Types and definitions of mathematical concepts in elementary mathematics. Open Library - open library of educational information · Give similar pairs of concepts in comparison and contrast
Testov Vladimir Afanasyevich,
Doctor of Pedagogical Sciences, Professor of the Department of Mathematics and Methods of Teaching Mathematics, Vologda State University, Vologda [email protected]
Features of the formation of basic mathematical concepts among schoolchildren in modern conditions
Annotation. The article examines the features of the formation of mathematical concepts among schoolchildren in the modern educational paradigm 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 mathematics teaching at school, bringing it closer to modern sections and practical application, and the widespread use of project activities. It is possible to overcome the existing disunity of various mathematical disciplines, the isolation of individual topics and sections, and to ensure integrity and unity in teaching mathematics only on the basis of identifying the main cores in it. Such cores are mathematical structures. A necessary condition for the implementation of the principle of accessibility of learning is the stage-by-stage process of forming concepts about basic mathematical structures. The project method can be of great help in the step-by-step study of mathematical structures. The use of this method when schoolchildren study mathematical structures makes it possible to solve a whole range of problems in 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 the comprehensive development of individual abilities of schoolchildren. Key words: content of teaching mathematics , mathematical structures, stage-by-stage process of concept formation, project method. Section: (01) pedagogy; history of pedagogy and education; theory and methods of teaching and education (by subject areas).
Currently, the transition to the information society is being completed, and at the same time a new paradigm in education is taking shape, based on post-non-classical methodology, synergetic 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 mathematics teaching at school, bringing it closer to modern sections and practical applications. The features of educational material in the information society are the fundamental redundancy of information, the nonlinear nature of its deployment, the possibility of variability of educational material. The role of mathematical education as the basis of competitiveness, a necessary element of the country's security, is recognized by the leadership of Russia. In December 2013, the government approved the concept of the development of mathematical education. This concept raises many current problems in mathematics education. The main problem identified is the low educational motivation of schoolchildren, which is associated with the current underestimation of mathematical education in the public consciousness, as well as the overload of programs, assessment and teaching materials with technical elements and outdated content. The current state of students' mathematical preparation raises serious concerns. There is a formalism in the mathematical knowledge of secondary school graduates and their insufficient 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, being unable to independently structure and comprehend it.
As a result, a significant part of such information is quickly forgotten and the mathematical baggage of a significant part of high school graduates consists of a greater or lesser number of loosely interconnected, dogmatically acquired information and better or worse established skills in performing certain standard operations and typical tasks. They lack the idea of mathematics as a single science with its own subject and method. Excessive enthusiasm for the purely informational side of learning leads to the fact that many students do not perceive the rich content of mathematical knowledge embedded in the program. The content side of mathematical education should be focused not so much on the narrowly understood needs of today, but on strategic prospects, on a vision of the diversity of its applications, widespread use of mathematical models in modern society. Thus, the task is set to bring the content of teaching mathematics closer to modern science. It is possible to overcome the disunity of various mathematical disciplines, the isolation of individual topics and sections, and ensure integrity and unity in teaching mathematics only on the basis of identifying its origins, the main cores. Such cores in mathematics, as noted by A.N. Kolmogorov and other major scientists are mathematical structures, which are divided, according to N. Bourbaki, 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 a product of the internal development of mathematics. From this view of the subject of mathematics it follows that in any mathematics course mathematical structures must be studied. The idea of mathematical structures, which turned out to be very fruitful, served as one of the motivations for a radical reform of mathematical education in the 60-70s. Although this reform was later criticized, its basic idea remains very useful for modern mathematics education. Recently, new important sections have emerged in mathematics that require reflection in both university and school mathematics curricula (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 in all their depth and completeness from the very beginning. As shown in, the process of teaching mathematics should be considered as a multi-level system with mandatory reliance on lower, more specific levels, stages of scientific knowledge. Without such support, learning can become formal, providing knowledge without understanding. The step-by-step process of forming basic mathematical concepts is a necessary condition for implementing the principle of accessibility of learning.
