Mathematical ability of the child. Psychology of math skills Example develops math skills
6.11. A Brief Overview of Research on Mathematical Ability
In studies led by V.A. Krutetsky reflects different levels of studying the problem of mathematical, literary and constructive-technical abilities. However, all studies were organized and conducted according to the general scheme:
1st stage - study of the essence, structure of specific abilities;
2nd stage - study of age and individual differences in the structure of specific abilities, age-related dynamics of the development of the structure;
3rd stage - the study of the psychological foundations of the formation and development of abilities.
The works of V. A. Krutetsky, I. V. Dubrovina, S. I. Shapiro give a general picture of the age-related development of the mathematical abilities of schoolchildren throughout the school years.
A special study of the mathematical abilities of schoolchildren was carried out by V.A. Krutetskiy(1968). Under ability to study mathematics he understands the individual psychological characteristics (primarily the characteristics of mental activity) that meet the requirements of educational mathematical activity and determine, other things being equal, the success of the creative mastery of mathematics as an educational subject, in particular, the relatively quick, easy and deep mastery of knowledge, skills and abilities in the field mathematics. In the structure of mathematical abilities, he identified the following main components:
1) the ability to formalize the perception of mathematical material, grasp the formal structure of the problem;
2) the ability to quickly and broadly generalize mathematical objects, relationships and actions;
3) the ability to fold the process of mathematical reasoning and the system of corresponding actions - the ability to think in folded structures;
4) flexibility of mental processes in mathematical activity;
5) the ability to quickly and freely restructure the direction of the thought process, switch from direct to reverse thought;
6) striving for clarity, simplicity, economy and rationality of decisions;
7) mathematical memory (generalized memory for mathematical relations, reasoning and proof schemes, methods for solving problems and principles for approaching them). The methodology for studying abilities for mathematics belongs to V.A. Krutetsky (1968).
Dubrovina I.V. a modification of this technique has been developed in relation to students in grades 2-4.
An analysis of the materials presented in this work allows us to draw the following conclusions.
1. Mathematically capable pupils of primary school age quite clearly reveal such components of mathematical abilities as the ability to analytically and synthetically perceive the conditions of problems, the ability to generalize mathematical material, and the flexibility of thought processes. Less clearly expressed at this age are such components of mathematical abilities as the ability to curtail reasoning and a system of appropriate actions, the desire to find the most rational, economical (elegant) way to solve problems.
These components are most clearly represented only among the students of the "Very capable" (OS) group. The same applies to the peculiarities of the mathematical memory of younger students. Only students of the OS group can find signs of generalized mathematical memory.
2. All the above components of mathematical abilities are manifested on the mathematical material accessible to students of primary school age, therefore, in a more or less elementary form.
3. The development of all the above components is noticeable in students capable of mathematics from grades 2 to 4: over the years, the tendency towards a relatively complete analytic-synthetic perception of the condition of the problem increases; the generalization of mathematical material becomes wider, faster and more confident; there is a rather noticeable development of the ability to curtail reasoning and a system of appropriate actions, which is initially formed on the basis of exercises of the same type, and over the years more and more often manifests itself “from the spot”; by grade 4, students switch much more easily from one mental operation to another, qualitatively different, more often they see several ways to solve a problem at the same time; memory is gradually freed from the storage of specific private material, the memorization of mathematical relationships is becoming increasingly important.
4. In the studied low-capacity (MS) pupils of primary school age, all of the above components of mathematical abilities are manifested at a relatively low level of development (the ability to generalize mathematical material, the flexibility of thought processes) or are not detected at all (the ability to reduce reasoning and the system of corresponding actions, generalized mathematical memory).
5. It was possible to form the main components of mathematical abilities at a more or less satisfactory level in the process of experimental training in children of the MS group only as a result of persistent, persistent, systematic work both on the part of the experimenter and the students.
6. Age differences in the development of the components of mathematical abilities in junior schoolchildren who are incapable of mathematics are weakly and indistinctly expressed.
In the article S.I. Shapiro"Psychological analysis of the structure of mathematical abilities in senior school age" shows that, unlike less capable students, whose information is usually stored in memory in a narrowly specific form, in a fragmented and undifferentiated way, students capable of mathematics memorize, use and reproduce material in generalized, "folded" form.
Of considerable interest is the study of mathematical abilities and their natural prerequisites. I.A. Lyovochkina, who believes that although mathematical abilities were not the subject of special consideration in the works of B.M. Teplov, however, answers to many questions related to their study can be found in his works devoted to the problems of abilities. Among them, a special place is occupied by two monographic works - "The Psychology of Musical Abilities" and "The Mind of a Commander", which have become classic examples of the psychological study of abilities and have incorporated universal principles of approach to this problem, which can and should be used in the study of any kind of abilities.
In both works, B.M. Teplov not only gives a brilliant psychological analysis of specific types of activity, but also, using the examples of outstanding representatives of musical and military art, reveals the necessary components that make up bright talents in these areas. B.M. Teplov paid special attention to the issue of the ratio of general and special abilities, proving that success in any kind of activity, including music and military affairs, depends not only on special components (for example, in music - hearing, a sense of rhythm ), but also on the general features of attention, memory, and intelligence. At the same time, general mental abilities are inextricably linked with special abilities and significantly affect the level of development of the latter.
The role of general abilities is most clearly demonstrated in the work "The Mind of a Commander". Let us dwell on the main provisions of this work, since they can be used in the study of other types of abilities associated with mental activity, including mathematical abilities. After a deep study of the activities of the commander, B.M. Teplov showed what place intellectual functions occupy in it. They provide an analysis of complex military situations, the identification of individual significant details that can affect the outcome of upcoming battles. It is the ability to analyze that provides the first necessary step in making the right decision, in drawing up a battle plan. Following the analytical work, the stage of synthesis begins, which makes it possible to combine the diversity of details into a single whole. According to B.M. Teplov, the activity of a commander requires a balance between the processes of analysis and synthesis, with a mandatory high level of their development.
Memory occupies an important place in the intellectual activity of a commander. It doesn't have to be universal. It is much more important that it should be selective, that is, it should retain, first of all, the necessary, essential details. As a classic example of such memory, B.M. Teplov cites statements about the memory of Napoleon, who remembered literally everything that was directly related to his military activities, from unit numbers to soldiers' faces. At the same time, Napoleon was unable to memorize meaningless material, but had the important feature of instantly assimilating what was subject to classification, a certain logical law.
B.M. Teplov comes to the conclusion that "the ability to find and highlight the essential and the constant systematization of the material are the most important conditions for ensuring the unity of analysis and synthesis, the balance between these aspects of mental activity that distinguish the work of the mind of a good commander" . Along with an outstanding mind, the commander must have certain personal qualities. This is, first of all, courage, determination, energy, that is, what, in relation to military leadership, is usually denoted by the concept of “will”. An equally important personal quality is stress resistance. The emotionality of a talented commander is manifested in the combination of the emotion of combat excitement and the ability to assemble and concentrate.
A special place in the intellectual activity of the commander B.M. Teplov assigned to the presence of such a quality as intuition. He analyzed this quality of the commander's mind, comparing it with the intuition of a scientist. There is much in common between them. The main difference, according to B.M. Teplov, consists in the need for the commander to make an urgent decision, on which the success of the operation may depend, while the scientist is not limited by time frames. But in both cases, "insight" must be preceded by hard work, on the basis of which the only true solution to the problem can be made.
Confirmation of the provisions analyzed and generalized by B.M. Teplov from a psychological standpoint, can be found in the works of many prominent scientists, including mathematicians. So, in the psychological study "Mathematical Creativity" Henri Poincaré describes in detail the situation in which he managed to make one of the discoveries. This was preceded by a long preparatory work, a large share of which, according to the scientist, was the process of the unconscious. The stage of "insight" was necessarily followed by the second stage - careful conscious work to put the proof in order and check it. A. Poincaré came to the conclusion that the most important place in mathematical abilities is occupied by the ability to logically build a chain of operations that lead to the solution of the problem. It would seem that this should be available to any person capable of logical thinking. However, not everyone is able to operate with mathematical symbols with the same ease as when solving logical problems.
It is not enough for a mathematician to have a good memory and attention. According to Poincare, people capable of mathematics are distinguished by ability to catch order, in which the elements necessary for the mathematical proof should be located. The presence of this kind of intuition is the main element of mathematical creativity. Some people do not possess this subtle feeling and do not have a strong memory and attention, therefore they are not able to understand mathematics. Others have weak intuition, but are gifted with a good memory and the ability to pay attention, so they can understand and apply mathematics. Still others have such a special intuition and, even in the absence of an excellent memory, they can not only understand mathematics, but also make mathematical discoveries.
Here we are talking about mathematical creativity accessible to few. But, as J. Hadamard wrote, “between the work of a student solving a problem in algebra or geometry, and creative work, the difference is only in level, in quality, since both works are of a similar nature.” In order to understand what qualities are still required to achieve success in mathematics, the researchers analyzed mathematical activity: the process of solving problems, methods of proof, logical reasoning, and features of mathematical memory. This analysis led to the creation of various variants of the structures of mathematical abilities, complex in their component composition. At the same time, the opinions of most researchers agreed on one thing - that there is not and cannot be the only pronounced mathematical ability - this is a cumulative characteristic that reflects the features of various mental processes: perception, thinking, memory, imagination.
Among the most important components of mathematical ability are specific ability to generalize mathematical material, ability to spatial representations, ability to abstract thinking. Some researchers also single out as an independent component of mathematical abilities mathematical memory for reasoning and proof schemes, methods for solving problems and principles for approaching them. The study of mathematical abilities also includes the solution of one of the most important problems - the search for natural prerequisites, or inclinations, of this type of ability. For a long time, inclinations were considered as a factor fatally predetermining the level and direction of development of abilities. Classics of Russian psychology B.M. Teplov and S.L. Rubinshtein scientifically proved the illegitimacy of such an understanding of inclinations and showed that the source of the development of abilities is the close interaction of external and internal conditions. The severity of one or another physiological quality in no way indicates the mandatory development of a particular type of ability. It can only be a favorable condition for this development. The typological properties that make up the deposits and are an important part of them reflect such individual characteristics the functioning of the organism, as the limit of working capacity, the speed characteristics of the nervous response, the ability to restructure the reaction in response to changes in external influences.
