6.11. Short review research of mathematical abilities
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 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 individual psychological characteristics (primarily the characteristics mental activity) that meet the requirements of educational mathematical activity and determine, with other equal conditions success in creative mastery of mathematics as subject, in particular, relatively fast, easy and deep mastery of knowledge, skills and abilities in the field of 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, in contrast to less capable students, whose information is usually stored in memory in a narrowly specific form, scattered and undifferentiated, 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 special place occupy two monographic works - "Psychology musical ability” and “The Mind of a Commander”, which have become classic examples of the psychological study of abilities and have incorporated universal principles for approaching this problem, which can and should be used when studying any kind of ability.
In both works, B.M. Teplov not only gives a brilliant psychological analysis specific types of activity, but also on the examples of outstanding representatives of musical and military art reveals the necessary components that make up bright talents in these areas. Special attention B.M. Teplov paid 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 from common features attention, memory, 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 most prominent role general abilities demonstrated in the work "The Mind of the 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 the obligatory high level their development.
important place in intellectual activity commander takes memory. 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 classic example such a memory of 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 possessed important feature instantly assimilate 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 constant systematization of the material is essential conditions that ensure the unity of analysis and synthesis, then the balance between these aspects of mental activity that distinguish the work of the mind a good general» . 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, specific gravity in which, according to the scientist, he constituted 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 important place in mathematical ability 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 mathematical symbols with the same ease as when solving logic 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 don't own it 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 various options 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 mathematical abilities stand out 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 includes the solution of one of critical issues- 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 specific type abilities. It can only be a favorable condition for this development. The typological properties that make up the inclinations and are an important part of them reflect such individual features of the functioning of the body 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.
Properties nervous system, closely related to the properties of temperament, in turn, affect the manifestation of the characterological characteristics of the personality (V.S. Merlin, 1986). B.G. Ananiev, developing ideas about the general natural basis development of character and abilities, pointed to the formation in the process of activity of connections of abilities and character, 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 (EA 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-logical functions is a necessary condition for mathematical abilities. 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, which was 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 the owners of 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. Additional electives are held with them, 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 guesses, 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 intense thought, deep analysis, and a 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 solve 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 question they are discussing 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 they study geometry, while others are geometers, even when they study geometers. 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 the students there are those who have a certain balance of these characteristics. They equally smoothly master all sections of mathematics, using different principles 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. In competitive tasks, the share of tasks in geometry is low - out of 4 - 5 tasks in best case 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 comparison 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 specifically 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 medium type, with the equilibrium 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 indicators 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 carried out with younger students 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 logical thinking, can be considered central among the components that ensure the successful mastery of reading drawings.
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: training in projection drawing can be started at primary 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 a plane 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.
Views of foreign psychologists on mathematical abilities
Such outstanding representatives of certain trends in psychology as A. Binet, E. Trondike and G. Reves, and such outstanding mathematicians as A. Poincaré and J. Hadamard contributed to the study of mathematical abilities.
A wide variety of directions determined and big variety in the approach to the study of mathematical abilities, in methodological tools and theoretical generalizations.
The only thing that all researchers agree on is, perhaps, 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 independent creation original and social value product.
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 skills, then with regard to the second there is complete unity - the creative abilities of a mathematician are an innate formation, a favorable environment is necessary only for their manifestation and development. 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.
The main issue in the study of mathematical abilities (both educational and creative) abroad has been and remains the question of the essence of this complex psychological education. Three important issues can be identified in this regard.
1. The problem of the specificity of mathematical abilities. Are there any proper mathematical abilities as specific education, different from the category of general intelligence? Or is mathematical ability a qualitative specialization of general mental processes and personality traits, that is, general intellectual ability developed in relation to mathematical activity? In other words, is it possible to argue that mathematical talent is nothing more than general intelligence plus an interest in mathematics and a penchant for doing it?
2. The problem of the structure of mathematical abilities. Is mathematical giftedness a unitary (single indecomposable) or an integral (complex) property? In the latter case, one can raise the question of the structure of mathematical abilities, of the components of this complex mental formation.
3. The problem of typological differences in mathematical abilities. Are there different types mathematical talent or with the same basis there are differences only in interests and inclinations to certain branches of mathematics?
