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What is than mathematical concepts. Open Library - open library of educational information

Lecture #2

mathematics

Topic: "Mathematical concepts"

    Mathematical concepts

    Definition of concepts

    Requirements for the definition of concepts

    Some kinds of definitions

1. Mathematical concepts

The concepts that are studied in the initial course of mathematics are usually presented in the form of four groups. The first includes concepts related to numbers and operations on them: number, addition, term, more, etc. The second includes algebraic concepts: expression, equality, equation, etc. The third includes geometric concepts: straight line, segment, triangle, etc. d. The fourth group is formed by concepts related to quantities and their measurement.

How to study such an abundance of various concepts?

First of all, one must have an idea of ​​the concept as a logical category and the features of mathematical concepts.

In logic, concepts are considered as a form of thought that reflects objects (objects or phenomena) in their essential and general properties. The linguistic form of a concept is a word or a group of words.

To compose a concept about an object means to be able to distinguish it from other objects similar to it. Mathematical concepts have a number of features. The main one is that the mathematical objects about which it is necessary to form a concept do not exist in reality. Mathematical objects are created by the human mind. These are ideal objects that reflect real objects or phenomena. For example, in geometry, the shape and size of objects are studied, without taking into account their other properties: color, mass, hardness, etc. From all this they are distracted, abstracted. Therefore, in geometry, instead of the word "object" they say "geometric figure".

The result of abstraction are also such mathematical concepts as "number" and "value".

In general, mathematical objects exist only in human thinking and in those signs and symbols that form the mathematical language.

It can be added to what has been said that, in studying the spatial forms and quantitative relations of the material world, mathematics not only uses various methods of abstraction, but the abstraction itself acts as a multi-stage process. In mathematics, one considers not only concepts that have appeared in the study of real objects, but also concepts that have arisen on the basis of the former. For example, the general concept of a function as a correspondence is a generalization of the concepts of specific functions, i.e. abstraction from abstractions.

In order to master general approaches to the study of concepts in the initial course of mathematics, the teacher needs knowledge about the scope and content of the concept, about the relationship between concepts and about the types of definitions of concepts.

2. The scope and content of the concept. Relationships between concepts

Every mathematical object has certain properties. For example, a square has four sides, four right angles equal to the diagonal. You can specify other properties as well.

Among the properties of an object, essential and non-essential are distinguished. A property is considered essential for an object if it is inherent in this object and without it it cannot exist. For example, for a square, all the properties mentioned above are essential. The property "side AD is horizontal" is not essential for the square ABCD. If the square is rotated, then the side AD will be located differently (Fig. 26).

Therefore, in order to understand what a given mathematical object is, one must know its essential properties.

When talking about a mathematical concept, they usually mean a set of objects denoted by one term (a word or a group of words). So, speaking of a square, they mean all geometric shapes that are squares. It is believed that the set of all squares is the scope of the concept of "square".

Generally the scope of a concept is the set of all objects denoted by a single term.

Any concept has not only scope, but also content.

Consider, for example, the concept of a "rectangle".

The scope of the concept is a set of different rectangles, and its content includes such properties of rectangles as “have four right angles”, “have equal opposite sides”, “have equal diagonals”, etc.

There is a relationship between the volume of a concept and its content: if the volume of a concept increases, then its content decreases, and vice versa. So, for example, the scope of the concept of "square" is part of the scope of the concept of "rectangle", and the content of the concept of "square" contains more properties than the content of the concept of "rectangle" ("all sides are equal", "the diagonals are mutually perpendicular", etc.). ).

Any concept cannot be assimilated without realizing its relationship with other concepts. Therefore, it is important to know in what relationships concepts can be, and to be able to establish these connections.

The relationships between concepts are closely related to the relationships between their volumes, i.e. sets.

Let us agree to designate concepts by lowercase letters of the Latin alphabet: a, b, c, ..., z.

Let two concepts a and b be given. Let us denote their volumes as A and B, respectively.

If A B (A ≠ B), then they say that the concept a - specific in relation to the conceptb, and the concept b- generic in relation to the concept a.

For example, if a is a “rectangle”, b is a “quadrilateral”, then their volumes A and B are in relation to inclusion (A B and A ≠ B), since every rectangle is a quadrilateral. Therefore, it can be argued that the concept of "rectangle" is specific in relation to the concept of "quadrilateral", and the concept of "quadrilateral" is generic in relation to the concept of "rectangle".

If A = B, then we say that concepts a andbare identical.

For example, the concepts of "equilateral triangle" and "equiangular triangle" are identical, since their volumes are the same.

If the sets A and B are not connected by an inclusion relation, then they say that the concepts a and b are not in relation to the genus and species and are not identical. For example, the concepts of "triangle" and "rectangle" are not connected by such relations.

Let us consider in more detail the relation of genus and species between concepts. First, the concepts of genus and species are relative: the same concept can be generic in relation to one concept and species in relation to another. For example, the concept of "rectangle" is generic in relation to the concept of "square" and specific in relation to the concept of "quadrilateral".

Secondly, several generic concepts can often be specified for a given concept. So, for the concept of "rectangle" the concepts of "quadrilateral", "parallelogram", "polygon" are generic. Among them, you can specify the nearest. For the concept of "rectangle" the closest is the concept of "parallelogram".

Thirdly, the specific concept has all the properties of the generic concept. For example, a square, being a species concept in relation to the concept of a "rectangle", has all the properties inherent in a rectangle.

Since the scope of a concept is a set, it is convenient, when establishing relationships between the scopes of concepts, to depict them using Euler circles.

Let us establish, for example, the relationship between the following pairs of concepts a and b, if:

1) a - "rectangle", b - "rhombus";

2) a - "polygon", b - "parallelogram";

3) a - "straight line", b - "segment".

In case 1) the volumes of concepts intersect, but not one set is a subset of another (Fig. 27).

Therefore, it can be argued that these concepts a and b are not in relation to the genus and species.

In case 2), the volumes of these concepts are in relation to inclusion, but do not coincide - every parallelogram is a polygon, but not vice versa (Fig. 28). Therefore, it can be argued that the concept of "parallelogram" is specific in relation to the concept of "polygon", and the concept of "polygon" is generic in relation to the concept of "parallelogram".

In case 3), the volumes of concepts do not intersect, since no segment can be said to be a straight line, and no straight line can be called a segment (Fig. 29).