The properties of the nervous system, closely related to the properties of temperament, in turn, affect the manifestation of the characterological features of the personality (V.S. Merlin, 1986). B.G. Ananiev, developing ideas about the general natural basis for the development of character and abilities, pointed to the formation of connections between abilities and character in the process of activity, leading to new mental formations, denoted by the terms "talent" and "vocation" (Ananiev B.G., 1980). Thus, temperament, abilities and character form, as it were, a chain of interrelated substructures in the structure of personality and individuality, which have a single natural basis(E.A. Golubeva, 1993).
The basic principles of an integrated typological approach to the study of abilities and individuality are described in detail by E.A. Golubev in the corresponding chapter of the monograph. One of the most important principles is the use, along with qualitative analysis, of measuring methods for diagnosing various personality characteristics. Based on this, I.A. Lyovochkin built an experimental study of mathematical abilities. The specific task included diagnosing the properties of the nervous system, which were considered as the makings of mathematical abilities, studying the personal characteristics of mathematically gifted students and the characteristics of their intellect. The experiments were carried out on the basis of school No. 91 in Moscow, which has specialized mathematical classes. High school students from all over Moscow are accepted into these classes, mostly winners of regional and city olympiads who have passed an additional interview. Mathematics is taught here according to a more in-depth program, and an additional course of mathematical analysis is taught. The study was conducted jointly with E.P. Guseva and teacher-experimenter V.M. Sapozhnikov.
All the students with whom the researcher happened to work in grades 8-10 have already decided on their interests and inclinations. They associate their further study and work with mathematics. Their success in mathematics significantly exceeds the success of students in non-math classes. But despite the overall high success within this group of students, there are significant individual differences. The study was structured in the following way: students were observed during the lessons, their control work was analyzed with the help of experts, and experimental tasks were proposed for solving, aimed at identifying some components of mathematical abilities. In addition, a series of psychological and psychophysiological experiments were conducted with the students. The level of development and originality of intellectual functions were studied, their personal characteristics and typological features of the nervous system were revealed. In total, 57 students with strong mathematical abilities were examined over the course of several years.
results
An objective measurement of the level of intellectual development using the Wexler test in mathematically gifted children showed that most of them have a very high level of general intelligence. The numerical values of the general intelligence of many students surveyed by us exceeded 130 points. According to some normative classifications, values of this magnitude are found only in 2.2% of the population. In the vast majority of cases, there was a predominance verbal intelligence over the non-verbal. In itself, the fact of the presence of highly developed general and verbal intelligence in children with pronounced mathematical abilities is not unexpected. Many researchers of mathematical abilities noted that a high degree of development of verbal and logical functions is necessary condition for mathematical ability. I.A. Lyovochkina was interested not only in the quantitative characteristics of intelligence, but also in how it is related to the psychophysiological, natural characteristics of students. Individual features of the nervous system were diagnosed using an electroencephalographic technique. Background and reactive characteristics of the electroencephalogram recorded on a 17-channel encephalograph were used as indicators of the properties of the nervous system. According to these indicators, the diagnosis of strength, lability and activation of the nervous system was carried out.
I.A. Lyovochkina established, using statistical methods of analysis, that the higher level of verbal and general intelligence in this sample had a stronger nervous system. They also had higher grades in the subjects of the natural and humanitarian cycles. According to other researchers, obtained on adolescent high school students of general education schools, the owners of a weak nervous system had a higher level of intelligence and better academic performance (Golubeva E.A. et al. 1974, Kadyrov B.R. 1977). The reason for this discrepancy should probably be sought, first of all, in the nature of the learning activities. Students in math classes experience significantly greater learning loads compared to students in regular classes. With them, additional electives are held, in addition, in addition to compulsory home and class assignments, they solve many tasks related to preparation for higher educational institutions. The interests of these guys are shifted towards an increased constant mental load. Such conditions of activity impose increased demands on endurance, performance, and since the main, defining feature of the property of the strength of the nervous system is the ability to withstand prolonged excitation without entering a state of transcendental inhibition, then, apparently. therefore, those students who have such characteristics of the nervous system as endurance and working capacity demonstrate the greatest effectiveness.
V.A. Krutetsky, studying the mathematical activity of students capable of mathematics, drew attention to their characteristic feature - the ability to maintain tension for a long time, when the student can study for a long time and with concentration without revealing fatigue. These observations allowed him to suggest that such a property as the strength of the nervous system may be one of the natural prerequisites that favor the development of mathematical abilities. The relations we obtained partly confirm this assumption. Why only partly? Reduced fatigue in the process of doing mathematics was noted by many researchers in students capable of mathematics compared with those incapable of it. I.A. Lyovochkina examined a sample that consisted only of capable students. However, among them were not only owners of a strong nervous system, but also those who were characterized as owners of a weak nervous system. This means that not only high overall performance, which is a favorable natural basis for success in this type of activity, can ensure the development of mathematical abilities.
An analysis of personality traits showed that, in general, for a group of students with a weaker nervous system, such personality traits as reasonableness, prudence, perseverance (J+ factor according to Cattell), as well as independence, independence (Q2+ factor) turned out to be more characteristic. Persons with high scores on the factor J pay a lot of attention to planning behavior, analyze their mistakes, while showing "cautious individualism". High scores on the Q2 factor are people who are prone to independent decision-making and are able to bear responsibility for them. This factor is referred to as "thinking introversion." Probably, the owners of a weak nervous system achieve success in this type of activity, including through the formation of such qualities as action planning, independence.
It can also be assumed that different poles of this property of the nervous system can be associated with different components of mathematical abilities. So it is known that the property of weakness of the nervous system is characterized by increased sensitivity. It is she who can underlie the ability of intuitive, sudden comprehension of the truth, "insight" or conjecture, which is one of the important components of mathematical abilities. And although this is only an assumption, but its confirmation can be found in specific examples among mathematically gifted students. Here two the brightest example. Dima based on the results of objective psychophysiological diagnostics, it can be attributed to representatives of the strong type of the nervous system. He is the "star of the first magnitude" in the math class. It is important to note that he achieves brilliant success without any visible effort, with ease. Never complains of being tired. Lessons, mathematics lessons are for him a necessary constant mental gymnastics. Particular preference is given to solving non-standard, complex tasks that require tension of thought, deep analysis, and strict logical sequence. Dima does not allow inaccuracies in the presentation of the material. If the teacher makes logical omissions when explaining, Dima will definitely pay attention to this. It is distinguished by a high intellectual culture. This is also confirmed by the test results. Dima has the highest indicator of general intelligence in the examined group - 149 conventional units.
Anton- one of the brightest representatives of the weak type of the nervous system, which we happened to observe among mathematically gifted children. He gets tired very quickly in class, is unable to work long and concentrated, often leaves some things to take on others without sufficient deliberation. It happens that he refuses to solve a problem if he foresees that it will require great effort. However, despite these features, teachers highly appreciate his mathematical abilities. The fact is that he has excellent mathematical intuition. It often happens that he is the first to decide the most difficult tasks, giving the final result and omitting all the intermediate steps of the solution. It is characterized by the ability to "enlightenment". He does not bother explaining why such a solution was chosen, but on verification it turns out to be optimal and original.
Mathematical abilities are very complex and multifaceted in their structure. And yet, there are two main types of people with their manifestation - these are "geometers" and "analysts". In the history of mathematics, vivid examples of this can be such names as Pythagoras and Euclid (the largest geometers), Kovalevskaya and Klein (analysts, creators of the theory of functions). This division is based primarily on the individual characteristics of the perception of reality, including mathematical material. It is not determined by the subject on which the mathematician works: analysts remain analysts in geometry, while geometers prefer to perceive any mathematical reality figuratively. In this regard, it is appropriate to quote the statement of A. Poincaré: “It is by no means the issue discussed by them that makes them use one method or another. If some are often said to be analysts, while others are called geometers, this does not prevent the former from remaining analysts even when dealing with questions of geometry, while others are geometers even when they are engaged in pure analysis.
In school practice, when working with gifted students, these differences are manifested not only in different success in mastering different sections of mathematics, but also in a preferential attitude to the principles of problem solving. Some students strive to solve any problems with the help of formulas, logical reasoning, while others, if possible, use spatial representations. Moreover, these differences are very stable. Of course, among students there are those who have a certain balance of these characteristics. They equally master all sections of mathematics, using different principles of approach to solving different problems. Individual differences between students in approaches to solving problems and methods for solving them were identified by I.A. Lyovochkina, not only through observation of students while working in the classroom, but also experimentally. To analyze the individual components of mathematical abilities, the teacher-experimenter V.M. Sapozhnikov developed a series of special experimental problems. An analysis of the results of solving problems in this series made it possible to obtain an objective idea of the nature of the mental activity of schoolchildren and of the relationship between the figurative and analytical components of mathematical thinking.
Students were identified who were better at solving algebraic problems, as well as those who were better at solving geometric problems. The experiment showed that among students there are representatives of the analytical type of mathematical thinking, which are characterized by a clear predominance of the verbal-logical component. They have no need for visual schemes, they prefer to operate with iconic symbols. The thinking of students who prefer geometric tasks is characterized by a greater severity of the visual-figurative component. These students feel the need for visual representation and interpretation in the expression of mathematical relationships and dependencies.
Of the total number of mathematically gifted students who took part in the experiments, the brightest "analysts" and "geometers" were singled out, which made up the two extreme groups. The group of "analysts" included 11 people, the most prominent representatives of the verbal-logical type of thinking. The group of "geometers" consisted of 5 people, with a bright visual-figurative type of thinking. The fact that much fewer students were selected into the group of bright representatives of the "geometries" can be explained, in our opinion, by the following circumstance. When conducting mathematical competitions and Olympiads, the role of visual-figurative components of thinking is not sufficiently taken into account. AT competitive tasks the proportion of problems in geometry is low - out of 4 - 5 tasks, at best, one is aimed at identifying spatial representations in students. Thus, in the course of selection, as it were, potentially capable mathematician geometers with a vivid visual-figurative type of thinking are “cut off”. Further analysis was carried out using the statistical method of comparing group differences (Student's t-test) for all available psychophysiological and psychological indicators.
