Methods of teaching mathematics junior schoolchildren How 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 younger schoolchildren as 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 is customary in last 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 initial training arithmetic based on results didactic experiences"(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 formation 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 building methodological system, are changes in the views of mathematicians on the definition of a system of basic postulates for constructing school course mathematics. 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 methodological concepts school textbooks mathematics, 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 the child uses the 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 stages maturation of the subcortical zones of the brain helps to understand the mechanisms of acquisition of skills, abilities and habits during the learning process. Special meaning for the development of methods of teaching mathematics in recent decades, there 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 and others). 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 effective development 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 results from studies of psychology, higher physiology nervous activity, mathematics and other sciences.
For example, specific results of J. Piaget’s research into the process of children’s perception younger age conservation of quantity gave rise to a whole series of specific mathematical tasks in various programs for primary schoolchildren: in 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 can into a narrow bottle, its visual perception increases level, 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 comprehensive research development of the child 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.
Generalizing the question of the connection between the methodology of mathematical development and the formation mathematical representations in preschoolers, the following can be noted:
It is impossible to derive a system from any one science methodological knowledge and methodological technologies;
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 theoretical principles(methodology) 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 a 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 patterns. Scientific methodological knowledge (educational technologies), consisting of clearly defined concepts, reflects their most significant relationships, which makes it possible to formulate methodological patterns. For example, an experienced, highly professional teacher can often determine by the nature of a child’s mistake which methodological patterns of formation this concept were violated during the education of 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”. So this is the way to go a sensitive, attentive mother, experimenting and vigilantly noticing the slightest positive results (which is not difficult to do when spending a lot of time with 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 we have the same problems. It’s hereditary." math skills a person either has it or doesn’t, 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 is perceived with skepticism by most people. 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
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 delay correction groups mental development, teaching mathematics to compensation groups (visually impaired, hearing impaired, etc.), teaching mathematics to children with disabilities mental retardation, teaching math-capable schoolchildren, 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 in small facts a reflection 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 more more likely formation of this intuition, especially if this rich worldly 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 practical activities 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, skills and abilities of a subject nature, but rather the development of higher mental functions, his abilities and the disclosure of the child’s internal potential.
For progressively thinking teacher it's 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. Understanding the essence of the process of mathematical development, the teacher changes his attitude towards educational process(changes itself!), 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 assimilation process is obviously determined by the individual differences of children. Teachers are paying more and more attention productive methods work with children: search and partial search, children's experimentation, heuristic conversation, organization of 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 different sides and properties mathematical thinking of a person shows that many of its most important components actually coincide with the components of such a category as general intellectual abilities of a person - these are logic, breadth and flexibility of thinking, spatial mobility, laconicism and consistency, etc. And such character traits as determination, perseverance in achieving goals , the ability to organize oneself, “intellectual endurance”, which are formed with active activities mathematics are already personal characteristics 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 has at the initial stage, when the foundation is laid successful learning generally. Omissions at the initial stage of education are manifested by gaps in children’s knowledge, lack of development of general educational skills, negative attitude to 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, stimulation of cognitive activity, activation of cognitive activity.
That's why great importance acquires a deep disclosure of the essence of the principle of activity in learning, taking into account the individual, psychophysiological characteristics of younger 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 explored issues related to the problem of intensifying teaching, considering independent work as one of the effective means of enhancing 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 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 experience different levels school maturity because of individual characteristics 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), 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. Imperfection visual perception and the 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. Flaws speech development children, in particular, the poverty of their vocabulary, affects when solving problems: students do not always adequately understand some words and expressions contained in the text, which leads to an incorrect solution. At independently compiling For tasks, they come up with template texts containing the same type of situations and life actions, repeating the same questions and numerical data.
All these features of children with some developmental delay, together with the insufficiency of their initial mathematical knowledge and ideas, create increased difficulties in their mastery of school knowledge in mathematics. It is possible to achieve successful mastery of program material by students provided that special correction techniques are used in teaching, 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 pedagogical process and performing student development functions; 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 efficiently and produce more quality results, if the student has strong, vivid 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 developed positive attitude to learning, there is a 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 learning difficulties. One of the techniques for creating a situation of success can beselection of not one, but a small number of tasks for studentsincreasing complexity. The first task is chosen to be easy so that students who need stimulation can complete it and feel knowledgeable and proficient. Next come the big ones and difficult exercises. 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 performing educational assignments 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. Promptly praising a child at the moment of success and emotional upsurge, finding words for a short reprimand when he crosses the boundaries of what is permissible is a real art that allows you to manage emotional state 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 mark, 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 individual problems. specific tasks. 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 organization of educational activities) are longer in time. Most often they are limited to the scope of the lesson, but they can last a little longer. For example, in elementary school, such a game can cover the entire school day.