Therefore, these concepts are not related to genus and species.

About the concepts of "straight line" and "segment" it can be said that they are in relation to the whole and the part: A segment is a part of a line, not a type of it. And if the specific concept has all the properties of the generic concept, then the part does not necessarily have all the properties of the whole. For example, a segment does not have such a straight line property as its infinity.

Testov Vladimir Afanasyevich,

Doctor of Pedagogical Sciences, Professor of the Department of Mathematics and Methods of Teaching Mathematics FSBEI HPE ©Vologda State University, Vologda [email protected]

Features of the formation of basic mathematical concepts in schoolchildren in modern conditions

Annotation. The article discusses the features of the formation of mathematical concepts in schoolchildren in the modern paradigm of education and in the light of the requirements put forward in the concept of the development of mathematical education. These requirements involve updating the content of teaching mathematics at school, bringing it closer to modern sections and practical application, and the widespread use of project activities. To overcome the existing disunity of various mathematical disciplines, the isolation of individual topics and sections, to ensure integrity and unity in teaching mathematics is possible only on the basis of highlighting the main cores in it. Mathematical structures are such rods. A necessary condition for the implementation of the principle of accessibility of learning is the gradual process of forming concepts about the basic mathematical structures. The method of projects can be of great help in the phased study of mathematical structures. The use of this method in the study of mathematical structures by schoolchildren allows us to solve a whole range of tasks for expanding and deepening knowledge in mathematics, considering the possibilities of their application in practical activities, acquiring practical skills in working with modern software products, and comprehensively developing the individual abilities of schoolchildren. Key words: the content of teaching mathematics , mathematical structures, stages of the process of concept formation, project method. Section: (01) Pedagogy; history of pedagogy and education; theory and methodology of training and education (by subject areas).

Currently, the transition to the information society is being completed, at the same time a new paradigm in education is being formed, based on post-non-classical methodology, synergistic principles of self-education, the introduction of network technologies, project activities, and a competency-based approach. All these new trends require updating the content of teaching mathematics at school, bringing it closer to modern sections and practical applications. The peculiarities of educational material in the information society are the fundamental redundancy of information, the non-linear nature of its deployment, the possibility of variability of educational material. The role of mathematical education as the basis of competitiveness, an essential element of the country's security, was recognized by the Russian leadership. In December 2013, the Government approved the concept of development of mathematical education. This concept raises many topical problems of mathematical education. The main problem is the low educational motivation of schoolchildren, which is associated with the underestimation of mathematical education that exists in the public mind, as well as the overload of programs, assessment and methodological materials with technical elements and outdated content. The current state of mathematical training of students causes serious concern. There is a formalism of mathematical knowledge of secondary school graduates, their lack of effectiveness; insufficient level of mathematical culture and mathematical thinking. In many cases, the specific material being studied does not add up to a system of knowledge; the student finds himself “buried” under the mass of information falling on him from the Internet and other sources of information, being unable to structure and comprehend it on his own.

As a result, a significant part of this information is quickly forgotten, and the mathematical baggage of a significant part of secondary school graduates consists of a larger or smaller number of dogmatically assimilated information that is loosely interconnected and better or worse fixed skills for performing certain standard operations and typical tasks. They lack the idea of ​​mathematics as a single science with its own subject and method. Excessive interest in the purely informational side of education leads to the fact that many students do not perceive the rich content of mathematical knowledge embedded in the program. widespread use of mathematical models in modern society. Thus, the task is set to bring the content of teaching mathematics closer to modern science. To overcome the disunity of various mathematical disciplines, the isolation of individual topics and sections, to ensure integrity and unity in the teaching of mathematics is possible only on the basis of highlighting its sources, the main cores. Such rods in mathematics, as noted by A.N. Kolmogorov and other prominent scientists are mathematical structures, which, according to N. Bourbaki, are divided into algebraic, ordinal and topological. Some of the mathematical structures can be direct models of real phenomena, others are connected with real phenomena only through a long chain of concepts and logical structures. Mathematical structures of the second type are the product of the internal development of mathematics. From this view of the subject of mathematics, it follows that in any mathematical course, mathematical structures should be studied. The idea of ​​mathematical structures, which turned out to be very fruitful, served as one of the motives for a radical reform of mathematical education in the 6070s. Although this reform was later criticized, its main idea remains very useful for modern mathematics education. Recently, new important sections have arisen in mathematics that require their reflection both in the university and in the school curriculum in mathematics (graph theory, coding theory, fractal geometry, chaos theory, etc.). These new directions in mathematics have great methodological, developmental and applied potential. Of course, all these new branches of mathematics cannot be studied from the very beginning in all their depth and completeness. As shown in, the process of teaching mathematics should be considered as a multi-level system with a mandatory reliance on the underlying, more specific levels, stages of scientific knowledge. Without such a support, learning can become formal, giving knowledge without understanding. The stage-by-stage process of formation of basic mathematical concepts is a necessary condition for the implementation of the principle of accessibility of education.