It is known that the typological concept of I.P. Pavlova, in addition to the physiological theory of the properties of the nervous system, included a classification of specially human types of higher nervous activity, differing in the ratio of signaling systems. These are “artists”, with a predominance of the first signal system, “thinkers”, with a predominance of the second signal system, and the middle type, with a balance of both systems. For "thinkers" the most characteristic is the abstract-logical way of processing information, while "artists" have a vivid figurative holistic perception of reality. Of course, these differences are not absolute, but reflect only the predominant forms of response. The same principles underlie the differences between "analysts" and "geometers". The former prefer analytical methods for solving any mathematical problems, that is, they approach “thinkers” by type. "Geometers" tend to isolate figurative components in tasks, thereby acting in a way that is typical for "artists".
Recently, a number of works have appeared in which attempts were made to combine the doctrine of the basic properties of the nervous system with ideas about specially human types - "artists" and "thinkers". It has been established that the owners of a strong, labile and activated nervous system gravitate towards the “artistic” type, and those who have a weak, inert and inactivated nervous system tend to the “thinking” type (Pechenkov V.V., 1989). In the work of I.A. Lyovochkina, from the indicators of various properties of the nervous system, the most informative psychophysiological characteristic in diagnosing the types of mathematical thinking turned out to be the characteristic of the strength-weakness property of the nervous system. The group of "analysts" included the owners of a relatively weaker nervous system, compared to the group of "geometers", that is, the differences between the groups in terms of the strength-weakness property of the nervous system were in line with the previously obtained results. For two other properties of the nervous system (lability, activation), no statistically significant differences were found, and the emerging trends do not contradict the initial assumptions.
Held also comparative analysis the results of the diagnosis of personality traits obtained using the Cattell questionnaire. Statistically significant differences between the groups were established by two factors - H and J. According to the factor H, the group of "analysts" can be generally characterized as relatively more restrained, with a limited range of interests (H-). Usually people with low scores on this factor are closed, do not seek additional contacts with people. The group of "geometers" has large values for this personal factor (H +) and differs in it by a certain carelessness, sociability. Such people do not experience difficulties in communication, they make many and willing contacts, they do not get lost in unexpected circumstances. They are artistic, able to withstand significant emotional stress. According to the J factor, which generally characterizes such a personality trait as individualism, the group of "analysts" has high average group values. This means that they are characterized by reasonableness, prudence, perseverance. People who have a high weight on this factor pay a lot of attention to planning their behavior, while remaining closed and acting individually.
In contrast to them, the guys included in the group of "geometers" are energetic and expressive. They love joint actions, they are ready to join in group interests and show their activity at the same time. The emerging differences show that the studied groups of mathematically gifted students differ most in two factors, which, on the one hand, characterize a certain emotional orientation (restraint, prudence - carelessness, expressiveness), on the other hand, features in interpersonal relationships (isolation - sociability). Interestingly, the description of these traits largely coincides with the description of the types of extroverts-introverts proposed by Eysenck. In turn, these types have a certain psychophysiological interpretation. Extroverts are strong, labile, activated; introverts are weak, inert, inactivated. The same set of psychophysiological characteristics was obtained for specially human types of higher nervous activity - "artists" and "thinkers".
The results obtained by I.A. Lyovochkina, allow you to build certain syndromes of the relationship of psychophysiological, psychological signs and types of mathematical thinking.
"Analysts" "Geometers"
(abstract-logical (visual-figurative type of thinking)
mindset)
Weak n.s. Strong n.s. prudence carelessness withdrawn sociability introverts extroverts
Thus, carried out by I.A. Lyovochkina, a comprehensive study of mathematically gifted schoolchildren made it possible to experimentally confirm the presence of a certain combination of psychological and psychophysiological factors that make up a favorable basis for the development of mathematical abilities. This applies to both general and special moments in the manifestation of this type of ability.
A few words about the ability to reading drawings.
In the study N. P. Linkova"The ability to read drawings among younger students" proved that the ability to read and execute drawings is one of the conditions that ensure the success of activities in the field of technology. Therefore, the study of the ability to read drawings is included as an integral part of the study on technical creativity.
Typically, a designer uses drawings to express thoughts that arise in him in the process of solving a problem.
The designer needs such a level of skills in reading drawings, in which the very process of creating an image from its flat image turns from a special purpose into a tool that helps to solve some other problem.
The difference between these two levels of proficiency in reading drawings lies not only in what goal is set for this - to represent an object by its image or use the resulting image to solve any problem, but also in the very nature of the activity.
Experiments conducted with younger schoolchildren confirmed the results obtained in work with high school students.
For the successful mastery of reading drawings, the most important thing is the student's ability to perform certain logical operations. These, first of all, include the ability to conduct a logical analysis of images and correlate them with each other, put forward hypotheses that anticipate decisions, draw logical conclusions based on the available images and carry out the necessary verification of one's assumptions.
The ability to master this kind of operations, conventionally called the ability to think logically, can be considered central among the components that ensure successful mastery of drawing reading techniques.
It must be combined with flexibility of thinking, with the ability to reject the wrong path taken by the decision, or even the solution already received.
A mental representation of the image of an object based on its image can only arise as a result of such an analysis.
The appearance of an image is the result of certain actions. If the task is too easy for the student, these actions are folded, inconspicuous. But they immediately appear in the case of a complication of the task or the appearance of any difficulties in the course of solving.
The success of reading drawings is ensured both by the logical analysis of the image and by the activity of spatial imagination, without which the appearance of an image is impossible. However, logical analysis plays a leading role in this work. It determines the direction of the search for a solution - an unsuccessful or incomplete analysis leads to the appearance of an incorrect image.
The ability to create stable and vivid images in this situation will only complicate the situation.
2. Experiments have shown that for some pupils of primary school age, the components of abilities necessary for mastering the techniques of reading drawings have reached such a level that they can perform a wide variety of tasks from the school drawing course without any difficulty.
For the majority of students of this age, the need to conduct a logical analysis of images, draw conclusions and justify their decisions causes serious difficulties. We are talking about the degree of development of the ability to logical thinking.
Conclusion: teaching projection drawing can be started in elementary school. The possibility of organizing such training was tested in the course of a special experiment conducted jointly with E.A. Faraponova (Linkova, Faraponova, 1967).
But when organizing such training, serious changes must be made to the methodology.
These changes should, first of all, go along the line of weakening the requirements for logical analysis at the first stage of learning. It is equally important, if not to unload, then at least not to complicate the requirements for spatial imagination by introducing such techniques for explaining the material as designing points on the plane of a trihedral angle, mental rotation of models or their images.
This requirement is explained not so much by the poor development of spatial imagination in children of this age (for the most part it turns out to be quite developed), but by their unpreparedness for the simultaneous performance of several operations.
The study showed that there are very large individual differences between students in the degree of development of their abilities necessary to master the techniques of reading drawings, starting from the moment they enter school. The question of the causes of these differences and the ways of developing these abilities is not considered in the study by N.P. Linkova.
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The ability to do mathematics is one of the talents given by nature, which manifests itself from an early age and is directly related to the development of creative potential, the desire to know the world around the baby. But why is learning math so difficult for some children, and can these abilities be improved?
The opinion that mathematics is subject only to gifted children is erroneous. Mathematical ability, like other talents, is the result of the harmonious development of the child, and must begin from a very early age.
In the modern computer world with its digital technologies, the ability to be friends with numbers is essential. Many professions are based on mathematics, which develops thinking and is one of the most important factors influencing the intellectual growth of children. This exact science, whose role in the upbringing and education of the child is undeniable, develops logic, teaches to think consistently, determine the similarities, connections and differences of objects and phenomena, makes the child's mind fast, attentive and flexible.
In order for mathematics lessons for children of five to seven years old to be effective, a serious approach is needed, and the first thing to do is to diagnose their knowledge and skills - to assess the level of the child's logical thinking and basic mathematical concepts.
Diagnosis of mathematical abilities of children 5-7 years old according to the method of Beloshistaya A.V.
If a child with a mathematical mindset has mastered mental counting at an early age, this is not yet the basis for one hundred percent confidence in his future as a mathematical genius. Mental counting skills are only a small element of exact science and far from the most difficult. The presence of a child's ability in mathematics is evidenced by a special way of thinking, which is inherent in logic and abstract thinking, understanding of diagrams, tables and formulas, the ability to analyze, the ability to see figures in space (volumetric).
To determine whether children from primary preschool (4-5 years old) to primary school age have these abilities, there is an effective diagnostic system created by Dr. pedagogical sciences Anna Vitalievna Beloshista. It is based on the creation by the teacher or parent of certain situations in which the child must apply this or that skill.
Diagnostic steps:
- Checking a child of 5-6 years old for the skills of analysis and synthesis. At this stage, you can evaluate how the child is able to compare objects. various forms, separate them and generalize according to certain criteria.
- Testing the skills of figurative analysis in children aged 5-6 years.
- Testing the ability to analyze and synthesize information, the results of which reveal the ability of a preschooler (first grader) to determine the shapes of various figures and notice them in complex pictures with figures superimposed on each other.
- Testing to determine the child's understanding of the basic theses of mathematics - we are talking about the concepts of "more" and "less", ordinal counting, the shape of the simplest geometric shapes.
The first two stages of such a diagnosis are carried out at the beginning school year, the rest - at the end, which makes it possible to assess the dynamics of the child's mathematical development.
The material used for testing should be understandable and interesting for children - age-appropriate, bright and with pictures.
Diagnostics of the mathematical abilities of the child according to the method of Kolesnikova E.V.
Elena Vladimirovna created a lot teaching aids for the development of mathematical abilities in preschoolers. Her method of testing children aged 6 and 7 is widely used by teachers and parents from different countries and meets the requirements of the Federal State Educational Standard (Russia).
Thanks to the Kolesnikova method, it is possible to accurately determine the level of the main indicators of the development of children's mathematical skills, find out their readiness for school, and identify weaknesses in order to fill in the gaps in a timely manner. This diagnostic helps to find ways to improve the mathematical abilities of the baby.
Developing a Child's Mathematical Ability: Tips for Parents
With any science, even such a serious one as mathematics, it is better to acquaint the baby in game form– this will be the best teaching method that parents should choose. Listen to the words of the famous scientist Albert Einstein: "Play is the highest form of exploration." After all, with the help of the game you can get amazing results:
- knowledge of oneself and the world around;
– formation of a base of mathematical knowledge;
- development of thinking:
- the formation of personality;
- development of communication skills.
You can use different games:
- Counting sticks. Thanks to them, the baby remembers the shapes of objects, develops his attention, memory, ingenuity, skills of comparison and perseverance are formed.