The views of B.M. Teplov on mathematical abilities
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 is very selective, that is, it retains, 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, starting from unit numbers and ending with the faces of soldiers. 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 that ensure the unity of analysis and synthesis, then the balance between these sides mental activity that distinguish the work of the mind of a good commander ”(B.M. Teplov 1985, p. 249). Along with an outstanding mind, the commander must have certain personal qualities. First of all, this is courage, determination, energy, that is, what, in relation to military leadership, is usually denoted by the concept of “will”. No less important personal quality is stress tolerance. 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, is 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 proportion 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. Poincare came to the conclusion that the most important place in mathematical abilities is the ability to logically build a chain of operations that will lead to the solution of a 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 good memory and attention. According to Poincare, people capable of mathematics are distinguished by the ability to grasp the order in which the elements necessary for mathematical proof. 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, and therefore are not able to understand mathematics. Others have little intuition, but are gifted with a good memory and a capacity for intense attention, and therefore can understand and apply mathematics. Still others have such a special intuition and, even in the absence of an excellent memory, 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 the student, problem solving 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 abilities are the specific ability to generalize mathematical material, the ability to spatial representations ability to abstract thinking. Some researchers also distinguish mathematical memory for reasoning and proof schemes, problem solving methods and principles of approach to them as an independent component of mathematical abilities. Soviet psychologist who studied the mathematical abilities of schoolchildren, V.A. Krutetsky gives the following definition of mathematical abilities: “Under the ability to study mathematics, we mean individual psychological characteristics (primarily the characteristics of mental activity) that meet the requirements of educational mathematical activity and determine, on other equal conditions, the success of creative mastery of mathematics as an educational subject, in particular, relatively fast , easy and deep mastery of knowledge, skills and abilities in the field of mathematics.
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. The inclinations include the innate anatomical and physiological characteristics of the individual, which are considered as favorable conditions for the development of abilities. 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 favorable condition for this development. Typological properties, which are part of the inclinations and are an important part of them, reflect such individual features of the functioning of the body 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.
The general scheme of the structure of mathematical abilities at school age according to V.A. Krutetsky
The material collected by V. A. Krutetsky allowed him to build general scheme structures of mathematical abilities at school age.
1. Obtaining mathematical information.
The ability to formalize the perception of mathematical material, grasping the formal structure of the problem.
2. Processing of mathematical information.
1) The ability for logical thinking in the field of quantitative and spatial relations, numerical and sign symbolism. The ability to think in mathematical symbols.
2) The ability to quickly and broadly generalize mathematical objects, relationships and actions.
3) The ability to curtail 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) Striving for clarity, simplicity, economy and rationality of decisions.
6) The ability to quickly and freely change direction thought process, switching from direct to reverse course of thought (reversibility of the thought process in mathematical reasoning).
3. Storage of mathematical information.
1) math memory(generalized memory for mathematical relations, typical characteristics, schemes of reasoning and evidence, methods for solving problems and principles of approach to them).
4. General synthetic component.
1) Mathematical orientation of the mind. The selected components are closely connected, influence each other and form in their totality a single system, an integral structure, a kind of syndrome of mathematical talent, a mathematical mindset.
Not included in the structure of mathematical talent are those components whose presence in this system is not necessary (although useful). In this sense, they are neutral in relation to mathematical giftedness. However, their presence or absence in the structure (more precisely, the degree of their development) determines the type mathematical warehouse mind. The following components are not mandatory in the structure of mathematical talent:
1. The speed of thought processes as a temporal characteristic.
2. Computational abilities (the ability to quickly and accurately calculate, often in the mind).
3. Memory for numbers, numbers, formulas.
4. Ability for spatial representations.
5. The ability to visualize abstract mathematical relationships and dependencies.
Surely you have met people who seemed to be born with slide rule in hand. To what extent are math abilities predetermined by nature?
We all have an innate mathematical sense - it is this that allows us to roughly estimate and compare the number of objects without resorting to exact counting. It is with this feeling that we automatically choose the shortest line at the supermarket checkout without counting the number of people.
But some people have a better mathematical sense than others. Several studies published in 2013 suggest that this innate ability, which is the foundation for further successful study mathematical science can be greatly improved through practice and training.
The researchers found structural features in the brains of children who were most successful at math problems. Ultimately, these new discoveries could help find the most effective ways to teach math, says psychologist Elizabeth Brannon of Duke University.
How was the research done?
Is it possible to develop a mathematical sense?
But inborn abilities do not impose restrictions on us at all. Brannon and her colleague Junku Park recruited 52 adult volunteers to participate in a small experiment. During the experiment, the participants had to solve several arithmetic problems with double digits. Half of the group then went through 10 training sessions in which they mentally estimated the number of dots on the cards. Control group such a series of tests has not been carried out. After that, both groups were asked to solve arithmetic examples again. It was found that the results of the participants who underwent training sessions were significantly superior to those of the control group.
These two small studies show that innate mathematical feeling and acquired mathematical skills are inextricably linked; work on one quality will inevitably lead to the improvement of another. Children's games aimed at training math skills really play big role in subsequent teaching of mathematics.