Long educational games.Games of this type are designed for 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 emerging mental and personal qualities 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 has been adopted, which is based on quantitative characteristic a 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 the search process, research activities. The success of problem-based learning is ensured by the joint efforts of the teacher and students. The main task of the teacher is not so much to convey information as to introduce listeners to the objective contradictions of development 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 when problem-based learning- Creation problematic situation, which 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- game method 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 an active learning method, one of the organizational forms cognitive activity of students, which allows them to consolidate previously acquired knowledge, fill in missing information, develop problem-solving skills, strengthen positions, and teach a culture of discussion. Characteristic feature "round table"is a 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 identification problems and issues for discussion.
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. placing the 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 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 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 Latin discussio - research, consideration) is a comprehensive discussion 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 new topic I spend lesson-lecture using multimedia presentation. This allows students to focus on significant moments 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 cognitive interest among students in the subject, which contributes to a deeper and more lasting mastery of the material being studied, increases Creative skills 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 promotes the development of constructive abilities, the 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 current problems modern methods of teaching mathematics. I believe the application information technologies necessary and I motivate this by the fact that they contribute to:
Improvement practical skills and skills;
Allows you to effectively organize independent work and individualize the learning process;
Increase interest in lessons;
Activate the cognitive activity of students;
Modernize 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.
Teaching mathematics in primary school has a very important. It is this subject that, if successfully studied, will create the prerequisites for the mental activity of a student in middle and senior education.
Mathematics as a subject forms sustainable cognitive interest and skills logical thinking. Mathematical tasks contribute to the development of a child's thinking, attention, observation, strict consistency of reasoning and creative imagination.
Today's world is undergoing significant changes that place new demands on people. If a student in the future wants to actively participate in all spheres of society, then he needs to be creative, continuously improve himself and develop his individual abilities. But this is exactly what school should teach a child.
Unfortunately, education for younger schoolchildren is most often carried out according to traditional system, when the most common way in the lesson remains to organize the actions of students according to a model, that is, most mathematical tasks are training exercises that do not require the initiative and creativity of children. The priority tendency is for the student to memorize educational material, memorize calculation techniques and solve problems using a ready-made algorithm.
It must be said that many teachers are already developing technologies for teaching mathematics to schoolchildren, which involve children solving non-standard problems, that is, those that form independent thinking and cognitive activity. The main goal of school education at this stage is the development of children’s searching, investigative thinking.
Accordingly, the tasks of modern education today have changed greatly. Now the school focuses not only on giving the student a set of certain knowledge, but also on the development of the child’s personality. All education is aimed at realizing two main goals: educational and educational.
Educational includes the formation of basic mathematical skills, abilities and knowledge.
The developmental function of education is aimed at the development of the student, and the educational function is aimed at the formation of moral values in him.
What is the peculiarity of mathematical teaching? At the very beginning of his studies, the child thinks in specific categories. At the end of primary school, he should learn to reason, compare, see simple patterns and draw conclusions. That is, at first he has a general abstract idea of the concept, and at the end of training this general idea is concretized, supplemented with facts and examples, and, therefore, turns into a truly scientific concept.
Teaching methods and techniques must fully develop mental activity child. This is possible only when the child finds attractive aspects during the learning process. That is, technologies for teaching younger schoolchildren should affect the formation of mental qualities - perception, memory, attention, thinking. Only then will learning be successful.
On modern stage Methods are of primary importance for the implementation of these tasks. Here is an overview of some of them.
Based on the methodology according to L.V. Zankov, learning is based on the mental functions of the child, which have not yet matured. The method assumes three lines of development of the student’s psyche - mind, feelings and will.
The idea of L.V. Zankov was embodied in the curriculum for studying mathematics, the author of which was I.I. Arginskaya. The training material here involves significant independent activity student to acquire and assimilate new knowledge. Particular importance is attached to tasks with in different forms comparisons. They are given systematically and taking into account the increasing complexity of the material.