Views on the need to single out successive stages in the formation of concepts of mathematical structures are widespread among mathematicians and educators. Even F. Klein, in his lectures for teachers, noted the need for preliminary stages in the study of basic mathematical concepts: © We must adapt to the natural inclinations of young men, slowly lead them to higher questions, and only in conclusion acquaint them with abstract ideas; teaching must follow the same path along which all mankind, starting from its naive primitive state, has reached the heights of modern knowledge. ... How slowly all mathematical ideas arose, how they almost always surfaced at first, rather as a guess, and only after a long development acquired a motionless crystallized form of a systematic presentationª. According to A.N. Kolmogorov, teaching mathematics should consist of several stages, which he justified by the inclination of the psychological attitudes of students to discreteness and by the fact that “the natural order of increasing knowledge and skills always has the character of “development in a spiral”ª. The principle of "linear" construction of a multi-year course, in particular mathematics, in his opinion, is devoid of clear content. However, the logic of science does not require that the "spiral" necessarily be broken into separate "coils". As an example of such a step-by-step study, let's consider the process of forming the concept of such a mathematical structure as a group. The first stage in this process can be considered even preschool age, when children get acquainted with algebraic operations (addition and subtraction), which are carried out directly on sets of objects. This process then continues at school. We can say that the whole course of school mathematics is permeated with the idea of ​​a group. Students' acquaintance with the concept of a group begins, in fact, already in the 15th grade. During this period, at school, algebraic operations are already performed on numbers. Number-theoretic material is the most fertile material in school mathematics for the formation of the concept of algebraic structures. An integer, addition of integers, introducing zero, finding its opposite for each number, studying the laws of action - all these are, in essence, stages in the formation of the concept of basic algebraic structures (groups, rings, fields). In subsequent grades of the school, students are faced with questions that contribute to the expansion of knowledge of this nature. In the course of algebra, there is a transition from concrete numbers, expressed in numbers, to abstract literal expressions, denoting specific numbers only with a certain interpretation of the letters. Algebraic operations are already performed not only on numbers, but also on objects of a different nature (polynomials, vectors). Students begin to realize the universality of some properties of algebraic operations. Especially important for understanding the idea of ​​a group is the study of geometric transformations and the concepts of composition of transformations and inverse transformation. However, the last two concepts are not reflected in the current school curriculum (the sequential execution of movements and the reverse transformation are only briefly mentioned in the textbook by A.V. Pogorelov). In elective and optional courses, it is advisable to consider groups of self-combinations of some geometric shapes, groups of rotations, ornaments, borders, parquets and various applications of group theory in crystallography, chemistry, etc. These topics, where you have to get acquainted with the mathematical formulation of practical problems, cause students the greatest interest. When getting acquainted with the concept of a group in general terms, it is necessary to rely on previously acquired knowledge, which acts as a structure-forming factor in the system of mathematical training of students, which allows you to properly solve the problem of continuity between school and university mathematics. Although the study of modern concepts of mathematics and its applications increases interest in the subject, it is almost impossible for a teacher to find additional time for this in the classroom. Therefore, the introduction of project activities into the educational process can help here. This type of labor organization is also one of the main forms of implementation of the competency-based approach in education. This type of labor organization, as noted by A.M. Novikov, requires the ability to work in a team, often heterogeneous, sociability, tolerance, self-organization skills, the ability to independently set goals and achieve them. To briefly formulate what education is in a post-industrial society, it is the ability to communicate, learn, analyze, design, choose and create. Therefore, the transition from the educational paradigm of an industrial society to the educational paradigm of a post-industrial society means, according to a number of scientists, first of all, the main role of the projective beginning, the rejection of understanding education only as the acquisition of ready-made knowledge, the change in the role of the teacher, the use of computer networks to obtain knowledge. The teacher remains at the heart of the learning process, with two critical functions of supporting motivation, facilitating the formation of cognitive needs, and modifying the learning process of the class or individual student. The electronic educational environment contributes to the formation of its new role. In such a highly informative environment, the teacher and the student are equal in access to information, learning content, so the teacher can no longer be the main or only source of facts, ideas, principles and other information. His new role can be described as mentoring. He is a guide who introduces students to the educational space, to the world of knowledge and the world of ignorance. However, the teacher retains many of the old roles. In particular, when teaching mathematics, the student very often faces the problem of understanding and, as experience shows, the student cannot cope with it without dialogue with the teacher, even when using the most modern information technologies. The architecture of mathematical knowledge does not fit well with random buildings and requires a special culture, both assimilation and teaching. Therefore, a mathematics teacher has been and remains an interpreter of the meanings of various mathematical texts. Computer networks in education can be used to share software resources, implement interactive interaction, receive information in a timely manner, continuously monitor the quality of knowledge gained, etc. One of the types of project activities of students when using networking technology is an educational networking project. When studying mathematics, network projects are a convenient tool for jointly practicing problem-solving skills, checking the level of knowledge, and also forming interest in the subject. Such projects are especially useful for students in the humanities and others who are far from mathematics. As for project activities, the theoretical prerequisites for using projects in education were formed back in the industrial era and are based on the ideas of American educators and psychologists of the late 19th century. J. Dewey and W. Kilpatrick. At the beginning of the XX century. domestic teachers (P.P. Blonsky, P.F. Kapterev, S.T. Shatsky, etc.), who developed the ideas of project-based learning, noted that the project method can be used as a means of merging theory and practice in teaching; development of independence and preparation of schoolchildren for working life; all-round development of the mind and thinking; formation of creative abilities. But even then it became clear that project-based learning is a useful alternative to the classroom system, but it should by no means supplant it and become a kind of panacea. independently acquire them, navigate the information space. The researchers note that the effectiveness of the implementation of educational projects is achieved if they are interconnected, grouped according to certain characteristics, and also subject to their systematic use at all stages of mastering the content of the subject: from mastering basic mathematical knowledge to independently acquiring new knowledge to a deep understanding of mathematical patterns. and their use in various situations. The result of the implementation of educational projects involves the creation of a subjectively new, personally significant product, focused on the formation of strong mathematical knowledge and skills, the development of independence, an increase in interest in the subject. It is generally recognized that school mathematics involves specially organized activity for solving problems. However, the first thing that catches your eye when considering projects "in mathematics" is the almost complete absence of proper mathematical activity in most of them. The topics of such projects are very limited, mainly topics related to the history of mathematics (the "golden section", "Fibonacci numbers", "the world of polyhedra", etc.). In most projects, there is only the appearance of mathematics, there are some activities related to mathematics only indirectly. Access to modern sections of mathematics is difficult due to the lack of even a hint of such sections in the school curriculum. In project activities, not the assimilation of knowledge, but the collection and organizing some information. At the same time, in mathematical activity, the collection and systematization of information is only the first stage of work on solving a problem, and the simplest one at that, solving a mathematical problem requires special mental actions that are impossible without the assimilation of knowledge. Mathematical knowledge has specific features, ignoring which leads to their vulgarization. Knowledge in mathematics is reworked meanings that have passed the stages of analysis, checks for consistency, compatibility with all previous experience. This does not allow us to understand “knowledge” as simply facts, to consider the ability to reduce as a full-fledged assimilation. Mathematics as an academic subject has another specific feature: in it, problem solving acts as both an object of study and a method of personality development. Therefore, problem solving should remain the main type of learning activity in it, especially for students who have chosen profiles related to mathematics. The student must enter, notes I.I. Melnikov, to get inside the most complex skill bestowed on a person, the decision-making process. He is offered to understand what it means to “solve a problem”, how to formulate a problem, how to determine the means for solving it, how to break a complex problem into interconnected chains of simple problems. Solving problems constantly prompts the developing consciousness that there is nothing mystical, vague, unclear in the creation of new knowledge, in solving problems, that a person has been given the ability to destroy the wall of ignorance, and this ability can be developed and strengthened. Induction and deduction, the two whales on which the decision rests, call for help by analogy and intuition, that is, just what in "adult" life will give the future citizen the opportunity to determine his own behavior in a difficult situation.

