7.

I method II method

1) 248-8 \u003d 240 (m.) 1) 248 + 8 \u003d 256 (m.)

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

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

Answer: 120 marks; 128 marks.

No. 7(6), p. 7.

I method II method

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

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

3) 192 - 12 \u003d 180 (open) 3) 180 + 12 \u003d 192 (open)

Answer: 180 postcards; 192 postcards.

Check - according to the finished sample on the board.

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

1 team:

2 team:

3 team:

Team captains report on team performance.

8. The result of the lesson.

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

9. Homework.

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

Subject: COMPARISON OF ANGLES.

Grade 4, 3 hours (1-4)

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

2) To introduce students to the method of comparing angles using direct overlay.

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

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

During the classes:

1. Organizational moment.

2. Statement of the learning task.

a) - Continue the row:

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

b) - Calculate and arrange in descending order:

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

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

Cross out the 2 extra letters. What word came out? (FIGURE.)

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

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

I connect the center of the circle with a point lying on the circle, what happened? (A line segment is called a radius.)

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

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

What are the types of corners? (Straight, sharp, blunt.)

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

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

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

d) No. 1, page 1.

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

e) No. 2, page 1.

Children's opinions about the relationship between the angles are usually different. This serves as the basis for creating a problem situation.

3. “Discovery” of new knowledge by children.

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

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

The teacher summarizes the discussion:

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

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

4. Primary fastening.

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

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

5. Independent work with checking in the class.

Students do the practice in #3, page 2 on their own, then in pairs explain how they put the corners. After that, 2-3 pairs explain the solution to the whole class.

6. Physical education.

7. Solving problems for repetition.

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

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

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

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

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

Find 16% off with. (since: 100 16)

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

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

What fraction of a leap year is February? (29/366)

Check - according to the model of the decision on portable boards. Errors made during the execution of the task are analyzed according to the scheme: it is established that it is not known - the whole or the part.

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

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

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

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

Less than 8/5 because 8/5 is an improper fraction;

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

5) No. 10, p. 3. The first line is solved with commenting:

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

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

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

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

1072	560	210	102	75	72
D	F	O	With	E	R

6) No. 12(6), p. 3

The mass of a camel is 700 kg, and the mass of the load that he carries on his back is 40% of the mass of the camel. What is the mass of the camel with the load?

Students mark the condition of the problem on the diagram and conduct its independent analysis:

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

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

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

2) 700 + 280 = 980 (kg)

Answer: the mass of a camel with a load is 980 kg.

8. The result of the lesson.

What have you learned? What did you repeat?

What did you like? What was difficult?

9. Homework: Nos. 5, 12 (a), 16

Annex 2

training

Topic: “Solving Equations”

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

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

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

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

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

In the last task, you must choose X an unusual situation that children have not yet encountered. Thus, here the depth of assimilation of a new topic and the child's ability to apply the studied algorithm of actions in new conditions are checked.

Epigraph of the lesson : "Everything hidden becomes clear." Here are some statements of children when summing up the results in the resource circle:

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

Everything that is unknown can be found if the actions are performed correctly.

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

We realized that there is no need to hide anything.

We learn to be smart, to make the unknown known.

Expert review

job number
1	b
2	a
3	in
4	a
5	a and b

Annex 3

oral exercises

The purpose of this lesson is to introduce children to the concept of a number line. In the proposed oral exercises, not only is work on the development of mental operations, attention, memory, constructive skills, not only numeracy skills are practiced and advanced preparation is carried out for the study of subsequent topics of the course, but also a variant of creating a problem situation is proposed that can help the teacher organize of this topic, the stage of setting a learning task.

Topic: “Numerical segment”

Main goal :

1) Introduce the concept of a numerical segment, teach

one unit.

2) Strengthen counting skills within 4.

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

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

1… 3… 5… 7… 9

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

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

1 2 3 4 5 6 7 8 9 10

Break the squares into parts by color:

With

+=+=

-=-=

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

Insert the missing letters and numbers into the "windows". Explain your decision.

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

Name the whole and parts in numerical equalities.

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

How many green squares? How many blue?

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

By what other sign can these squares be divided into parts? (Sizes are large and small.)

Into what parts will the number 4 be divided then? (2 and 2.)

Make two triangles of 6 sticks.

Now make two triangles of 5 sticks.

Remove 1 stick to make a rectangle.

Name the meanings of numerical expressions:

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

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

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

Compare the expressions in the first column.

In case of difficulty, you can ask leading questions:

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

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

- Tasks in verse(problem solution is substantiated):

Anya has two balls, Tanya has two balls. (Looking for the whole. To find

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

How many of them, can you imagine? 2 + 2 = 4.)

Four magpies came to the lessons. (Looking for a part. To find

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

How many diligently worked forty? other part: 4 -1 = 3.)

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

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

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

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

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

The material of oral exercises should not be used formally - “everything in a row”, but should be correlated with specific working conditions - the level of preparation of children, their number in the class, the technical equipment of the classroom, the level of pedagogical skill of the teacher, etc. In order to use this material correctly, in work should be guided by the following principles.

1. The atmosphere in the classroom should be calm and friendly.“Race”, overload of children should not be allowed - it is better to sort out one task with them fully and efficiently than seven, but superficially and chaotically.

2. Forms of work must be diversified. They should change every 3-5 minutes - a collective dialogue, work with object models, cards or a cash register of numbers, mathematical dictation, work in pairs, an independent answer at the blackboard, etc. The thoughtful organization of the lesson allows significantly increase the amount of material, which can be considered with children without overload.

3. The introduction of new material should begin no later than at the 10-12th minute of the lesson. Exercises that precede the study of something new should be aimed mainly at updating the knowledge that is necessary for its full assimilation.