Views on the need to identify successive stages in the formation of concepts about mathematical structures among mathematician teachers are widespread. 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 finally introduce them to abstract ideas; teaching must follow the same path along which all humanity, 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 in the form of a guess and only after a long development acquired a fixed, crystallized form of systematic presentationª. According to A.N. Kolmogorov, teaching mathematics should consist of several stages, which he justified by the tendency of students’ psychological attitudes towards discreteness and the fact that “the natural order of increasing knowledge and skills always has the character of “development in a spiral.” 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 divided into separate “turns.” As an example of such a step-by-step study, let us consider the process of forming the concept of such a mathematical structure as a group. The first stage in this process can be considered preschool age, when children become familiar 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 entire 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 performed on numbers. In school mathematics, number theoretical material is the most fertile material for developing the concept of algebraic structures. An integer, addition of integers, introducing zero, finding its opposite for each number, studying the laws of actions - all these are essentially stages in the formation of the concept of basic algebraic structures (groups, rings, fields). In subsequent grades of school, students are faced with questions that contribute to the expansion of knowledge of this nature. In an algebra course, a transition is made from concrete numbers, expressed in numbers, to abstract alphabetic expressions, denoting specific numbers only with a certain interpretation of the letters. Algebraic operations are 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. Particularly important for understanding the idea of a group is the study of geometric transformations and the concepts of composition of transformations and inverse transformations. 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 elective courses, it is advisable to consider groups of self-combinations of some geometric figures, groups of rotations, ornaments, borders, parquets and various applications of group theory in crystallography, chemistry, etc. These topics, where one has to get acquainted with the mathematical formulation of practical problems, cause the greatest concern among students interest. When becoming familiar with the concept of a group in general, 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 us 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 the teacher to find additional time for this in the lessons. Therefore, the introduction of project activities into the educational process can help here. This type of work 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 diverse, communication skills, 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, access to the main role of the projective principle, refusal to understand education only as the acquisition of ready-made knowledge, changing the role of the teacher, the use of computer networks to obtain knowledge. The teacher remains central to the learning process, with the two most important 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 informational environment, the teacher and student are equal in access to information and 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 into the educational space, into the world of knowledge and the world of ignorance. However, the teacher retains many of the old roles. In particular, when teaching mathematics, a 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 was and remains an interpreter of the meanings of various mathematical texts. Computer networks in teaching can be used to share software resources, implement interactive interactions, timely obtain information, continuously monitor the quality of acquired knowledge, etc. One of the types of project activities of students when using network technologies is an educational network project. When studying mathematics, network projects are a convenient means for students to jointly practice problem-solving skills, test their level of knowledge, and also develop 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 the use of projects in teaching developed in the industrial era and are based on the ideas of American teachers and psychologists of the late 19th century. J. Dewey and W. Kilpatrick. At the beginning of the 20th 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; developing independence and preparing schoolchildren for working life; comprehensive 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 not at all displace it and become a kind of panacea. Modern research into the use of projects in teaching has revealed the wide possibilities of educational projects using ICT, allowing one to deepen, update knowledge, and develop skills independently acquire them and navigate the information space. 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 independent acquisition of new knowledge to a deep understanding of mathematical laws and using them in various situations. The result of completing 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, and increased interest in the subject. It is generally accepted that school mathematics involves specially organized activity in solving problems. However, the first thing that catches your eye when considering “mathematics” projects is the almost complete absence of actual mathematical activity in most of them. The topics of such projects are very limited, mainly topics related to the history of mathematics (the “golden ratio”, “Fibonacci numbers”, “the world of polyhedra”, etc.). In most projects there is only the appearance of mathematics, there is some activity related to mathematics only indirectly. Access to modern branches of mathematics is difficult due to the absence of even a hint of such sections in the school curriculum. In project activities, the focus is not on the assimilation of knowledge, but on the collection and systematization of some information. At the same time, in mathematical activity, collecting and systematizing information is only the first stage of working on solving a problem, and the simplest one at that; solving a mathematical problem requires special mental actions that are impossible without acquiring knowledge. Mathematical knowledge has specific features, ignoring which leads to their vulgarization. Knowledge in mathematics is processed meanings that have gone through stages of analysis, testing for consistency, and compatibility with all previous experience. This does not allow us to understand “knowledge” as simply facts, or 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 personal development. Therefore, in it, problem solving should remain the main type of educational activity, especially for students who have chosen profiles related to mathematics. The student must enter, notes I.I. Melnikov, to penetrate into the most complex skill given to man, the decision-making process. He is asked to understand what it means to “solve a problem,” how to formulate a problem, how to identify means for solution, how to break a complex problem into interconnected chains of simple tasks. Solving problems constantly prompts the developing consciousness that in creating new knowledge, in solving problems there is nothing mystical, vague, unclear, that a person is given the ability to destroy the wall of ignorance, and this skill can be developed and strengthened. Induction and deduction, the two pillars on which the decision rests, call on analogy and intuition for help, that is, exactly what in “adult” life will give the future citizen the opportunity to determine his own behavior in a difficult situation.
As A.A. wrote. Carpenter, teaching mathematics through problems has long been a known problem. Problems should serve as both a motive for further development of the theory and an opportunity for its effective application. Considering the problem-based approach to be the most effective means of developing students’ educational and mathematical activity, he set the task of constructing a pedagogically appropriate system of tasks, with the help of which it would be possible to guide the student consistently through all aspects of mathematical activity (identifying problematic situations and problems, mathematizing specific situations, solving problems that motivate the expansion theories, etc.) . It has been established that solving 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 exception, perhaps, 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, problem solving should apparently remain the main type of educational activity, and projects should only be an addition to it. This most important type of educational activity allows schoolchildren 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 ways of organizing students’ activities to solve them, i.e. problem solving techniques. Educators, psychologists and methodologists have proven that in order to effectively implement the goals of mathematics education, it is necessary to use in the educational process systems 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 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 an individual approach, etc.