As A.A. Carpenter, teaching mathematics through tasks has long been a known problem. Problems should serve both as a motive for further development of the theory and as an opportunity for its effective application. Considering the task-based approach to be the most effective means of developing the educational and mathematical activity of students, he set the task of constructing a pedagogically expedient system of tasks, with the help of which it would be possible to lead the student consistently through all aspects of mathematical activity (identifying problem situations and tasks, mathematizing specific situations, solving problems that motivate the expansion theories, etc.). It has been established that the solution of traditional problems in mathematics teaches a young person to think, independently model and predict the world around him, i.e. ultimately pursues almost the same goals as project activities, with the possible exception of acquiring communication skills, since more often In general, teachers do not impose requirements on the presentation of solutions to the problem. Therefore, in teaching mathematics, the solution of problems, apparently, should remain the main type of educational activity, and projects are only an addition to it. This most important type of learning activity allows students to master mathematical theory, develop creative abilities and independent thinking. As a result, the effectiveness of the educational process largely depends on the choice of tasks, on the methods of organizing the activities of students to solve them, i.e. problem solving techniques. Teachers, psychologists and methodologists have proven that for the effective implementation of the goals of mathematical education, it is necessary to use in the educational process a system of tasks with a scientifically based structure, in which the place and order of each element are strictly defined and reflect the structure and functions of these tasks. Therefore, in his professional activity, a mathematics teacher should strive to present the content of teaching mathematics to a large extent precisely through systems of tasks. A number of requirements are imposed on such systems: hierarchy, rationality of volume, increasing complexity, completeness, purpose of each task, the possibility of implementing an individual approach, etc.

If a student has solved a difficult problem, then in principle there is not much difference how the student will draw up the result: in the form of a presentation, report, or simply scribble the solution on a sheet in a cage. It is considered sufficient that he solved the problem. Therefore, the general requirements put forward for the presentation of project results: the relevance of the problem and the presentation of the results (artistry and expressiveness of the performance) are not very suitable for evaluating those projects in mathematics, which are based on the solution of complex problems. However, based on the requirements of modern society, the activity of solving problems needs to be improved, paying more attention to the initial stage (realization of the place of this problem in the system of mathematical knowledge) and the final stage (presentation of the solution to the problem). If we talk about project activities, then the most appropriate is the use in the practice of teaching interdisciplinary projects that implement an integrative approach in teaching mathematics and several natural sciences or humanitarian disciplines at once. Such projects have more diverse and interesting topics, such projects in four-five-six disciplines are the longest-term, since their creation involves the processing of a large amount of information. Examples of such interdisciplinary projects are given in the book by P.M. Gorev and O.L. Luneeva. The result of such a macroproject can be a website dedicated to the topic of the project, a database, a brochure with the results of the work, etc. When working on such macro-projects, the student carries out educational activities in cooperation with other network users, i.e. educational activities become not individual, but joint. Because of this, we need to look at such learning as a process taking place in the learning community. In a community in which both students and teachers perform their well-defined functions. And the result of learning can be regarded precisely from the point of view of the performance of these functions, and not according to one or another external, formal parameters that characterize purely subject knowledge of individual students. It must be admitted that the practice of using the “project method” in school teaching of mathematics is still quite poor, everything often comes down to finding some information on the Internet on a given topic and to designing a “project”. In many cases, it turns out just an imitation of project activities. Due to these features, many teachers are very skeptical about the use of the project method in teaching students their subject: someone simply cannot understand the meaning of such student activity, someone does not see the effectiveness of this educational technology in relation to their discipline. However, the effectiveness of the project method for most school subjects is already undeniable. Therefore, it is very important that the content of the projects is not just related to mathematics, but contributes to overcoming the isolation of individual topics and sections in it, ensuring integrity and unity in teaching mathematics, which is possible only on the basis of highlighting it contains the cores of mathematical structures. Let us consider in more detail the application of the project method in the study of mathematical material by younger students. Due to the age characteristics of such students, the study of mathematical material, in particular geometric, is purely exploratory in nature. At the same time, the projects allow younger students to understand the role of geometry in real life situations, to arouse interest in the further study of geometry. When performing these projects, it is quite possible to use various software for educational purposes. Various computer environments are suitable for the implementation of most projects on geometric material. In primary school, it is advisable to use the PervoLogo integrated computer environment, the Microsoft Office PowerPoint program, as well as the electronic textbook "Mathematics and Design" and IISS "Geometric Design on a Plane and in Space", which are presented in the Electronic Collection of Digital Educational Resources and are intended for free use in the educational process. Choice of these software products is justified by the fact that they correspond to the age characteristics of elementary school students, are available for use in the educational process, and provide great opportunities for implementing the project method. The teacher of the Vologda Pedagogical College O.N. Kostrova developed a program of extracurricular activities containing a set of projects on geometric material and methodological recommendations for teachers on organizing work on projects. The main goal of the exemplary program is the formation of geometric representations of younger students based on the use of the method of educational projects. Work on the implementation of a set of projects is aimed at deepening and expanding students' knowledge of geometric material, understanding the world around them from geometric positions, developing the ability to apply the acquired knowledge in the course of solving educational, cognitive and practical problems using software, the formation of spatial and logical thinking. An exemplary program provides for an in-depth study of such topics as "Polygons", "Circle". Krugª, ©Plan. Scaleª, "Three-dimensional figures", the study of additional topics familiarity with axial symmetry, the presentation of numerical data of area and volume in the form of diagrams. Work on some projects involves the use of historical and local history material, which contributes to an increase in cognitive interest in the study of geometric material. The set of projects is represented by the following topics: ©World of Linesª, Old Units of Length Measurementª, Beauty of Patterns from Polygonsª, © Flags of the Vologda Oblast Regionsª, © Geometric fairy taleª(2nd grade); ©Ornaments of the Vologda Oblastª, ©Parquetª, ©A note in the newspaper about a circle or a circleª, ©Meanderª, ©Dachny plotª(3rd grade);©Anglesª, ©The Mystery of the Pyramidª, constructionª, work with designers (4th grade).

