Development of mathematical abilities
among younger schoolchildren
Abilities are formed and developed in the process of learning, mastering relevant activities, therefore it is necessary to form, develop, educate and improve the abilities of children. In the period from 3-4 years to 8-9 years, rapid development of intelligence occurs. Therefore, during primary school age the opportunities for developing abilities are the highest.
The development of the mathematical abilities of a junior schoolchild is understood as the purposeful, didactically and methodically organized formation and development of a set of interrelated properties and qualities of the child’s mathematical thinking style and his abilities for mathematical knowledge of reality.
The problem of ability is a problem of individual differences. With the best organization of teaching methods, the student will progress more successfully and faster in one area than in another.
Naturally, success in learning is determined not only by the student’s abilities. In this sense, the content and methods of teaching, as well as the student’s attitude to the subject, are of key importance. Therefore, success and failure in learning do not always provide grounds for making judgments about the nature of the student’s abilities.
The presence of weak abilities in students does not relieve the teacher from the need, as far as possible, to develop the abilities of these students in this area. At the same time, there is an equally important task - to fully develop his abilities in the area in which he demonstrates them.
It is necessary to educate the capable and select the capable, while not forgetting about all schoolchildren, and to raise the overall level of their training in every possible way. In this regard, various collective and individual working methods are needed in their work in order to intensify the activities of students.
The learning process should be comprehensive, both in terms of organizing the learning process itself, and in terms of developing in students a deep interest in mathematics, problem-solving skills, understanding the system of mathematical knowledge, solving with students a special system of non-standard problems, which should be offered not only in lessons, but also on tests. Thus, a special organization of the presentation of educational material and a well-thought-out system of tasks help to increase the role of meaningful motives for studying mathematics. The number of result-oriented students is decreasing.
In the lesson, not just problem solving, but the unusual way of solving problems used by students should be encouraged in every possible way; in this regard, special importance is placed not only on the result in solving the problem, but on the beauty and rationality of the method.
Teachers successfully use the method of “composing tasks” to determine the direction of motivation. Each task is assessed according to a system of the following indicators: the nature of the task, its correctness and relation to the source text. The same method is sometimes used in a different version: after solving the problem, students were asked to create any problems that were somehow related to the original problem.
To create psychological and pedagogical conditions for increasing the efficiency of organizing the learning process system, the principle of organizing the learning process in the form of substantive communication using cooperative forms of student work is used. This is group problem solving and collective discussion of grading, pair and team forms of work.
The methodology for using the system of long-term assignments was considered by E.S. Rabunsky when organizing work with high school students in the process of teaching German at school.
A number of pedagogical studies have considered the possibility of creating systems of such tasks in various subjects for high school students, both to master new material and to eliminate knowledge gaps. In the course of research, it was noted that the vast majority of students prefer to perform both types of work in the form of “long-term tasks” or “delayed work.” This type of organization of educational activities, traditionally recommended mainly for labor-intensive creative work (essays, abstracts, etc.), turned out to be the most preferable for the majority of schoolchildren surveyed. It turned out that such “deferred work” satisfies the student more than individual lessons and assignments, since the main criterion for student satisfaction at any age is success at work. The absence of a sharp time limit (as happens in a lesson) and the possibility of freely returning to the content of the work many times allows you to cope with it much more successfully. Thus, tasks designed for long-term preparation can also be considered as a means of cultivating a positive attitude towards the subject.
For many years, it was believed that everything said applies only to older students, but does not correspond to the characteristics of the educational activities of primary school students. Analysis of the procedural characteristics of the activities of capable children of primary school age and the work experience of Beloshista A.V. and teachers who took part in the experimental testing of this methodology, showed the high efficiency of the proposed system when working with capable children. Initially, to develop a system of tasks (hereinafter we will call them sheets in connection with the form of their graphic design, convenient for working with a child), topics related to the formation of computational skills were selected, which are traditionally considered by teachers and methodologists as topics that require constant guidance at the stage acquaintance and constant monitoring at the consolidation stage.
During the experimental work, a large number of printed sheets were developed, combined into blocks covering an entire topic. Each block contains 12-20 sheets. The worksheet is a large system of tasks (up to fifty tasks), methodically and graphically organized in such a way that as they are completed, the student can independently approach the understanding of the essence and method of performing a new computational technique, and then consolidate the new way of activity. A worksheet (or a system of sheets, i.e. a thematic block) is a “long-term task”, the deadlines for which are individualized in accordance with the desires and capabilities of the student working on this system. Such a sheet can be offered in class or instead of homework in the form of a task with a “delayed deadline” for completion, which the teacher either sets individually or allows the student (this path is more productive) to set a deadline for himself (this is a way to form self-discipline, since independent planning of activities in connection with independently determined goals and deadlines is the basis of human self-education).
The teacher determines tactics for working with worksheets for the student individually. At first, they can be offered to the student as homework (instead of a regular assignment), individually agreeing on the timing of its completion (2-4 days). As you master this system, you can move on to the preliminary or parallel method of work, i.e. give the student a sheet before learning the topic (on the eve of the lesson) or during the lesson itself for independent mastery of the material. Attentive and friendly observation of the student in the process of activity, “contractual style” of relationships (let the child decide for himself when he wants to receive this sheet), perhaps even exemption from other lessons on this or the next day to concentrate on the task, advisory assistance (on one question can always be answered immediately when passing a child in class) - all this will help the teacher to fully individualize the learning process of a capable child without spending a lot of time.
Children should not be forced to copy assignments from the sheet. The student works with a pencil on a sheet of paper, writing down answers or completing actions. This organization of learning evokes positive emotions in the child - he likes to work on a printed basis. Freed from the need for tedious copying, the child works with greater productivity. Practice shows that although the worksheets contain up to fifty tasks (the usual homework norm is 6-10 examples), the student enjoys working with them. Many children ask for a new sheet every day! In other words, they exceed the work quota for the lesson and homework several times over, while experiencing positive emotions and working at their own discretion.
During the experiment, such sheets were developed on the topics: “Oral and written calculation techniques”, “Numbering”, “Quantities”, “Fractions”, “Equations”.
Methodological principles for constructing the proposed system:
- The principle of compliance with the mathematics program for primary grades. The content of the sheets is tied to a stable (standard) mathematics program for primary grades. Thus, we believe it is possible to implement the concept of individualizing mathematics teaching for a capable child in accordance with the procedural features of his educational activities when working with any textbook that corresponds to the standard program.
- Methodically, each sheet implements the principle of dosage, i.e. in one sheet only one technique or one concept is introduced, or one connection, but essential for a given concept, is revealed. This, on the one hand, helps the child clearly understand the purpose of the work, and on the other hand, helps the teacher to easily monitor the quality of mastery of this technique or concept.
- Structurally, the sheet represents a detailed methodological solution to the problem of introducing or introducing and consolidating one or another technique, concept, connections of this concept with other concepts. The tasks are selected and grouped (i.e., the order in which they are placed on the sheet matters) in such a way that the child can “move” along the sheet independently, starting from the simplest methods of action already familiar to him, and gradually master a new method, which in the first steps fully revealed in smaller actions that are the basis of this technique. As you move through the sheet, these small actions are gradually arranged into larger blocks. This allows the student to master the technique as a whole, which is the logical conclusion of the entire methodological “construction”. This structure of the sheet allows you to fully implement the principle of a gradual increase in the level of complexity at all stages.
- This structure of the worksheet also makes it possible to implement the principle of accessibility, and to a much deeper extent than can be done today when working only with a textbook, since the systematic use of sheets allows you to learn the material at an individual pace convenient for the student, which the child can regulate independently.
- The system of sheets (thematic block) allows you to implement the principle of perspective, i.e. gradual inclusion of the student in the activities of planning the educational process. Tasks designed for long-term (delayed) preparation require long-term planning. The ability to organize your work, planning it for a certain period of time, is the most important educational skill.
- The system of worksheets on the topic also makes it possible to implement the principle of individualization of testing and assessing students’ knowledge, not on the basis of differentiating the level of difficulty of tasks, but on the basis of the unity of requirements for the level of knowledge, skills and abilities. Individualized deadlines and methods for completing tasks make it possible to present all children with tasks of the same level of complexity, corresponding to the program requirements for the norm. This does not mean that talented children should not be held to higher standards. Worksheets at a certain stage allow such children to use material that is more intellectually rich, which in a propaedeutic way will introduce them to the following mathematical concepts of a higher level of complexity.
Belarusian State Pedagogical University named after Maxim Tank
Faculty of Pedagogy and Primary Education Methods
Department of Mathematics and Methods of Its Teaching
USING EDUCATIONAL TECHNOLOGY “SCHOOL 2100” IN TEACHING MATHEMATICS TO JUNIOR SCHOOL CHILDREN
Graduate work
INTRODUCTION… 3
CHAPTER 1. Features of the mathematics course of the general education program “School 2100” and its technology... 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 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 working on educational technology “School 2100” in mathematics lessons... 20
2.1. Using the activity method in teaching mathematics to primary schoolchildren... 20
2.1.1. Setting a learning task... 21
2.1.2. “Discovery” of new knowledge by children... 21
2.1.3. Primary consolidation… 22
2.1.4. Independent work with testing in class... 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. Model of a training lesson... 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. Educational experiment... 37
3.3. Control experiment... 40
Conclusion... 43
Literature… 46
Appendix 1… 48
Appendix 2… 69
2.2. The essence of educational technology
Before defining educational technology, it is necessary to reveal the etymology of the word “technology” (the science of skill, art, because from the Greek - techne- craftsmanship, art and logos- the science). The concept of technology in its modern meaning is used primarily in production (industrial, agricultural), various types of scientific and production human activities and presupposes a body of knowledge about methods (a set of methods, operations, actions) of carrying out production processes that guarantee obtaining 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 results.
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 interiorized content of education (the result of educational activity) in relation to the subject of interiorization is also called education(Sometimes - education).
Thus, the concept of “education” has three meanings: a social institution of society, the activities of this institution and the result of its activities.
There is a two-level nature of interiorization: interiorization that does not affect the subconscious will be called assimilation, and internalization, affecting the subconscious (forming automatisms of actions), - assignment .
It is logical to name the learned facts representations, assigned- knowledge, learned methods of activity - skills, assigned - skills, and learned value orientations and emotional-personal relationships - standards, assigned - beliefs or meanings .
In a specific educational process, the object of internalization is the target group. The relationship of power in the target group corresponds to the internalization of the corresponding components by the subject of the study: the primary elements must be appropriated, the secondary elements must be assimilated. We will call the pedagogical target groups interpreted in the described way targets. For example, a target group with the primary elements of “facts and methods of action” and the secondary element of “values” sets a target setting for knowledge, skills and norms. The assignment of primary goals occurs explicitly as a result of specially organized and controlled educational activities (education), and the assimilation of secondary goals occurs implicitly, as a result of uncontrolled 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 based on known scientific laws and tested experimentally). In the first case, it may relate to the transmission of some specific content or be generalized to various types of content. In the second case, it is contentless by definition and can be adjusted to various specific content options.
An empirically derived system of rules for transmitting specific content is called teaching methodology .
An empirically derived or theoretically designed system of rules for educational activities that is not related to specific content is a educational technology .
A set of rules of educational activity that do not have signs of systematicity 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 goals of educational activity are a system-forming factor in relation to educational technologies, considered as systems of rules for this activity.
Classification of educational technologies according to technological targets, that is, in a pedagogical sense, according to objects of appropriation:
· Informational.
· Information and value.
· Activity.
· Activity-value.
· Value-based.
· Value-informational.
· Value-based activity.
Unfortunately, the first of these names has been assigned to technologies that are not related to educational activities. Information 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 facts are the primary element of activity goals, that is, knowledge constitutes the technological target setting, are usually called information-perceptual .
The final classification of educational technologies according to technological targets (objects of assignment) looks like this:
· Information-perceptual.
· Information and activity.
· Information and value.
· Activity.
· Activity and information.
· Activity-value.
· Value-based.
· Value-informational.
· Value-based activity.
Really existing educational technologies have yet to be sorted into classes. Apparently some classrooms are currently empty. The choice of classes of educational technologies used by one or another society (one or another humanitarian system) in a specific historical situation depends on what components of the accumulated spiritual culture of the society in this situation considers the most important for its survival and development. They define goals external to educational technology 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 targets when designing 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 learned 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 educational technology “School 2100”
Modern approaches to organizing the school education system, including mathematics education, are determined, first of all, by the rejection of a uniform, unitary secondary school. The guiding vectors of this approach are humanization and humanitarization school education.
This determines the transition from the principle of “all mathematics for everyone” 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 wants and/or can master it, to design a course of “mathematics for everyone,” or, more precisely, “mathematics for everyone.”
One of the main goals of the academic subject “Mathematics” as a component of general secondary education related to to each for 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, logical and algorithmic thinking, many qualities of thinking, such as strength and flexibility, constructiveness and criticality, etc., can be formed in its purest form.
These qualities of thinking in themselves are not associated with any mathematical content or with mathematics in general, but teaching mathematics introduces an important and specific component into their formation, which currently cannot be effectively implemented even by the entire set of individual school subjects.
At the same time, specific mathematical knowledge that lies beyond, relatively speaking, the arithmetic of natural numbers and the primary foundations of geometry, are not“a subject of basic necessity” for the vast majority of people and cannot, therefore, form the target basis for teaching mathematics as a subject of general education.
That is why, as a fundamental principle of the educational technology “School 2100” in the aspect of “mathematics for everyone,” the principle of priority of the developmental function in teaching mathematics comes to the fore. In other words, teaching mathematics is focused not so much on mathematics education itself, in in the narrow sense of the word, how much for education with using mathematics.
In accordance with this principle, the main task of teaching mathematics is not the study of the fundamentals 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 individual human activity, based on acquired specific mathematical knowledge, for knowledge and awareness of the surrounding world by means of mathematics remains, naturally, an equally essential component of school mathematical education.
From the point of view of the priority of the developmental function, specific mathematical knowledge in “mathematics for everyone” is considered not so much as a goal of learning, but as a base, a “testing ground” for organizing intellectually valuable activities of students. For the formation of a student’s personality, for achieving a high level of his development, it is precisely this activity, if we talk about a mass school, that, as a rule, 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 resulting idea of priority in “mathematics for everyone” of the developmental function of teaching in relation to its purely educational function requires a reorientation of the methodological system of teaching mathematics from increasing 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 mathematics education in educational technology, “School 2100” occupies a central place development of the abstract thinking, which includes not only the ability to perceive specific abstract objects and structures inherent in mathematics, but also the ability to operate with such objects and structures 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 symbols as a fragment of a global artificial language that plays a significant role in the communication process and is currently necessary are also considered as general goals of mathematical education every educated person.
The humanitarian orientation of teaching mathematics as a general education subject determines the specification of general goals in building a methodological system for teaching mathematics, reflecting the priority of the developmental 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:
Mastery of a complex of mathematical knowledge, abilities and skills 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 school subjects in the natural sciences and humanities at a modern level; c) to continue studying mathematics in any form of continuous education (including, at the appropriate stage of education, upon transition to training in any profile at the senior level of school);
Formation and development of the qualities of thinking necessary for an educated person to function fully 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 in students of moral and ethical personality traits adequate to full-fledged mathematical activity;
Realization of the possibilities of mathematics in the formation of the scientific worldview of students, in their mastery of the scientific picture of the world;
Formation of a mathematical language and mathematical apparatus as a means of describing and studying the surrounding world and its patterns, in particular as a basis for computer literacy and culture;
Familiarization 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;
Familiarization with the nature of scientific knowledge, with the principles of constructing scientific theories in the unity and opposition of mathematics and the natural and human sciences, with the criteria of truth in various forms of 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 has been experiencing in recent decades have radically changed not only people’s living conditions, but also the educational situation. In this regard, the task of creating a new concept of education that reflects both the interests of society and the interests of each individual has become urgent.
Thus, in recent years, society has developed a new understanding of the main goal of education: the formation readiness for self-development, ensuring the integration of the individual into 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 your activities to achieve them and evaluate the results of your 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 to the modern level of knowledge and the level of the educational program.
It should be emphasized that the focus on developmental education is completely does not mean a refusal to develop knowledge, skills and abilities, without which personal self-determination and self-realization are impossible.
That is why the didactic system of Ya.A. Comenius, which has absorbed the centuries-old traditions of the system of transmitting knowledge about the world to students, and today forms the methodological basis of the so-called “traditional” school:
· Didactic principles - clarity, accessibility, scientific character, systematicity, and conscientiousness in mastering educational material.
· Teaching method - explanatory and illustrative.
· Form of training - class lesson.