If a student has solved a complex problem, then in principle there is not much difference in how the student presents the result: in the form of a presentation, report, or simply scribbles the solution on a sheet of paper in a square. It is considered sufficient that he has 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 presentation") are of little relevance to the assessment of those projects in mathematics, which are based on the solution of complex problems. However, based on the requirements of modern society, problem solving activities need to be improved, paying more attention to the initial stage (awareness of the place of a given 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 it seems most appropriate to use interdisciplinary projects in teaching practice that implement an integrative approach to teaching mathematics and several natural science or humanities disciplines at once. Such projects have more diverse and interesting topics; such projects in four-five-six disciplines are the most long-term, since their creation involves processing 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 macro-project could be a website dedicated to the project topic, a database, a brochure with the results of the work, etc. When working on such macro-projects, the student carries out learning activities in interaction with other network users, i.e. learning activities become not individual, but joint. Because of this, we need to look at such learning as a process taking place in a learning community. In a community in which both students and teachers perform their very specific functions. And the result of learning can be assessed 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 mathematics teaching is still quite poor; everything often comes down to the student finding some information on a given topic on the Internet and drawing up a “project”. In many cases, the result is simply an imitation of project activity. Due to these features, many teachers are very skeptical about the use of the project method in teaching schoolchildren their subject: some simply cannot understand the meaning of such student activities, others do 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 projects is not just related to mathematics, but helps to overcome 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 use of the project method when studying mathematical material by junior schoolchildren. Due to the age characteristics of such students, the study of mathematical material, in particular geometric material, is purely educational in nature. At the same time, projects make it possible to develop in younger schoolchildren an understanding of the role of geometry in real life situations and to arouse interest in further study of geometry. When carrying out 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 elementary school, it is advisable to use the integrated computer environment PervoLogo, the Microsoft Office PowerPoint program, as well as the electronic textbook “Mathematics and Design” and the 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 These software products are justified by the fact that they correspond to the age characteristics of primary school students, are accessible for use in the educational process, and provide great opportunities for implementing the project method. 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 approximate program is the formation of geometric concepts of primary schoolchildren based on the use of the method of educational projects. The work on implementing a set of projects is aimed at deepening and expanding students’ knowledge of geometric material, understanding the world around them from a geometric perspective, developing the ability to apply acquired knowledge in solving educational, cognitive and educational-practical problems using software, and developing spatial and logical thinking. The approximate program provides for an in-depth study of such topics as “Polygons”, “Circle”. Circleª, ©Plan. Scaleª, ©Volume figuresª, study of additional topics, introduction to axial symmetry, 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 helps to increase cognitive interest in the study of geometric material. The set of projects is represented by the following topics: ©World of linesª, ©Ancient units of length measurementª, ©Beauty of patterns from polygonsª, ©Flags of the regions of the Vologda regionª, © Geometric fairy taleª (2nd grade); ©Ornaments of the Vologda regionª, ©Parquetª, ©Note in the newspaper about a circle or circleª, ©Meanderª, ©Dacha plotª (3rd grade); ©Anglesª, ©The mystery of the pyramidª, ©Streets of our cityª, ©Calculation work for constructionª, work with designers (4th grade).
In the process of working on projects, students construct flat and three-dimensional geometric figures, construct and model other figures and various objects from geometric figures, and conduct small studies on geometric material. Using the project method in studying 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-based approach to learning, since learning occurs in the process of activity of younger schoolchildren; contributes to the development of skills in planning one's educational activities, problem solving, competence in working with information, and communicative competence. Thus, the use of the project method when teaching schoolchildren geometric material makes it possible to solve a whole range of problems in expanding and deepening knowledge of the elements of geometry, considering the possibilities of their use in practical activities, acquiring practical skills in working with modern software products, and comprehensively developing the individual abilities of schoolchildren. Projects on mathematical material for primary schoolchildren 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 students’ knowledge of basic mathematical structures. In addition, when using the project method in teaching mathematics, one must not forget that problem solving should remain the main type of educational activity. This specific feature of an educational subject should be taken into account when developing projects, therefore educational projects should be a means for schoolchildren to practice problem-solving skills, test their level of knowledge, and develop cognitive interest in the subject.
Links to sources 1. Testov V. A. Updating the content of teaching mathematics: historical and methodological aspects: monograph. Vologda, VSPU, 2012. 176 pp. 2. Testov V. A. Mathematical structures as a scientific and methodological basis for constructing mathematical courses in the system of continuous education (university school): dis. ...dra ped. Sci. Vologda, 1998.3. Kolmogorov A. N. 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.: Publishing house ©Egvesª, 2008.5. Education that we can lose: collection. / under general ed. Rector of Moscow State University Academician V.A. Sadovnichego M.: Moscow State University named after. M. V. Lomonosova, 2002. P. 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 for high school students. Mathematical and natural science cycles: educational methodological manual. Kirov: Publishing house MTSITO, 2014. 58 p. 8. Ibid. 9. Kostrova O.N. Software tools in the implementation of the project method in studying the elements of geometry by junior schoolchildren // Scientific review: theory and practice. 2012. No. 2. P.4148.