In the process of working on projects, students build flat and three-dimensional geometric shapes, construct and model other shapes, various objects from geometric shapes, conduct small studies on geometric material. The use of the project method in the study of geometric material involves the use of knowledge and skills from other subject areas, which contributes to the all-round development of students. This method implements an activity approach to learning, since learning takes place in the process of the activity of younger students; contributes to the development of skills in planning their educational activities, problem solving, competence in working with information, communicative competence. Thus, the use of the project method in teaching geometric material to schoolchildren makes it possible to solve a whole range of tasks for expanding and deepening knowledge of the elements of geometry, considering the possibilities of their application in practice, acquiring practical skills in working with modern software products, and comprehensively developing the individual abilities of schoolchildren. mathematical material for younger students represent only the first stage of project activities in mathematics. At the next stages of education, it is necessary to continue this activity, developing and deepening the knowledge of schoolchildren about the basic mathematical structures. In addition, using the project method in teaching mathematics, one should not forget that problem solving should remain the main type of educational activity. This specific feature of the subject should be taken into account when developing projects, so educational projects should be a means for students to develop their skills in solving problems, checking the level of knowledge, and forming a cognitive interest in the subject.

Links to sources 1. Testov V. A. Updating the content of teaching mathematics: historical and methodological aspects: monograph. Vologda, VGPU, 2012. 176 p.2. Testov V. A. Mathematical structures as a scientific and methodological basis for constructing mathematical courses in the system of continuous learning (school university): dis. ... drag ped. Sciences. Vologda, 1998.3. Kolmogorov AN To discuss the work on the problem "Prospects for the development of the Soviet school for the next thirty years" // Mathematics at school. 1990. No. 5. S. 5961.4. Novikov A. M. Post-industrial education. M.: Izdvo ©Egvesª, 2008.5. Education that we may lose: Sat. / under total ed. Rector of Moscow State University Academician V.A. Sadovnichy M.: Moscow State University. M. V. Lomonosov, 2002. S. 72.6. Stolyar A. A. Pedagogy of mathematics: a course of lectures. Minsk: Highest. School, 1969.7. Gorev P.M., Luneeva O.L. Interdisciplinary projects of secondary school students. Mathematical and natural science cycles: textbook.method.allowance. Kirov: Izdvo MCITO, 2014. 58 p. 8. Ibid. 9. Kostrova O.N. Software tools in the implementation of the project method in the study of elements of geometry by younger students // Scientific Review: Theory and Practice. 2012. No. 2. S.4148.

Vladimir Testov,

Doctor of Padagogic Sciences, Professor at the chair of Mathematics and Methods of Teaching Mathematics, Vologda State University, Vologda, Russia [email protected] ofpupils’main mathematical notionsformation in modern conditionsAbstract.The paperdiscusses the peculiarities of pupils’mathematical notions the formation in the modern paradigm of education and in the light of the demands,made in the concept of mathematical education. These requirements imply updating the content of teaching mathematics at school, bringing it closer to the modern sections and practical applications, the widespread using of project activities. To overcome the existing fragmentation of various mathematical disciplines and the isolation of individual sections, to ensure the integrity and unity in the teaching of mathematics is possible only on by allocatingthemain lines in it. Mathematical structures are therods, the main construction lines of mathematical courses. Phased process of formation of concepts about the basic mathematical structures is a prerequisite for the implementation of the principle of availability of training. Method of projects can be of great help in a phased study of mathematical structures. Application of this method in the study of mathematical structures allows you to solve a number of tasks to expand and deepen the knowledge of mathematics, consider the possibilities of their application inpractice, the acquisition of practical skills to work with modern software products, the full development of the individual abilities of pupils.Keywords: content of teaching mathematics, mathematical structures, phased process of formation of notions, project method.

References1.Testov,V. A. (2012) Obnovlenie soderzhanija obuchenija matematike: istoricheskie i metodologicheskie aspekty: monografija, VGPU, Vologda, 176 p.(in Russian).2.Testov,V. A. (1998) Matematicheskie struktury kak nauchnometodicheskaja osnova postroenija matematicheskih kursov v sisteme nepreryvnogo obuchenija (shkola vuz): dis. …draped. nauk, Vologda(in Russian).3.Kolmogorov,A. N. (1990) “K obsuzhdeniju raboty po probleme 'Perspektivy razvitija sovetskoj shkoly na blizhajshie tridcat" let'”, Matematika v shkole, no. 5, pp. 5961(in Russian).4.Novikov,A. M.(2008) Postindustrial "noe obrazovanie, Izdvo "Jegves",Moscow(in Russian).5.V. A. Sadovnichij (ed.)(2002) Obrazovanie, kotoroe my mozhem poterjat": sb. MGU im. M. V. Lomonosova, Moscow, p.72(in Russian). 6. Stoljar, A. A. (1969) Pedagogika matematiki: kurs lekcij, Vyshjejsh. shk., Minsk(in Russian). 7. Gorev, P. M. & Luneeva, O. L. (2014) Mezhpredmetnye proekty uchashhihsja srednej shkoly. ).8.Ibid.9.Kostrova,O.N. (2012) “Programmnye sredstva v realizacii metoda proektov pri izuchenii jelementov geometria mladshimi shkol"nikami”, Nauchnoe obozrenie: teorija i praktika, No. 2, pp.4148 (in Russian).

Nekrasova G.N., doctor of pedagogical sciences, professor, member of the editorial board of the magazine "Concept"

Formation of elementary mathematical concepts of a younger student

E.Yu. Togobetskaya, master student of the Department of Pedagogy and Teaching Methods

Togliatti Pedagogical University, Togliatti (Russia)

Keywords: mathematical concepts, absolute concepts, relative concepts, definitions.

Annotation: In school practice, many teachers force students to memorize the definitions of concepts and require knowledge of their basic properties to be proved. However, the results of such training are usually insignificant. This happens because the majority of students, when applying the concepts learned at school, rely on unimportant signs, while students realize and reproduce the essential signs of concepts only when answering questions that require a definition of the concept. Often students accurately reproduce concepts, that is, they discover knowledge of its essential features, but they cannot apply this knowledge in practice, they rely on those random features identified through direct experience. The process of assimilation of concepts can be controlled, they can be formed with the given qualities.

keywords: mathematical concepts, absolute concepts, relative concepts, definitions.

