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Теория и методика обучения (из областей знаний)
The structure of conception " mathematical abilities”
 
Author: Visitaevа Mаret Balaudinovna Chechen institute of professional development of educators
 
It is natural to teach pupils not to learn ready material but "open” mathematic truth (open for themselves what is already opened in the science, to organize logically, experimentally gained mathematic material (though it is already organized in the science) and eventually to apply the theory in different specific situations [4]. In accordance with the Federal educational standard of the second generation there is reorientation of methodical system of training to a priority of developing education and information function.
It is necessary to emphasize that every activity is "bipolar: aims to create objective values and the development of the individual" [3, pp. 91]. We believe that every activity is bipolar: aims to create objective values, including their perception of the dynamic, and the development of the individual.
O.V. Guscin (2004) pointed out that the dynamic perception is an attempt not to see just the object, but the perception itself in its internal dynamics, which constantly updates perceived images and objects in observer’s sight.
 "Mathematical activity" is an activity aimed to master mathematics, able to increase knowledge, perceived or created by the subject.
In psychology, math skills mean individual-psychological characteristics that contribute to the success of  mathematical activity.
O.V. Guscin (2004) pointed out that the dynamic perception is an attempt not to see just the object, but the perception itself in its internal dynamics, which constantly updates perceived images and objects in observer’s sight.
 "Mathematical activity" is an activity aimed to master mathematics, able to increase knowledge, perceived or created by the subject.
In psychology, math skills mean individual-psychological characteristics that contribute to the success of  mathematical activity.
As V.A. Krutetskiy, N.V. Metelskiy noted, there is still no single definition of mathematical abilities, which would satisfy all researchers, but they all  define them as normal "school" ability to assimilate mathematical knowledge, their self-dependent use, and creative math skills related to self-creation of the original and  valued product.
E.J. Gingulis after A.N. Kolmogorov considered the three components of mathematical ability: algorithmic, geometrical and logical.
The most comprehensive study of the mathematical structure had V. Haecker and Th. Ziehen [10], in their view, the structure of a "mathematical thinking" consists of four components: spatial, logical, numerical and symbolic.
Research on mathematical abilities are important for understanding the structure of a "mathematical thinking" (J. Hadamard, A. Bine, A. Blackwell, E.R. Dunkan, K. Duncker, H.R. Hemley, V.A. Krutetskiy, A.N. Kolmogorov, A.I. Markushevich, F.W. Mitchell, D.D. Mordukhai-Boltovskoi, J. Piaget, A. Poincare, I.S. Yakimanskaya etc.).
An analysis of research on mathematical abilities mentioned above and of many other authors allows doing about the complex nature of mathematical activity; different authors refer to as the most important variety of structural and functional elements of the latter. The essential elements are such common mental operations as (comparison, deduction, analysis and synthesis). As a specific mathematical skills the following abilities are most often reported: 1) the ability to spatial concepts; 2) the geometric vision; 3) the manipulation of ideas and concepts in the abstract, without reference to a specific; 4) classification; 5) an understanding of symbols and manipulation of them; 6) intellectual curiosity; 7) a strong visual imagery; 8) the ability to apply knowledge to new situations; 9) the ability to retrieve similar in remote areas.
A structural approach to understanding the nature of mathematical ability was earlier scheduled by K. Duncker [5]. He marked characteristics that distinguish individuals who are able and unable to mathematics.
According to his opinion the structure of mathematical abilities are as follows:
1. Getting the mathematical information
2. Processing of mathematical information
3. Storage of mathematical information
4. General synthetic components.
                The following  components are not required in the structure of mathematical abilities: speed of thought processes as the temporal characteristics of the individual rate is not critical; computational power (the ability to fast and accurate calculations, often in the mind), memory for numbers, formulas, and the ability to space ideas, and the ability to visualize abstract mathematical relationships and dependencies [7].
Thus, according to V.A. Krutetskiy, mathematical ability "is a complex structural mental formation, a synthesis of the properties, integral quality of the mind, encompassing a variety of it and developed in the course of mathematical activity" [7, pp. 93]
This scheme of the structure of mathematical ability has a number of disadvantages, to which the researchers of psychological and pedagogical foundations of mathematics teaching point. In particular, the N.V Metelskiy notes retreat the  above-mentioned principle of construction of the scheme on the basis of stages of the task: "... the structure of these skills is inadequate overall structure of the making task process" [9, pp. 37]. What is more, there are some missing components, which are, according to psychologists and trainers, an integral part of the mathematical abilities of students, in particular, the capacity for abstract thinking, mathematical intuition. Mathematical intuition - the ability to find quickly a way to an optimal solution.
French mathematician A. Poincare 'classified mathematical ability of people in power of mathematical intuition "[2, pp. 8]. J. Hadamard [1] also spoke of a kind peculiar to mathematicians mathematical intuition, about the subconscious creative work, about the specifics of mathematician thinking.
Of particular importance is the work of mathematicians, which treat various aspects of the mathematical abilities of A.I. Markushevich, V.A. Testov, S.I. Shvartsburd and others: 1) the dominance of logic reasoning; 2) conciseness; 3) a clear division of the progress of reasoning; 4) the accuracy of the symbolism; 5) the ability of students to isolate the root of the matter, apart from the non-essential, the ability to think abstractly; 6) the ability to deduce the logical consequences of these assumptions, the possession of skills of deductive reasoning; 7) the accuracy, conciseness and clarity of verbal expression of thought, possession sufficiently developed mathematical speech, etc.
The problem of mathematical skills developed in a number of dissertation research, some of which addressed issues related to the analysis and development of common and some particular aspects of the development and diagnosis of mathematical ability.
As it is known, the mathematical ability of students appear in the speed, depth and strength of learning in Mathematics. Each of the parameters (speed, depth, strength) is not strictly required indicator. The depth of assimilation of knowledge characterizes the number of conscious connections of this knowledge with others which correlate with them. The strength of the assimilation of knowledge means the duration of storing it in memory, reproducibility when necessary (M.N. Skatkin, 1978).
"Mathematical ability is the ability to mathematical creativity, ie ability to independently obtain the solution of mathematical problems that go beyond the application of known algorithms and theorems "[8, pp. 18].
V.V. Kertanova in his study determines mathematical ability as "individual psychological personality traits that determine the success of learning and productivity performance of the individual cognitive actions required to solve mathematical problems" [6, pp. 12].
The ways to a deeper understanding of the structure of a "mathematical thinking" and mathematical skills necessary to search in the direction of penetration into the complex dialectical nature of the relevant productive processes and mental structures as a unity ("alloy", a special combination) of general and special (special) (G. Glaser, 1984).
Starting from the model proposed by psychologists, methodists and making adjustments to it, we consider mathematical ability as individual psychological characteristics that contribute to success of the implementation of activities aimed at the mastery of mathematics, able to increase their knowledge, perceived or created dynamically by the subject.
Dedicated mathematical ability (1) the ability to spatial concepts, and; 2) the manipulation of ideas and concepts in the abstract, without reference to a specific, and; 3) the mathematical intuition; 4) flexibility of thought processes in mathematical activity, and; 5) the geometric vision; 6) classification; 7) intellectual curiosity; 8) a strong visual imagery; 9) the ability to apply knowledge to new situations; 10) conciseness; 11) the accuracy of symbols; 12) the ability to think abstractly; 13) the possession of skills of deductive reasoning; 14) possession of a well-developed mathematical and speech etc.) and consider their future dynamics are responsible for finding ways and methods of forming mathematical aptitudes.
 