However, it is obvious to everyone that the existing didactic system, although it has not exhausted its significance, at the same time does not allow for the effective implementation of the developmental function of education. In recent years, in the works of L.V. Zankova, V.V. Davydova, P.Ya. Galperin and many other teacher-scientists and practitioners have formed new didactic requirements that solve modern educational problems taking into account the needs of the future. The main ones:
1. Operating principle
The main conclusion of psychological and pedagogical research in recent years is that The formation of a student’s personality and his advancement in development takes place not when he perceives ready-made knowledge, but in the process of his own activity aimed at “discovering” new knowledge.
Thus, the main mechanism for realizing the goals and objectives of developmental education is inclusion of the child in educational and cognitive activities. IN that's what it's all about operating principle, Education that implements the principle of activity is called an activity approach.
2. The principle of a holistic view of the world
Also Y.A. Comenius noted that phenomena need to be studied in mutual connection, and not separately (not like a “pile of firewood”). Nowadays, this thesis acquires even greater significance. It means that The child must form a generalized, holistic idea of the world (nature - society - himself), about the role and place of each science in the system of sciences. Naturally, 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 scientificity in the traditional system, but is 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 acquired, as well as ability to apply them in their practical activities. For example, if we are talking about environmental knowledge, then the student should not just to know that it is not good to pick certain flowers, leave garbage behind in the forest, etc., and 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 technique .
The idea of continuity is also not new for pedagogy, however, until now it is most often limited to the so-called “propaedeutics”, and is not solved systematically. The problem of continuity has acquired particular relevance 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 “preschool preparation - school - university” model have been developed in recent years by V.N. Prosvirkin.
4. Minimax principle
All children are different, and each of them develops at their own pace. At the same time, education in mass schools is focused on 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 distinguished. level. However, there are exactly as many real levels in a class as there are children! Is it possible to accurately determine them? Not to mention that it is practically difficult to account for even four - after all, for a teacher this means 20 preparations a day!
The solution is simple: select only two levels - maximum, determined by the zone of proximal development of children, and necessary minimum. The minimax principle is as follows: the school must offer the student educational content at the maximum level, and the student must master this content at the minimum level(see Appendix 1) .
The minimax system is apparently optimal for implementing an individual approach, since it self-regulating system. A weak student will limit himself to the minimum, while a strong student will take everything and move on. Everyone else will be placed between these two levels in accordance with their abilities and capabilities - they will choose their level themselves to its maximum possible.
The work is carried out at a high level of difficulty, but Only the required result and success are evaluated. This will allow students to develop an attitude towards achieving success, rather than avoiding getting a bad grade, 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 relaxes children and in which they feel “at home.”
No academic success will be of any use if it is “involved” in fear of adults and suppression of the child’s personality.
However, psychological comfort is necessary not only for the assimilation of knowledge - it depends on physiological state children. Adaptation to specific conditions, creating an atmosphere of goodwill will help relieve tension and neuroses that destroy health children.
6. The principle of variability
Modern life requires a person to be able to make a choice - from choosing goods and services to choosing friends and choosing a life path. The principle of variability presupposes the development of variable thinking among students, that is understanding the possibility of various options for solving a problem and the ability to systematically enumerate options.
Education, which implements the principle of variability, removes the fear of mistakes in students and teaches them to perceive failure not as a tragedy, but as a signal for its correction. This approach to solving problems, especially in difficult situations, is also necessary in life: in case of failure, do not become discouraged, but look for 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, and the degree of their adaptation in the educational process. However, this right also gives rise to greater responsibility for the teacher for the final result of his activities - the quality of teaching.
7. The principle of creativity (creativity)
The principle of creativity presupposes maximum orientation towards creativity in the educational activities of schoolchildren, their acquisition of their own experience of creative activity.
We are not talking here about simply “inventing” tasks by analogy, although such tasks should be welcomed in every possible way. Here, first of all, we mean the formation in students of the ability to independently find solutions to problems that have not been encountered before, their independent “discovery” of new ways of action.
The ability to create something new and find a non-standard solution to life’s problems has become an integral part of the real life success of any person today. Therefore, the development of creative abilities is acquiring general educational importance these days.
The teaching principles outlined above, developing the ideas of traditional didactics, integrate useful and non-conflicting ideas from new concepts of education from the standpoint of continuity of scientific views. They don't reject, but continue and develop traditional didactics towards solving modern educational problems.
In fact, it is obvious that the knowledge that the child himself “discovered” is visual for him, accessible and consciously assimilated by him. However, the inclusion of a child in activities, in contrast to traditional visual learning, activates his thinking and 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 being scientific, but at the same time it also implements new approaches, such as humanization and humanitarization of education (G.V. Dorofeev, A.A. Leontyev, L.V. Tarasov).
The minimax system effectively promotes the development of personal qualities and forms the motivational sphere. Here the problem of multi-level teaching is solved, which makes it possible to promote the development of all children, both strong and weak (L.V. Zankov).
The requirements of psychological comfort ensure that the child’s psychophysiological state is taken into account, promotes the development of cognitive interests and the preservation of children’s health (L.V. Zankov, A.A. Leontyev, Sh.A. Amonashvili).
The principle of continuity gives a systemic character to the solution of succession issues (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 educational technology “School 2100” to a certain extent necessary and sufficient to achieve modern educational goals and can already be carried out today in secondary schools.
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 emphasis is justified by a specific historical, cultural and social application.
CHAPTER 2. Features of working on educational technology “School 2100” in mathematics lessons
2.1. Using the activity method in teaching mathematics to primary schoolchildren
Practical adaptation of the new didactic system requires updating traditional forms and methods of teaching, and developing new educational content.
Indeed, the inclusion of students in activities - the main type of knowledge acquisition in the activity approach - is not included in the technology of the explanatory-illustrative method on which education in a “traditional” school is based today. The main stages 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 activity, which are:
· setting a learning task;
· learning activities;
· actions of self-control and self-esteem.
Thus, communicating the topic and purpose of the lesson does not provide a statement of the problem. A teacher’s explanation cannot replace children’s learning activities, as a result of which they independently “discover” new knowledge. The differences between control and self-control of knowledge are also fundamental. Consequently, the explanatory and illustrative method cannot fully achieve the goals of developmental education. A new technology is needed, which, on the one hand, will allow the implementation of the principle of activity, and on the other, will ensure the passage of the necessary stages of knowledge acquisition, namely:
· motivation;
· creation of an indicative basis of action (IBA):
· material or materialized action;
· external speech;
· inner speech;
· automatic mental action(P.Ya. Galperin). These requirements are satisfied by the activity method, the main stages of which are presented in the following diagram:
(Steps included in a lesson on introducing a new concept are marked with a dotted line).
Let us describe in more detail the main stages of working on a concept in this technology.
2.1.1. Setting a learning task
Any process of cognition begins with an impulse that encourages action. Surprise is necessary, coming from the impossibility of momentarily ensuring this or that phenomenon. What is needed is delight, an emotional surge that comes from participation in this phenomenon. In a word, motivation is needed to encourage the student to enter into activity.
The stage of setting a learning task is the stage of motivation and goal-setting of activity. Students complete tasks that update their knowledge. The list of tasks includes a question that creates a “collision,” that is, a problematic situation that is personally significant for the student and shapes his need mastering this or that concept (I don’t know what’s happening. I don’t know how it’s happening. But I can find out - I’m interested in it!). The cognitive target.
2.1.2. “Discovery” of new knowledge by children
The next stage of work on the concept is solving the problem, which is carried out teach yourself going on during a discussion, discussion based on substantive actions with material or materialized objects. The teacher organizes a leading or stimulating dialogue. Finally, he concludes by introducing common terminology.
This stage includes students in active work in which there are no disinterested people, because the teacher’s dialogue with the class is the teacher’s dialogue with each student, focusing on the degree and speed of mastering the sought concept and adjusting the quantity and quality of tasks that will help ensure a solution to the problem. The dialogical form of searching for truth is the most important aspect of the activity method.
2.1.3. Primary consolidation
Primary consolidation is carried out through commenting on each sought-after situation, speaking out loud the established action algorithms (what I am doing and why, what follows what, what should happen).
At this stage, the effect of mastering the material is enhanced, since the student not only reinforces written speech, but also voices internal speech, through which 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 repeated playback of educational material in the independent actions of students.
2.1.4. Independent work with testing in class
The task of the fourth stage is self-control and self-esteem. Self-control encourages students to take a responsible attitude to the work they do and 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 moves to the internal plane. The student pronounces the algorithm of action “to himself,” as if conducting a dialogue with his intended opponent. It is important that at this stage a situation is created for each student success(I can, I can do it).
It is better to go through the four stages of working on a concept listed above in one lesson, without separating them over time. This usually takes about 20-25 minutes of a lesson. The remaining time is devoted, on the one hand, to consolidating the knowledge, skills and abilities accumulated earlier and their integration with new material, and on the other hand, to advanced preparation for the following topics. Here, errors on a new topic that could arise at the self-control stage are individually refined: positive self-esteem is important for every student, so we must do everything possible to correct the situation in the same lesson.
You should also pay attention to organizational issues, setting general goals and objectives at the beginning of the lesson and summing up the activities at the end of the lesson.
Thus, lessons for introducing new knowledge in the activity approach have the following structure:
1) Organizational moment, general lesson plan.
2) Statement of the educational task.
3) “Discovery” of new knowledge by children.
4) Primary consolidation.
5) Independent work with testing in class.
6) Repetition and consolidation of previously studied material.
7) Lesson summary.
(See Appendix 2.)
The principle of creativity determines the nature of consolidating new material in homework. Not reproductive, but productive activity is the key to lasting assimilation. Therefore, as often as possible, homework assignments should be offered in which it is necessary to correlate the particular and the general, to identify stable connections and patterns. Only in this case does knowledge become thinking and acquire consistency and dynamics.
2.1.5. Training exercises
In subsequent lessons, the learned material is practiced and consolidated, bringing it to the level of automated mental action. Knowledge undergoes a qualitative change: a revolution occurs in the process of cognition.
According to L.V. Zankov, consolidation of material in the system of developmental education should not be merely reproducing in nature, but should be carried out in parallel with the study of new ideas - deepen the learned properties and relationships, broaden the horizons of children.
Therefore, the activity method, as a rule, does not provide lessons for “pure” consolidation. Even in lessons whose main goal is to practice the studied material, some new elements are included - this can be the expansion and deepening of the material being studied, advanced preparation for the study of subsequent topics, etc. This “layer cake” allows every child move forward at your own pace: children with a low level of preparation have enough time to “slowly” master 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 test should be offered to students based on the minimax principle (readiness at the top level of knowledge, control at the bottom). Under this condition, the negative reaction of schoolchildren to grades and the emotional pressure of the expected result in the form of a grade will be minimized. The teacher’s task is to evaluate the mastery of educational material according to the bar necessary for further advancement.
Described teaching technology - activity method- developed and implemented in a mathematics course, but can, in our opinion, be used in the study of any subject. This method creates favorable conditions for multi-level learning and practical implementation of all 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 the educational task;
2) educational activities of children - at the stage of “discovery” of new knowledge;
3) actions of self-control and self-esteem - at the stage of independent work, which children check here in the classroom.
On the other hand, the activity method ensures completion of all necessary stages of mastering concepts, which allows you to significantly increase the strength of knowledge. Indeed, setting a learning task ensures the motivation of the concept and the construction of an indicative basis for action (IBA). The “discovery” of new knowledge by children is carried out through 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 carry out established action algorithms in written form. In independent learning work, the action is no longer accompanied by speech; students pronounce the action algorithms “to themselves”, internal speech (see Appendix 3). And finally, in the process of performing the final training exercises, the action moves to the internal plane and becomes automated (mental action).
Thus, The activity method meets the necessary requirements for teaching technologies that implement modern educational goals. It makes it possible to master subject content in accordance with a unified approach, with a unified focus on activating both external and internal factors that determine the development of the child.
New education goals require updating content education and search forms training that will enable their optimal implementation. The entire body of information should be subordinated to an orientation toward life, toward the ability to act in any situation, toward getting out of crisis and conflict situations, which include situations of searching for knowledge. A student at school learns not only to solve mathematical problems, but through them also life problems, not only the rules of spelling, but also the rules of social life, not only the perception of culture, but also its creation.
The main form of organizing educational and cognitive activity of students in the activity approach is collective dialogue. It is through collective dialogue that “teacher-student” and “student-student” communication takes place, in which learning material is learned at the level of personal adaptation. 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 teaching practice, can be effectively used within the framework of the activity approach.
2.2. Lesson-training
This is a lesson in active mental and verbal activity of students, the form of organization of which is group work. In 1st grade it is work in pairs, from 2nd grade it is work in fours.
Trainings can be used to study new material and consolidate what has been learned. However, it is especially advisable to use them when generalizing and systematizing students’ knowledge.
Conducting training is not an easy task. Special skill is required from the teacher. In such a lesson, the teacher is a conductor, whose task is to skillfully switch and concentrate the attention of students.
The main character in the training lesson is the student.
2.2.1. Structure of training lessons
1. Setting a goal
The teacher, together with the students, determines the main goals of the lesson, including the sociocultural position, which is inextricably linked with “revealing the secrets of words.” The fact is that each lesson has an epigraph, the words of which reveal their special meaning for each 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 and hold hands. The teacher’s task is to make every child feel supported and treated kindly. A feeling of unity with the class and the teacher helps create an atmosphere of trust and mutual understanding.
2. Independent work. Making your own decision
Each student receives a task card. The question contains a question and three possible answers. One, two, or all three options may be correct. The choice hides possible common mistakes made by students.
Before starting to complete tasks, children pronounce the “rules” of work that will help them organize a dialogue. They may be different in each class. Here is one option: “Everyone should speak out and listen to everyone.” Pronouncing these rules out loud helps create a mindset for all children in the group to participate in the dialogue.
At the stage of independent work, the student must consider all three answer options, comparing and contrasting them, make a choice and prepare to explain his choice to a friend: why he thinks this way and not otherwise. To do this, everyone needs to delve into their knowledge base. The knowledge acquired by students in lessons is built into a system and becomes a means for evidence-based choice. The child learns to systematically search through options, compare them, and find the best option.
In the process of this work, not only systematization, but also generalization of knowledge occurs, 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 must explain which answer option he chose and why. Thus, working in pairs (fours) necessarily requires active speech activity from each child and develops listening and hearing skills. Psychologists say: students retain 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 becomes in demand.
At the moment of logical comprehension and structuring of speech, concepts are adjusted and knowledge is structured.
An important point at this stage is the adoption of a group decision. The very process of making such a decision contributes to the adjustment of personal qualities and creates conditions for the development of the individual and the group.
4. Listen to different opinions as a class
By giving the floor to different groups of students, the teacher has an excellent opportunity to track how well the concepts are formed, how strong the knowledge is, how well the children have mastered the terminology, and 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 convincing speech.
5. Expert assessment
After the discussion, the teacher or students voice the correct choice.
6. Self-esteem
The child learns to evaluate the results of his activities himself. This is facilitated by a system of questions:
Did you listen carefully to your friend?
Were you able to prove the correctness of your choice?
If not, why not?
What happened, what was difficult? Why?
What needs to be done to make the work successful?
Thus, the child learns to evaluate his actions, plan them, realize his understanding or misunderstanding, his progress.
Students open a new card with the task, and the work again proceeds in 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 express) their attitude to the epigraph, as they understand it. At this stage, the “mystery of the words” of the epigraph is revealed. This technique allows the teacher to address problems of morality, the relationship of educational activities with real problems of the surrounding world, and allows students to perceive educational activities as their own social experience.
Trainings should not be confused with practical lessons, where strong skills and abilities are formed through a variety of training exercises. They also differ from testing, although they also provide for a choice of answer. However, during testing, it is difficult for the teacher to monitor how justified the choice was made by the student; a choice at random is not excluded, since the student’s reasoning remains at the level of internal speech.
The essence of training lessons is in the development of a unified conceptual apparatus, in students’ awareness of their achievements and problems.
The success and efficiency of this technology is possible with a high level of lesson organization, the necessary conditions of which are the thoughtfulness of working pairs (fours) and the experience of students working together. Pairs or fours 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 a holistic perception of the material and self-development of each child.
The training lessons were developed in accordance with the thematic planning of L.G. Peterson and are conducted through reserve lessons. Subjects of training lessons: numbering, the meaning of arithmetic operations, methods of calculations, order of actions, quantities, solving problems and equations. During the academic year, from 5 to 10 trainings are conducted depending on the class.
Thus, 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 for the goals of mathematics education have significantly affected the process of teaching mathematics. The main idea is the priority of the developmental function in teaching. Oral exercises are one of the means in the educational and cognitive process that makes it possible to realize the idea of development.
Oral exercises contain enormous potential for developing thinking and activating students’ cognitive activity. They allow you to organize the educational process in such a way that as a result of their implementation, students form a holistic picture of the phenomenon under consideration. This provides the opportunity not only to retain in memory, but also to reproduce exactly those fragments that turn out to be necessary in the process of passing subsequent steps of cognition.