Another published study helps explain why some children learn better than others. Scientists from Stanford University taught 24 third graders for 8 weeks in a special curriculum With mathematical bias. The level of improvement in mathematical skills of this group of children ranged from 8% to 198% and did not depend on the results of tests for intellectual development, the level of memory and cognitive abilities.
Calculators can be surprisingly useful, but they're not always readily available. In addition, not everyone is comfortable getting out calculators or phones to calculate how much you need to pay in a restaurant, or calculate the size of a tip. Here are ten tips that can help you do all those mental calculations. In fact, it is not at all difficult, especially if you remember a few simple rules.
Add and subtract from left to right
Remember how in school we were taught to add and subtract in a column from right to left? This addition and subtraction is convenient when a pencil and a piece of paper are at hand, but in the mind these mathematical operations easier to do by counting from left to right. In the number on the left there is a figure that defines large values, for example, hundreds and tens, and on the right, smaller ones, that is, units. From left to right, counting is more intuitive. Thus, when adding 58 and 26, start with the first digits, first 50 + 20 = 70, then 8 + 6 = 14, then add both results and get 84. Easy and simple.
Make it easy for yourself
If you are faced with a complex example or task, try to find a way to simplify it, such as adding or subtracting certain number to do general calculation easier. If, for example, you need to calculate how much 593 + 680 will be, first add 7 to 593 to get a more convenient number 600. Calculate how much 600 + 680 will be, and then subtract the same 7 from the result 1280 to get the correct answer - 1273.
You can do the same with multiplication. To multiply 89 x 6, calculate how much 90 x 6 will be, and then subtract the remaining 1 x 6. So 540 - 6 = 534.
Remember building blocks
Memorizing multiplication tables is an important and necessary part of mathematics, which is great for solving examples in your head.
Memorizing the basic "building blocks" of mathematics, such as the multiplication table, square roots, percentages decimal and ordinary fractions, we can immediately get answers to simple tasks hidden in the more difficult.
Remember useful tricks
To get through multiplication faster, it's important to remember a few simple tricks. One of the most obvious rules is multiplying by 10, that is, simply adding zero to the number being multiplied, or moving the comma one decimal point. When multiplied by 5, the answer will always end with a 0 or 5.
Also, when multiplying a number by 12, first multiply it by 10 and then by 2, then add the results. For example, to calculate 12 x 4, first multiply 4 x 10 = 40, then 4 x 2 = 8, and add 40 + 8 = 48. When multiplying by 15, simply multiply the number by 10, and then add another half of the result, for example, 4 x 15 = 4 x 10 = 40 plus half (20) makes 60.
There is also a trick to multiplying by 16. First, multiply the number in question by 10 and then multiply half the number by 10. Then add both results to the number to get the final answer. So, to calculate 16 x 24, first calculate 10 x 24 = 240, then half of 24, i.e. 12, multiply by 10 and get 120. And the last step: 240 + 120 + 24 = 384.
Squares and their roots are very useful
Almost like a multiplication table. And they can help with the multiplication of larger numbers. A square is obtained by multiplying a number by itself. Here's how multiplication works using squares.
Let's assume for a moment that we don't know the answer to 10 x 4. First, figure out the average between these two numbers, which is 7 (i.e. 10 - 3 = 7, and 4 + 3 = 7, with the difference between the average the number is 3 - this is important).
We then determine the square of 7, which is 49. We now have a number that is close to the final answer, but it's not close enough. To get the correct answer, go back to the difference between the average (in this case 3), square it gives us 9. The last step involves a simple subtraction, 49 - 9 = 40, now you have the correct answer.
It's like devious and over the top hard way calculate how much 10 x 4 will be, but the same technique works great for large numbers. Let's take 15 x 11 for example. First we have to find the middle number between these two (15 - 2 = 13, 11 + 2 = 13). The square of 13 is 169. The square of the difference of the average 2 is 4. We get 169 - 4 = 165, that's the correct answer.
Sometimes an approximate answer is enough
If you're trying to decide challenging tasks in your mind, it's no wonder it takes a lot of time and effort. If you don't need an absolutely exact answer, it may be enough to calculate an approximate number.
The same applies to tasks in which you do not know all the exact data. For example, during the Manhattan Project, physicist Enrico Fermi wanted to roughly calculate the force of an atomic explosion before scientists had accurate data. To this end, he threw scraps of paper on the floor and watched them from a safe distance, at the moment when he reached the pieces of paper. blast wave. After measuring the distance over which the fragments moved, he suggested that the force of the explosion was approximately 10 kilotons of TNT. This estimate turned out to be quite accurate for offhand guessing.