- Puzzles that develop logic and ingenuity, attention and memory. Logic tasks help children learn better space perception, thoughtful planning, simple and backwards, and ordinal counting.
- Mathematical puzzles are a great way to develop the basic aspects of thinking: logic, analysis and synthesis, comparison and generalization. While searching for a solution, children learn to make their own conclusions, cope with difficulties and defend their point of view.
The development of mathematical abilities through the game forms educational excitement, adds vivid emotions, helps the baby to fall in love with the subject of study that interests him. It is also worth noting that play activity promotes the development of creative abilities.
The role of fairy tales in the development of mathematical abilities of preschoolers
Children's memory has its own characteristics: it captures bright emotional moments, that is, the child remembers the information that is associated with surprise, joy, admiration. And learning “under pressure” is an extremely inefficient way. In the search for effective teaching methods, adults should remember such a simple and mundane element as a fairy tale. It is a fairy tale that is one of the first means of introducing a baby to the outside world.
For children, a fairy tale and reality are closely connected, magical characters are real and alive. Thanks to fairy tales, the child's speech, his imagination and ingenuity develop; they give the concept of kindness, honesty, broaden one's horizons, and also provide an opportunity to develop mathematical skills.
For example, in the fairy tale “Three Bears”, the baby in an unobtrusive form gets acquainted with the count to three, the concepts of “small”, “medium” and “large”. “Turnip”, “Teremok”, “The kid who could count up to 10”, “The wolf and the seven kids” - in these tales you can learn simple and ordinal counting.
When discussing fairy tale characters, you can offer the baby to compare them in width and height, “hide” them in geometric shapes that are suitable in size or shape, which contributes to the development of abstract thinking.
You can use fairy tales not only at home, but also in the classroom at school. Children are very fond of lessons based on the plots of their favorite fairy tales, with the use of riddles, labyrinths, fingering. Such classes will become a real adventure in which the kids will take a personal part, which means that the material will be learned better. The main thing is to involve children in the process of the game and arouse their interest.
The problem of the formation and development of mathematical abilities of younger students is relevant at the present time, but at the same time it is given insufficient attention among the problems of pedagogy. Mathematical abilities refer to special abilities that are manifested only in a separate type of human activity.
Often teachers try to understand why children studying in the same school, with the same teachers, in the same class, achieve different success in mastering this discipline. Scientists explain this by the presence or absence of certain abilities.
Abilities are formed and developed in the process of learning, mastering the relevant activity, 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 there is a rapid development of intelligence. Therefore, during the period of primary school age, the possibilities for developing abilities are the highest. The development of the mathematical abilities of a junior schoolchild is understood as a purposeful, didactically and methodically organized formation and development of a set of interrelated properties and qualities of the child's mathematical style of thinking and his abilities for mathematical knowledge of reality.
The first place among academic subjects, which represent a particular difficulty in learning, is given to mathematics, as one of the abstract sciences. For children of primary school age, it is extremely difficult to perceive this science. An explanation for this can be found in the works of L.S. Vygotsky. He argued that in order “to understand the meaning of a word, it is necessary to create a semantic field around it. To build a semantic field, a projection of meaning into a real situation must be carried out. It follows from this that mathematics is complex, because it is an abstract science, for example, it is impossible to transfer a number series into reality, because it does not exist in nature.
From the foregoing, it follows that it is necessary to develop the child's abilities, and this problem must be approached individually.
The problem of mathematical abilities was considered by the following authors: Krutetsky V.A. "Psychology of mathematical abilities", Leites N.S. "Age giftedness and individual differences", Leontiev A.N. "Ability Chapter", Zak Z.A. "Development of intellectual abilities in children" and others.
To date, the problem of developing the mathematical abilities of younger students is one of the least developed problems, both methodological and scientific. This determines the relevance of this work.
The purpose of this work: systematization of scientific points of view on this issue and identification of direct and indirect factors affecting the development of mathematical abilities.
When writing this paper, the following tasks:
1. The study of psychological and pedagogical literature in order to clarify the essence of the concept of ability in the broad sense of the word, and the concept of mathematical ability in the narrow sense.
2. Analysis of psychological and pedagogical literature, materials of periodicals devoted to the problem of studying mathematical abilities in historical development and at the present stage.
ChapterI. The essence of the concept of ability.
1.1 General concept of abilities.
The problem of abilities is one of the most complex and least developed in psychology. Considering it, first of all, it should be taken into account that the real subject of psychological research is the activity and behavior of a person. There is no doubt that the source of the concept of abilities is the indisputable fact that people differ in the quantity and quality of the productivity of their activities. The variety of human activities and the quantitative and qualitative difference in productivity makes it possible to distinguish between types and degrees of abilities. A person who does something well and quickly is said to be capable of this work. The judgment about abilities is always comparative in nature, that is, it is based on a comparison of productivity, the ability of one person with the ability of others. The criterion of ability is the level (result) of activity, which one manages to achieve, while others do not. The history of social and individual development teaches that any skillful skill is achieved as a result of more or less intense work, various, sometimes gigantic, "superhuman" efforts. On the other hand, some achieve high mastery of activity, skill and skill with less effort and faster, others do not go beyond average achievements, and others are below this level, even if they try hard, study and have favorable external conditions. It is the representatives of the first group that are called capable.
Human abilities, their different types and degrees, are among the most important and most complex problems of psychology. However, the scientific development of the question of abilities is still insufficient. Therefore, in psychology there is no single definition of abilities.
V.G. Belinsky understood the potential natural forces of the individual, or his capabilities, as abilities.
According to B.M. Teplov, abilities are individual psychological characteristics that distinguish one person from another.
S.L. Rubinstein understands abilities as suitability for a certain activity.
The psychological dictionary defines ability as a quality, opportunity, skill, experience, skill, talent. Abilities allow you to perform certain actions at a given time.
Ability is the readiness of an individual to perform some action; suitability - the available potential to perform any activity or the ability to achieve a certain level of ability development.
Based on the foregoing, we can give a general definition of abilities:
Ability is an expression of the correspondence between the requirements of activity and a complex of neuropsychological properties of a person, which ensures high qualitative and quantitative productivity and the growth of his activity, which is manifested in a high and rapidly growing (compared to the average person) ability to master this activity and own it.
1.2 The problem of developing the concept of mathematical abilities abroad and in Russia.
A wide variety of directions also determined a wide variety in the approach to the study of mathematical abilities, in methodological tools and theoretical generalizations.
The study of mathematical abilities should begin with the definition of the subject of study. The only thing that all researchers agree on is the opinion that one should distinguish between ordinary, “school” abilities for mastering mathematical knowledge, for their reproduction and independent application, and creative mathematical abilities associated with the independent creation of an original and socially valuable product.
Back in 1918, Rogers noted two aspects of mathematical abilities, reproductive (associated with the function of memory) and productive (associated with the function of thinking). In accordance with this, the author built a well-known system of mathematical tests.
The well-known psychologist Reves, in his book "Talent and Genius", published in 1952, considers two main forms of mathematical abilities - applicative (as the ability to quickly detect mathematical relationships without preliminary tests and apply relevant knowledge in similar cases) and productive (as the ability to discover relationships, not directly derived from existing knowledge).
Foreign researchers show great unity of views on the question of innate or acquired mathematical abilities. If here we distinguish two different aspects of these abilities - “school” and creative abilities, then with respect to the second there is complete unity - the creative abilities of a scientist - mathematician are an innate education, a favorable environment is necessary only for their manifestation and development. Such, for example, is the point of view of mathematicians who were interested in questions of mathematical creativity - Poincaré and Hadamard. Betz also wrote about the innateness of mathematical talent, emphasizing that we are talking about the ability to independently discover mathematical truths, "because probably everyone can understand someone else's thought." The thesis about the innate and hereditary nature of mathematical talent was vigorously promoted by Reves.
With regard to "school" (educational) abilities, foreign psychologists are not so unanimous. Here, perhaps, the theory of the parallel action of two factors - the biological potential and the environment - dominates. Until recently, ideas of innateness also dominated school mathematical abilities.
Back in 1909-1910. Stone and independently Curtis, studying achievements in arithmetic and ability in this subject, came to the conclusion that one can hardly speak of mathematical ability as a whole, even in relation to arithmetic. Stone pointed out that children who are good at calculations often lag behind in arithmetic reasoning. Curtis also showed that it is possible to combine a child's success in one branch of arithmetic and his failure in another. From this they both concluded that each operation required its own special and relatively independent ability. Some time later, a similar study was conducted by Davis and came to the same conclusions.
One of the significant studies of mathematical abilities must be recognized as the study of the Swedish psychologist Ingvar Verdelin in his book Mathematical Ability. The main intention of the author was to analyze the structure of the mathematical abilities of schoolchildren, based on the multifactorial theory of intelligence, to identify the relative role of each of the factors in this structure. Werdelin accepts as a starting point the following definition of mathematical abilities: “Mathematical ability is the ability to understand the essence of mathematical (and similar) systems, symbols, methods and proofs, memorize, retain them in memory and reproduce, combine them with other systems, symbols, methods and proofs, use them in solving mathematical (and similar) problems. The author analyzes the question of the comparative value and objectivity of measuring mathematical abilities by teachers' educational marks and special tests and notes that school marks are unreliable, subjective and far from the real measurement of abilities.
A great contribution to the study of mathematical abilities was made by the famous American psychologist Thorndike. In The Psychology of Algebra, he gives a host of all kinds of algebraic tests to determine and measure abilities.
Mitchell, in his book on the nature of mathematical thinking, lists several processes that he believes characterize mathematical thinking, in particular:
1. classification;
2. ability to understand and use symbols;
3. deduction;
4. manipulation with ideas and concepts in an abstract form, without relying on the concrete.
Brown and Johnson in the article "Ways to identify and educate students with potentialities in the sciences" indicate that practicing teachers have identified those features that characterize students with potentialities in mathematics, namely:
1. extraordinary memory;
2. intellectual curiosity;
3. ability for abstract thinking;
4. ability to apply knowledge in a new situation;
5. the ability to quickly "see" the answer when solving problems.
Concluding the review of the works of foreign psychologists, it should be noted that they do not give a more or less clear and precise idea of the structure of mathematical abilities. In addition, it must also be borne in mind that in some works the data were obtained by a slightly objective introspective method, while others are characterized by a purely quantitative approach while ignoring the qualitative features of thinking. Summarizing the results of all the studies mentioned above, we will get the most general characteristics of mathematical thinking, such as the ability for abstraction, the ability for logical reasoning, good memory, the ability for spatial representations, etc.