The use of oral exercises reduces the number of tasks in the lesson that require full written documentation, which leads to more effective development of speech, mental operations and creative abilities of students.
Oral exercises destroy stereotypical thinking by constantly involving the student in the analysis of initial information and predicting errors. The main thing when working with information is to involve students themselves in creating an indicative basis, which shifts the emphasis of the educational process from the need for memorization to the need for the ability 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 carry out systematic work on the formation of computational skills and skills in solving word 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) formation of spatial representations.
2.4. Knowledge control
Modern learning technologies can significantly increase the efficiency of the learning process. At the same time, most of these technologies leave out of the scope 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' training currently used at 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 advisable to submit for control is discussed. Issues related to the forms of control, and even more so the methods of processing and storing educational information received during control, remain without due attention from teachers. At the same time, in modern society, an information revolution has occurred quite a long time ago; new methods of analysis, collection and storage of data have appeared, making 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. Monitoring 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 information received during this communication reliable, comprehensive and reliable, The effectiveness of the decisions made also depends. 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 activities, directing them to achieve learning goals. If management at this level is absent or is not coordinated with learning goals, then a situation occurs when the student is taught, but he himself does not learn. Accordingly, in order to effectively manage his activities, a student must have all the necessary information about the learning results he achieves. Naturally, at the lower stages of education, the student mainly receives this information from the teacher in ready-made form.
The second level is the teacher. This is the main figure directly responsible for 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 students the information they need so that they can consciously take part in the educational process.
The third level is public education authorities. This level represents a hierarchical system of institutions for managing public education. Management bodies deal both with information that they receive independently and independently of the teacher, and with information transmitted to them by teachers.
The information that the teacher transmits to students and to higher authorities is the school grade assigned by the teacher based on the results of students’ activities during the educational process. It is advisable to distinguish between two types: current and final grade. The current assessment, as a rule, takes into account the results of students’ performance of certain types of activities; the final assessment is, as it were, a derivative of the current assessments. Thus, the final grade may not directly reflect the final level of student preparation.
Assessment of students' achievements by the teacher is a necessary component of the educational process, ensuring its successful functioning. Any attempts to ignore knowledge assessment (in one form or another) lead to disruption of the normal course of the educational process. Evaluation, on the one hand serves as a guide For students, showing them how their efforts meet the teacher's requirements. On the other hand, the presence of assessment allows educational authorities, as well as parents of students, to monitor the success of the educational process and 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 identified properties of this object or process with some given criterion. An example of an assessment would be the award of a rank in sports. The category is assigned based on measuring the athlete's performance results by comparing them with given standards. (For example, the running result in seconds is compared with the standards corresponding to a particular category.)
Evaluation is secondary to measurement and Maybe be obtained only after the measurement has been carried out. In modern schools, these two processes are often not distinguished, since the measurement process takes place as if in a compressed form, and the assessment itself has the form of a number. Teachers do not think about the fact that, by recording the number of actions correctly performed by a student (or the number of errors made by him/her) when performing this or that work, they thereby measure the results of the students’ activities, and when giving a grade to the student, they correlate the identified quantitative indicators with those available in the their disposal of evaluation criteria. Thus, teachers themselves, having, as a rule, the results of measurements that they use to grade students, rarely inform other participants in the educational process about them. This significantly narrows the information available to students, their parents and governing bodies.
Knowledge assessment can be in either numerical or verbal form, which in turn creates additional confusion that often exists between measurements and assessments. The measurement results can only be in numerical form, since in general measurement is establishing a correspondence between an object and a number. The form of the assessment is an unimportant characteristic of it. So, for example, a judgment like “student fully has mastered the taught material” may be equivalent to the statement “the student knows the covered material in Great” or “the student has a grade of 5 for the completed course material.” The only thing that researchers and practitioners should remember is that in the latter case the assessment 5 is not a number in the mathematical sense and with it no arithmetic operations are allowed. A score of 5 serves to classify a given student into a certain category, the meaning of which can be deciphered unambiguously only taking into account the adopted assessment system.
The modern school assessment system suffers from a number of significant shortcomings that do not allow it to be fully used as a high-quality source of information about the level of student preparation. School assessment is usually 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.
Standard tests and independent work, common to all students, are used as measuring tools in the educational process. The results of these tests are assessed 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 stated learning goals. At the same time, the issues of processing test results, measuring student performance results and their evaluation in most of the methodological literature are studied at an insufficiently high level of development and formalization. This leads to the fact that teachers often give different grades to students for the same work results. There may be even greater differences in the results of assessing the same work by different teachers. The latter occurs due to the fact that in the absence of strictly formalized rules defining algorithm measurement and assessment, different teachers may perceive the measurement algorithms and assessment criteria proposed to them differently, replacing them with their own.
The teachers themselves explain it as follows. When evaluating work, they have in mind first of all student's reaction on the rating he received. The main task of the teacher is to encourage the student to new achievements, and here the assessment function as an objective and reliable source of information about the level of students’ preparation is of less importance to them, but to a greater extent teachers are aimed at implementing the control function of assessment.
Modern methods for measuring the level of student preparation, focused on the use of computer technologies, fully meeting the realities of our time, provide the teacher with fundamentally new opportunities and increase the efficiency of his activities. 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, with traditional control, information about the level of students’ preparation was owned and completely 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 comrades in the same important matter, in the results of which they are equally interested.
Traditional control is represented by independent and test work (12 workbooks that make up a set of mathematics for primary school).
When carrying out independent work, the goal is primarily to identify the level of mathematical preparation of children and promptly eliminate existing knowledge gaps. At the end of each independent work there is a space for work on bugs. At first, the teacher should help children choose tasks that allow them to correct their mistakes in a timely manner. Throughout the year, independent work with corrected errors is collected in a folder, which helps students track their path in mastering knowledge.
Tests summarize this work. Unlike independent work, the main function of control work is precisely the control of knowledge. From the very first steps, a child should be taught to be especially attentive and precise in his actions while monitoring knowledge. Test results, as a rule, are not corrected - you need to prepare for knowledge testing before him, and not after. But this is exactly how any competitions, exams, administrative tests are conducted - after they are carried out, the result cannot be corrected, and children need to be gradually psychologically prepared for this. At the same time, preparatory work and timely correction of errors during independent work provide a certain guarantee that the test will be written successfully.
The basic principle of 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 improvement and elimination. A calm atmosphere during tests is determined by the extensive preparatory work that has been done in advance and which removes all reasons for concern. In addition, the child must clearly feel the teacher’s faith in his strength and 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 of them, without exception, cope with the proposed variants of tasks.
Independent work usually takes 7-10 minutes (sometimes up to 15). If the child does not have time to complete the independent work assignment within the allotted time, after checking the work by the teacher, he finalizes these assignments at home.
Grading for independent work is given after the errors have been corrected. What is assessed is not so much what the child managed to do during the lesson, but how he ultimately worked on the material. Therefore, even those independent works that are not written very well in class can be given a good or excellent score. In independent work, the quality of work on oneself is fundamentally important and only success is assessed.
Test work takes from 30 to 45 minutes. If one of the children does not complete the tests within the allotted time, then at the initial stages of training you can allocate some additional time for him to give him the opportunity to calmly finish the work. Such “adding” to work is excluded when carrying out independent work. But in the control work there is no provision for subsequent “revision” - the result is evaluated. The grade for the test work is corrected, as a rule, in the next test work.
When grading, you can rely on the following scale (tasks with an asterisk are not included in the mandatory part and are assessed with an additional mark):
“3” - if at least 50% of the work has been 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 arbitrary, since when giving a grade, the teacher must take into account many different factors, including the level of preparedness of the children, and their mental, physical, and emotional state. In the end, assessment should not be a sword of pre-Mocles in the hands of a teacher, but a tool that helps a child learn to work on himself, overcome difficulties, and believe in himself. Therefore, first of all, you 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 1st grade, grades are given only for works written as “good” and “excellent”. You can say to the rest: “We need to catch up, we will succeed too!”
In most cases, work is carried out on a printed basis. But in some cases, they are offered on cards or can even be written on the board to accustom children to different forms of presentation of material. The teacher can easily determine in what form the work is being carried out by whether there is space left for writing in answers or not.
Independent work is offered approximately 1-2 times a week, and tests are offered 2-3 times a quarter. At the end of the year children first they write the translation work, determining the ability to continue education in the next grade in accordance with the state knowledge standard, and then - the final test.
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 working conditions, the level of the final test may be reduced. In any case, a 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 identify 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, and to emotionally experience the joy of victory.
The high level of testing 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 must increase. Administrative control is carried out in the same way as in classes taught according to any other programs and textbooks. You should only take into account that the material on topics is sometimes distributed differently (for example, the methodology adopted in this textbook assumes a later introduction of the first ten numbers). Therefore, it is advisable to carry out administrative control at the end educational of the year .
Chapter 3. Analysis of the experiment
How do schoolchildren perceive the simplest tasks? Is the approach proposed by the School 2100 program more effective in teaching problem solving compared to the traditional one?
To answer these questions, we conducted an experiment in gymnasium No. 5 and secondary school No. 74 in Minsk. Preparatory school students took part in the experiment. The experiment consisted of three parts.
Stater. Simple tasks were proposed that needed to be solved according to plan:
1. Condition.
2. Question.
4. Expression.
5. Solution.
A system of exercises was proposed using the activity method in order to develop skills 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 in total?
2. The cat Murka has 7 kittens. Of these, 3 are white and the rest are variegated. How many motley kittens does Murka have?
3. There were 5 passengers on the bus. At the stop, some of the passengers got off, only 1 passenger remained. How many passengers got off?
The purpose of the ascertaining experiment: check the initial level of knowledge, skills and abilities of preparatory school students when solving simple problems.
Conclusion. The result of the ascertaining experiment is reflected in the graph.
Decided: 25 problems - students of gymnasium No. 5
24 problems - students of secondary 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.
The highest results were achieved when solving problem No. 1. The lowest results were achieved when solving problem No. 3.
The general level of students in the two groups who coped with solving 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 identify elements of a task (condition, question);
b) the ability to model the text of a problem using segments (constructing 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 problem (“dressing” the diagram) and composing an expression.
3.2. Educational experiment
Purpose of the experiment: continue work on solving problems using the activity method with students from gymnasium No. 5 studying under the “School 2100” program. To develop stronger knowledge, skills and abilities when solving problems, special attention was paid to drawing up a diagram (“dressing” the diagram) and composing an expression according to the scheme.
The following tasks were offered.
1. Game “Part or whole?”
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In another version of the proposed game, the situation is closer to the one in which students will find themselves when modeling the problem. Schemes are drawn up on the board in advance. 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 following conventions:
¾ - whole
The technique of mutual verification and the technique of reconciliation with the correct execution of the task on the board can be used.
2. Game "What changed?"
The diagram is in front of the students:
It turns out what is known: a part or a whole. Then the students close their eyes, the diagram takes the form 2), the students answer the same question, close their eyes again, the diagram is transformed, etc. - as many times as the teacher considers necessary.
Similar tasks in a game form can be offered to students with a question mark. Only the task will be formulated somewhat differently: “What unknown: part or whole?”
In previous assignments, students “read” the diagram; It is equally important to be able to “dress” the scheme.
3. Game “Wear the scheme”
Before the start of the lesson, each student receives a small piece of paper with diagrams that are “dressed up” according to the teacher’s instructions. Tasks can be like this:
- A- Part;
- b– whole;
Unknown whole;
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 a group of “a” boys and “b” 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 placed in the second diagram.
When modeling the text of a problem, the student must clearly imagine what needs to be found in the problem: a part or a whole. For this purpose, the following work can be carried out.
5. Game “What is unknown?”
The teacher reads the text of the problem, and the students answer the question about what is unknown in the problem: part or whole. A card that looks like this can be used as a means of feedback:
on the one hand, on the other: .
For example: in one bunch there are 3 carrots, and in the other there are 5 carrots. How many carrots are there in two bunches? (the whole is unknown).
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 problem: a part or a whole, the question is asked about how to do this (by what action). Students are prepared to make informed choices of arithmetic operations based on the relationship between the whole and its parts.
Show the whole, show the parts. What is known, what is unknown?
I show - do you name what it is: a whole or a part, is it known or not?
What is greater, the part or the whole?
How to find the whole?
How to find a part?
What can you find if you know the whole and the part? How? (What action?).
What can you find if you know the parts of a 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 the part? How? (What action?).
Write an expression for each diagram?
The reference diagrams used at this stage of working on the task can look like this:
During the experiment, students came up with their own problems, illustrated them, “dressed up” diagrams, used commenting, and worked independently with various types of testing.
3.3. Control experiment
Target: check the effectiveness of the approach to solving simple problems proposed by the educational program “School 2100”.
The following 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 were playing in the yard, 5 of them boys. How many girls were there?
6 birds were sitting on a birch tree. Several birds flew away, 4 birds remained. How many birds flew away?
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 - 4 pages. How many pages did Dima read on the third day?
Conclusion. The result of the control experiment is reflected in the graph.
Decided: 63 problems – students of gymnasium No. 5
50 problems – students of school No. 74
As you can see, the results of students from gymnasium No. 5 in solving problems are higher than those of students from secondary school No. 74.
So, the results of the experiment confirm the hypothesis that if the educational program “School 2100” (an activity method) is used when teaching mathematics to primary schoolchildren, 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. Students have not previously been offered such problems. When solving such problems, it was necessary, using a certain base of knowledge, skills and abilities, to independently find solutions to more complex problems. Students from gymnasium No. 5 completed them more successfully (21 problems solved) than students from secondary school No. 74 (14 problems solved).
I would like to present 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 board 100%
Able to express their thoughts more clearly and clearly 100%
Not afraid to make a mistake 100%
Became more active and independent 86.7%
93.3% are not afraid to express their point of view
Better justify their answers 100%
Calmer 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, cautious and serious in their actions;
· children are at ease and bold in communicating with adults, they easily come into contact with them;
· they have excellent self-control skills, including in the area of relationships and rules of behavior.
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. It should be emphasized that the “School 2100” program was created specifically for mass secondary schools. The following can be distinguished benefits of this program:
1. The principle of psychological comfort embedded in the program is based on the fact that each student:
· is an active participant in cognitive activities in the classroom and can demonstrate his creative abilities;
· progresses while studying the material at a pace convenient for him, gradually assimilating the material;
· masters the material to the extent that is accessible and necessary to him (the minimax principle);
· feels interest in what is happening in each lesson, learns to solve problems that are interesting in content and form, learns new things not only from the mathematics course, 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 acts not as an informant, but as an organizer search activity of students. A specially selected system of tasks, during which students analyze the situation, express their suggestions, listen to others and find the right answer, helps the teacher in this.
The teacher often offers tasks during which the children cut out, measure, color, and trace. This allows you not to memorize the material mechanically, but to study it consciously, “passing it through your hands.” Children draw their own conclusions.
The exercise system is designed in such a way that it also contains a sufficient set of exercises that require actions according to a given pattern. In such exercises, skills and abilities are not only developed, 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. One cannot fail to mention special exercises aimed at developing the creative abilities of students. The important thing is that these tasks are given in the system, starting from the first lessons. Children come up with their own examples, problems, equations, etc. They really enjoy this activity. It is no coincidence that children’s creative works on their own initiative are usually brightly and colorfully designed.
Textbooks are multi-level, allow you to organize differentiated work with textbooks in the lesson. Assignments typically include both practice of mathematics education standards and questions that require 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 performance in studying mathematics.
5. The program provides effective preparation for studying algebra and geometry courses in high school.
From the very beginning of the mathematics course, students are accustomed to working with algebraic expressions. Moreover, the work is carried out in two directions: composing and reading expressions.
The ability to compose letter expressions is honed in an unconventional type of task - blitz tournaments. These tasks arouse great interest in children and are successfully completed by them, despite the fairly high level of complexity.
Early use of algebra elements provides a solid foundation for the study of mathematical models and for exposing advanced students to the role and significance of mathematical modeling.
This program provides an opportunity through activities 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 triangle, and 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 learning results if students have low interest in mathematics. To develop and consolidate it, the course offers quite a lot of exercises that are interesting in content and form. A large number of numerical crosswords, puzzles, ingenuity tasks, and decodings help the teacher make lessons truly exciting and interesting. In the course of completing these tasks, children decipher either a new concept or a riddle... Among the deciphered words are the names of literary characters, titles of works, names of historical figures that are not always familiar to children. This stimulates learning new things; there is a desire to work with additional sources (dictionaries, reference books, encyclopedias, etc.)
7. Textbooks have a multi-linear structure, giving the ability to systematically work on repeating material. It is well known that knowledge that is not included in work for a certain time is forgotten. It is difficult for a teacher to independently work on selecting knowledge for repetition, because searching for them takes considerable time. These textbooks provide the teacher with great assistance in this matter.