The emphasis of teaching is on the classroom activities of the students themselves. Moreover, schoolchildren do not just solve and discuss tasks, but compare, classify, generalize, and find patterns. It is precisely this kind of activity that strains the mind, awakens intellectual feelings, and, therefore, gives children pleasure from the work done. In such lessons, it becomes possible to reach a point where students learn not for grades, but to gain new knowledge.
A feature of I. I. Arginskaya’s methodology is its flexibility, that is, the teacher uses every thought expressed by the student in the lesson, even if it was not planned by the teacher. In addition, it is expected to actively include weak schoolchildren in productive activities, providing them with measured assistance.
N.B. Istomina’s methodological concept is also based on the principles of developmental education. The course is based on systematic work to develop in schoolchildren such techniques for studying mathematics as analysis and comparison, synthesis and classification, and generalization.
N.B. Istomina’s technique is aimed not only at developing necessary knowledge, skills and abilities, but also to improve logical thinking. A special feature of the program is the use of special methodological techniques for practicing common methods mathematical operations that will take into account the individual abilities of the individual student.
The use of this educational and methodological complex allows you to create in the classroom favorable atmosphere, in which children freely express their opinions, participate in discussions and receive teacher help if necessary. For the development of the child, the textbook includes tasks of a creative and exploratory nature, the implementation of which is associated with the child’s experience, previously acquired knowledge, and, possibly, with a guess.
In the methodology of N. B. Istomina, work is systematically and purposefully carried out to develop the student’s mental activity.
One of the traditional methods is the course of teaching mathematics to junior schoolchildren by M. I. Moro. The leading principle of the course is a skillful combination of training and education, the practical orientation of the material, and the development of the necessary skills and abilities. The methodology is based on the assertion that in order to successfully master mathematics, it is necessary to create a solid foundation for learning in the elementary grades.
The traditional method develops in students conscious, sometimes even automatic, computational skills. Much attention The program focuses on the systematic use of comparison, comparison, and generalization of educational material.
A special feature of M.I. Moro’s course is that the concepts, relationships, and patterns studied are applied in solving specific problems. After all, the decision word problems is a powerful tool for developing children’s imagination, speech, and logical thinking.
Many experts highlight the advantage of this technique - it is the prevention of student mistakes by performing numerous training exercises with the same techniques.
But a lot is said about its shortcomings - the program does not fully ensure the activation of schoolchildren’s thinking in the classroom.
Teaching mathematics to primary schoolchildren assumes that each teacher has the right to independently choose the program in which he will work. And yet, we must take into account that today’s education requires increased active thinking of students. But not every task requires thinking. If the student has mastered the solution method, then memory and perception are sufficient to cope with the proposed task. It’s another matter if a student is given a non-standard task that requires a creative approach, when the accumulated knowledge must be applied in new conditions. Then mental activity will be fully realized.
Thus, one of important factors, ensuring mental activity is the use of non-standard, entertaining tasks.
Another way to awaken a child’s thoughts is to use interactive learning in mathematics lessons. Dialogue teaches a student to defend his opinion, pose questions to a teacher or classmate, review peers’ answers, explain incomprehensible points to weaker students, find several different ways solving a cognitive problem.
A very important condition for activating thought and developing cognitive interest is the creation of a problem situation in a mathematics lesson. It helps to attract the student to educational material, put him in front of some complexity, which can be overcome by activating mental activity.
Activation of students' mental work will also occur if such developmental operations as analysis, comparison, synthesis, analogy, and generalization are included in the learning process.
Primary school students find it easier to find differences between objects than to determine what they have in common. This is due to their predominantly visual-figurative thinking. In order to compare and find commonality between objects, the child must move from visual methods of thinking to verbal-logical ones.
Comparison and comparison will lead to the discovery of differences and similarities. This means that it will be possible to classify according to some criteria.
Thus, for a successful result in teaching mathematics, the teacher must include a number of techniques in the process, the most important of which are solving entertaining problems, analyzing various types educational tasks, the use of a problem situation and the use of “teacher-student-student” dialogue. Based on this, we can highlight the main task of teaching mathematics - to teach children to think, reason, and identify patterns. The lesson should create an atmosphere of search in which every student can become a pioneer.