Abstract: In school practice many teachers achieve from pupils of learning of definitions of concepts and the knowledge of their basic proved properties demands. However, the results of such training are usually insignificant. It occurs because the majority of pupils, applying the concepts acquired at school, pupils lean against the unimportant signs, essential signs of concepts realize and reproduce only at the answer to the questions demanding definition of concept. Often pupils unmistakably reproduce concepts, that is find out knowledge of its essential signs, but put this knowledge into practice cannot, lean against those casual signs allocated thanks to a first-hand experience. Process of mastering of concepts it is possible to operate, form them with the set qualities.

When mastering scientific knowledge, elementary school students are faced with different types of concepts. The student's inability to differentiate concepts leads to their inadequate assimilation.

Logic in concepts distinguishes volume and content. The volume is understood as the class of objects that belong to this concept, are united by it. So, the scope of the concept of a triangle includes the entire set of triangles, regardless of their specific characteristics (types of angles, size of sides, etc.).

The content of concepts is understood as the system of essential properties, according to which these objects are combined into a single class. In order to reveal the content of a concept, it is necessary to establish by comparison what signs are necessary and sufficient to highlight its relationship to other objects. As long as the content and features are not established, the essence of the object reflected by this concept is not clear, it is impossible to accurately and clearly distinguish this object from those adjacent to it, confusion of thinking occurs.

For example, the concept of a triangle, such properties include the following: a closed figure, consists of three line segments. The set of properties by which objects are combined into a single class are called necessary and sufficient features. In some concepts, these features complement each other, forming together the content, according to which objects are combined into a single class. An example of such concepts is a triangle, an angle, a bisector, and many others.

The set of these objects to which this concept applies constitutes a logical class of objects. A logical class of objects is a collection of objects that have common features, as a result of which they are expressed by a common concept. The logical class of objects and the scope of the corresponding concept are the same. Concepts are divided into types according to content and scope, depending on the nature and number of objects to which they apply. By volume, mathematical concepts are divided into singular and general. If the scope of the concept includes only one object, it is called singular.

Examples of single concepts: “the smallest two-digit number”, “number 5”, “square with a side length of 10 cm”, “circle with a radius of 5 cm”. The general concept displays the features of a certain set of objects. The volume of such concepts will always be greater than the volume of one element. Examples of general concepts: “a set of two-digit numbers”, “triangles”, “equations”, “inequalities”, “numbers that are multiples of 5”, “primary school mathematics textbooks”. According to the content, the concepts of conjunctive and disjunctive, absolute and concrete, irrelative and relative are distinguished.

Concepts are called conjunctive if their features are interconnected and none of them individually allows you to identify objects of this class, the features are connected by the union "and". For example, objects related to the concept of a triangle must necessarily consist of three line segments and be closed.

In other concepts, the relationship between necessary and sufficient features is different: they do not complement each other, but replace. This means that one feature is equivalent to the other. An example of this type of relationship between signs can serve as signs of equality of segments, angles. It is known that the class of equal segments includes such segments that: a) either coincide when superimposed; b) or separately equal to the third; c) or consist of equal parts, etc.

In this case, the listed features are not required all at the same time, as is the case with the conjunctive type of concepts; here it is enough to have one of all the features listed: each of them is equivalent to any of the others. Because of this, the signs are connected by the union "or". Such a connection of attributes is called disjunction, and the concepts are respectively called disjunctive. It is also important to take into account the division of concepts into absolute and relative.

Absolute concepts combine objects into classes according to certain characteristics that characterize the essence of these objects as such. Thus, the concept of angle reflects the properties that characterize the essence of any angle as such. The situation is similar with many other geometric concepts: circle, ray, rhombus, etc.

Relative concepts combine objects into classes according to properties that characterize their relationship to other objects. So, in the concept of perpendicular lines, what characterizes the relationship of two lines to each other is fixed: the intersection, the formation of a right angle. Similarly, the concept of number reflects the ratio of the measured value and the accepted standard. Relative concepts cause students more serious difficulties than absolute concepts. The essence of the difficulties lies precisely in the fact that schoolchildren do not take into account the relativity of concepts and operate with them as with absolute concepts. So, when a teacher asks students to draw a perpendicular, some of them draw a vertical. Particular attention should be paid to the concept of number.

The number is the ratio of what is being quantified (length, weight, volume, etc.) to the standard that is used for this assessment. Obviously, the number depends both on the measured value and on the standard. The larger the measured value, the larger the number will be with the same standard. On the contrary, the larger the standard (measure), the smaller the number will be when evaluating the same value. Therefore, students should understand from the very beginning that comparison of numbers in magnitude can only be done when they are backed by the same standard. Indeed, if, for example, five is obtained when measuring length in centimeters, and three when measured in meters, then three denotes a greater value than five. If students do not learn the relative nature of number, then they will experience serious difficulties in learning the number system. Difficulties in the assimilation of relative concepts persist among students in the middle and even in the upper grades of school. There is a relationship between the content and scope of the concept: the smaller the scope of the concept, the greater its content.

For example, the concept of "square" has a smaller scope than the scope of the concept of "rectangle" since any square is a rectangle, but not every rectangle is a square. Therefore, the concept of "square" has a greater content than the concept of "rectangle": a square has all the properties of a rectangle and some others (for a square, all sides are equal, the diagonals are mutually perpendicular).

In the process of thinking, each concept does not exist separately, but enters into certain connections and relationships with other concepts. In mathematics, an important form of connection is generic dependence.

For example, consider the concepts of "square" and "rectangle". The scope of the concept "square" is part of the scope of the concept "rectangle". Therefore, the first is called species, and the second - generic. In genus-species relations, one should distinguish between the concept of the nearest genus and the next generic steps.

For example, for the view "square" the closest genus will be the genus "rectangle", for the rectangle the closest genus will be the genus "parallelogram", for the "parallelogram" - "quadrilateral", for the "quadrilateral" - "polygon", and for "polygon" - " flat figure.