8. Printed textbook base in elementary school, it 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 developing students’ skills is solved self-control.
The work carried out confirmed the hypothesis put forward. The use of an activity-based approach to teaching mathematics to junior schoolchildren has shown that cognitive activity, creativity, and liberation of students increase, and fatigue decreases. The “School 2100” program meets the challenges of modern education and lesson requirements. For several years, children did not have unsatisfactory grades in 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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Annex 1
Topic: SUBTRACTING TWO-DIGITUAL NUMBERS WITH TRANSITION THROUGH THE DIGIT
2nd grade. 1 hour (1 - 4)
Target: 1) Introduce the technique of subtracting two-digit numbers with transition through the digit.
2) Consolidate the learned computational techniques, the ability to independently analyze and solve compound problems.
3) Develop thinking, speech, cognitive interests, creative abilities.
During the classes:
1. Organizational moment.
2. Statement of the educational task.
2.1. Solving subtraction examples with transition through digits within 20.
The teacher asks the children to solve examples:
Children verbally name the answers. The teacher writes the children's answers on the board.
Divide 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 equal to 8 and not equal to 8, etc.)
What do all the examples have in common? (The same calculation method is subtraction with transition through the digit.)
What other subtraction examples can you solve? (For subtracting two-digit numbers.)
2.2. Solving examples on subtracting two-digit numbers without jumping through the place value.
Let's see who can solve these examples better! What's interesting about the differences: *9-64, 7*-54, *5-44,
It is better to place examples one below the other. Children should notice that in the minuend one digit is unknown; unknown tens and ones alternate; all known digits in the minuend are odd and are in descending order: in the subtrahend, the number of tens is reduced by 1, but the number of units does not change.
Solve the minuend if you know that the difference between the digits denoting tens and units is 3. (In the 1st example - 6 d., 12 d. cannot be taken, since only one digit can be put in a digit; in the 2nd example - 4 units, since 10 units are not suitable; in the 3rd - 6 units, 3 units 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 closed numbers and asks 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 =. From 9 units. subtract 4 units, we get 5 units. From 6 d. subtract 6 d., we get O d. Answer: 5.
2.3. Formulation of the problem. Goal setting.
When solving the last example, children experience difficulty (different answers are possible, some will not be able to solve it at all): 41-24 = ?
The goal of our lesson is to invent a subtraction technique that will help us solve this example and examples like it.
Children lay out the example model on the desk and on the demonstration canvas:
How to subtract two-digit numbers? (Subtract tens from tens, and ones from units.)
Why did the difficulty arise here? (The minuend is missing units.)
Is our minuend less than our subtrahend? (No, the minuend is greater.)
Where are the few hiding? (In the top ten.)
What need to do? (Replace 1 ten with 10 units. - Discovery!)
Well done! Solve the example.
Children replace the tens triangle in the minuend with a triangle on which 10 units are drawn:
11e -4e = 7e, Zd-2d=1d. In total it turned out to be 1 d. and 7 e. or 17.
So. “Sasha” offered us a new method of calculations. It is as follows: split ten and take from his missing units. Therefore, we could write down our example and solve it like this (the entry is commented):
Can you think of what you should always remember when using this technique, where an error is possible? (The number of tens is reduced by 1.)
4. Physical education minute.
5. Primary consolidation.
1) No. 1, page 16.
Comment on the first example using the following example:
32 - 15. From 2 units. You cannot subtract 5 units. Let's split ten. From 12 units. subtract 5 units, and from the remaining 2 tenths. subtract 1 dec. We get 1 dec. and 7 units, that is 17.
Solve the following examples with explanation.
Children complete the graphical models of the examples and at the same time comment on the solution aloud. Lines connect pictures with equalities.
2) No. 2, p. 16
Once again, the solution and commentary on the example are clearly stated in a column:
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 cannot subtract 9 units. I borrow 1 day and put an end to it. 11-9 = 2 units. I write under units.
I subtract the 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, p. 16.
Let's play a guessing game:
82 - 6 41 -17 74-39 93-45
82-16 51-17 74-9 63-45
Children write down and solve examples in squared notebooks. 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 copy examples from the board in a column. for a new computing technique
98-19, 64-12, 76 - 18, 89 - 14, 54 - 17.
Children write down the necessary examples in their notebooks in a square, and then check the accuracy of their notes using the finished sample:
19 18 17
They then solve the written examples on their own. After 2-3 minutes the teacher shows the correct answers. Children check them themselves, mark correctly solved examples with a plus, and correct mistakes.
Find a pattern. (The numbers in the minuends are written in order from 9 to 4, the subtrahends themselves go in decreasing order, etc.)
Write your own example that would continue this pattern.
7. Repetition tasks.
Children who have completed their independent work come up with and solve problems in their notebooks, and those who have made mistakes refine their mistakes individually together with the teacher or consultants. then they solve 1-2 more examples on a new topic on their own.
Come up with a problem and solve according to the options:
Option 1 Option 2
Perform cross-check. What did you notice? (The answers to the problems are the same. These are mutually inverse problems.)
8. Lesson summary.
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 technique!
Children offer several options. One is selected. Children. write it down and solve it in a notebook, and one of the children does it on the board.
9. Homework.
No. 5, p. 16. (Unravel the name of the fairy tale and the author.)
Compose your own example of a new computational technique and solve it graphically and columnarly.
Topic: MULTIPLICATION BY 0 AND 1.
2kl., 2h. (1-4)
Target: 1) Introduce special cases of multiplication with 0 and 1.
2) Reinforce the meaning of multiplication and the commutative property of multiplication, practice 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:
| 2 | 5 | 8 | ||||
| 10 | 4 | |||||
| 3 | 5 | ||||
| 1 | 9 | 6 |
What's interesting about the numbers written down? (Write in different colors; all “red” numbers are even, and “blue” numbers are odd.)
Which number is the odd one out? (10 is round, and the rest are not; 10 is two-digit, and the rest are single-digit; 5 is repeated twice, and the rest - one at a time.)
I’ll close the number 10. Is there an extra one among the other numbers? (3 - he doesn’t have a pair until 10, but the rest do.)
Find the sum of all the “red” numbers and write it in the red square. (thirty.)
Find the sum of all the “blue” numbers and write it 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 did you use? (By subtraction.)
2.2. Tasks for the development of memory and speech. Updating knowledge.
a) -Repeat in order the words that I will name: addend, addend, sum, minuend, subtrahend, difference. (Children try to reproduce the order of words.)
What components of actions were named? (Addition and subtraction.)
What new action are we introduced to? (Multiplication.)
Name the components of multiplication. (Multiplier, multiplier, product.)
What does the first factor mean? (Equal terms in the sum.)
What does the second factor mean? (The number of such terms.)
Write down the definition of multiplication.
b) -Look at the notes. What task will you be doing?
12 + 12 + 12 + 12 + 12
33 + 33 + 33 + 33
(Replace the sum with the product.)
What will happen? (The first expression has 5 terms, each equal to 12, so it is equal to
12 5. Similarly - 33 4, and 3)
c) - Name the inverse 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) Equalities are written on the board:
21 3 = 21+22 + 23
44 + 44 + 44 + 44 = 44 + 4
17 + 17-17 + 17-17 = 17 5
Next to each equation, the teacher places pictures of a chicken, a baby elephant, a frog and a mouse, respectively.
The animals from the forest school were completing a task. Did they do it correctly?
Children establish that the baby elephant, frog and mouse made a mistake, and explain what their mistakes were.
e) - Compare the expressions:
8 – 5… 5 – 8 34 – 9… 31 2
5 6… 3 6 a – 3… a 2 + a
(8 5 = 5 8, since the sum does not change from rearranging the terms; 5 6 > 3 6, since there are 6 terms on the left and right, but there are more terms on the left; 34 9 > 31 - 2. since there are more terms on the left and themselves the terms are greater; a 3 = a 2 + a, since on the left and right there are 3 terms equal to a.)
What property of multiplication was used in the first example? (Commutative.)
2.3. Formulation of the problem. Goal setting.
Look at the picture. Are the equalities true? Why? (Correct, since the sum is 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.)
What does the expression 5 1 mean? 50? (? Problem!) Bottom line 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 to do this, we need to check whether we will violate the commutative property of multiplication. So, the goal of our lesson is determine whether we can count equalities 5 1 = 5 and 5 0 = 0 true? - Lesson problem!
3. “Discovery” of new knowledge by children.
1) No. 1, page 80.
a) - Follow 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
Draw a conclusion: 1 a -? (1 a = a.) The teacher puts out a card: 1 a = a
b) - Do the expressions 7 1, 4 1, 5 1 make sense? Why? (No, because the sum cannot have one term.)
What should they be equal to so that the commutative property of multiplication is not violated? (7 1 must also equal 7, so 7 1 = 7.)
4 1 = 4 are considered similarly. 5 1 = 5.
Draw a conclusion: and 1 =? (a 1 = a.)
The card is displayed: 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 number line? (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) The case of multiplication from 0 in No. 4, p. 80 is studied in a similar way. Conclusion - multiplying a number by 0 or 0 by a number produces zero:
a 0 = 0 a = 0.
Compare both equalities: what do 0 and 1 remind you of?
Children express their versions. You can draw their attention to those images that are given in the textbook: 1 - “mirror”, 0 - “terrible beast” or “invisible hat”.
Well done! So, when multiplied by 1, the same number is obtained (1 is a “mirror”), and when multiplied by 0, the result is 0 (0 is an “invisible hat”).
4. Physical education minute.
5. Primary consolidation.
Examples written on the board:
23 1 = 0 925 = 364 1 =
1 89= 156 0 = 0 1 =
Children solve them in a notebook with the resulting rules spoken out loud, for example:
3 1 = 3, since when a number is multiplied 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 an unknown number, the result was 145. This means that they multiplied by 1 x= 1. Etc.
3) No. 6, p. 81.
a) 8 x = 0; b) x 1= 0.
When multiplying 8 by an unknown number, the result was 0. So, multiplied by 0 x = 0. Etc.
6. Independent work with testing in 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 written examples. Then, based on the finished sample, they check their answers with pronunciation in loud speech, mark correctly solved examples with a plus, and correct the mistakes made. Those who made mistakes receive a similar task on a card and refine it individually with the teacher while the class solves repetition problems.
7. Repetition tasks.
a) - We are invited to visit today, but to whom? You will find out by deciphering the recording:
[P] (18 + 2) - 8 [O] (42+ 9) + 8
[A] 14 - (4 + 3) [H] 48 + 26 - 26
[F] 9 + (8 - 1) [T] 15 + 23 - 15
Who are we invited to visit? (To Fortran.)
b) - Professor Fortran is a computer expert. 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 like the one on page 56, M-2, part 1.) We set off according to X's program. Which house did we come to?
One student follows the poster on the board, and the rest follow the program in their textbooks and find the Fortran house.
c) - Professor Fortran meets us with his students. His best student, the caterpillar, has prepared a task for you: “I thought of a number, subtracted 7 from it, added 15, then added 4 and got 45. What number did I think of?”
Reverse operations must be done in reverse order: 45-4-15 + 7 = 31.
G) Game-competition.
- The Fortran professor himself invited us to play the game “Computing Machines”.
| A | 1 | 4 | 7 | 8 | 9 |
| x |
Table in students' notebooks. They independently perform calculations and fill out the table. The first 5 people who complete the task correctly win.
8. Lesson summary.
Did you do everything you planned in the lesson?
What new rules have you met?
9. Homework.
1) №№ 8, 10, p. 82 - in a squared notebook.
2) Optional: № 9 or № 11 on p.82 - on a printed basis.
Topic: PROBLEM SOLVING.
2nd grade, 4 hours (1 - 3).
Target: 1) Learn to solve problems using sum and difference.
2) Strengthen computational skills, composing letter expressions for word problems.
3) Develop attention, mental operations, speech, communication skills, interest in mathematics.
During the classes:
1. Organizational moment .
2. Statement of the educational 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 in front.
Front work:
Reduce the number 244 by 2 times (122)
Find the product of 57 and 2 (114)
Reduce the number 350 by 230 (120)
How much is 134 greater than 8? (126)
Reduce the number 1280 by 10 times (128)
What is the quotient of 363 and 3? (121)
How many centimeters are in 1 m 2 dm 4 cm? (124)
Arrange the resulting numbers in ascending order:
| 114 | 120 | 121 | 122 | 124 | 126 | 128 |
| Z | A | Y | H | A | T | A |
Individual work at the board:
- Three The trickster 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 left without a pair? (Number 7.)
Describe this number. (Single digit, odd, multiples of 1 and 7.)
2.2. Setting a learning task.
Each team receives 4 “Blitz Tournament” problems, a plate and a diagram.
“Blitz tournament”
a) One hare put on a rings, and the other one put on 2 more rings than the first. How many rings do they both have?
b) The mother hare had rings. She gave her three daughters each b rings How many rings does she have left?
c) There were red rings, b white rings and pink rings. They were distributed equally to 4 bunnies. How many rings did each hare receive?
d) The mother hare had rings. She gave them to her two daughters so that one of them got n more rings than the other. How many rings did each daughter receive?
For the 1st team:
For the 2nd team:
For the III team:
It has become fashionable among rabbits to wear rings in their ears. Read the problems on your sheets of paper and determine which problem your diagram and your expression fit into?
Students discuss problems in groups and find the answer together. One person from the group “defends” the team’s opinion.
What problem did I not choose a diagram and expression for?
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 conditions could we consider it correct? (If both hare had the same number of rings.)
We encountered 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.) Difference of rings? (Letter n .)
Is it possible to equalize the number of rings on both hare? 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 there are? (a-n)
Is it twice the smaller number or the larger number? (Less.)
How to find the smaller number? ((a-n): 2.)
Have we answered the problem question? (No.)
What else should you know? (Larger number.)
How to find a larger number? (Add difference: (a-n): 2 + n)
Tablets with the obtained expressions are recorded on the board:
(a-n): 2 - smaller number,
(a-n): 2 + n - greater number.
We first found twice the smaller number. How else could one reason? (Find twice the number.)
How to do it? (a + n)
How then to answer the questions of the task? ((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 double the smaller number - by subtraction, or find first double a larger number by 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 minute.
5. Primary consolidation.
Students work with a textbook-notebook. Tasks are solved with comments, the solution is written down on a printed basis.
a) - Read the problem to yourself № 6(a), p. 7.
What do we know about the problem and what do we need to find? (We know that there are 56 people in two classes, and in class 1 there are 2 more people than in class two. We need to find the number of students in each class.)
- “Dress” the diagram and analyze the problem. (We know the sum - 56 people, and the difference - 2 students. First, we will find twice the smaller number: 56 - 2 = 54 people. Then we will find out how many students are in the second grade: 54: 2 = 27 people. Now we will find out how many students are in first class - 27 + 2 = 29 people.)
How else can you find out how many students are in first grade? (56 – 27 = 29 people.)
How to check if a problem has been solved correctly? (Calculate the sum and difference: 27 + 29 = 56, 29 – 27 = 2.)
How could the problem be solved differently? (First find the number of students in the first grade and subtract 2 from it.)
b) - Read the problem to yourself № 6 (b), page 7. Analyze which quantities are known and which are not and come up with a solution plan.
After a minute of discussion in the teams, a representative of the team that was ready first speaks. Both ways 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 testing in class.
Students, using 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 = 240(m.) 1) 248 +8 = 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 =180 (open) 3)180+12 = 192 (open)
Answer: 180 postcards; 192 postcards.
Check - according to the finished sample on the board.
Each team receives a sign with the task: “Find a pattern and enter the required numbers instead of question marks.”
1 team:
2 team:
3 team:
Team captains report on the team's performance.
8. Lesson summary.
Explain how you reason when solving problems if the following operations are performed:
9. Homework.
Come up with your own new type of problem and solve it in two ways.
Topic: COMPARISON OF ANGLES.
4th grade, 3 hours (1-4)
Target: 1) Review the concepts: point, ray, angle, vertex of an angle (point), sides of an angle (rays).
2) Introduce students to the method of comparing angles using direct superposition.
3) Repeat problems into 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 educational task.
a) - Continue the series:
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 [H] 68: 4
Cross out the extra 2 letters. What word did you get? (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 happens? (The segment is called the 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, cubic ball, cylinder, cone, pyramid, etc.)
What types of angles are there? (Straight, sharp, blunt.)
Show with pencils a model of an acute angle, a right angle, an obtuse one.
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, p. 1.
Children must determine that all corners in the drawing have the side formed by the large arrow in common. The more the arrows are “spread apart,” the greater the angle.
e) No. 2, p. 1.