Vladimir Testov,
Doctor of Pedagogic 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 solve a number of tasks to expand and deepen the knowledge of mathematics, consider the possibilities of their application in practice, 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. …dra ped. 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. Matematicheskij i estestvennonauchnyj cikly: ucheb.metod. posobie, Izdvo MCITO, Kirov, 58 p. (in Russian ).8.Ibid.9.Kostrova, O. N. (2012) “Programmnye sredstva v realizacii metoda proektov pri izuchenii jelementov geometrii mladshimi shkol"nikami”, Nauchnoe obozrenie: teorija i praktika, No. 2, pp. 4148 (inRussian).
Nekrasova G.N., Doctor of Pedagogical Sciences, Professor, member of the editorial board of the journal ©Conceptª
Lecture No. 2
mathematics
Topic: "Mathematical Concepts"
Mathematical concepts
Definition of concepts
Requirements for defining concepts
Some types of definitions
1. Mathematical concepts
The concepts that are studied in an initial mathematics course are usually presented in the form of four groups. The first includes concepts related to numbers and operations on them: number, addition, term, greater, etc. The second includes algebraic concepts: expression, equality, equation, etc. The third consists of geometric concepts: line, segment, triangle, etc. d. The fourth group consists of concepts related to quantities and their measurement.
How to study such an abundance of different concepts?
First of all, you need to 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 group of words.
To form 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 thing is that the mathematical objects about which it is necessary to formulate 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 they study the shape and size of objects without taking into account their other properties: color, mass, hardness, etc. They are distracted from all this, abstracted. Therefore, in geometry, instead of the word “object” they say “geometric figure”.
The result of abstraction is such mathematical concepts as “number” and “magnitude”.
In general, mathematical objects exist only in human thinking and in those signs and symbols that form mathematical language.
To what has been said, we can add that, when studying spatial forms and quantitative relations of the material world, mathematics not only uses various abstraction techniques, but abstraction itself acts as a multi-stage process. In mathematics, they consider not only concepts that appeared during the study of real objects, but also 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. an 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, the relationships between concepts and the types of definitions of concepts.
2. 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, and equal diagonals. You can specify its other properties.
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 a square ABCD. If the square is rotated, then side AD will be located differently (Fig. 26).
Therefore, in order to understand what a given mathematical object is, you need to know its essential properties.
When people talk 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, we mean all geometric figures that are squares. It is believed that the set of all squares constitutes the scope of the concept “square”.
At all the scope of a concept is the set of all objects denoted by one term.
Any concept has not only volume, but also content.
Consider, for example, the concept of “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 “square” is part of the scope of the concept “rectangle”, and the content of the concept “square” contains more properties than the content of the concept “rectangle” (“all sides are equal”, “diagonals are mutually perpendicular”, etc. ).
Any concept cannot be learned without realizing its relationship with other concepts. Therefore, it is important to know in what relationships concepts can be found 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 denote concepts with 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 “quadrangle”, then their volumes A and B are in the inclusion relation (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 “quadrangle”, and the concept of “quadrangle” is generic in relation to the concept of “rectangle”.
If A = B, then they say that concepts a andbare identical.
For example, the concepts “equilateral triangle” and “equiangular triangle” are identical, since their volumes coincide.
If the sets A and B are not related by the inclusion relation, then they say that the concepts a and b are not in the relation of genus and species and are not identical. For example, the concepts “triangle” and “rectangle” are not connected by such relations.
Let us consider in more detail the relationship of genus and species between concepts. Firstly, the concepts of genus and species are relative: the same concept can be generic in relation to one concept and specific 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 “quadrangle”.
Secondly, for a given concept it is often possible to specify several generic concepts. Thus, for the concept of “rectangle” the generic concepts are “quadrangle”, “parallelogram”, “polygon”. Among them, you can indicate the closest one. For the concept of “rectangle” the closest concept is “parallelogram”.
Thirdly, a species concept has all the properties of a generic concept. For example, a square, being a specific concept in relation to the concept of “rectangle”, has all the properties inherent in a rectangle.
Since the volume of a concept is a set, it is convenient, when establishing relationships between the volumes 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”, b - “segment”.
In case 1) the volumes of concepts intersect, but neither one set is a subset of the other (Fig. 27).
Consequently, it can be argued that these concepts a and b are not in the relation of genus and species.
In case 2) the volumes of the given concepts are in the relation of inclusion, but do not coincide - every parallelogram is a polygon, but not vice versa (Fig. 28). Consequently, 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 not a single segment can be said to be a straight line, and not a single straight line can be called a segment (Fig. 29).
Consequently, these concepts are not in relation to genus and species.
About the concepts “straight line” and “segment” we can say that they are in relation to the whole and the part: A segment is a part of a straight line, not its type. And if a species concept has all the properties of a generic concept, then a part does not necessarily have all the properties of the whole. For example, a segment does not have the same property of a straight line as its infinity.
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 at school is the concept of number. If this concept is not mastered, students will have serious problems in further learning mathematics.