In Russian pedagogy and psychology, only a few works are devoted to the psychology of abilities in general and the psychology of mathematical abilities in particular. It is necessary to mention the original article by D. Mordukhai-Boltovsky "Psychology of Mathematical Thinking". The author wrote the article from an idealistic position, giving, for example, special meaning"unconscious thought process", stating that "the thinking of a mathematician ... is deeply embedded in the unconscious sphere." The mathematician is not aware of every step of his thought "sudden appearance in the mind ready solution any task that we could not solve for a long time, - the author writes, - we explain by unconscious thinking, which ... continued to deal with the task, ... and the result pops up beyond the threshold of consciousness.
The author notes the specific nature of mathematical talent and mathematical thinking. He argues that the ability to do mathematics is not always inherent even in brilliant people, that there is a difference between a mathematical and non-mathematical mind.
Of great interest is Mordukhai-Boltovsky's attempt to isolate the components of mathematical abilities. These components include, in particular:
1. “strong memory”, it was stipulated that “mathematical memory” is meant, memory for “an object of the type that mathematics deals with”;
2. “wit”, which is understood as the ability to “embrace in one judgment” concepts from two loosely connected areas of thought, to find in the already known something similar to the given;
3. speed of thought (speed of thought is explained by the work done by unconscious thinking in favor of the conscious).
D. Mordukhai-Boltovsky also expresses his views on the types of mathematical imagination that underlie different types of mathematicians - "geometers" and "algebraists". "Arithmeticians, algebraists, and analysts in general, whose discovery is made in the most abstract form of discontinuous quantitative symbols and their interrelations, cannot express like a geometer." He also expressed valuable thoughts about the peculiarities of the memory of "geometers" and "algebraists".
The theory of abilities was created for a long time by the joint work of the most prominent psychologists of that time: B.M. Teplov, L.S. Vygotsky, A.N. Leontiev, S.L. Rubinstein, B.G. Anafiev and others.
In addition to general theoretical studies of the problem of abilities, B.M. Teplov, with his monograph “Psychology of Musical Abilities,” laid the foundation for an experimental analysis of the structure of abilities for specific types of activity. The significance of this work goes beyond the narrow question of the essence and structure of musical abilities, it found a solution to the main, fundamental questions of research into the problem of abilities for specific types of activity.
This work was followed by studies of abilities similar in idea: to visual activity - V.I. Kireenko and E.I. Ignatov, literary abilities - A.G. Kovalev, pedagogical abilities - N.V. Kuzmin and F.N. Gonobolin, structural and technical abilities - P.M. Jacobson, N.D. Levitov, V.N. Kolbanovsky and mathematical abilities - V.A. Krutetsky.
Row experimental studies thinking was carried out under the guidance of A.N. Leontiev. Some issues of creative thinking were clarified, in particular, how a person comes to the idea of solving a problem, the method of solving which does not directly follow from its conditions. An interesting pattern was established: the effectiveness of exercises leading to right decision, is different depending on at what stage of solving the main problem auxiliary exercises are presented, i.e., the role of suggestive exercises was shown.
Directly related to the problem of abilities is a series of studies by L.N. Landes. In one of the first works of this series - "On some shortcomings in the study of students' thinking" - he raises the question of the need to reveal the psychological nature, the internal mechanism of "the ability to think." Cultivate abilities, according to L.N. Landa means “to teach the technique of thinking”, to form the skills and abilities of analytical and synthetic activity. In his other work - “Some data on the development of mental abilities” - L. N. Landa found significant individual differences in the assimilation by schoolchildren of a new method of reasoning for them when solving geometric problems for proof - differences in the number of exercises necessary to master this method, differences in the pace of work, differences in the formation of the ability to differentiate the application of operations depending on the nature of the conditions of the task and differences in the assimilation of operations.
Of great importance for the theory of mental abilities in general and mathematical abilities in particular are the studies of D.B. Elkonin and V.V. Davydova, L.V. Zankova, A.V. Skripchenko.
It is usually believed that the thinking of children aged 7-10 has a figurative character, is distinguished by a low ability to distract and abstract. Experiential learning led by D.B. Elkonin and V.V. Davydov, showed that already in the first grade, with a special teaching methodology, it is possible to give students in alphabetical symbolism, that is, in a general form, a system of knowledge about the relationships of quantities, dependencies between them, to introduce them into the field of formally symbolic operations. A.V. Skripchenko showed that students of the third - fourth grades, under appropriate conditions, can form the ability to solve arithmetic problems by compiling an equation with one unknown.
1.3 Mathematical ability and personality
First of all, it should be noted that characterizing capable mathematicians and necessary for successful activity in the field of mathematics "unity of inclinations and abilities in vocation", expressed in a selectively positive attitude towards mathematics, the presence of deep and effective interests in the relevant field, the desire and need to engage in it, passionate passion for the job.Without an aptitude for mathematics, there can be no genuine aptitude for it. If the student does not feel any inclination towards mathematics, then even good abilities are unlikely to ensure a completely successful mastery of mathematics. The role that inclination and interest play here boils down to the fact that a person interested in mathematics is intensively engaged in it, and, consequently, vigorously exercises and develops his abilities.
Numerous studies and the characteristics of gifted children in the field of mathematics indicate that abilities develop only in the presence of inclinations or even a peculiar need for mathematical activity. The problem is that often students are capable of mathematics, but have little interest in it, and therefore do not have much success in mastering this subject. But if the teacher can awaken their interest in mathematics and the desire to do it, then such a student can achieve great success.Such cases are not uncommon at school: a student capable of mathematics has little interest in it, and does not show much success in mastering this subject. But if the teacher can awaken his interest in mathematics and the inclination to do it, then such a student, "captured" by mathematics, can quickly achieve great success.
From this follows the first rule of teaching mathematics: the ability to interest in science, to push for the independent development of abilities. Emotions experienced by a person are also an important factor in the development of abilities in any activity, not excluding mathematical activity. The joy of creativity, the feeling of satisfaction from intense mental work, mobilize his strength, make him overcome difficulties. All children who are capable of mathematics are distinguished by a deep emotional attitude to mathematical activity, they experience real joy caused by each new achievement. Awakening a creative streak in a student, teaching him to love mathematics is the second rule of a mathematics teacher.Many teachers point out that the ability to quickly and deeply generalize can manifest itself in any one subject without characterizing the student's learning activity in other subjects. An example is that a child who is able to generalize and systematize material in literature does not show similar abilities in the field of mathematics.
Unfortunately, teachers sometimes forget that general in nature mental capacity, in some cases act as specific abilities. Many teachers tend to apply an objective assessment, that is, if a student is weak in reading, then in principle he cannot reach heights in the field of mathematics. This opinion is typical for primary school teachers who lead a complex of subjects. This leads to an incorrect assessment of the child's abilities, which in turn leads to a lag in mathematics.
1.4 Development of mathematical abilities in younger students.
The problem of ability is the problem of individual differences. With the best organization of teaching methods, the student will advance more successfully and faster in one area than in another.
Naturally, success in learning is determined not only by the abilities of the student. In this sense, the content and methods of teaching, as well as the attitude of the student to the subject, are of primary importance. Therefore, success and failure in learning do not always give grounds for judgments about the nature of the student's abilities.
The presence of weak abilities in students does not relieve the teacher of 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 shows them.
It is necessary to educate those who are capable and select those who are capable, while not forgetting about all schoolchildren, in every possible way to raise general level their preparation. In this regard, in their work, various collective and individual methods of work are needed in order to activate the activity of students in this way.
The learning process should be comprehensive both in terms of organizing the learning process itself, and in terms of developing students' deep interest in mathematics, skills and abilities in solving problems, understanding the system of mathematical knowledge, solving a special system of non-standard tasks with students, which should be offered not only on lessons, but control work. Thus, a special organization of the presentation of educational material, a well-thought-out system of tasks, contribute to an increase in the role of meaningful motives for studying mathematics. The number of results-oriented students is decreasing.
In the lesson, not just solving problems, 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 the course of solving the problem, but on the beauty and rationality of the method.
Teachers successfully use the technique of "setting tasks" to determine the direction of motivation. Each task is evaluated according to the system of the following indicators: the nature of the task, its correctness and attitude to source code. The same method is sometimes used in the wine version: after solving the problem, the students were asked to compose any problems somehow related to the original problem.
To create psycho-pedagogical conditions for increasing the effectiveness of the organization of the learning process system, the principle of organizing the learning process in the form of subject communication using cooperative forms of work of students is used. This is a group problem solving and collective discussion of grading, pair and team work.
Chapter II. The development of mathematical abilities in younger schoolchildren as a methodological problem.
2.1 General features of capable and talented children
The problem of developing children's mathematical abilities is one of the least developed methodological problems of teaching mathematics in primary school today.
The extreme heterogeneity of views on the very concept of mathematical ability leads to the absence of any conceptually sound methods, which in turn creates difficulties in the work of teachers. Perhaps that is why not only among parents, but also among teachers there is a widespread opinion: mathematical abilities are either given or not given. And there's nothing you can do about it.
Undoubtedly, abilities for one or another type of activity are due to individual differences in the human psyche, which are based on genetic combinations of biological (neurophysiological) components. However, today there is no evidence that certain properties of nerve tissues directly affect the manifestation or absence of certain abilities.
Moreover, purposeful compensation for unfavorable natural inclinations can lead to the formation of a personality with pronounced abilities, of which there are many examples in history. Mathematical abilities belong to the group of so-called special abilities (as well as musical, visual, etc.). For their manifestation and further development, the assimilation of a certain stock of knowledge and the presence of certain skills, including the ability to apply existing knowledge in mental activity, are required.
Mathematics is one of those subjects where the individual characteristics of the psyche (attention, perception, memory, thinking, imagination) of the child are crucial for its assimilation. Behind the important characteristics of behavior, behind the success (or failure) of educational activity, those natural dynamic features that were mentioned above are often hidden. Often they give rise to differences in knowledge - their depth, strength, generalization. According to these qualities of knowledge, related (along with value orientations, beliefs, skills) to the content side of a person's mental life, they usually judge the giftedness of children.