Children's opinions about the relationship between angles usually vary. This serves as the basis for creating a problematic 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 continuation of the sides of the corners none of the corners is inside the other. Then, based on the third method - “which fits”, a rule for comparing angles is derived: the angles must be superimposed on one another 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 coincides. Then the angle whose side is inside the other angle is smaller.
The resulting output is compared with the textbook text on page 1.
4. Primary consolidation.
Task No. 4, page 2 of the textbook is solved with commentary, aloud the rule for comparing angles is spelled out.
In task No. 4, page 2, the angles must be compared “by eye” and arranged in ascending order. The name of the pharaoh is CHEOPS.
5. Independent work with testing in class.
Students do the practice work in No. 3, page 2 independently, then in pairs explain how they made the angles. After this, 2-3 pairs explain the solution to the whole class.
6. Physical education minute.
7. Solving repetition problems.
1) - I have a difficult task. Who wants to try to solve it?
During a mathematical dictation, two volunteers together must come up with a solution to the problem: “Find 35% of 4/7 of the number x” .
2) The mathematical dictation was recorded on a tape recorder. Two write down the task on individual boards, the rest - in a notebook “in a column”:
Find 4/9 of number a. (a: 9 4)
Find a number if 3/8 of it is b. (b: 3 8)
Find 16% of the village. (from: 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 part of a leap year is February? (29/366)
Check - according to the sample solution on portable boards. Errors made while completing a task are analyzed according to the scheme: it is established what is unknown - the whole or the part.
3) Analysis of the solution to the additional task: (x: 7 4): 100 35.
Students recite 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 it by its numerator.
4) No. 9, p. 3 - orally with justification for the decision:
- A greater than 2/3, since 2/3 is a proper fraction;
Bless than 8/5, since 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, page 3. The first line is solved with commentary:
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, you need to 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 first deciphers the name of the pharaoh in whose honor the very first pyramid was built:
| 1072 | 560 | 210 | 102 | 75 | 72 |
| D | AND | ABOUT | WITH | E | R |
6) No. 12(6), page 3
The camel's mass is 700 kg, and the mass of the load it carries on its back is 40% of the camel's mass. What is the mass of the camel with its load?
Students mark the condition of the problem on the diagram and analyze it independently:
To find the mass of a camel with a load, you need to add the mass of the load to the mass of the camel (we are looking for the 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 camel's mass. Therefore, in the first step we find 40% of 700 kg, and then add the resulting number to 700 kg.
The solution to the problem with explanations is written down in a notebook:
1) 700: 100 40 = 280 (kg) - mass of the load.
2) 700 + 280 = 980 (kg)
Answer: the mass of a loaded camel is 980 kg.
8. Lesson summary.
What have you learned? What did they repeat?
What did you like? What was difficult?
9. Homework: No. 5, 12 (a), 16
Appendix 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 operations of addition and subtraction, determine which component expresses the part and which the whole.
In the second task, having determined what the unknown is, children choose a rule to solve 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, from 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 need to choose X an unusual situation that the children have not yet encountered. Thus, here the depth of mastery of a new topic and the child’s ability to apply the learned algorithm of actions in new conditions are tested.
Epigraph of the lesson : “Everything secret becomes clear.” Here are some of the children's statements when summing up the results in the resource circle:
In this lesson, I remembered that the whole is found by addition, and the parts are found by subtraction.
Everything that is unknown can be found if you follow the right steps.
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 so that the unknown becomes known.
Expert review
| Job No. | |
| 1 | b |
| 2 | A |
| 3 | V |
| 4 | A |
| 5 | a and b |
Appendix 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 work is being done to develop mental operations, attention, memory, constructive skills, not only are counting skills being developed and advanced preparation is being made for studying subsequent topics of the course, but also an option is offered for creating a problem situation, which can help the teacher organize when studying This topic is the stage of setting a learning task.
Topic: “Number segment”
Main target :
1) Introduce the concept of a number line, 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 location of each missing number? (For example, 2 is 1 more than 1, but 1 less than 3.)
1… 3… 5… 7… 9
Establish 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
Divide the squares into parts by color:
|
|
+=+=
-=-=
How are all the figures labeled? How are the parts labeled? Why?
Fill in the missing letters and numbers in the boxes. 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 equations.
How to find the whole? How to find a part?
How many green squares? How many blue ones?
Which squares are larger - green or blue - and by how many? Which squares are smaller and by how many? (The answer can be explained in the figure by making pairs.)
On what other basis can these squares be divided into parts? (By size - large and small.)
What parts will the number 4 be broken into then? (2 and 2.)
Make two triangles from 6 sticks.
Now make two triangles from 5 sticks.
Remove 1 stick to form a quadrangle.
Name the meanings of numerical expressions:
3 + 1 = 2-1 = 2 + 2 =
1 + 1 = 2 + 1 = 1 + 2 + 1 =
Which expression is “superfluous”? Why? (“Expression 2-1 may be superfluous, since this is a 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 guiding questions:
What do these numerical expressions have in common? (The same sign of the action, the second term is less than the first and equal to 1.)
What is the difference? (Different first terms; in the second expression, both terms are equal, and in the first, one term is 2 more than the other.)
- Problems in verse(the solution to the problems is justified):
Anya has two goals, Tanya has two goals. (We are looking for the whole. To find
Two balls and two, baby, the whole, the parts must be added:
How many are there, can you imagine? 2 + 2 = 4.)
Four magpies came to class. (We are looking for a part. To find
One of the forty did not know the lesson. part must be subtracted from the whole
How diligently did forty work? other part: 4 -1 = 3.)
Today we are waiting for a meeting with our favorite heroes: Boa Constrictor, Monkey, Baby Elephant and Parrot. The boa constrictor really wanted to measure its length. All attempts by Monkey and Baby Elephant to help him were in vain. Their trouble was that they did not know how to count, they did not know how to add and subtract numbers. And so the smart Parrot advised me to measure the length of the boa constrictor with my own steps. He took the first step, and everyone shouted in unison... (One!)
The teacher lays out a red segment on the flannelgraph and puts the number 1 at the end of it. Students draw a red segment 3 cells long in their notebooks and write down the number 1. The blue, yellow and green segments are completed in the same way, each with 3 cells. A colored drawing appears on the board and in students’ notebooks - a numerical segment:
Did the Parrot take the same steps? (Yes, all steps are equal.)
- What does each number show? (How many steps taken.)
How do numbers change when moving left and right? (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. To use this material correctly, in work must be guided by the following principles.
1. The atmosphere in the lesson should be calm and friendly. You shouldn’t allow “races,” overloading children - it’s better to deal with one task fully and efficiently than seven, but superficially and chaotically.
2. Forms of work need to be diversified. They should change every 3-5 minutes - collective dialogue, work with subject models, cards or numbers, mathematical dictation, work in pairs, independent answer at the board, etc. Thoughtful organization of the lesson allows significantly increase the volume of material, which can be considered with children without overload.
3. The introduction of new material should begin no later than 10-12 minutes into the lesson. Exercises prior to learning something new should be aimed primarily at updating the knowledge that is necessary for its full assimilation.
Methods of teaching mathematics to junior schoolchildren as an academic subject
Lecture 2. Subject, objectives and goals of studying the course on methods of teaching mathematics at a university
1. Methods of teaching mathematics to junior schoolchildren as an academic subject
2. Methods of teaching mathematics to junior schoolchildren as a pedagogical science and as a field of practical activity
Let's consider the purpose of studying the course “Methods of teaching mathematics in primary school” in the process of preparing a future primary school teacher.
Lecture discussion with students
Considering the methodology of teaching mathematics to primary schoolchildren as a science, it is necessary, first of all, to determine its place in the system of sciences, outline the range of problems that it is designed to solve, and determine its object, subject and features.
In the system of sciences, methodological sciences are considered in the block didactics. As is known, didactics is divided into education theory And theory training. In turn, in the theory of learning, general didactics (general issues: methods, forms, means) and particular didactics (subject-specific) are distinguished. Private didactics are called differently - teaching methods or, as has become common in recent years - educational technologies.
Thus, methodological disciplines belong to the pedagogical cycle, but at the same time, they represent purely subject areas, since the methods of teaching literacy will certainly be very different from the methods of teaching mathematics, although both of them are private didactics.
The methodology of teaching mathematics to primary 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. Rock paintings from the Paleolithic era tell stories about learning to count. The first textbooks for teaching children mathematics include Magnitsky’s “Arithmetic” (1703) and the book by V.A. Laya “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 “The 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 mastering arithmetic content by a child of primary school age. Thus, the emergence of this science in its modern form was preceded not only by the development of mathematics as a science, but also by the development of two large areas of knowledge: general didactics of learning and the psychology of learning and development. Recently, the psychophysiology of child brain development has begun to play an important role in the development of teaching methods. At the intersection of these areas, answers to three “eternal” questions in the methodology of teaching subject content are being born today:
1. Why teach? What is the purpose of teaching math to a young child? Is this necessary? And if necessary, then why?
2. What to teach? What content should be taught? What should be the list of mathematical concepts to be taught to your child? Are there any criteria for selecting this content, a hierarchy of its construction (sequence) and how are they justified?
3. How to teach? What are the ways to organize a child’s activities?
(methods, techniques, means, forms of teaching) should be selected and applied so that the child can usefully assimilate the selected content? What is meant by “benefit”: the amount of knowledge and skills of the child or something else? How to take into account the psychological characteristics of age and individual differences of children when organizing training, but at the same time “fit” within the allotted time (curriculum, pro
grams, daily routine), and also take into account the actual content of the class in connection with the system of collective education adopted in our country (classroom-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 specific content, selection and ordering of it in accordance with the set learning goals, on the other hand, to the pedagogical methodological activity of the teacher and the educational (cognitive) activity of the child in the lesson, to the process of mastering the selected material. content managed by the teacher.
Object of study of this science - 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 teaching (Why teach?), content (What to teach?) and the activity of the teacher and the activity of the child (How to teach?) . These components form methodological system in which a change in one of the components will cause a change in the other. The modifications of this system that resulted from a change in the purpose of primary education due to a change in the educational paradigm in the last decade were discussed above. Later we will consider the modifications of this system that entail 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 constructing a methodological system is changing the views of mathematicians on defining 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 focus of initial mathematical training. In recent decades, mathematicians have increasingly talked 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 a child’s initial mathematical preparation 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's consider the relationship between the methods of teaching mathematics in elementary school and other sciences.
1. The method of mathematical development of a child uses basic ideas, theoretical principles and research results from other sciences.
For example, philosophical and pedagogical ideas play a fundamental and guiding role in the process of developing a methodological theory. In addition, borrowing ideas from 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 research are widely used by the methodology to substantiate the content of training and the sequence of its study, to develop methodological techniques and systems of exercises that organize children’s assimilation of various mathematical knowledge, concepts and ways of acting with them. Physiological ideas about conditioned reflex activity, two signaling systems, feedback and age-related stages of maturation of the subcortical zones of the brain help to understand the mechanisms of acquisition of skills, abilities and habits in the learning process. Of particular importance for the development of methods of teaching mathematics in recent decades are the results of psychological and pedagogical research and theoretical research in the field of constructing the theory of developmental learning (L.S. Vygotsky, J. Piaget, L.V. Zankov, V.V. Davydov, D. B. Elkonin, P.Ya. Galperin, N.N. Poddyakov, L.A. Wenger, etc.). This theory is based on the position of L.S. Vygotsky that learning is built not only on completed cycles of child 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 conditions of integration of sciences, research methods very quickly become general scientific. Thus, the method of literature analysis familiar to students (composing bibliographies, taking notes, summarizing, drawing up theses, 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, and conversation; from mathematics - methods of statistical analysis, etc.
3. The technique uses specific research results from psychology, physiology of higher nervous activity, mathematics and other sciences.
For example, the specific results of J. Piaget’s research into the process of young children’s perception of the conservation of quantity gave rise to a whole series of specific mathematical tasks in various programs for primary schoolchildren: using specially designed exercises, the 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 is more water in the bottle than there was in the jar).
4. The technique is involved in complex studies of child development in the process of his education and upbringing.
For example, in 1980-2002. A number of scientific studies have appeared on the process of personal development of a child of primary school age in the course of teaching him mathematics.
Summarizing the question of the connection between the methods of mathematical development and the formation of mathematical concepts in preschoolers, we can note the following:
It is impossible to derive a system of methodological knowledge and methodological technologies from any one science;
Data from other sciences are necessary for the development of methodological theory and practical guidelines;
The technique, 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 simply borrow and use data from other sciences, but processes them in order to develop ways to optimally organize the learning process;
The methodology is determined by the corresponding concept of the child’s mathematical development; Thus, concept - This is not something abstract, far from life and real educational practice, but a theoretical basis that determines the construction of the totality of all components of the methodological system: goals, content, methods, forms and means of teaching.
Let us consider the relationship between modern scientific and “everyday” ideas about teaching mathematics to primary schoolchildren.
The basis of any science is the experience of people. For example, physics relies 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 shapes of objects in the surrounding world, their location in space, quantitative characteristics and relationships between parts of real sets and individual objects. The first harmonious mathematical theory - Euclid's geometry (IV century BC) was born from practical land surveying.
The situation is completely different with the methodology. Each of us has a store of life experience in teaching someone something. However, it is possible to engage in the mathematical development of a child only with special methodological knowledge. With what different special (scientific) methodological knowledge and skills from life Thayan ideas that to teach mathematics to a primary school student, it is enough to have some understanding of counting, calculations and solving simple arithmetic problems?
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 her child’s perception, through repeated repetitions teaches the child to name numerals in the correct order and recognize specific geometric figures. If the mother is persistent enough, the child learns to name numerals fluently, recognizes a fairly large number of geometric shapes, recognizes and even writes numbers, etc. Many people believe that this is exactly what the child should be taught before going to school. Does this training guarantee the development of a child's mathematical abilities? Or at least this child’s continued success in math? Experience shows that it does not guarantee. Will this mother be able to teach the same to another child who is different from her child? Unknown. Will this mother be able to help her child learn other math material? Most likely not. Most often, you can observe a picture when the mother herself knows, for example, how to add or subtract numbers, solve this or that problem, but cannot even explain to her child so that he learns the method of solution. Thus, everyday methodological knowledge is characterized by specificity, limitation of the task, situations and persons to which it applies,
Scientific methodological knowledge (knowledge of educational technology) tends to to generality. They use scientific concepts and generalized psychological and pedagogical principles. 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 which methodological patterns in the formation of a given concept were violated when teaching this child.
2. Everyday methodological knowledge is intuitive. This is due to the method of obtaining them: they are acquired through practical trials and “adjustments”. A sensitive, attentive mother follows this path, experimenting and vigilantly noticing the slightest positive results (which is not difficult to do after spending a lot of time with the child. Often the subject “mathematics” itself leaves specific imprints on the perception of parents. You can often hear: “I myself struggled with mathematics at school , he has the same problems. It’s hereditary for us." Or vice versa: “I didn’t have any problems with mathematics at school, I don’t understand who he was born like!” It is a common opinion that a person either has mathematical abilities or no, and nothing can be done about it. The idea that mathematical abilities (as well as musical, visual, sports and others) can be developed and improved by most people is perceived with skepticism. This position is very convenient for justifying doing nothing, but from the point of view of general methodological scientific knowledge about the nature, character and genesis of a child’s mathematical development, it is, of course, inadequate.
We can say that, in contrast to intuitive methodological knowledge, scientific methodological knowledge rational And conscious. A professional methodologist will never blame heredity, “planidas”, lack of materials, poor quality of teaching aids and insufficient attention of parents to the child’s educational problems. He has a fairly large arsenal of effective methodological techniques; you just need to select from it those that are most suitable for a given 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, patterns, methodological theories and recorded in scientific literature, educational and methodological manuals that future teachers read, which allows them to come even to their first practice in their lives with a fairly large amount of generalized methodological knowledge.
4. Everyday knowledge about teaching methods and techniques is gained
usually through observation and reflection. In scientific activity, these methods are supplemented methodical experiment. The essence of the experimental method is that the teacher does not wait for a combination of circumstances as a result of which the phenomenon of interest to him arises, but causes the phenomenon himself, creating the appropriate conditions. He then purposefully varies these conditions in order to identify the patterns that govern the phenomenon.
obeys. This is how any new methodological concept or methodological pattern 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 and more diverse than everyday knowledge; it possesses unique factual material, inaccessible in its volume to any bearer of everyday methodological knowledge. This material is accumulated and comprehended in separate sections of the methodology, for example: methods of teaching problem solving, methods of forming the concept of a natural number, methods of forming ideas about fractions, methods of forming ideas about quantities, etc., as well as in certain branches of methodological science, for example : teaching mathematics in groups for 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 methods 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 teaching mathematics.