From the very beginning, students encounter concepts when studying various mathematical disciplines. Thus, when starting to study geometry, students immediately encounter the concepts: point, line, angle, and then with a whole system of concepts associated with types of geometric objects.
The teacher’s task is to ensure a complete understanding of concepts. However, in school practice this problem is not solved as successfully as the goals of a comprehensive school require.
“The main disadvantage of mastering concepts in school 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 on the application of this concept. Therefore, the formation of concepts is an important current problem.
Object of study: the process of forming mathematical concepts in grades 5-6.
Goal of the work: develop methodological recommendations for studying mathematical concepts in grades 5-6.
Job objectives:
1. Study mathematical, methodological, pedagogical literature on this topic.
2. Identify the main ways of defining concepts in textbooks for 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, we take into account the following features:
· concepts are mostly defined through construction, and often the formation of a correct understanding of the concept among students is achieved with the help of explanatory descriptions;
· concepts are introduced in a concrete inductive way;
· throughout the entire process of concept formation, much attention is paid to clarity, then this process will be more effective.
Research methods:
· study of methodological and psychological literature on the topic;
· comparison of different mathematics textbooks;
· Experienced teaching.
Fundamentals of methods for studying mathematical concepts
Mathematical concepts, their content and scope, classification of concepts
A concept is a form of thinking about a holistic set of essential and non-essential properties of an object.
Mathematical concepts have their own characteristics: they often arise from the needs 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 needs of practice or from the needs of 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. Let us consider the connection between the volume and content of a concept. If the content corresponds to reality and does not include contradictory features, then 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 according to one criterion;
o classes must be disjoint;
o the union of all classes should give the entire set;
o classification must be continuous (classes must be the closest species concepts in relation to the concept that is subject to classification).
The following types of classification are distinguished:
1. According to a modified characteristic. Objects to be classified may have several characteristics, so they can be classified in different ways.
Example. The concept of "triangle".
2. Dichotomous. Dividing the scope of a concept into two specific concepts, one of which has a given feature and the other does not.
Example .
Let us highlight the objectives of classification training:
1) development of logical thinking;
2) by studying species differences, we get a clearer idea of the generic concept.
Both types of classification are used in school. As a rule, first dichotomous, and then on a modified basis.
Formation of elementary mathematical concepts of primary schoolchildren
E.Yu. Togobetskaya, Master's student at the Department of Pedagogy and Teaching Methods
Tolyatti Pedagogical University, Tolyatti (Russia)
Keywords: mathematical concepts, absolute concepts, relative concepts, definitions.
Annotation: In school practice, many teachers force students to memorize definitions of concepts and require knowledge of their basic provable properties. However, the results of such training are usually insignificant. This happens because most students, when applying concepts learned at school, rely on unimportant signs, while students are aware of and reproduce the essential signs of concepts only when answering questions that require defining the concept. Often students accurately reproduce concepts, that is, they discover knowledge of its essential features, but cannot apply this knowledge in practice; they rely on those random features identified through direct experience. The process of mastering concepts can be controlled and formed with given qualities.
Keywords: mathematical concepts, absolute concepts, relative concepts, definitions.
Annotation: In school practice many teachers achieve from pupils of learning of definitions of concepts and the knowledge of their basic proven 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 encounter different types of concepts. The student’s inability to differentiate concepts leads to their inadequate assimilation.
Logic in concepts distinguishes between volume and content. By volume we mean the class of objects that relate to this concept and are united by it. Thus, the scope of the concept of 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 by which these objects are combined into a single class. To reveal the content of a concept, it is necessary to establish by comparison what features are necessary and sufficient to highlight its relationship to other objects. Until the content and characteristics are 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, and confusion of thinking occurs.
For example, for the concept of a triangle, such properties include the following: a closed figure, consisting of three straight segments. The set of properties by which objects are combined into a single class are called necessary and sufficient characteristics. In some concepts, these features complement each other, forming together the content by which objects are united into a single class. Examples of such concepts are triangle, angle, bisector and many others.
The collection 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 characteristics, 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 by content and scope depending on the nature and number of objects to which they apply. Based on their scope, mathematical concepts are divided into individual and general. If the scope of a concept includes only one object, it is called singular.
Examples of single concepts: “the smallest two-digit number”, “the number 5”, “a square with a side length of 10 cm”, “a circle with a radius of 5 cm”. General concept reflects the characteristics 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: “set of two-digit numbers”, “triangles”, “equations”, “inequalities”, “numbers multiples of 5”, “mathematics textbooks for primary school”. According to their content, concepts are distinguished between conjunctive and disjunctive, absolute and concrete, non-relative and relative.
Concepts are called conjunctive if their features are interrelated and individually none of them allows identifying objects of this class; the features are connected by the conjunction “and”. For example, objects related to the concept of a triangle must necessarily consist of three straight segments and be closed.