Individuality and giftedness are interrelated concepts. Researchers dealing with the problem of mathematical abilities, the problem of the formation and development of mathematical thinking, with all the differences of opinion, note first of all the specific features of the psyche of a mathematically capable child (as well as a professional mathematician), in particular, the flexibility of thinking, i.e. unconventionality, originality, the ability to vary the ways of solving a cognitive problem, the ease of transition from one solution to another, the ability to go beyond the usual way of activity and find new ways to solve a problem under changed conditions. Obviously, these features of thinking directly depend on the special organization of memory (free and connected associations), imagination and perception.
Researchers distinguish such a concept as the depth of thinking, i.e. the ability to penetrate into the essence of each studied fact and phenomenon, the ability to see their relationship with other facts and phenomena, to identify specific, hidden features in the material being studied, as well as the purposefulness of thinking, combined with breadth, i.e. the ability to form generalized methods of action, the ability to cover the problem as a whole, without missing details. Psychological analysis of these categories shows that they should be based on a specially formed or natural inclination to a structural approach to the problem and extremely high stability, concentration and a large amount of attention.
Thus, the individual typological features of the personality of each student individually, which include temperament, character, inclinations, and the somatic organization of the personality as a whole, etc., have a significant (and perhaps even decisive!) influence on the formation and the development of the mathematical style of thinking of the child, which, of course, is a necessary condition for the preservation natural potential(inclinations) of the child in mathematics and its further development in pronounced mathematical abilities.
Experienced subject teachers know that mathematical abilities are a “piece of goods”, and if such a child is not dealt with individually (individually, and not as part of a circle or elective), then abilities may not develop further.
That is why we often observe how a first-grader with outstanding abilities “levels out” by the third grade, and in the fifth grade completely ceases to differ from other children. What's this? Psychological research shows that there may be different types age mental development:
. "Early rise" (at preschool or primary school age) - due to the presence of bright natural abilities and inclinations of the appropriate type. In the future, consolidation and enrichment of mental merits may occur, which will serve as a start for the formation of outstanding mental abilities.
At the same time, the facts show that almost all scientists who proved themselves before the age of 20 were mathematicians.
But “alignment” with peers can also occur. We believe that such “leveling” is largely due to the lack of a competent and methodically active individual approach to the child in the early period.
"Slow and extended rise", i.e. gradual accumulation of intelligence. The absence of early achievement in this case does not mean that the prerequisites for great or outstanding ability will not emerge later. Such a possible "rise" is the age of 16-17 years, when the factor of the "intellectual explosion" is the social reorientation of the individual, directing his activity in this direction. However, such a "rise" can occur in more mature years.
For a primary school teacher, the most urgent problem is the "early rise", which falls on the age of 6-9 years. It is no secret that one such brightly capable child in the class, who also has a strong type of nervous system, is capable, in the literal sense of the word, of not letting any of the children open their mouths in the lesson. And as a result, instead of stimulating and developing the little “wunderkind” as much as possible, the teacher is forced to teach him to be silent (!) And “keep his brilliant thoughts to himself until asked.” After all, there are 25 other children in the class! Such “slowing down”, if it occurs systematically, can lead to the fact that in 3-4 years the child “levels out” with his peers. And since mathematical abilities belong to the group of “early abilities”, then, perhaps, it is the mathematically capable children that we lose in the process of this “slowing down” and “leveling out”.
Psychological studies have shown that although the development of learning abilities and creative gifts in typologically different children proceeds differently, it is equally high degree development of these abilities can be achieved (achieved) by children with opposite characteristics of the nervous system. In this regard, it may be more useful for the teacher to focus not on the typological features of the nervous system of children, but on some general features of capable and talented children, which are noted by most researchers of this problem.
Different authors single out a different “set” of common features of capable children within the framework of those types of activities in which these abilities were studied (mathematics, music, painting, etc.). We believe that it is more convenient for the teacher to rely on certain purely procedural characteristics of the activity of capable children, which, as a comparison of a number of special psychological and pedagogical research on this topic, turn out to be the same for children with different types of abilities and giftedness. Researchers note that most capable children are characterized by:
Increased propensity for mental action and a positive emotional response to any new mental challenge. These kids don't know what boredom is - they always have something to do. Some psychologists generally interpret this trait as an age factor of giftedness.
The constant need to renew and complicate the mental load, which entails a constant increase in the level of achievements. If this child is not loaded, then he finds a load for himself and can master chess, a musical instrument, radio work, etc., study encyclopedias and reference books, read special literature, etc.
The desire for independent choice of affairs and planning of their activities. This child has his own opinion about everything, stubbornly defends the unlimited initiative of his activity, has a high (almost always adequate at the same time) self-esteem and is very persistent in self-affirmation in the chosen area.
Perfect self-regulation. This child is capable of full mobilization of forces to achieve the goal; is able to repeatedly resume mental efforts, striving to achieve the goal; has, as it were, an “original” attitude to overcome any difficulties, and his failures only make him strive to overcome them with enviable persistence.
Increased performance. Prolonged intellectual loads do not tire this child, on the contrary, he feels good precisely in the situation of a problem that needs to be solved. Purely instinctively, he knows how to use all the reserves of his psyche and his brain, mobilizing and switching them at the right time.
It is clearly seen that these general procedural characteristics of the activity of capable children, recognized by psychologists as statistically significant, are not uniquely inherent in any one type of the human nervous system. Therefore, pedagogically and methodically, the general tactics and strategy of an individual approach to a capable child, obviously, should be based on such psychological and didactic principles that ensure that the above procedural characteristics of the activities of these children are taken into account.
From a pedagogical standpoint, a capable child is most in need of an instructive style of relations with the teacher, which requires greater information content and validity of the requirements put forward by the teacher. The instructive style, as opposed to the imperative style that prevails in elementary school, involves appealing to the personality of the student, taking into account his individual characteristics and focusing on them. This style of relationship contributes to the development of independence, initiative and creativity, which is noted by many research educators. It is equally obvious that from a didactic point of view, capable children need, at a minimum, to ensure the optimal pace of progress in the content and the optimal amount of teaching load. Moreover, it is optimal for oneself, for one's abilities, i.e. higher than for normal children. If we take into account the need for a constant complication of the mental load, the persistent craving for self-regulation of their activities and the increased efficiency of these children, it can be stated with sufficient confidence that these children are by no means "prosperous" students at school, since their educational activity constantly takes place not in zone of proximal development (!), but far behind this zone! Thus, in relation to these students, we (willingly or unwittingly) constantly violate our proclaimed credo, the basic principle of developmental education, which requires the child to be taught taking into account the zone of his proximal development.
Working with talented children in primary school today is no less a "sore" problem than working with underachieving ones.
Its lesser “popularity” in special pedagogical and methodological publications is explained by its lesser “strikingness”, since a loser is an eternal source of trouble for a teacher, and only the teacher knows that Petya’s five does not even half reflect his capabilities (and then not always), yes, Petya's parents (if they deal with this issue on purpose). At the same time, the constant “underload” of a capable child (and the norm for everyone is an underload for a capable child) will contribute to insufficient stimulation of the development of abilities, not only to the “failure to use” the potential of such a child (see paragraphs above), but also to the possible extinction of these abilities as unclaimed in educational activities (leading during this period of the child's life).
There is also a more serious and unpleasant consequence of this: it is too easy for such a child to learn at the initial stage; transition from primary to secondary.
In order for a teacher of a mass school to be able to successfully cope with work with a capable child in mathematics, it is not enough to indicate the pedagogical and methodological aspects of the problem. As the thirty-year practice of implementing the developmental education system has shown, in order for this problem to be solved in the conditions of education in a mass primary school, a specific and fundamentally new methodological solution is needed, which is fully presented to the teacher.
Unfortunately, today there are practically no special methodological manuals for primary school teachers designed to work with capable and gifted children in mathematics lessons. We cannot cite a single such manual or methodological development, except for various collections of the Mathematical Box type. To work with capable and gifted children, tasks that are not entertaining are needed, this is too poor food for their minds! We need a special system and special "parallel" to the existing teaching aids. The lack of methodological support for individual work with a capable child in mathematics leads to the fact that elementary school teachers do not do this work at all (it cannot be considered individual circle or optional work, where a group of children solve entertaining tasks with a teacher, as a rule, not systematically selected). One can understand the problems of a young teacher who does not have enough time or knowledge to select and organize the relevant materials. But a teacher with experience is not always ready to solve such a problem. Another (and, perhaps, the main!) constraint here is the presence of a single textbook for the entire class. Work on the same for all children study guide, according to a single calendar plan, simply does not allow the teacher to realize the requirement of individualization of the learning pace of a capable child, and the content of the textbook, which is the same for all children, does not allow the requirement of individualization of the volume of the teaching load (not to mention the requirement of self-regulation and independent planning of activities).
We believe that the creation of special teaching materials in mathematics to work with capable children is the only possible way to implement the principle of individualization of education in relation to these children in the conditions of teaching the whole class.
2.2 Methodology for long-term assignments
The methodology for using the system of long-term tasks was considered by E.S. Rabunsky in organizing work with high school students in the learning process German at school.
In a number of pedagogical studies, the possibility of creating systems of such tasks for various subjects for high school students both in mastering new material and in eliminating gaps in knowledge. 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 works(essays, essays, etc.), turned out to be the most preferable for the majority of the students surveyed. It turned out that such “delayed work” satisfies the student more than individual lessons and assignments, since the main criterion for student satisfaction at any age is success in work. The absence of a sharp time limit (as happens in the classroom) and the possibility of free multiple return to the content of the work 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 all of the above 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 experience of Beloshistaya A.V. and teachers who took part in the experimental verification 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 their 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 acquaintances and constant control at the stage of consolidation.
During the experimental work, a large number of printed sheets were developed, combined into blocks covering the whole topic. Each block contains 12-20 sheets. The sheet 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 come to an understanding of the essence and method of performing a new computational technique, and then consolidate the new method of activity. A sheet (or sheet system, i.e. a thematic block) is a “long-term task”, the deadlines for which are individualized in accordance with the desire 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 execution, which the teacher either sets individually or allows the student (this way is more productive) to set the deadline for its implementation for himself (this is the way to form self-discipline, since independent planning of activities in connection with independently defined goals and timing is the basis of human self-education).