One should not think, however, that everyday methodological knowledge is an unnecessary or harmful thing. The “golden mean” is to see small facts as reflections 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 and constant practice in them can form in the teacher what is called “methodological intuition.” Experience shows that the more everyday methodological knowledge a teacher has, the greater the likelihood of forming this intuition, especially if this rich everyday methodological experience is constantly accompanied by scientific analysis and comprehension.
The methodology for teaching mathematics to primary schoolchildren 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 was already noted above that the methodology of mathematical development as a science is actually taking its first steps, although the methodology of teaching mathematics has a thousand-year history. Today there is not a single primary (and preschool) education program 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 20th century. began to talk about a new methodological direction - theory and practice mathematical development child.
This direction became possible in connection with the emergence of the theory of developmental education for young children. This direction in traditional methods of teaching mathematics is still debatable. Not all teachers today support the need to implement developmental education in progress teaching mathematics, the purpose of which is not so much the formation in the child of a certain list of knowledge, abilities and skills of a subject nature, but rather the development of higher mental functions, his abilities and the disclosure of the child’s internal potential.
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 simply teaching methods of teaching primary 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 manage the process of mathematical development (i.e., 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 showing that the teacher’s new knowledge 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 teaching, and to the methodology. By 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 activities today, this is reflected in modifications in the forms of work with children: teachers are paying more and more attention to individual work, since the effectiveness of the learning process is obviously determined by the individual differences of children. Teachers are paying more and more attention to productive methods of working with children: search and partial search, children's experimentation, heuristic conversation, organizing problem situations in lessons. Further development of this direction may lead to significant substantive modifications in mathematics education programs for primary schoolchildren, since many psychologists and mathematicians in recent decades have expressed doubts about the correctness of the traditional content of primary school mathematics programs primarily with arithmetic material.
There is no doubt about the fact that the process of teaching a child mathematics is constructive for the development of his personality . The process of teaching any subject content leaves its mark on the development of the child’s cognitive sphere. However, the specificity of mathematics as an academic subject is such that its study can significantly influence the overall personal development of the child. 200 years ago this idea was expressed by M.V. Lomonosov: “Mathematics is good because it puts the mind in order.” The formation of systematic thought processes is only one side of the development of a 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 general human intellectual abilities - these are logic, breadth and flexibility of thinking, spatial mobility, laconicism and consistency, etc. And such character traits as determination, perseverance in achieving a goal, the ability to organize oneself, “intellectual endurance”, which are formed through active mathematics, are already personal characteristics of a person.
Today, there are a number of psychological studies showing that a systematic and specially organized system of mathematics classes actively influences the formation and development of an internal action plan, reduces the child’s level of anxiety, developing a sense of confidence and mastery of the situation; increases the level of development of creativity (creative activity) and the general level of mental development of the child. All of these studies support the idea that math content is 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, is implemented when teaching specific mathematical content in the practical activities of the teacher in the classroom.
Modern requirements of society for personal development dictate the need to more fully implement the idea of individualization of education, taking into account the readiness of children for school, their state of health, individual typological characteristics of students. Construction of the educational process taking into account the individual development of the student is important for all levels of education, but of particular importance the implementation of this principle occurs at the initial stage, when the foundation for successful learning as a whole is laid. Omissions at the initial stage of education are manifested by gaps in children’s knowledge, lack of development of general educational skills, and a negative attitude towards school, which can be difficult to correct and compensate for. Observations of underachieving schoolchildren have shown that among them there are children whose learning difficulties are caused by mental retardation.
Learning difficulties are characterized by cognitive passivity, increased fatigue during intellectual activity, a slow pace of formation of knowledge, abilities, skills, poor vocabulary and an insufficient level of development of oral coherent speech.
The lack of cognitive activity during learning is manifested in the fact that these students do not strive to effectively use the time allotted for completing a task, make few conjectural judgments before starting to solve problems, and need special work aimed at developing cognitive interest, stimulating cognitive activity, and intensifying cognitive activity. .
Therefore, it is of great importance to deeply disclose the essence of the principle of activity in learning, taking into account the individual, psychophysiological characteristics of younger schoolchildren with learning difficulties and determining ways of its implementation in the conditions of school education.
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Explanatory note
Modern requirements of society for personal development dictate the need to more fully implement the idea of individualization of education, taking into account the readiness of children for school, their state of health, individual typological characteristics of students. Construction of the educational process taking into account the individual development of the student is important for all levels of education, but of particular importance the implementation of this principle occurs at the initial stage, when the foundation for successful learning as a whole is laid. Omissions at the initial stage of education are manifested by gaps in children’s knowledge, lack of development of general educational skills, and a negative attitude towards school, which can be difficult to correct and compensate for. Observations of underachieving schoolchildren have shown that among them there are children whose learning difficulties are caused by mental retardation.
Learning difficulties are characterized by cognitive passivity, increased fatigue during intellectual activity, a slow pace of formation of knowledge, abilities, skills, poor vocabulary and an insufficient level of development of oral coherent speech.
The lack of cognitive activity during learning is manifested in the fact that these students do not strive to effectively use the time allotted for completing a task, make few conjectural judgments before starting to solve problems, and need special work aimed at developing cognitive interest, stimulating cognitive activity, and intensifying cognitive activity. .
Therefore, it is of great importance to deeply disclose the essence of the principle of activity in learning, taking into account the individual, psychophysiological characteristics of younger schoolchildren with learning difficulties and determining ways of its implementation in the conditions of school education.
Pedagogical science has accumulated quite a lot of experience on the problem of intensifying learning.
In the 60s of the last century in our country, independence and activity were proclaimed as the leading didactic principle. Work to intensify learning has led to the need to find ways to intensify the educational and cognitive activity of students, as well as methods of stimulating their learning. In the School Law of 1958, the development of cognitive activity and independence of students was considered as the main task of restructuring the comprehensive school.
Scientists and teachers Z.A. studied cognitive activity. Abasov, B.I. Korotyaev, N.A. Tomin and others, who revealed the content and structure of this concept.
B.P. Esipov, O.A. Nilsson investigated issues related to the problem of intensifying learning, considering independent work as one of the effective means of intensifying cognitive activity.
Modern scientists and methodologists have been developing ways to enhance and develop students’ cognitive activity: V.V. Davydov, A.V. Zankov, D.B. Elkonin and others.
Relevance The identified problem determined the choice of topic: “Active methods of teaching mathematics as a means of stimulating the cognitive activity of primary schoolchildren with learning difficulties.”
Target - identify, theoretically substantiate and experimentally test the effectiveness of using active teaching methods for primary schoolchildren with learning difficulties in mathematics lessons.
An object research - the process of teaching primary schoolchildren with learning difficulties in primary school.
Item research - active learning methods as a means of stimulating the cognitive activity of primary schoolchildren with learning difficulties.
Hypothesis research: the process of teaching primary schoolchildren with learning difficulties will be more successful if:
During mathematics lessons, active teaching methods will be used for primary schoolchildren with learning difficulties;
active teaching methods will act as a means of stimulating the cognitive activity of primary schoolchildren with learning difficulties.
Tasks :
To identify active teaching methods in mathematics lessons that stimulate the cognitive activity of primary schoolchildren with learning difficulties.
Use a variety of forms and methods of work to stimulate the cognitive activity of primary schoolchildren with learning difficulties.
To determine, justify and test the effectiveness of using active teaching methods for primary schoolchildren with learning difficulties in mathematics lessons.
The practical significance of the work lies in the identification of active teaching methods that stimulate the cognitive activity of primary schoolchildren with learning difficulties in mathematics lessons.
Cognitive activity is a qualitative characteristic of the effectiveness of teaching primary schoolchildren.
Cognitive activity is a socially significant quality of personality and is formed in schoolchildren in educational activities. The problem of developing the cognitive activity of younger schoolchildren, as research shows, has been the focus of attention of teachers for a long time. Pedagogical reality proves every day that the learning process is more effective if the student shows cognitive activity. This phenomenon is recorded in pedagogical theory as the principle of “activity and independence of students in learning.” The means of implementing the leading pedagogical principle are determined depending on the content of the concept of “cognitive activity”. In the content of the concept of “cognitive activity”, a number of scientists consider cognitive activity as a natural desire of schoolchildren to learn.
Cognitive activity reflects a certain interest of younger schoolchildren in acquiring new knowledge, abilities and skills, internal determination and a constant need to use different methods of action to fill knowledge, expand knowledge, and broaden their horizons.
Cognitive interest is a form of manifestation of needs, expressed in the desire to learn.
Interest depends on:
The level and quality of acquired knowledge, skills, development of methods of mental activity;
The student's relationship with the teacher.
The most important components of teaching as an activity are its content and form.
Features of the formation of mathematical knowledge, skills and abilities in younger schoolchildren with learning difficulties
One of the most important conditions for the effectiveness of the educational process is the prevention and overcoming of the difficulties that primary schoolchildren experience in their studies.
Among secondary school students, there are a significant number of children who have insufficient mathematical preparation. Already by the time they enter school, students have different levels of school maturity due to individual characteristics of psychophysical development. The lack of preparedness of some children for schooling is often aggravated by health and other unfavorable factors.
Difficulties in learning mathematics cannot but be affected by such characteristics of students as reduced cognitive activity, fluctuations in attention and performance, insufficient development of basic mental operations (analysis, synthesis, comparison, generalization, abstraction), and some underdevelopment of speech. Reduced perceptual activity is expressed in the fact that children do not always recognize familiar geometric figures if they are presented from an unusual angle or in an inverted position. For the same reason, some students cannot find numerical data in the text of a problem if they are written in words, or highlight the question of the problem if it is not at the end, but in the middle or at the beginning. Imperfect visual perception and motor skills of younger schoolchildren cause increased difficulties when teaching them to write numbers: children take much longer to master this skill, often mix up numbers, write them in mirror images, and are poorly oriented in the cells of a notebook. Deficiencies in children's speech development, in particular poor vocabulary, affect problem solving: students do not always adequately understand some words and expressions contained in the text, which leads to incorrect solutions. When composing tasks independently, they come up with template texts containing similar situations and life actions, repeating the same questions and numerical data.
All these characteristics of children with some developmental delay, together with the insufficiency of their initial mathematical knowledge and ideas, create increased difficulties in their mastery of school knowledge in mathematics. It is possible to achieve successful mastery of program material by students provided that special correctional techniques are used in teaching, a differentiated approach to children, taking into account the characteristics of their mental development.
Methods and means of stimulating the cognitive activity of primary schoolchildren
Teaching methods - a system of consistent, interconnected actions of the teacher and students, ensuring the assimilation of the content of education, the development of mental strength and abilities of students, and their mastery of the means of self-education and self-study. Teaching methods indicate the purpose of training, the method of assimilation and the nature of interaction between the subjects of training.
Facilities - material objects and objects of spiritual culture, intended for the organization and implementation of the pedagogical process and performing the functions of student development; substantive support for the pedagogical process, as well as a variety of activities in which students are involved: work, play, learning, communication, cognition.
Technical training aids (TSO)- devices and instruments used to improve the pedagogical process, increase the efficiency and quality of teaching by demonstrating audiovisual aids.
The effectiveness of mastering any type of activity largely depends on the child’s motivation for this type of activity. Activities proceed more effectively and produce better results if the student has strong, vibrant and deep motives that evoke a desire to act actively, overcome inevitable difficulties, persistently moving towards the intended goal.
Learning activities are more successful if students have formed a positive attitude towards learning, have cognitive interest and a need for cognitive activity, and also if they have developed a sense of responsibility and commitment.
Stimulation methods.
Creating situations for learning successrepresents the creation of a chain of situations in which the student achieves good results in learning, which leads to the emergence of a sense of self-confidence and ease of the learning process.This method is one of the most effective means of stimulating interest in learning.
It is known that without experiencing the joy of success it is impossible to truly count on further success in overcoming educational difficulties. One of the techniques for creating a situation of success can beselection of not one, but a small number of tasks for studentsof increasing complexity. The first task is chosen to be easy so that students who need stimulation can complete it and feel knowledgeable and proficient. Larger and more complex exercises follow. For example, you can use special double tasks: the first is available to the student and prepares him the basis for solving a subsequent, more complex problem.
Another technique that helps create a situation of success isdifferentiated assistance to schoolchildren in completing educational tasks of the same complexity.Thus, low-performing schoolchildren can receive advice cards, analogous examples, plans for the upcoming answer and other materials that will allow them to cope with the presented task. Next, you can invite the student to perform an exercise similar to the first, but independently.
Reward and reprimand in learning.Experienced teachers often achieve success as a result of widespread use of this particular method. Praising a child in a timely manner at the moment of success and emotional upsurge, and finding words for a short reprimand when he crosses the boundaries of what is acceptable is a real art that allows you to manage the emotional state of a student.
The range of incentives is very diverse. In the educational process, this can be praising the child, a positive assessment of some particular quality, encouraging the child’s chosen direction of activity or method of completing a task, giving an increased grade, etc.
The use of reprimands and other types of punishment is an exception in the formation of teaching motives and, as a rule, is used only in forced situations.
The use of games and game forms of organizing educational activities.A valuable method of stimulating interest in learning is the method of using various games and playful forms of organizing cognitive activity. It can use ready-made ones, for example, board games with educational content or game shells of ready-made educational material. Game shells can be created for one lesson, a separate discipline, or an entire educational activity over a long period of time. In total, there are three groups of games suitable for use in educational institutions.
Short games. By the word “game” we most often mean games of this particular group. These include subject-based, role-playing and other games used to develop interest in educational activities and solve certain specific problems. Examples of such tasks are mastering a specific rule, practicing a skill, etc. Thus, for practicing mental calculation skills in mathematics lessons, chain games are suitable, built (like the well-known city game) on the principle of transferring the right to answer along the chain.
Game shells. These games (more likely not even games, but game forms of organizing educational activities) last longer. Most often they are limited to the scope of the lesson, but can last a little longer. For example, in elementary school, such a game can cover the entire school day.
Long educational games.Games of this type are designed for different time periods and can last from several days or weeks to several years. They are oriented, in the words of A.S. Makarenko, to the distant promising line, i.e. towards a distant ideal goal, and are aimed at the formation of slowly developing mental and personal qualities of the child. The peculiarity of this group of games is seriousness and efficiency. The games of this group are no longer like games as we imagine them - with jokes and laughter, but like a task done responsibly. Actually, they teach responsibility - these are educational games. To create cognitive interest among students, we used tasks in the form of “Joke Problems.”
1.Who has a little money but can’t buy anything with it? (At the piglet).
2. When a heron stands on one leg, it weighs 3 kg. How much will a heron weigh if it stands on two legs? (Weight will not change).
There were 3 glasses with cherries on the table. Kostya ate cherries from one glass. How many glasses are left? (Three).
During the evaluation, for each correctly solved problem, the team received two tokens.. In didactics, the following classification of forms of educational activity is adopted, which is based on the quantitative characteristics of the group of students interacting with the teacher at a given moment in the lesson:
general or frontal (work with the whole class);
individual (with a specific student);
group (link, brigade, pair, etc.).
The first involves the joint actions of all students in the class under the guidance of the teacher, the second - the independent work of each student individually; group - students work in groups of three to six people or in pairs. Tasks for groups can be the same or different.basic active learning methods
Problem-based learning- a form in which the process of student cognition approaches search and research activity. The success of problem-based learning is ensured by the joint efforts of the teacher and students. The main task of the teacher is not so much to convey information as to introduce listeners to the objective contradictions in the development of scientific knowledge and ways to resolve them. In collaboration with the teacher, students “discover” new knowledge and comprehend the theoretical features of a particular science.
The main didactic technique of “involving” students’ thinking during problem-based learning is the creation of a problem situation that has the form of a cognitive task, fixing some contradiction in its conditions and ending with a question (questions) that objectifies this contradiction. The unknown is the answer to the question that resolves the contradiction.
Case Study Analysis- one of the most effective and widespread methods of organizing active cognitive activity of students. The case study method develops the ability to analyze unrefined life and production problems. When faced with a specific situation, the student must determine whether there is a problem in it, what it is, and determine his attitude to the situation.
Role-playing- a gaming method of active learning, characterized by the following main features:
O the presence of a task and problem and the distribution of roles between the participants in solving them. For example, using the role-playing method, a production meeting can be simulated;
"Round table" - This is a method of active learning, one of the organizational forms of students’ cognitive activity, which allows them to consolidate previously acquired knowledge, fill in missing information, develop problem-solving skills, strengthen positions, and teach a culture of discussion. A characteristic feature of the round table is the combination of a thematic discussion with a group consultation. Along with the active exchange of knowledge, students develop professional skills to express thoughts, argue their ideas, justify proposed solutions and defend their beliefs. At the same time, information and independent work with additional material are consolidated, as well as problems and issues for discussion are identified.