In other concepts, the relationship between necessary and sufficient characteristics is different: they do not complement each other, but replace each other. This means that one attribute is equivalent to another. An example of this type of relationship between characteristics can be the signs of equality of segments and angles. It is known that the class of equal segments includes those 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 characteristics are not required all at the same time, as is the case with the conjunctive type of concepts; here, just one attribute from all those listed is sufficient: each of them is equivalent to any of the others. Because of this, the signs are connected by the conjunction “or”. Such a connection of features is called disjunction, and concepts are accordingly called disjunctive. It is also important to take into account the division of concepts into absolute and relative.
Absolute concepts unite 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 unite objects into classes according to properties that characterize their relationship to other objects. Thus, the concept of perpendicular lines captures what characterizes the relationship of two lines to each other: intersection, the formation of a right angle. Similarly, the concept of number reflects the relationship between the measured quantity and the accepted standard. Relative concepts cause more serious difficulties for students 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 absolute concepts. So, when the teacher asks students to draw a perpendicular, some of them draw a vertical. Particular attention should be paid to the concept of number.
A 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 on both the quantity being measured and 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 estimating the same value. Consequently, students must understand from the very beginning that comparisons of numbers by magnitude can only be made when they have the same standard behind them. In fact, if, for example, five is obtained when measuring length in centimeters, and three is obtained when measuring in meters, then three denotes a greater value than five. If students do not understand the relative nature of numbers, they will have serious difficulties in learning the number system. Difficulties in mastering relative concepts persist among students in middle and even high school. There is a relationship between the content and scope of a concept: the smaller the scope of the concept, the greater its content.
For example, the concept “square” has a smaller scope than the scope of the concept “rectangle” since any square is a rectangle, but not every rectangle is a square. Therefore, the concept of “square” has more content than the concept of “rectangle”: a square has all the properties of a rectangle and some others (all sides of a square 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 genus-specific dependence.
For example, consider the concepts “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 closest genus and the following generic stages.
For example, for the type “square” the closest genus will be the genus “rectangle”, for a rectangle the closest genus will be the genus “parallelogram”, for a “parallelogram” - “quadrilateral”, for a “quadrilateral” - “polygon”, and for a “polygon” - “ flat figure."
In the elementary grades, for the first time, each concept is introduced visually, through observation of specific objects or practical operation (for example, when counting them). The teacher relies on the knowledge and experience of children that they acquired in preschool age. Familiarization with mathematical concepts is fixed using 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 characteristics 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 identifies and distinguishes 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 profound. It depends on the level of knowledge about the concept that is meant. The better we know it, the more likely we are to be able to define it better. In the practice of teaching primary schoolchildren, 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 primary schoolchildren, contextual and ostensive definitions are of particular interest among implicit definitions.
Any passage from a text, whatever the context, in which the concept that interests us occurs is, in some sense, its implicit definition. Context puts a concept in connection with other concepts and thereby reveals its content.
For example, when working with children, using such expressions as “find the meaning of the expression”, “compare the meaning of the expressions 5 + a and (a - 3) 2, if a = 7”, “read expressions that are sums”, “read expressions , and then read the equations,” we expand on the concept of “mathematical expression” as a record that consists of numbers or variables and action signs. Almost all definitions that we encounter in everyday life are contextual definitions. Having heard an unknown word, we try to establish its meaning ourselves based on everything that has been said. A similar thing happens in teaching younger students. Many math concepts in elementary school are defined through context. These are, for example, concepts such as “big - small”, “any”, “any”, “one”, “many”, “number”, “arithmetic operation”, “equation”, “problem” and etc.
Contextual definitions remain largely incomplete and incomplete. They are used due to the unpreparedness of younger schoolchildren to master the full, and especially scientific, definition.
Ostensive definitions are definitions by demonstration. They resemble ordinary contextual definitions, but the context here is not a passage of any text, but the situation in which the object designated 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 primary school, ostensive definitions are used when considering such concepts as “red (white, black, etc.) color”, “left - right”, “left to right”, “digit”, “preceding and following number”, “signs” arithmetic operations", "comparative signs", "triangle", "quadrangle", "cube", etc.
Based on the ostensive assimilation of the meanings of words, it is possible to introduce the verbal meaning of new words and phrases into the child’s dictionary. Ostensive definitions - and only they - connect words with things. Without them, language is just a verbal lace that has no objective, substantive content. Note that in elementary grades, acceptable definitions like “We will use the word “pentagon” to mean a polygon with five sides.” This is the so-called “nominal definition”. In mathematics, different explicit definitions are used. The most common of them is determination through the nearest genus and species characteristic. The generic definition is also called classical.
Examples of definitions through genus and specific characteristics: “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 whose sides equal”, “A rhombus with right angles is called a square.”
Let's look at the definitions of a square. In the first definition, the closest genus would be “rectangle,” and the specific feature would be “all sides are equal.” In the second definition, the closest genus is “rhombus”, and the specific character is “right angles”. If we take not the closest genus (“parallelogram”), then there will be two specific characteristics of a square: “A square is a parallelogram in which all sides are equal and all angles are right.”
In the generic relation there 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 genus-species relationships among the many mathematical concepts that are discussed in the primary grades. But taking into account the importance of definition through genus and species in further education, it is advisable to ensure that students understand the essence of the definition of this species already in the elementary grades.