References:
1. Hadamard J. Psychology research process inventions in the field of mathematics: Trans. c France. – Moscow: Soviet Radio, 1970.
2. Poincare A. About science. – Moscow: Nauka, 1983.
3. Serikov V.V. Formation of students ready to work (Teaching science - reform school). – M.: Education, 1988.
4. Joiner A. Pedagogy of mathematics. – Minsk.: Higher School, 1986.
5. Dunker K. Psychology productive (creative) thinking / cognitive psychology / ed. A.M. Matyushkina. – Moscow: Progress Publishers, 1965. – S. 86-234.
6. Kertanova V.V. The development of mathematical skills of students in the context of their future careers: avtoref. dis. ... Candidate of ped. science. – M., 2007.
7. Krutetskiy V.A. Psychology of mathematical abilities of students. – M.: Education, 1968.
8. Kulikova O.S. Geometric problems on construction as a means to develop mathematical skills of students: Thesis. ... Candidate of ped. science. – M., 1998.
9. Metelskiy N.V. Psycho-pedagogical foundations of mathematics. – Minsk.: Higher School, 1977.
10. Haecker V., Ziehen Th. The contribution to the theory of heredity and analysis of drawing and mathematical talent, especially with regard to the correlation of musical talent // Journal for pedagogy, 1931. – № 121.

Категория: Педагогические науки | Добавил: Иван155 (17.05.2013)
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