An important condition when organizing a “round table”: it must be truly round, i.e. the process of communication, communication, took place “eye to eye.” The “round table” principle (it is no coincidence that it was adopted at the negotiations), i.e. arrangement of participants facing each other, and not at the back of the head, as in a regular lesson, generally leads to an increase in activity, an increase in the number of statements, the possibility of personally including each student in the discussion, increases the motivation of students, includes non-verbal means of communication, such as facial expressions, gestures , emotional manifestations.
The teacher also sits in the general circle, as an equal member of the group, which creates a less formal environment compared to the generally accepted one, where he sits separately from the students, who face him. In the classic version, the participants in the discussion address their statements primarily to him, and not to each other. And if the teacher sits among the children, the group members’ addresses to each other become more frequent and less constrained, this also helps to create a favorable environment for discussion and the development of mutual understanding between teachers and students. The main part of a round table on any topic is discussion. Discussion (from the Latin discussio - research, consideration) is a comprehensive discussion of a controversial issue in a public meeting, in a private conversation, in a dispute. In other words, a discussion consists of a collective discussion of any issue, problem or comparison of information, ideas, opinions, proposals. The purposes of the discussion can be very diverse: education, training, diagnostics, transformation, changing attitudes, stimulating creativity, etc.
One of the effective ways to activate the educational activities of younger schoolchildren isnon-traditional lessons.
In my work I often use:
- Lesson - fairy tale
- Lesson-KVN
- Lesson-travel
- Quiz lesson
- Relay lesson
- Lesson-competition
Application of multimedia technologies in mathematics lessons
In my teaching practice, along with traditional ones, I use educational information technologies in order to create conditions for each student to choose an individual educational path; I strive to inspire students to satisfy their cognitive interest, therefore, I consider my main task to be the creation of conditions for the formation of motivation in students, the development of their abilities , increasing the effectiveness of training.
When teaching mathematics lessons I use multimedia presentations. In such lessons, the principles of accessibility and clarity are more clearly implemented. Lessons are effective due to their aesthetic appeal. Presentation lessons provide a large amount of information and assignments in a short period. You can always return to the previous slide (a regular blackboard cannot accommodate the volume that can be put on a slide).
When studying a new topic, I conduct a lesson-lecture using a multimedia presentation. This allows students to focus their attention on significant points of the information presented. The combination of oral lecture material with slide demonstrations allows you to concentrate visual attention on particularly significant moments of educational work.
Multi-slide presentations are effective in any lesson due to significant time savings, the ability to demonstrate a large amount of information, clarity and aesthetics. Such lessons arouse students' cognitive interest in the subject, which contributes to a deeper and more lasting mastery of the material being studied, and increases the creative abilities of schoolchildren.
I also use the presentation to systematically check that all students in the class have completed their homework correctly. When checking homework, a lot of time is usually spent reproducing the drawings on the board and explaining those fragments that caused difficulties.
I use presentation for oral exercises. Working from a finished drawing contributes to the development of constructive abilities, development of speech culture skills, logic and consistency of reasoning, and teaches the preparation of oral plans for solving problems of varying complexity. This is especially good to use in high school geometry lessons. You can offer students examples of how to write solutions, write down the conditions of a problem, repeat demonstrations of some fragments of constructions, and organize oral solutions to problems that are complex in content and formulation.
Experience shows that the use of computer technologies in teaching mathematics makes it possible to differentiate educational activities in the classroom, activates the cognitive interest of students, develops their creative abilities, stimulates mental activity, and encourages research activities.
The use of multimedia technologies is one of the promising areas of informatization of the educational process and is one of the pressing problems of modern methods of teaching mathematics. I consider the use of information technologies necessary and motivate this by the fact that they contribute to:
Improving practical skills;
Allows you to effectively organize independent work and individualize the learning process;
Increase interest in lessons;
Activate the cognitive activity of students;
Updating the lesson.
Conclusions:
I note that the systematic use of active teaching methods for younger schoolchildren with learning difficulties in mathematics lessons forms the level of cognitive activity, and this helps to increase the efficiency of the learning process in mathematics lessons.
All this allows us to confirm the correctness of the chosen path in using active methods in lessons in primary school.
Ministry of Education, Science and Youth Policy of the Republic of Dagestan
GBOUSPO "Republican Pedagogical College" named after. Z.N. Batymurzaeva.
Course work
on TONKM with teaching methods
on the topic of: " Active methods of teaching mathematics in primary school"
Completed by: St. 3 "v" course
Ezerkhanova Zalina
Scientific adviser:
Adilkhanova S.A.
Khasavyurt 2014
Introduction
Chapter I.
Chapter II
Conclusion
Literature
Introduction
“The mathematician takes pleasure in the knowledge 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 organizing the educational process. In my work I give preference to active learning methods. Active learning methods are a set of methods for organizing and managing the educational and cognitive activities of students, which have the following main features:
forced learning activity;
independent development of solutions by students;
high degree of involvement of students in the educational process;
constant processing of communication between students and teachers, and control of independent learning.
The main point of developing federal state educational standards, solving 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 the 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 for primary general education of the new generation is the system-activity approach.
The system-activity approach is aimed at personal development and the formation of civic identity. Training must be organized in such a way as to purposefully lead development. Since the main form of organization of learning is the lesson, it is necessary to know the principles of lesson construction, an approximate typology of lessons and criteria for assessing a lesson within the framework of a systemic activity approach and active methods of work used in the lesson.
Currently, the student has great difficulty setting goals and drawing conclusions, synthesizing material and connecting complex structures, generalizing knowledge, and even more so finding connections in it. Teachers, noting students’ indifference to knowledge, reluctance to learn, and low level of development of cognitive interests, try to design more effective forms, models, methods, and learning conditions.
Creating didactic and psychological conditions for the meaningfulness of learning and the inclusion of students 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 learning faced new tasks: 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 targeted activation of students’ mental processes, 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 the need for independent acquisition of knowledge.
A chain of failures can turn talented children away from mathematics; on the other hand, learning should proceed close to the ceiling of the student’s capabilities: a 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
To organize active cognitive activity of students in the classroom, the optimal combination of active learning methods is crucial. It is very important for me to evaluate the work and psychological climate in my lessons. Therefore, we need to try to ensure that children are not only actively engaged in their studies, but also feel confident and comfortable.
The problem of individual activity in learning is one of the most pressing in educational practice.
Taking this into account, I chose the research topic: “Active methods of teaching mathematics in primary school.”
Purpose of the study: to identify and theoretically substantiate the effectiveness of using active teaching methods for primary schoolchildren with learning difficulties in mathematics lessons.
Research problem: what methods contribute to the activation of cognitive activity in students during the learning process.
Object of study: the process of teaching mathematics to junior schoolchildren.
Subject of research: studying active methods of teaching mathematics in primary school.
Research hypothesis: the process of teaching mathematics to junior schoolchildren will be more successful under the following conditions if:
During mathematics lessons, active teaching methods will be used for younger students.
Research objectives:
)study the literature on the problem of using active methods of teaching mathematics in primary school;
2)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 primary school;
observation of younger schoolchildren.
Structure of the work: the work consists of an introduction, 2 chapters, a conclusion, and a list of references.
Chapter I
1.1 Introduction to active learning methods
Method (from the Greek methodos - path of research) - a way to achieve.
Active teaching methods are a system of methods that ensure activity and diversity in the mental and practical activities of students in the process of mastering educational material.
Active methods provide solutions to educational problems in various aspects:
A teaching method is an ordered set of didactic techniques and means by which the goals of teaching and education are realized. Teaching methods include interrelated, sequentially alternating methods of purposeful activity between the teacher and students.
Any teaching method presupposes a goal, a system of actions, learning tools 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 educational or research purposes. Usually the teacher combines various teaching methods.
Today there are different approaches to the modern theory of teaching methods.
Active learning methods are methods that encourage students to engage in active mental and practical activity in the process of mastering educational material. Active learning involves the use of a system of methods that is aimed primarily not at the teacher presenting ready-made knowledge, memorizing and reproducing it, but at students’ independent acquisition of knowledge and skills in the process of active mental and practical activity. The use of active methods in mathematics lessons helps to develop not just reproduction knowledge, but the skills and needs to apply this knowledge to analyze, assess the situation and make the right decision.
Active methods ensure interaction between participants in the educational process. When using them, the distribution of “responsibilities” is carried out when receiving, processing and applying information between the teacher and the student, between the students themselves. It is clear that a large developmental load is borne by the learning process, which is active on the part of the student.
When choosing active learning methods, you should be guided by a number of criteria, namely:
· compliance with goals and objectives, principles of training;
· compliance with the content of the topic being studied;
· compliance with the capabilities of the trainees: age, psychological development, level of education and upbringing, etc.
· compliance with the conditions and time allocated for training;
· compliance with the teacher’s capabilities: his experience, desires, level of professional skill, personal qualities.
· Student activity can be ensured if the teacher purposefully and makes maximum use of tasks in the lesson: formulate a concept, prove, explain, develop an alternative point of view, etc. In addition, the teacher can use techniques for correcting “intentionally made” errors, formulating and developing tasks for friends.
· An important role is played by developing the skill of asking questions. Analytical and problematic questions like “Why? What does it follow from? What does it depend on? require constant updating in work and special training in their production. The methods of this training are varied: from tasks to pose a question to a text in class to the game “Who can ask the most questions on a certain topic in a minute.
· Active methods provide solutions to educational problems in various aspects:
· formation of positive learning motivation;
· increasing the cognitive activity of students;
· active involvement of students in the educational process;
· stimulation of independent activity;
· development of cognitive processes - speech, memory, thinking;
· effective assimilation of a large volume of educational information;
· development of creative abilities and innovative 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 independent mental work skills;
· development of universal skills.
Let's talk about the effectiveness of teaching methods in more detail.
Active learning methods place 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 teaching and upbringing is directly related to the interaction of thinking processes and the formation of a student’s conscious knowledge, strong skills, and active learning methods.
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 general 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 make the learning process easy and accessible to every child.
The activity of students is possible only if there are incentives. Therefore, among the principles of activation, the motivation of educational and cognitive activity acquires a special place. An important factor of motivation is encouragement. 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 how to develop a child’s sustainable interest in learning, knowledge and the need for independent search, in other words, how to intensify cognitive activity in the learning process.
If a habitual and desirable form of activity for a child is a game, then it is necessary to use this form of organizing activities for learning, combining the game and the educational process, or more precisely, using a game form of organizing the activities of students to achieve educational goals. Thus, the motivational potential of the game will be aimed at more effective development of the educational program by schoolchildren. And the role of motivation in successful learning can hardly be overestimated. Conducted studies of student motivation have revealed interesting patterns. It turned out that the importance of motivation for successful study is higher than the importance of the student’s intelligence. High positive motivation can play the role of a compensating factor in the case of a student’s insufficiently high abilities, but this principle does not work in the opposite direction - no abilities can compensate for the absence of a learning motive or its low expression and ensure significant academic success.
The goals of school education, which are set for the school by the state, society and family, in addition to acquiring a certain set of knowledge and skills, are to reveal and develop the child’s potential, to create favorable conditions for the realization of his natural abilities. A natural play environment, in which there is no coercion and there is an opportunity for each child to find his place, show initiative and independence, and freely realize his abilities and educational needs, is optimal for achieving these goals.
To create such an environment in the classroom, I use active learning methods.
Using active learning methods in the classroom allows you to:
provide positive motivation for learning;
conduct a lesson at a high aesthetic and emotional level;
ensure a high degree of differentiation of training;
increase the volume of work performed in class 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 - primary acquisition of knowledge. This could be a problem lecture, a heuristic conversation, an educational discussion, etc.
stage - knowledge control (consolidation). Methods such as collective mental activity, testing, etc. can be used.
stage - the formation of skills based on knowledge and the development of creative abilities; It is possible to use simulated learning, game and non-game methods.
In addition to intensifying the development of educational information, active teaching methods make it possible to carry out the educational process just as effectively during the lesson and in extracurricular activities. Team work, joint project and research activities, defending one’s position and a tolerant attitude towards other people’s opinions, taking responsibility for oneself and the team form the personality traits, moral attitudes and value guidelines 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 solve problems, communication skills and qualities, the ability to clearly formulate messages and clearly set tasks, the ability to listen and take into account different points of view and opinions of other people, leadership skills and qualities, the ability to work in a team and 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 is an important feature of modern education. Education changes in content, forms, methods, responds to changes in society, and takes into account global trends.
Educational innovation is 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 force him to drink.” This proverb reflects the basic principle of learning - you can create all the necessary conditions for learning, but knowledge itself will happen only when the student wants to know. How can we make sure that the student feels needed at every stage of the lesson and is a full-fledged member of the class team? Another wisdom teaches: “Tell me - I will forget. Show me - I will remember. Let me act on my own - and I will learn.” According to this principle, one’s own active activity is the basis for learning. And therefore, one of the ways to increase effectiveness in studying school subjects is to introduce 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 conventionally divided into two classes: traditional and active. The fundamental difference between these methods is that when they are used, students are created conditions under which they cannot remain passive and have the opportunity for active exchange of knowledge and work experience.
The goal of using active learning methods in elementary school is to develop 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, it is necessary to use more logical problems related to mathematics, language, knowledge of the world around us, 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's consider several examples of the use of active methods in the educational process.
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 specificity of this method. The essence of the conversation is that the teacher, through skillfully posed questions, encourages students to reason, to analyze the facts and phenomena being studied in a certain logical sequence, and to independently formulate appropriate theoretical conclusions and generalizations.
A conversation is not a reporting, but a question-and-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” conclusions, ideas, laws, etc. that are new to them. Therefore, when conducting a conversation to comprehend new material, it is necessary to pose questions so 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 ones. knowledge. It is equally important that the questions have a clear sequence and focus, allowing students to deeply comprehend the internal logic of the knowledge they are learning.
These specific features of conversation make it a very active learning method. However, the use of this method also has its limitations, because not all 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 that allow them to comprehend and assimilate knowledge in a heuristic (from the Greek heurisko - I find) way.
Active methods involve conducting classes through the organization of gaming activities for students. The pedagogy of the game collects ideas that facilitate contacts 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 purpose of play pedagogy is to provide techniques that support group work and create an atmosphere that makes participants feel safe and good.
The pedagogy of the game helps the presenter 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 timidity, shyness, as well as existing social stereotypes.
For active teaching methods, a special place is occupied by forms of organizing the educational process - non-standard lessons: a lesson - a fairy tale, a game, a journey, a scenario, a quiz, lessons - knowledge reviews.
During such lessons, children's activity increases; they are happy to help Kolobok escape from the fox, save ships from attacks by pirates, and store food for the squirrel for the winter. In such lessons, children are in for a surprise, so they try to work fruitfully and complete as many different tasks as possible. The very beginning of such lessons captivates children from the first minutes: “We’re going to the forest for science today” or “The floorboard is creaking about something...” Books from the series “I’m going to a lesson in elementary school” and, of course, the creativity of the student himself help to teach such lessons. teachers. They help the teacher prepare for lessons in less time and conduct them in a more meaningful, modern, and interesting way.
In my work, feedback tools 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. Feedback tools are used to monitor the quality of acquisition of knowledge, skills and abilities. Every student has feedback tools (we make them ourselves during labor lessons or purchase them in stores), they are an essential logical component of his cognitive activity. These are signal circles, cards, number and letter 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 is carried out systematically.
One of the new means of checking the quality of training is tests. This is a qualitative way of checking learning outcomes, characterized by such parameters as reliability and objectivity. Tests test theoretical knowledge and practical skills. With the arrival of a computer at school, new methods for intensifying educational activities open up for teachers.
Modern teaching methods are mainly focused on teaching not ready-made knowledge, but activities for 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, prompt feedback in teaching is provided: the teacher sees all the shortcomings in the students’ knowledge and eliminates them in a timely manner. You can refrain from recording grades “2” and “3” in the class journal for now (by posting them in the student’s notebook or diary). This assessment system is quite humane, mobilizes students well, helps them better understand their difficulties and overcome them, and helps improve the quality of knowledge. Students find themselves better prepared for the test; their fear of such work and the fear of getting a bad mark disappear. The number of unsatisfactory grades, as a rule, is sharply reduced. Students develop a positive attitude towards business-like, rhythmic work, and rational use of lesson time.
Don't forget the restorative power of relaxation in the classroom. After all, sometimes a few minutes are enough to shake yourself up, relax cheerfully and actively, 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 primary school
· using an activity-based approach to learning;
· practical orientation of the activities of participants in the educational process;
· playful and creative nature of learning;
· interactivity of the educational process;
· inclusion of various communications, dialogue and polylogue in the work;
· using the knowledge and experience of students;
· reflection of the learning process by its participants
Another necessary quality of a mathematician is an interest in patterns. Regularity is the most stable characteristic of a constantly changing world. Today cannot be like yesterday. You cannot see the same face twice from the same angle. Regularities are found already at the very beginning of arithmetic. The multiplication table contains many elementary examples of patterns. Here's one of them. Typically, children like to multiply by 2 and 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 simpler, 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, then we see that the difference between the next and previous digits is: - 3; +7; - 3; - 3; +7; - 3; - 3, - 3. There is a very definite rhythm in this row.