Homework plays a very important role in children's mathematical development. Many teachers are of the opinion that the number of homework should be reduced to a minimum or even abolished. Thus, the student’s workload, which has a negative impact on health, is reduced.
On the other hand, in-depth research and creativity require leisurely reflection, which should be carried out outside the lesson. And, if a student’s homework involves not only educational functions, but also developmental ones, then the quality of learning the material will significantly increase. Thus, the teacher should design homework so that students can engage in creative and exploratory activities both at school and at home.
When a student completes homework, parents play a big role. Therefore, the main advice to parents is that the child should do his math homework himself. But this does not mean that he should not receive help at all. If a student cannot cope with solving a task, then you can help him find the rule with which the example is solved, give a similar task, give him the opportunity to independently find the error and correct it. Under no circumstances should you complete the task for your child. The main educational goal of both the teacher and the parent is the same - to teach the child to obtain knowledge himself, and not to receive ready-made ones.
Parents need to remember that the purchased book “Ready Homework” should not be in the hands of the student. The purpose of this book is to help parents check the correctness homework, and not give the student the opportunity, using it, to rewrite ready-made solutions. In such cases, you can completely forget about the child’s good performance in the subject.
The formation of general educational skills is also facilitated by proper organization schoolchild's work at home. The role of parents is to create conditions for their child to work. The student must do homework in a room where the TV is not on and there are no other distractions. You need to help him plan his time correctly, for example, specifically choose an hour to do his homework and never put off this work until the very last moment. Helping your child with homework is sometimes simply necessary. And skillful help will show him the relationship between school and home.
Thus, for the successful education of the student, parents are also given important role. In no case should they reduce the child’s independence in learning, but at the same time skillfully come to his aid if necessary.
LECTURE 1.
Methods of primary teaching mathematics as an academic subject.
Primary mathematics teaching methods answer questions
· For what? –
· To what? –
The methodology of primary teaching of mathematics as an academic subject is associated with
Essay “Is teaching mathematics a science, an art or a craft?”
Objectives of elementary mathematics education.
1. Educational purposes.
2. Developmental goals.
3. Educational goals.
Features of the construction of an initial mathematics course.
1. The main content of the course is arithmetic material.
2. Elements of algebra and geometry do not constitute special sections of the course. They are organically connected with arithmetic material.
The initial mathematics course is structured in such a way that elements of algebra and geometry are included simultaneously with the study of arithmetic material. Consequently, in one lesson, in addition to arithmetic material, algebraic and geometric material is often considered. The inclusion of material from different sections of the course certainly influences the structure of the mathematics lesson and the methodology for its delivery.
4. Connection between practical and theoretical issues. Therefore, in every mathematics lesson, work on mastering knowledge goes simultaneously with the development of skills and abilities.
5. Many theoretical questions are introduced inductively.
6. Mathematical concepts, their properties and patterns are revealed in their interrelation. Each concept receives its own development.
7. Convergence in time of studying some questions of the course, for example, addition and subtraction are introduced simultaneously.
1. Arithmetic material.
Concept natural number, formation of a natural number.
Visual representation of fractions
The concept of the number system.
The concept of arithmetic operations.
2. Elements of algebra.
3.Geometric material.
4.The concept of quantity and the idea of measuring quantities.
5. Tasks. (As a goal and means of teaching mathematics).
Messages.
Analysis of various mathematics programs
1. Elkonin-Davydov
2. Zankov (Arginskaya)
3. Peterson L.G.
4. Istomina N.B.
5. Chekin
Methods and techniques for teaching mathematics to primary schoolchildren.
1. Define the concepts of “teaching method”, “teaching method”.
The problem of teaching methods is formulated briefly with the question how to teach?
To solve the question of how to teach something to students, it is necessary
When talking about methods of teaching mathematics, it is natural to first clarify this concept.
The method is
The description of each teaching method should include:
1) description of the teacher’s teaching activities;
2) description of the student’s educational (cognitive) activity and
3) the connection between them, or the way in which the teacher’s teaching activity controls cognitive activity students.
The subject of didactics, however, is only general teaching methods, that is, methods that generalize a certain set of systems of sequential actions of the teacher and student in the interaction of teaching and learning, which do not take into account the specifics of individual academic subjects.