Fortunately, we do not have to regularly evaluate the approximate strength atomic explosions, but it doesn't hurt to make a rough estimate if, for example, you need to guess how many piano tuners there are in the city. To do this, it is easiest to operate with numbers that are easy to divide and multiply. So you first estimate the population of your city (say, one hundred thousand people), then you estimate the estimated number of pianos (say, ten thousand), and then the number of piano tuners (say, 100). You won't get an exact answer, but you can quickly guess an estimate.
Rearrange the examples
The basic rules of mathematics help to rebuild complex examples into simpler ones. For example, mentally calculating an example of 5 x (14 + 43) seems like a daunting and even overwhelming task, but the example can be “broken down” into three fairly simple calculations. For example, this overwhelming problem can be rearranged as follows: (5 x 14) + (5 x 40) + (5 x 3) = 285. Not that hard, right?
Simplify your tasks
If a task seems difficult, simplify it. It's always easier to deal with multiple simple tasks than with one complex. Solution of many difficult examples in the mind lies in the ability to correctly divide them into more simple examples, the solution of which is not difficult.
For example, multiplying by 8 is easiest by doubling the number three times. So instead of trying to figure out how much 12 x 8 would be the traditional way, just double 12 three times: 12 x 2 = 24, 24 x 2 = 48, 48 x 2 = 96.
Or when multiplying by 5, first multiply by 10 since that's easy, then divide the result by 2, since that's pretty easy too. For example, to solve 5 x 18, calculate 10 x 18 and divide by 2, where 180:2 = 90.
Use exponentiation
When calculating large amounts in your head, remember that you can convert them to smaller numbers multiplied by 10 to the desired power. For example, how much will it be if 44 billion is divided by 400 thousand? An easy way to solve this problem is to convert 44 billion into the next number - 44 x 10 9 , and from 400 thousand to make 4 x 10 5 . Now we can transform the problem like this: 44: 4 and 10 9: 10 5 . According to the mathematical rules, it all looks like this: 44: 4 x 10(9-5), so we get 11 x 10 4 = 110,000.
The easiest way to calculate required tips
Mathematics is necessary even during dinner in a restaurant, or rather after it. Depending on the institution, the tip can range from 10% to 20% of the bill value. For example, in the USA it is customary to tip waiters 15%. And there, as in many European countries, tips are required.
If we calculate 10% of total amount relatively easy (just divide by 10), 15% and 20% seem to be more difficult. But in fact, everything is just as simple and very logical.
When calculating a 10 percent tip for a dinner that cost $112.23, just move the decimal point to the left one digit, you get $11.22. When calculating the 20% tip, do the same and just double the amount (20% is just twice the 10%), in which case the tip is $22.44.
For a 15% tip, first determine 10% of the amount and then add half of the amount received (an additional 5% is half of the 10% amount). Don't worry if you can't get an exact answer down to the last cent. If we don't bother too much with decimals, we can quickly figure out that a 15 percent tip of $112.23 is $11 + $5.50, which gives us $16.50. Pretty accurate. If you don't want to offend the waiter by missing a few cents, round up the amount to the nearest whole number and pay $17.
Mathematical abilities provide direct influence on the mental development of the preschooler. The child is much more have 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 understand their characteristics and relationships.
What abilities in preschool age are related to mathematical
Many parents think that it is too early to develop the mathematical abilities of children at preschool age. And by this concept they mean some special abilities, allowing children to operate with large 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.
It relies 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 reverse order(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 simple qualitative ratios: largest, smallest, lowest, etc.
- Understand complicated relationship: “larger than the smallest, but smaller than others”, “ahead and above others”, etc.
- Be able to identify an extra object that is not suitable for a group of others.
- line up 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 the object with numerical sign(On the example of the previous task: cubes with points 3, 5 and 8 are placed. Where to put the 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 basic skills there are completely simple ones that are available already in 3-4 years, but there are also those (9-12 points) that use 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. Value lies not only in the accumulation mathematical representations and skills, but also in the general mental development of a preschooler.
AT practical psychology There are three categories of gaming activities aimed at developing individual components of mathematical abilities.
- Exercises to determine the properties of objects, identify objects according to a designated feature (analytical and synthetic abilities).
- Games for comparing various properties, identifying essential features, abstraction from secondary, 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 single-color strips of cardboard of the same width and various lengths and arrange them randomly in front of a 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 big 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 be started with simple groups objects and gradually complicate. 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. It is necessary to prepare at least 20 strips of equal length and width or counting sticks in two colors. For example: of blue color- S, 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 should surround the child with educational toys that require elemental analysis and search for regular connections. These are various pyramids, mosaics, insert toys, sets of cubes and others. 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 about 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.