If we read the final digits of the answers when multiplying by 7 in reverse order, then we get the final digits from multiplying by 3. Even in elementary school, you can develop the skill of observing mathematical patterns.
During the adaptation period of first-graders, you must try to be attentive to the little person, support her, worry about her, try to interest her in learning, help so that further education for the child is successful and brings mutual joy to the teacher and student. The quality of teaching and upbringing is directly related to the interaction of thinking processes and the formation of a student’s conscious knowledge, strong skills, and active learning methods.
The key to quality education is love for children and 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 general 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 make the learning process easy and accessible to every child. The activity of students is possible only if there are incentives. Therefore, among the principles of activation, the motivation of educational and cognitive activity acquires a special place. An important factor of motivation is encouragement. 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 schoolchildren indicate the need to use incentives to achieve 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 involves the factor of recognition and assessment of the child’s achievements, if necessary, correction of knowledge, statement of success, stimulating further achievements. Encouragement promotes the development of memory, thinking, and creates cognitive interest.
The success of learning also depends on visual aids. These are tables, supporting diagrams, didactic and handouts, individual teaching aids that help make the lesson interesting, joyful, and ensure deep assimilation of the program material.
Individual teaching aids (mathematical pencil cases, letter boxes, abaci) ensure that children are involved in the active learning process, they become active participants in the educational process, and activate children’s attention and thinking.
1Using information technology in a mathematics lesson in elementary school .
In elementary school, it is impossible to conduct a lesson without using visual aids, and 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:
· play activities;
· creating positive emotional situations;
· work in pairs;
· problem-based learning.
Over the past 10 years, there has been a radical change in the role and place of personal computers and information technology in the life of society. Proficiency in information technology is ranked in the modern world on a par with such qualities as the ability to read and write. A person who skillfully and effectively masters technology and information has a different, new style of thinking and has a fundamentally different approach to assessing the problem that has arisen and to organizing his activities. As practice shows, it is no longer possible to imagine a modern school without new information technologies. It is obvious that in the coming decades the role of personal computers will increase and, in accordance with this, the requirements for computer literacy of entry-level students will increase. The use of ICT in primary school lessons 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 can act not only according to a model, but also independently, receiving the necessary information from as many sources as possible; 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 and multimedia technology;
development of communication abilities;
developing the skills to make the optimal decision or propose solutions in a difficult situation (the use of situational computer games aimed at 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:
increasing the efficiency and quality of the learning process through the implementation of ICT tools;
providing incentives (stimuli) that determine the activation of cognitive activity;
deepening interdisciplinary connections through the use of modern information processing tools, including audiovisual, when solving problems from various subject areas.
Using information technology in primary school lessonsis one of the most modern means of developing the personality of a junior schoolchild and forming his information culture.
Teachers are increasingly beginning to use computer capabilities in preparing and conducting lessons in primary school.Modern computer programs make it possible to demonstrate vivid clarity, offer various interesting dynamic types of work, and identify the level of knowledge and skills of students.
The role of the teacher in culture is also changing - he must become a coordinator of information flow.
Today, when information becomes a strategic resource for the development of society, and knowledge becomes a relative and unreliable subject, as it quickly becomes outdated 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 implementation in our country have left their mark on the development of the personality of the modern child. Today, a new link is being introduced into the traditional scheme “teacher - student - textbook” - a computer, and computer education is being introduced into school consciousness. One of the main parts of informatization of education is the use of information technologies in educational disciplines.
For primary schools, this means a change in priorities in setting educational goals: one of the results of training and education in a first-level 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 different strategies for teaching younger schoolchildren in the practice of primary school teachers, and, first of all, the use of information and communication technologies in the teaching and educational process.
Lessons using computer technology make them more interesting, thoughtful, and mobile. Almost any material is used, there is no need to prepare a lot of encyclopedias, reproductions, audio accompaniments 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. Students in grades 1-4 have visual-figurative thinking, so it is very important to build their education using as much high-quality illustrative material as possible, involving not only vision, but also hearing, emotions, and imagination in the process of perceiving new things. Here, the brightness and entertainment of computer slides and animation comes in handy.
The organization of the educational process in primary 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, not undermine their health. ICT should act as an auxiliary element of the educational process, and not the main one. Taking into account the psychological characteristics of a primary school student, work using ICT should be clearly thought out and dosed. Thus, the use of ITC in the classroom should be gentle. When planning a lesson (work) in primary school, the teacher must carefully consider the purpose, place and method of using ICT. Consequently, 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
By the nature of cognitive activity:
explanatory and illustrative (story, lecture, conversation, demonstration, etc.);
reproductive (solving problems, repeating experiments, etc.);
problematic (problematic tasks, cognitive tasks, etc.);
partially search - heuristic;
research.
By activity components:
organizational-effective - methods of organizing and implementing educational and cognitive activities;
stimulating - methods of stimulating and motivating educational and cognitive activity;
control and evaluation - methods of monitoring and self-control of the effectiveness of educational and cognitive activities.
For didactic purposes:
methods of studying new knowledge;
methods of consolidating knowledge;
control methods.
By way of presenting educational material:
monologue - informational and informative (story, lecture, explanation);
dialogical (problem presentation, conversation, debate).
By sources of knowledge transfer:
verbal (story, lecture, conversation, instruction, discussion);
visual (demonstration, illustration, diagram, display of material, graph);
practical (exercise, laboratory work, workshop).
Taking into account the personality structure:
consciousness (story, conversation, instruction, illustration, etc.);
behavior (exercise, training, etc.);
feelings - stimulation (approval, praise, blame, 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 be effective or ineffective depending on the conditions under which it is applied. New content of education gives rise to new methods in teaching mathematics. An integrated approach to the application of teaching methods, their flexibility and dynamism are required.
The main methods of mathematical research are: observation and experience; comparison; analysis and synthesis; generalization and specialization; abstraction and concretization.
Modern methods of teaching mathematics: problem-based (prospective), laboratory, programmed learning, heuristic, building mathematical models, axiomatic, etc.
Let's consider the classification of teaching methods:
Information and development methods are divided into two classes:
Transmission 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 technologies).
Problem-based 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 small groups, organizational activity game, research work.
Reproductive methods: retelling educational material, performing exercises according to a model, laboratory work according to instructions, exercises on simulators.
Creative and reproductive methods: essays, variable exercises, analysis of production situations, business games and other types of imitation of professional activities.
An integral part of teaching methods are the methods of educational activity of the teacher and students. Methodological techniques - actions, methods of work aimed at solving a specific problem. Hidden behind the methods of educational work are the methods of mental activity (analysis and synthesis, comparison and generalization, proof, abstraction, concretization, identification of the essential, formulation of conclusions, concepts, techniques 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 learning 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 sequentially assigned tasks, “guides” 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, schoolchildren often encounter various difficulties. However, in heuristically structured learning, these difficulties often become a kind of stimulus for learning. So, for example, if schoolchildren are found to have an insufficient supply of knowledge to solve a problem or prove a theorem, then they themselves strive to fill this gap by independently “discovering” this or that property and thereby immediately discovering the usefulness of studying it. In this case, the teacher’s role comes down to organizing and directing the student’s work so that the difficulties that the student overcomes are within his capabilities. Often the heuristic method appears in teaching practice in the form of a so-called heuristic conversation. The experience of many teachers who widely use the heuristic method has shown that it influences students' attitudes towards learning activities. Having acquired a “taste” for heuristics, students begin to regard working according to “ready-made instructions” as uninteresting and boring work. The most significant moments of their learning activities in the classroom and at home are the independent “discoveries” of one or another way to solve a problem. Students' interest in those types of work in which heuristic methods and techniques are used is clearly increasing.
Modern experimental studies conducted in Soviet and foreign schools indicate the usefulness of the widespread use of the heuristic method in the study of mathematics by secondary school students, starting from primary school age. Naturally, in this case, students can be presented with only those educational problems that can be understood and resolved by students at this stage of training.
Unfortunately, the frequent use of the heuristic method in the process of teaching posed educational problems requires much more educational time than studying the same issue by the method of the teacher communicating a ready-made solution (proof, result). Therefore, the teacher cannot use the heuristic teaching method 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 “time spent on fundamental issues, worked out with the personal participation of students, is not wasted time: new knowledge is acquired almost effortlessly thanks to previous deep thinking experience.” Heuristic activity or heuristic processes, although they include mental operations as an important component, at the same time have some specificity. That is why heuristic activity should be considered as a type of human thinking that creates a new system of actions or discovers previously unknown patterns of objects surrounding a person (or objects of the science being studied).
The beginning of the use of the heuristic method as a method of teaching mathematics can be found in the book of the famous French teacher and 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 “to maintain the appearance of play, respect the child’s freedom, maintaining the illusion (if there is one) of his own discovery of the truth”; “to avoid in the initial upbringing of a child the dangerous temptation of abusing memory exercises,” because this kills his innate qualities; teach based on interest in what is being studied.
The famous methodologist-mathematician V.M. Bradis defines the heuristic method as follows: “A teaching method is called heuristic when the teacher does not inform students of ready-made information to be learned, but leads students to independently rediscover the relevant proposals and rules.”
But the essence of these definitions is the same - an independent, planned only in general terms, search for a solution to the problem posed.
The role of heuristic activity in science and in the practice of teaching mathematics is covered in detail in the books of the American mathematician D. Polya. The purpose of heuristics is to explore 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 tries 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 that you must have ability, and along with it, luck. The second rule is to hold firm and not give up until a happy idea appears.” The problem solving diagram given at the end of the book is interesting. The diagram indicates the sequence in which actions must be taken to achieve success. It includes four stages:
Understanding the problem statement.
Drawing up a solution plan.
Implementation of the plan.
Looking back (studying the resulting solution).
During these steps, the problem solver must answer the following questions: What is unknown? What is given? What is the condition? Haven't I encountered this problem before, at least in a slightly different form? Is there any related task to this one? Is it possible to use it?
The book “Prelude to Mathematics” by the American teacher W. Sawyer is very interesting from the point of view of using the heuristic method in school.
“All mathematicians,” writes Sawyer, “are characterized by audacity of mind. A mathematician does not like to be told about something; he wants to figure it out himself.”
This “boldness of 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-BASED LEARNING Problem-based learning is a didactic system based on the patterns of creative assimilation of knowledge and methods of activity, including a combination of techniques and methods of teaching and learning, which have the main features of scientific research.
Problem-based teaching method is training that takes place in the form of removing (resolving) problem situations that are consistently created for educational purposes.
A problematic situation is a conscious difficulty generated by a discrepancy between existing knowledge and the knowledge that is necessary to solve the proposed problem.
A task that creates a problematic situation is called a problem, or a problematic task.
The problem should be understandable to students, and its formulation should arouse students’ interest and desire 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 problematic task can be considered as the simplest, special case of a problem consisting of one task. Problem-based learning is focused on the formation and development of students’ ability for creative activity and the need for it. It is advisable to start problem-based learning with problematic tasks, thereby preparing the ground for setting educational goals.
PROGRAMMED TRAINING
Programmed training is such training when the solution to 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 using training programs that determine not only the content, but also the learning process. There are two different systems for programming educational material - linear and branched.
The advantages of programmed training include: dosage of educational material, which is absorbed accurately, which leads to high learning results; individual assimilation; constant monitoring of assimilation; possibility of using technical automated teaching devices.
Significant disadvantages of using this method: not all educational material is amenable 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 and sensory component of learning.
2.4 Interactive methods of teaching mathematics and their advantages
The learning process is inextricably linked with such a concept as teaching methodology. 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 current learning conditions, the learning process is considered 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 until today, only three forms of interaction between teacher and students have developed, established themselves and become widespread. Methodological approaches to teaching 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 teachers 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, independent work, tests, tests, etc. The passive method is considered the most ineffective from the point of view of students’ assimilation of educational material, but its advantages are the relatively easy preparation of a lesson and the ability to present a relatively large amount of educational material in a limited time frame. Given these advantages, many teachers prefer it to other methods. Indeed, in some cases this approach works successfully in the hands of a skilled and experienced teacher, especially if students already have clear goals aimed at thorough learning of the subject.
An active methodological approach is a form of interaction between students and teachers, in which the teacher and students interact with each other during the lesson and students are no longer passive listeners, but active participants in the lesson. If in a passive lesson the main character was the teacher, then here the teacher and students are on equal terms. If passive lessons assumed an authoritarian teaching style, then active ones assumed a democratic style. Active and interactive methodological approaches have much in common. In general, the interactive method can be considered as the most modern form of active methods. It’s just that, unlike active methods, interactive ones are focused on broader 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 engage and reflect on what they know and think. The teacher's place in interactive lessons often comes down to directing students' activities to achieve the lesson's goals. He also develops a lesson plan (as a rule, this is a set of interactive exercises and tasks, during which the student learns the material).
Thus, the main components of interactive lessons are interactive exercises and tasks that students complete.
The fundamental difference between interactive exercises and tasks is that during their implementation, not only and not so much the already learned material is consolidated, but new material is learned. And then interactive exercises and tasks are designed for so-called interactive approaches. Modern pedagogy has accumulated a rich arsenal of interactive approaches, among which the following can be distinguished:
Creative tasks;
Work in small groups;
Educational games (role-playing games, simulations, business games and educational games);
Use of public resources (invitation of a specialist, excursions);
Social projects, classroom teaching methods (social projects, competitions, radio and newspapers, films, performances, exhibitions, performances, songs and fairy tales);
Warm-ups;
Studying and consolidating new material (interactive lecture, working with visual video and audio materials, “student in the role of 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 - two - 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 educational tasks that require students not to simply reproduce information, but to create creativity, since tasks contain a greater or lesser element of uncertainty and, as a rule, have several approaches.
The creative task constitutes 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 and 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; related to students' lives; arouses interest among students; serves learning purposes as best as possible. If students are not used to working creatively, then they should gradually introduce simple exercises first, and then more and more complex tasks.
Small group work - This is one of the most popular strategies, as it gives all students (including shy ones) the opportunity to participate in work, practice cooperation and interpersonal communication skills (in particular, the ability to listen, develop a common opinion, resolve disagreements). All this is often impossible in a large team. Small group work is an integral part of many interactive methods, such as mosaics, debates, public hearings, almost all types of simulations, etc.
At the same time, working in small groups requires a lot of time; this strategy should not be overused. Group work should be used when there is a problem to be solved that students cannot solve on their own. You should start group work slowly. You can organize pairs first. Pay special attention to students who have difficulty adjusting to small group work. When students learn to work in pairs, move on to working in a group of three students. Once we are confident 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 multiple perspectives. In such groups, more constructive relationships between participants are built.
Interactive learning helps a child not only learn, but also live. Thus, interactive learning is undoubtedly an interesting, creative, promising direction in our pedagogy.
Conclusion
Lessons using active learning methods are interesting not only for students, but also for teachers. But their unsystematic, ill-considered use does not give good results. Therefore, it is very important to actively develop and implement your own gaming methods into the lesson in accordance with the individual characteristics of your class.
It is not necessary to use these techniques all in one lesson.
In the classroom, quite acceptable work 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 motivation to learn and develops the best sides of the student. At the same time, there is no need to use these methods without searching for an answer to the question: why are we using them and what consequences may result from this (both for the teacher and for the students).
Without well-thought-out teaching methods, it is difficult to organize the assimilation of program material. That is why it is necessary to improve those methods and means of teaching that help to involve students in cognitive search, in the work of learning: they help teach students to actively, independently obtain knowledge, stimulate their thoughts and develop interest in the subject. There are many different formulas in a mathematics course. In order for students to be able to operate them freely when solving problems and exercises, they must know the most common ones, often encountered in practice, by heart. Thus, the teacher’s task is to create conditions for the practical application of abilities for each student, to choose teaching methods that would allow each student to show their activity, and also to intensify the student’s cognitive activity in the process of learning mathematics. Correct selection of types of educational activities, various forms and methods of work, search for various resources to increase students’ motivation to study mathematics, orient students towards acquiring competencies necessary for life and
activities in a multicultural world will provide the required
learning result.
The use of active teaching methods not only increases the effectiveness of the lesson, but also harmonizes personal development, which is possible only through active activity.
Thus, active teaching methods are ways of activating the educational and cognitive activity of students, which encourage them to active mental and practical activity in the process of mastering the material, when not only the teacher is active, but the students are also active.
To summarize, I will note that each student is interesting for his uniqueness, and my task is to preserve this uniqueness, grow a self-valued personality, develop inclinations and talents, and expand the capabilities of each self.
Literature
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