In the elementary grades, for the first time, each concept is introduced visually, by observing specific objects or by practical operation (for example, when counting them). The teacher relies on the knowledge and experience of children, which they acquired in preschool age. Familiarization with mathematical concepts is fixed with the help of a term or a term and a symbol. This method of working on mathematical concepts in elementary school does not mean that various types of definitions are not used in this course.

To define a concept is to list all the essential features of objects that are included in this concept. The verbal definition of a concept is called a term. For example, "number", "triangle", "circle", "equation" are terms.

The definition solves two problems: it singles out and separates a certain concept from all others and indicates those main features without which the concept cannot exist and on which all other features depend.

The definition can be more or less deep. It depends on the level of knowledge about the concept that is meant. The better we know it, the more likely we will be able to give it a better definition. In the practice of teaching younger students, explicit and implicit definitions are used. Explicit definitions take the form of equality or coincidence of two concepts.

For example: "Propaedeutics is an introduction to any science." Here, two concepts are equated one to one - “propaedeutics” and “entry into any science”. In the definition "A square is a rectangle in which all sides are equal" we have a coincidence of concepts. In teaching younger students, contextual and ostensive definitions are of particular interest among implicit definitions.

Any passage from the text, whatever the context, in which the concept that interests us occurs, is, in some sense, an implicit definition of it. The context puts the concept in connection with other concepts and thereby reveals its content.

For example, when working with children such expressions as “find the values ​​of the expression”, “compare the value of the expressions 5 + a and (a - 3) 2 if a = 7”, “read expressions that are sums”, “read the expressions , and then read the equations”, we reveal the concept of “mathematical expression” as a record that consists of numbers or variables and signs of actions. Almost all definitions that we encounter in everyday life are contextual definitions. Having heard an unknown word, we try to establish its meaning ourselves on the basis of everything that has been said. The same is true in teaching younger students. Many mathematical concepts in elementary school are defined through context. These are, for example, such concepts as “big - small”, “any”, “any”, “one”, “many”, “number”, “arithmetic operation”, “equation”, “task” and etc.

Contextual definitions remain for the most part incomplete and incomplete. They are used in connection with the unpreparedness of the younger student to assimilate the full and, all the more so, the scientific definition.

Ostensive definitions are definitions by demonstration. They resemble ordinary contextual definitions, but the context here is not a passage of some text, but the situation in which the object denoted by the concept finds itself. For example, the teacher shows a square (drawing or paper model) and says "Look - it's a square." This is a typical ostensive definition.

In elementary grades, ostensive definitions are used when considering such concepts as “red (white, black, etc.) color”, “left - right”, “left to right”, “number”, “preceding and following number”, “signs arithmetic operations", "comparison signs", "triangle", "quadrilateral", "cube", etc.

Based on the assimilation of the meanings of words in an ostensive way, it is possible to introduce into the child's dictionary the already verbal meaning of new words and phrases. Ostensive definitions - and only they - connect the word with things. Without them, language is just a verbal lace that has no objective, substantive content. Note that in elementary grades, acceptable definitions are like "The word 'pentagon' we will refer to as a polygon with five sides." This is the so-called "nominal definition". Various explicit definitions are used in mathematics. The most common of them is the definition through the nearest genus and species character. The generic definition is also called the classical one.

Examples of definitions through a genus and a specific feature: “A parallelogram is a quadrilateral whose opposite sides are parallel”, “A rhombus is a parallelogram whose sides are equal”, “A rectangle is a parallelogram whose angles are right”, “A square is a rectangle in which the sides are equal”, “A square is a rhombus with right angles”.

Consider the definitions of a square. In the first definition, the closest genus would be "rectangle", and the species trait would be "all sides are equal". In the second definition, the closest genus is "rhombus", and the specific feature is "right angles". If we take not the nearest genus (“parallelogram”), then there will be two specific signs of a square “A parallelogram is called a square, in which all sides are equal and all angles are right.”

In the generic relation are the concepts of "addition (subtraction, multiplication, division)" and "arithmetic operation", the concept of "acute (right, obtuse) angle" and "angle". There are not so many examples of explicit generic relations among the many mathematical concepts that are considered in elementary grades. But taking into account the importance of definition through the genus and species trait in further education, it is desirable to achieve students' understanding of the essence of the definition of this species already in the primary grades.

Separate definitions can consider the concept and the method of its formation or occurrence. This type of definition is called genetic. Examples of genetic definitions: "Angle is the rays that come out from one point", "The diagonal of a rectangle is a segment that connects the opposite vertices of the rectangle." In elementary grades, genetic definitions are used for such concepts as "segment", "broken line", "right angle", "circle". The definition through the list can also be attributed to genetic concepts.

For example, "The natural series of numbers is the numbers 1, 2, 3, 4, etc." Some concepts in elementary grades are introduced only through the term. For example, the units of time are year, month, hour, minute. There are concepts in elementary grades that are presented in a symbolic language in the form of equality, for example, a 1 = a, and 0 = 0

From the above, we can conclude that in the primary grades, many mathematical concepts are first acquired superficially, vaguely. At the first acquaintance, schoolchildren learn only about some properties of concepts, they have a very narrow idea of ​​their scope. And this is natural. Not all concepts are easy to grasp. But it is indisputable that the teacher's understanding and timely use of certain types of definitions of mathematical concepts is one of the conditions for the formation of solid knowledge about these concepts in students.

Bibliography:

1. Bogdanovich M.V. Definition of mathematical concepts // Primary school 2001. - No. 4.

2. Gluzman N. A. Formation of generalized methods of mental activity in younger schoolchildren. - Yalta: KSGI, 2001. - 34 p.

3. Drozd V.L. Urban M.A. From small problems to big discoveries. //Primary School. - 2000. - No. 5.


Lecture 7. Mathematical concepts

1. Groups of concepts studied in the initial course of mathematics. Features of mathematical concepts.

2. The scope and content of the concept.

3. Relations between concepts.

4. Operations with concepts: generalization, restriction, definition and division of a concept.

5. Rules necessary for formulating the definition of concepts through the genus and specific difference.

6. Contextual and ostensive definitions. Description, comparison.

Groups of concepts studied in the initial course of mathematics. Features of mathematical concepts.

The concepts that are studied in the elementary course of mathematics are usually presented in the form of four groups. First concepts related to numbers and operations on them are included: number, addition, summand, more, etc. Second includes algebraic concepts: expression, equality, equation, etc. Third make up geometric concepts: a straight line, a segment, a triangle, etc. fourth the group is formed by concepts related to quantities and their measurement.