The teacher determines the tactics of working with sheets for the student individually. At first, they can be offered to the student as homework (instead of the usual assignment), individually agreeing on the timing of its implementation (2-4 days). As you master this system, you can switch to a preliminary or parallel way of working, i.e. give the student a sheet before getting to know the topic (on the eve of the lesson) or at the lesson itself for independent mastering of the material. Attentive and friendly observation of the student in the process of activity, “contractual style” of relations (let the child decide 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, passing by the child in the lesson) - all this will help the teacher to fully make the learning process of a capable child individualized without spending a lot of time.
Children should not be forced to rewrite tasks from a sheet. The student works with a pencil on a sheet, writing down answers or adding actions. Such an organization of learning causes the child positive emotions He likes to work on a printed basis. Saved from the need for tedious rewriting, the child works with greater productivity. Practice shows that although the sheets contain up to fifty tasks (the usual homework norm is 6-10 examples), the student works with them with pleasure. Many children ask for a new leaf every day! In other words, they exceed the working norm of the lesson and homework several times, while experiencing positive emotions and working on their own.
During the experiment, such sheets were developed on the topics: "Oral and written computational techniques", "Numbering", "Values", "Fractions", "Equations".
Methodological principles for constructing the proposed system:
1. The principle of compliance with the program in mathematics for elementary grades. Content sheets are tied to a stable (standard) program in mathematics for elementary grades. Thus, we believe that it is possible to implement the concept of individualization of teaching mathematics to a capable child in accordance with the procedural features of his educational activity when working with any textbook that corresponds to a standard program.
2. 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 this concept, is revealed. This, on the one hand, helps the child to clearly understand the purpose of the work, and on the other hand, it helps the teacher to easily monitor the quality of assimilation of this technique or concept.
3. Structurally, the sheet is a detailed methodological solution to the problem of introducing or getting to know and fixing one or another technique, concept, connections of this concept with other concepts. The tasks are selected and grouped (that is, 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 at the first steps fully disclosed in smaller actions that are the basis of this technique. As you move along the sheet, these small actions are gradually assembled into larger blocks. This allows the student to master the technique as a whole, which is the logical conclusion of the entire methodological "construction". Such a structure of the sheet allows you to fully implement the principle of a gradual increase in the level of complexity at all stages.
4. Such a sheet structure also makes it possible to implement the principle of accessibility, and to a much deeper extent than it is possible to do today when working only with a textbook, since the systematic use of sheets allows you to assimilate the material at an individual pace convenient for the student, which the child can regulate independently.
5. The system of sheets (thematic block) allows you to implement the principle of perspective, i.e. gradual inclusion of the student in planning activities educational process. Tasks designed for long (delayed) preparation require long-term planning. The ability to organize one's work, planning it for a certain period of time, is the most important learning skill.
6. The system of sheets on the topic also makes it possible to implement the principle of individualization of testing and assessing students' knowledge, and not on the basis of differentiation of the level of complexity of tasks, but on the basis of the unity of requirements for the level of knowledge, skills and abilities. Individualized terms and methods of 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 do not need to make higher demands. Sheets at a certain stage allow such children to use more intellectually rich material, which in a propaedeutic plan will introduce them to the following mathematical concepts of a higher level of complexity.
Conclusion
An analysis of the psychological and pedagogical literature on the problem of the formation and development of mathematical abilities shows that all researchers (both domestic and foreign) without exception associate it not with the content side of the subject, but with the procedural side of mental activity.
Thus, many teachers believe that the development of a child's mathematical abilities is possible only if there are significant natural data for this, i.e. most often in the practice of teaching it is believed that it is necessary to develop abilities only in those children who already have them. But the experimental studies of Beloshistaya A.V. showed that work on the development of mathematical abilities is necessary for every child, regardless of his natural giftedness. It's just that the results of this work will be expressed in varying degrees of development of these abilities: for some children it will be a significant advance in the level of development of mathematical abilities, for others it will be a correction of natural insufficiency in their development.
A great difficulty for the teacher in organizing work on the development of mathematical abilities is that today there is no specific and fundamentally new methodological solution that can be presented to the teacher in full. The lack of methodological support for individual work with capable children leads to the fact that elementary school teachers do not do this work at all.
With my work, I wanted to draw attention to this problem and emphasize that the individual characteristics of each gifted child are not only his characteristics, but, possibly, the source of his giftedness. And the individualization of the education of such a child is not only a way of his development, but also the basis for his preservation in the status of “capable, gifted”.
Bibliographic list.
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4. Zaitseva, S.A. Activation of the mathematical activity of younger schoolchildren [Text] / S.A. Zaitseva // Primary education. - 2009. - No. 1. - S. 12 - 19
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11. Reverse, J.. Talent and Genius [Text] / J. Reverse. - M., 1982. - S. 512
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13. Thorndike, E.L. Principles of teaching based on psychology [electronic resource]. - Access mode. - http://metodolog.ru/vigotskiy40.html
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16. Volkov, I.P. Are there many talents in the school? [Text] / I.P. Volkov. - M.: Knowledge, 1989. - P.78
17. Dorofeev, G.V. Does teaching math improve the level intellectual development schoolchildren? [Text] /G.V. Dorofeev // Mathematics at school. - 2007. - No. 4. - S. 24 - 29
18. Istomina, N.V. Methods of teaching mathematics in elementary grades [Text] / N.V. Istomin. - M.: Academy, 2002. - S. 288
19. Savenkov, A.I. A gifted child in a mass school [Text] / ed. M.A. Ushakov. - M.: September, 2001. - S. 201
20. Elkonin, D.B. Questions of psychology of educational activity of junior schoolchildren [Text] / Ed. V. V. Davydova, V. P. Zinchenko. - M.: Enlightenment, 2001. - S. 574
Mathematical abilities have a direct impact on the mental development of a preschooler. The child has much more to look at the world"mathematical eye" than an adult. The reason is that in a short period, the child's brain needs to figure out the shapes and sizes, geometric shapes and spatial orientation, to understand their characteristics and relationships.
What abilities in preschool age are related to mathematical
Many parents think that developing children's mathematical abilities in preschool age it is too early. And they mean by this concept some special abilities that allow children to operate big numbers, or passion for formulas and algorithms.
In the first case, abilities are confused with natural giftedness, and in the other case, a pleasing result may have nothing to do with mathematics. Perhaps the child liked the rhythm of counting or remembered the images of numbers in an arithmetic example.
To dispel this misconception, it is important to clarify what abilities are called mathematical.
Mathematical abilities are the features of the flow of the thought process with the severity of analysis and synthesis, rapid abstraction and generalization in relation to mathematical material.
Based on the same mental operations. They develop in all children with varying efficiency. It is possible and necessary to stimulate their development. This does not mean at all that the child will awaken mathematical talent, and he will grow up to be a real mathematician. But, if you develop the ability to analyze, highlight signs, generalize, build a logical chain of thoughts, then this will contribute to the development of the preschooler's mathematical abilities and more general intellectual ones.
Elementary mathematical representations of preschoolers
So, abilities for mathematics go far beyond arithmetic and develop on the basis of mental operations. But, just as the word is the basis of speech, so in mathematics there are elementary ideas, without which it is pointless to talk about development.
Toddlers need to be taught to count, to introduce quantitative relationships, to expand their knowledge of geometric shapes. By the end of preschool age, the child should have basic mathematical representations:
- Know all the numbers from 0 to 9 and recognize them in any form of writing.
- Count from 1 to 10, both forward and backward (starting with any number).
- Have an idea about simple ordinal numbers and be able to operate with them.
- Perform addition and subtraction operations within 10.
- Be able to equalize the number of items in two sets (There are 5 apples in one basket, 7 pears in the other. What needs to be done to make the fruits in the baskets equally?).
- Know the basic geometric shapes and name the features that distinguish them.
- Operate with quantitative ratios "more-less", "further-closer".
- Operate with simple qualitative ratios: the largest, smallest, lowest, etc.
- Understand complex relationships: “more than the smallest, but less than others”, “ahead and above others”, etc.
- Be able to identify an extra object that is not suitable for a group of others.
- Arrange simple rows in ascending and descending order (The cubes show dots in the amount of 3, 5, 7, 8. Arrange the cubes so that the number of dots on each subsequent one decreases).
- Find the corresponding place of an object with a numerical sign (On the example of the previous task: cubes with points 3, 5 and 8 are placed. Where to put a cube with 7 points?).
This mathematical "baggage" is to be accumulated by the child before entering school. The listed representations are elementary. It is impossible to study mathematics without them.
Among the basic skills, there are quite simple ones that are already available at 3-4 years old, but there are also those (9-12 points) that use the simplest analysis, comparison, generalization. They have to be formed in the process of playing lessons at the senior preschool age.
The list of elementary representations can be used to identify the mathematical abilities of preschoolers. Having offered the child to complete the task corresponding to each item, they determine which skills have already been formed and which ones need to be worked on.
We develop the mathematical abilities of the child in the game
Completing tasks with a mathematical bias is especially useful for children, as it develops. The value lies not only in the accumulation of mathematical concepts and skills, but also in the fact that the general mental development of the preschooler takes place.
In practical psychology, there are three categories of gaming activities aimed at developing individual components of mathematical abilities.
- Exercises to determine the properties of objects, identifying objects according to a designated feature (analytical and synthetic abilities).
- Games for comparing various properties, identifying essential features, abstracting from secondary ones, generalization.
- Games for the development of logical conclusions based on mental operations.
The development of mathematical abilities in preschool children should be carried out exclusively in a playful way.
Exercises for the development of analysis and synthesis
1.Get in order! A game to sort objects by size. Prepare 10 one-color strips of cardboard of the same width and different lengths and arrange them randomly in front of the preschooler.
Instruction: "Arrange the "athletes" in height from the shortest to the tallest." If the child is at a loss with the choice of the strip, invite the "athletes" to measure their height.
After completing the task, invite the child to turn away and swap some of the strips. The preschooler will have to return the "hooligans" to their places.
2.Make a square. Prepare two sets of triangles. 1st - one large triangle and two small ones; 2nd - 4 identical small ones. Invite the child to first fold a square of three parts, then of four.
Picture 1.
If a preschooler spends less time compiling the second square, then understanding has come. Capable children complete each of these tasks in less than 20 seconds.
Abstraction and generalization exercises
1.The fourth is redundant. You will need a set of cards that show four items. On each card, three objects should be interconnected by a significant feature.