In addition to specifying and modifying general teaching methods taking into account the specifics of mathematics, the subject of the methodology is also the addition of these methods with private (special) teaching methods that reflect the basic methods of cognition used in mathematics itself.
Thus, the system of methods of teaching mathematics consists of general teaching methods developed by didactics, adapted to teaching mathematics, and private (special) methods of teaching mathematics, reflecting the basic methods of cognition used in mathematics.
1. EMPIRICAL METHODS: OBSERVATION, EXPERIENCE, MEASUREMENTS.
Observation, experience, measurements - empirical methods, used in experimental natural sciences.
Observation, experience and measurements should be aimed at creating special situations in the learning process and providing students with the opportunity to extract from them obvious patterns, geometric facts, ideas of proof, etc. Most often, the results of observation, experience and measurements serve as premises for inductive conclusions, using in which new truths are discovered. Therefore, observation, experience and measurement are also classified as heuristic teaching methods, that is, methods that promote discovery.
Observation.
2. COMPARISON AND ANALOGY - logical thinking techniques used both in scientific research and in teaching.
By using comparisons the similarities and differences of the compared objects are revealed, i.e., the presence of common and non-common (different) properties between them.
The comparison leads to the correct conclusion if following conditions:
1) the concepts being compared are homogeneous and
2) comparison is carried out according to such characteristics that are of significant importance.
By using analogies the similarity of objects revealed as a result of their comparison extends to a new property (or new properties).
The reasoning by analogy is as follows general scheme:
A has properties a, b, c, d;
B has properties a, b, c;
Probably (possibly) B also has property d.
A conclusion by analogy is only probable (plausible), and not reliable.
3. GENERALIZATION AND ABSTRACT - two logical techniques that are almost always used together in the process of cognition.
Generalization- this is a mental selection, fixation of some common essential properties that belong only to this class objects or relationships.
Abstraction- this is a mental distraction, the separation of general, essential properties, isolated as a result of generalization, from other unimportant or non-general properties of the objects or relations under consideration and discarding (within the framework of our study) the latter.
Under o bobbing They also understand the transition from the individual to the general, from the less general to the more general.
Under specification understand the reverse transition - from the more general to the less general, from the general to the individual.
If generalization is used in the formation of concepts, then specification is used when describing specific situations using previously formed concepts.
4. SPECIFICATION is based on a known rule of inference
called the instantiation rule.
5. INDUCTION.
The transition from the particular to the general, from individual facts established through observation and experience, to generalizations is a pattern of knowledge. Integral logical form Such a transition is induction, which is a method of reasoning from the particular to the general, drawing a conclusion from particular premises (from the Latin inductio - guidance).
Usually, when they say “inductive teaching methods,” they mean the use of incomplete induction in teaching. Further, when we say “induction”, we will mean incomplete induction.
At certain stages of education, in particular in primary school, mathematics is taught primarily by inductive methods. Here the inductive conclusions are quite convincing psychologically and for the most part remain so far (at this stage of training) unproven. Only isolated “deductive islands” can be found, consisting of the use of simple deductive reasoning as evidence for individual propositions.
6. DEDUCTION (from Latin deductio - deduction) in in a broad sense is a form of thinking that consists in the fact that a new sentence (or rather, the thought expressed in it) is derived in a purely logical way, that is, according to certain rules of logical inference (following) from certain known sentences (thoughts).
Special Development taking into account the needs of mathematics, it received in the form of a theory of proof in mathematical logic.
By teaching proof, we mean teaching the mental processes of searching and constructing a proof, rather than reproducing and memorizing ready-made proofs. Learning to prove means, first of all, learning to reason, and this is one of the main tasks of learning in general.
7. ANALYSIS - a logical technique, a research method, consisting in the fact that the object being studied is mentally (or practically) divided into component elements (signs, properties, relationships), each of which is studied separately as part of a dissected whole.
SYNTHESIS is a logical technique by which individual elements are combined into a whole.
In mathematics, most often, analysis is understood as reasoning in the “reverse direction”, i.e. from the unknown, from what needs to be found, to the known, to what has already been found or given, from what needs to be proven, to what has already been proven or accepted as true.
In this understanding, the most important for learning, analysis is a means of finding a solution, a proof, although in most cases it is not a solution or a proof in itself.
Synthesis, based on data obtained during analysis, provides a solution to a problem or a proof of a theorem.