Separate definitions can consider a concept according to the method of its formation or emergence. This type of determination is called genetic. Examples of genetic definitions: “An angle is the rays that come out from one point,” “The diagonal of a rectangle is a segment that connects the opposite vertices of a rectangle.” In elementary grades, genetic definitions are used for such concepts as “segment”, “broken line”, “right angle”, “circle”. Genetic concepts can also be defined through a list.
For example, “The natural series of numbers are the numbers 1, 2, 3, 4, etc.” Some concepts in elementary grades are introduced only through the term. For example, time units are year, month, hour, minute. There are concepts in elementary grades that are presented in symbolic language in the form of an equality, for example, a 1 = a, and 0 = 0
From the above, we can conclude that in the elementary grades many mathematical concepts are first acquired superficially and vaguely. At the first acquaintance, schoolchildren learn only about some properties of concepts and have a very narrow idea of their scope. And this is natural. Not all concepts are easy to grasp. But there is no doubt that the teacher’s understanding and timely use of certain types of definitions of mathematical concepts is one of the conditions for students to develop solid knowledge about these concepts.
Bibliography:
1. Bogdanovich M.V. Definition of mathematical concepts //Primary school 2001. - No. 4.
2. Gluzman N. A. Formation of generalized techniques of mental activity in primary schoolchildren. - Yalta: KGGI, 2001. - 34 p.
3. Drozd V.L. Urban M.A. From small problems to big discoveries. //Elementary School. - 2000. - No. 5.
Methods for studying mathematical concepts
1. The essence of the concept. Content and scope of the concept.
2. Definition of mathematical concepts.
3. Classification of mathematical concepts.
4. Methodology for introducing new mathematical concepts.
Any science is a system of concepts, therefore in mathematics, as in other academic subjects, considerable attention is paid to teaching concepts. The concept refers to forms of theoretical thinking, which is a rational stage of cognition.
1. The essence of the concept. Content and scope of the concept. With the help of concepts, we express general, essential features of things and phenomena of objective reality.
Perception is called the direct sensory reflection of reality in human consciousness.
Submission is the image of an object or phenomenon imprinted in our consciousness that is not currently perceived by us.
Perception disappears as soon as the impact of the object on the human senses ends. What remains is the show. For example, we show a cube and then remove it. We know different cubes, different colors, etc., but we abstract from this, preserving the general and essential.
Concept abstracts from individual traits and characteristics of individual perceptions and ideas and is the result of generalization of perceptions and ideas of a very large number of homogeneous objects and phenomena, for example: number, pyramid, circle, straight line. Concepts are formed through such logical techniques as analysis and synthesis, abstraction and generalization. Concept we will call the thought about an object that highlights its essential features.
Significant features concepts are such features, each of which is necessary, and all together are sufficient to distinguish objects of a given kind from other objects (for example, a parallelogram).
In each concept, its content and volume are distinguished.
Scope of concept is the collection of objects to which this concept applies.
For example, the concept of “person”. Contents: a living being, creates tools of production, has the ability of abstract thinking. Volume: all people.
The concept of "tetrahedron". Contents: a polyhedron bounded by four triangle-shaped faces. Volume: the set of all tetrahedra.
There is a relationship between the volume and content of a concept: the greater the content of the concept, the smaller its volume. Reducing the content of a concept entails expanding its scope. This operation is called generalization concepts. For example, if the property “equality of all sides” is removed from the content of the concept “equilateral triangle,” then the set of triangles that satisfy the new content will become “wider” - it will contain the set of equilateral triangles as a subset. Expanding the content of a concept leads to a narrowing of its scope and is called limitation(specialization) concepts. An example of such an operation is the transition from the concept of identity transformations to the concept of reduction of fractions.
If the scope of one concept is included as part of the scope of another concept, then the first concept is called species, and the second – ancestral.
The concepts of genus and species are relative character. For example, the concept of “prism” is generic in relation to the concept of “straight prism”, but a specific concept in relation to the concept of “polyhedron”.
Euler circles.
2. Definition of mathematical concepts. The content of the concept is revealed using a definition.
Definition(definition) concepts– this is a logical operation with the help of which the main content of a concept or the meaning of a term is revealed.
Define concept- this means listing the essential characteristics of the objects displayed in this concept.
The task of listing characteristics can be difficult, but it is simplified if we rely on concepts that have already been previously established. The concept is fixed in speech using a word or phrase called name or term concepts. In mathematics, a concept is often denoted not only by a name, but also symbol. For example, others.
Thus, the definition first indicates the genus into which the concept being defined is included as a species, and then indicates those characteristics that distinguish this species from other species of the closest genus. This method of defining a concept is called definition of a concept through the nearest genus and species difference.
Concept = genus + species difference.
Types of definitions
Explicit Implicit
Through genus and species
differences Axiomatic Descriptive
(described by the system
Explicit definitions are called in which the meaning of the term being defined is completely conveyed through the meaning of the defining terms, i.e. explicit definitions contain a direct indication of the essential features of the concept being defined. Determination through the nearest genus and species difference refers to obvious ones.