Instructions: “Find what is odd in the picture. What does not suit everyone else and why?
Figure 2.
Such exercises should start with simple groups of objects and gradually complicate them. For example, a card with the image of a table, a chair, a kettle and a sofa can be used in classes with 4-year-old children, and sets with geometric shapes can be offered to older preschoolers.
2.Build a fence. Prepare at least 20 strips equal length and widths or counting sticks in two colors. For example: blue - C, and red - K.
Instruction: “Let's build a beautiful fence where colors alternate. The first will be a blue stick, followed by a red one, then ... (we continue to lay out the sticks in the sequence SKSSKKSK). And now you continue to build a fence so that there is the same pattern.
In case of difficulty, pay the child's attention to the rhythm of the alternation of colors. The exercise can be performed several times with a different rhythm of the pattern.
Logical and mathematical games
1.We're going, we're going, we're going. It is necessary to select 10-12 rectangular pictures depicting objects well known to the child. A child plays with an adult.
Instruction: “Now we will make a train of wagons, which will be firmly interconnected by an important feature. There will be a cup in my trailer (puts the first picture), and in order for your trailer to join, you can select a picture with a picture of a spoon. The cup and spoon are connected because they are dishes. I will complete our train with a picture of a scoop, since the scoop and spoon have a similar shape, etc.”
The train is ready to go if all the pictures have found their place. You can mix pictures and start the game again, finding new relationships.
2. Tasks for finding a suitable “patch” for a rug are of great interest to preschoolers different ages. To play the game, you need to make several pictures that show a rug with a cut out circle or rectangle. Separately, it is necessary to depict options for “patches” with a characteristic pattern, among which the child will have to find a suitable one for the rug.
You need to start completing tasks with the color shades of the rug. Then offer cards with simple patterns of rugs, and as the skills of logical choice develop, complicate the tasks on the model of the Raven test.
Figure 3
“Repairing” the rug simultaneously develops a number of important aspects: visual-figurative representations, mental operations, the ability to recreate the whole.
Recommendations for parents on the development of mathematical abilities of the child
Oftentimes, liberal arts parents tend to ignore the development of math skills in their children, and this is a misguided approach. At preschool age, these abilities are used by the child to learn about the world around them.
A preschooler needs to be stimulated by a mathematical approach in order to understand the patterns, the cause-and-effect and logical way of real life.
From early childhood, the child should be surrounded by educational toys that require elementary analysis and the search for regular connections. These are various pyramids, mosaics, insert toys, sets of cubes and other geometric bodies, LEGO constructors.
Upon reaching the age of three, it is necessary to supplement cognitive activity child with games that stimulate the formation of mathematical abilities. In this case, several important points should be taken into account:
- Educational games should be short. Preschoolers with the right inclinations show curiosity towards such games, therefore, they should last as long as there is interest. Other children need to be skillfully lured to complete the task.
- Games of an analytical and logical nature should be carried out using visual material - pictures, toys, geometric shapes.
- It is easy to prepare stimulus material for the game yourself, focusing on the examples in this article.
Scientists substantiated that the use of geometric material is most effective in the development of mathematical abilities. The perception of figures is based on sensory abilities that are formed in the child earlier than others, allowing the baby to capture the connections and relationships between objects or their details.
Developing logical and mathematical games and exercises contribute to the formation of independent thinking of a preschooler, his ability to highlight the main thing in a significant amount of information. And these are the qualities that are necessary for successful learning.
Federal Agency for Education
Smolensky State University
Department of Methods of Teaching Mathematics, Physics and Informatics
DEVELOPMENT OF MATHEMATICAL ABILITIES OF STUDENTS IN THE BASIC SCHOOL
5th year students
Faculty of Physics and Mathematics
full-time department
MAKSIMOVICH Ulyana Anatolyevna
Scientific adviser:
Candidate of Pedagogical Sciences
Professor of the Department of Methodology
teaching mathematics and physics
SENKINA Gulzhan Yerzhanovna
Smolensk
Introduction ………………………………………………………………..…. ..….3
Chapter I Theoretical basis problems of mathematical abilities ...... 6
Section 1 general characteristics abilities.
1.1.1. The concept of ability…..………………………………………….………6
1.1.2. General and special abilities…...……………………………..…...8
1.1.3. Abilities and inclinations……………….…………….…………………….10
Section 2. Mathematical ability.
1.2.1. The study of mathematical abilities in foreign psychology……………………………………………………………………….....13
1.2.2. Study of the problem of mathematical abilities in domestic psychology……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
1.2.3. Classification of mathematical abilities……………………….22
Chapter II. Methodology for the development of mathematical abilities………………....24
Section 1. General methodology.
2.1.1. General provisions of the theory of development of abilities……..………...…….24
2.1.2. Principles of work on the development of mathematical abilities of students………………………………………………………………………….....28
2.1.3. Development of mathematical talent…………………………………………………………………………………………………………………………………………………………………….
Section 2. Private technique.
2.2.1. The development of mathematical abilities in mathematics lessons………37
2.2.2. The development of mathematical abilities in extracurricular activities ...... 44
Chapter III. Development of a database for the development of mathematical abilities……………………………………………………………………..54
3.1. Organization of data in the database………………………………………..54
3.2. Description of work in the database…………………………………………...56
Conclusion……………...………………………...……………………………..62
Literature………………...…………………………………………………...…....64
Applications……………………………………………………………………………67
Introduction.
Recently, in many countries there has been a significant increase in interest in the problems of mathematical education. This is due to the fact that the importance of mathematics in the life of human society is increasing every day. A high level of development of mathematics is a necessary condition for the rise and effectiveness of a number of important areas of knowledge. As scientists emphasize, the development of sciences has recently been characterized by a tendency towards their mathematization, and this applies not only to physics, astronomy or chemistry, but also to such sciences as modern biology, medicine, meteorology, economics, linguistics and others. Mathematical methods and the mathematical style of thinking permeate everywhere. It is difficult to find a field of knowledge to which mathematics would have nothing to do. Every year mathematics will find more and more widespread application in various fields of human activity. Fundamentally, the scope of mathematics is unlimited, points out Academician A.N. Kolmogorov.
In this regard, the need for mathematicians in our country increases every year. Recently, this need has clearly not been satisfied, "mathematicians have become scarce."
It is well known that the main contribution to the development of a particular science is made by people who demonstrate abilities in the relevant field. All this puts before the school the task of developing in every possible way the mathematical abilities, inclinations and interests of students, the task of raising the level of mathematical culture, the level of mathematical development of schoolchildren. Along with this, the school should pay special attention to students who show a high level of ability in mathematics, to promote the mathematical development of students who show a special inclination to study mathematics.
Some believe that instead of selecting students capable of mathematics, it is necessary to look for opportunities for the maximum mathematical development of all students. But one will always complement the other, since even with the most perfect methods of teaching, individual differences in mathematical abilities will always take place - some will then be more capable, others less capable. An equation in this respect will never be reached.
Consequently, mathematics teachers should carry out systematic work to develop the mathematical abilities of all schoolchildren, to nurture their interests and aptitudes for mathematics, and along with this, they should pay special attention to schoolchildren who show increased abilities in mathematics, organize special work with them, aimed at further development of these abilities.
Despite the need of society for people who can contribute to the development of mathematical science, and the task assigned to the school to develop mathematical abilities, in modern school the following situation occurs:
Reducing the hours of teaching mathematics;
formalism of mathematical knowledge;
lack of motivation for learning;
inability to apply the acquired knowledge in practice;
lack of independent and creative activity of students;
· the absence in the vast majority of textbooks and didactic manuals of tasks that help prepare students for this creative activity.
How to help the teacher in organizing educational activities to develop mathematical abilities?
The object of study of my work is the process of developing abilities at school.
The subject of research in my work is the process of developing mathematical abilities in elementary school.
The purpose of my research is a theoretical and methodological analysis of the problem of developing the mathematical abilities of schoolchildren, and on its basis, the development and description of a software tool that allows the best way for the teacher to process data on the development of mathematical abilities.
Hypothesis: software tools contribute to the development of mathematical abilities if
They offer a system of methodological developments for the development of mathematical abilities,
Take into account the age of students, types of mathematical abilities and types of classes for their development,
Focused on reducing the time spent by the teacher in preparing for classes,
Ensure that the stored information is up to date.
To achieve the set goal and confirm the hypotheses put forward, it is necessary to perform the following tasks:
Give a literary-critical review on this issue;
Consider the possibilities, principles, features of working methods for the development of mathematical abilities;
Develop a database that would provide input, storage and automated search for information necessary for the development of mathematical abilities;
Carry out the initial filling of the database methodological developments.
Theoretical foundations of the problem of mathematical abilities.
Section 1. General characteristics of students' abilities.
1.1.1. The concept of ability.
Naturally, in my work I will talk mainly about mathematical abilities, but in order to understand the complex problems of this theory, it is necessary to highlight some fundamental questions of the theory of abilities.
First of all, one should understand how psychology interprets the very concept of “ability” and its relationship with the process of forming an integral, comprehensively developed personality.
The concept of “ability” is used by the teacher in a variety of combinations: “capable student”, “gifted student”, “talented student”, “this student has natural abilities”, “he has great inclinations”, etc. In didactics and methodology teaching mathematics, we are talking about creative, research, cognitive abilities, about the ability to count or other types of mathematical activity.
All this variety of terminology makes us think about the essence of the concept.
The Russian Pedagogical Encyclopedia gives the following definition:
"Abilities are individual psychological characteristics of a person, which are the conditions for the successful implementation of a certain activity."
The problem of abilities has been widely studied and is being studied by psychologists in Russia.
One of the founders of this theory in our country was Rubinstein. He wrote: "Abilities are usually understood as the properties or qualities of a person that make him suitable for the successful implementation of any of the types of socially useful activities that have developed in the course of socio-historical development."
B.M. Teplov included three features in the concept of “abilities”: “Firstly, abilities are understood as individual psychological characteristics that distinguish one person from another ... Secondly, not all, in general, individual characteristics are called abilities, but only those that are related to the essence of performing any activity or many activities ... Thirdly, the concept of "ability" is not limited to those knowledge, skills or abilities that have already been developed by this person". The last remark is debatable, since the knowledge, skills and abilities that students have already developed also require certain abilities from them.
Very interesting is the conclusion of B.M. Teplova: "It's not that abilities are manifested in activity, but that they are created in this activity."