How to study such an abundance of various concepts?

First of all, one must have an idea of ​​the concept as a logical category and the features of mathematical concepts.

In the logic of the concept consider as a form of thought, reflecting objects(objects or phenomena) in their essential and general properties. The linguistic form of the concept is word or group of words.

Make a concept about the object- it means to be able to distinguish it from other objects similar to it.

Mathematical concepts have a number of features. The main one is that the mathematical objects about which it is necessary to form a concept do not exist in reality. Mathematical objects are created by the human mind. These are ideal objects that reflect real objects or phenomena. For example, in geometry, the shape and size of objects are studied, without taking into account their other properties: color, mass, hardness, etc. From all this they are distracted, abstracted. Therefore, in geometry, instead of the word "object" they say "geometric figure".



The result of abstraction are also such mathematical concepts as "number" and "value".

Generally mathematical objects exist only in human thinking and in those signs and symbols that form the mathematical language.

It can be added to what has been said that studying spatial forms and quantitative relationships material world, mathematics not only uses various abstraction techniques, but the abstraction itself acts as a multi-stage process. In mathematics, not only the concepts that appeared in the study of real objects are considered, but also the concepts that arose on the basis of the former. For example, the general concept of a function as a correspondence is a generalization of the concepts of specific functions, i.e. abstraction from abstractions.

In order to master general approaches to the study of concepts in the initial course of mathematics, the teacher needs knowledge about the scope and content of the concept, about the relationship between concepts and about the types of definitions of concepts.

2. Scope and content of the concept

Every mathematical object has certain properties. For example, a square has four sides, four right angles equal to the diagonal. You can specify other properties as well.

Among object properties distinguish significant and non-essential.

Property feel essential for an object if it is inherent in this object and without it it cannot exist. For example, for a square, all the properties mentioned above are essential. The property "side AD is horizontal" is not essential for the square ABCD. If the square is rotated, then the side AD will be located differently (Fig. 26). Therefore, in order to understand what a given mathematical object is, one must know its essential properties.

When talking about a mathematical concept, they usually mean a set of objects denoted by one term (a word or a group of words). So, speaking of a square, they mean all geometric shapes that are squares. It is believed that the set of all squares is the scope of the concept of "square".

Any concept is characterized by a word, volume and content.

The scope of the concept a is the set of all objects that can be called a given word (term)

Example. Let us highlight the scope and content of the concept of "rectangle".

The scope of the concept is a set of different rectangles, and in its content includes such properties of rectangles as "have four right angles", "have equal opposite sides", "have equal diagonals", etc.

There is a relationship between the scope of a concept and its content.: if the volume of a concept increases, then its content decreases, and vice versa. So, for example, the scope of the concept of "square" is part of the scope of the concept of "rectangle", and the content of the concept of "square" contains more properties than the content of the concept of "rectangle" ("all sides are equal", "the diagonals are mutually perpendicular", etc.). ).

Any concept cannot be assimilated without realizing its relationship with other concepts. Therefore, it is important to know in what relationships concepts can be, and to be able to establish these connections.

Introduction

The concept is one of the main components in the content of any academic subject, including mathematics.

One of the first mathematical concepts that a child encounters in school is the concept of number. If this concept is not mastered, the students will have serious problems in the further study of mathematics.

From the very beginning, students encounter concepts while studying various mathematical disciplines. So, starting to study geometry, students immediately meet with the concepts: point, line, angle, and then with a whole system of concepts associated with the types of geometric objects.

The task of the teacher is to ensure the full assimilation of concepts. However, in school practice, this problem is not solved as successfully as required by the goals of the general education school.

“The main drawback of the school assimilation of concepts is formalism,” says psychologist N.F. Talyzina. The essence of formalism is that students, while correctly reproducing the definition of a concept, that is, realizing its content, do not know how to use it when solving problems for the application of this concept. Therefore, the formation of concepts is an important, up-to-date problem.

Object of study: the process of forming mathematical concepts in grades 5-6.

Objective: to develop guidelines for the study of mathematical concepts in grades 5-6.

Work tasks:

1. Study mathematical, methodical, pedagogical literature on this topic.

2. Identify the main ways of defining concepts in textbooks of grades 5-6.

3. Determine the features of the formation of mathematical concepts in grades 5-6.

Research hypothesis : If, in the process of forming mathematical concepts in grades 5-6, the following features are taken into account:

concepts are mostly determined by construction, and often the formation of a correct understanding of the concept in students is achieved with the help of explanatory descriptions;

concepts are introduced in a concrete-inductive way;

· Throughout the process of concept formation, much attention is paid to visibility, then this process will be more effective.

Research methods:

study of methodological and psychological literature on the topic;

comparison of various textbooks in mathematics;

Experienced teaching.

Fundamentals of the methodology for studying mathematical concepts

Mathematical concepts, their content and scope, classification of concepts

A concept is a form of thinking about an integral set of essential and non-essential properties of an object.

Mathematical concepts have their own characteristics: they often arise from the need of science and have no analogues in the real world; they have a high degree of abstraction. Because of this, it is desirable to show students the emergence of the concept being studied (either from the need for practice or from the need for science).

Each concept is characterized by volume and content. Content - many essential features of the concept. Volume - a set of objects to which this concept is applicable. Consider the relationship between the scope and content of the concept. If the content is true and does not include contradictory features, then the volume is not an empty set, which is important to show students when introducing the concept. The content completely determines the volume and vice versa. This means that a change in one entails a change in the other: if the content increases, then the volume decreases.

o should be carried out on one basis;

o classes must be non-overlapping;

o the union of all classes should give the whole set;

o the classification should be continuous (classes should be the closest specific concepts in relation to the concept that is subject to classification).

There are the following types of classification:

1. On a modified basis. Objects to be classified may have several features, so they can be classified in different ways.

Example. The concept of a triangle.

2. Dichotomous. The division of the scope of the concept into two specific concepts, one of which has this feature, and the other does not.

Example .

Let's single out the goals of training classification:

1) development of logical thinking;

2) by studying specific differences, we get a clearer idea of ​​the generic concept.

Both types of classification are used in the school. As a rule, first dichotomous, and then on a modified basis.