IN implicit In definitions, the meaning of the term being defined is not fully conveyed by the defining terms. An example of an implicit definition is the definition of initial concepts using a system of axioms. Such definitions are called axiomatic. Examples of axiomatic definitions are the definitions of a group, a ring and a field, etc. (Hilbert's, Weyl's axiomatics, the Peano axiom system for natural numbers).
Genetic is called the definition of an object by indicating the method of its construction, formation, origin. For example, “a truncated cone is a body resulting from the rotation of a rectangular trapezoid around the side perpendicular to the bases of the trapezoid.” Or the definition of the concept of “linear dihedral angle”.
IN inductive(recurrent) definition, an object is defined as a function of a natural number ..gif" width="56" height="21"> and. For example, by induction in mathematics, the definition of a natural number is introduced.
Ostensive definitions and descriptive describe objects using models, consideration of special cases, highlighting individual essential properties, introduced through direct demonstration, demonstration of objects. Often used in primary grades and partially in grades 5-6. The teacher, depicting triangles on the board, introduces students to the concept of triangle. In high school, verbal definitions predominate.
To give a logically correct definition, one must observe definition rules:
1. The definition should be proportionate, that is, the defined and defining concepts must be equal in volume. To check proportionality, you need to make sure that the concept being defined satisfies the characteristics of the defining concept and vice versa.
For example, the definition is given: “A parallelogram is a polygon whose opposite sides are parallel.” Let's check it: “Every polygon whose opposite sides are parallel is a parallelogram” - this is false. Or: “parallel lines are lines that do not intersect” (wrong, these can also be intersecting lines).
2. The definition should not contain " vicious circle" This means that it is impossible to construct a definition in such a way that the defining concept is one that is itself defined by the concept being defined.
For example, “a right angle is an angle containing , and a degree is 1/90 of a right angle.” Sometimes the "vicious circle" takes the form of a tautology (same by means of the same) - the use of a word that has the same meaning.
3. Definition if possible must not be negative. The definition must indicate the essential features of the subject, and not what the subject is not.
For example, “a rhombus is not a triangle,” “an ellipse is not a circle.” In mathematics, in some cases negative definitions are acceptable, for example, “any non-algebraic function is called a transcendental function.”
4. The definition should be clear And clear, not allowing ambiguous or metamorphic expressions.
For example, “arithmetic is the queen of mathematics” is a figurative comparison, not a definition; the statement “laziness is the mother of all vices” is instructive, but does not define the concept of laziness.
3. Classification of mathematical concepts. The scope of a concept is revealed through classification. Classification is a systematic distribution of a certain set into classes, resulting from sequential division based on the similarity of objects of one type and their difference from objects of other types.
The division operation is a logical operation that reveals the scope of a concept by highlighting possible types of an object in it. For example, all students at a pedagogical university can be divided into those who intend to go to work at school and those who do not. The basis of division is the property according to which species are distinguished. In our example, the basis is the property: “to have the intention to work at school.”
When making a classification, the choice of basis is important: different reasons give different classifications. Classification can be made according to essential properties (natural) and non-essential ones (auxiliary). With natural classification, knowing which group an element belongs to, we can judge its properties.
Two types of division:
1. division by modification of a characteristic is a division in which the property - the basis of division - is inherent in objects of the selected types to varying degrees
2. dichotomous division is a division in which a given concept is divided into two types according to the presence or absence of a certain property.
The division operation obeys the following rules:
1. The division must be proportionate, that is, the union of the selected classes must form the original set (the sum of the volumes of specific concepts is equal to the volume of the generic concept).
2. Division must be carried out using only one base.
3. The intersection of classes must be empty.
4. division must be continuous.
4. Methodology for introducing new mathematical concepts. In the methodology of teaching mathematics, there are two methods of introducing concepts: concrete-inductive And abstract-deductive(the terms were introduced by a Russian methodologist).
Application diagram concrete-inductive method.
1. Examples are reviewed and analyzed (analysis, comparison, abstraction, generalization,...).
2. The general features of the concept that characterize it are clarified.
3. A definition is formulated.
4. The definition is reinforced by giving examples and counterexamples.
Application diagram abstract-deductive method.
The definition of the concept is formulated. Examples and counterexamples are given. The concept is reinforced by performing various exercises.
For example, the introduction of a quadratic equation, the concept of Cartesian coordinates, etc.
When forming concepts, it is advisable to apply the recommendations of psychological and pedagogical sciences, for example, the theory of the gradual formation of mental actions.
Stage 1. They explain the purpose of the introduced concept and provide orientation.
Stage 2. Students formulate a definition based on the picture.
Stage 3. Students formulate a definition using loud (external) speech without relying on a drawing.
Stage 4. The definition is pronounced in the form of external speech to oneself.
Stage 5. The definition is spoken in the form of inner speech.
When studying concepts, it is necessary to vary non-essential features (principles of variation) - this is a varied arrangement of pictures and drawings on the board, for example, a triangle, its height, a perpendicular to a line, etc. (not only the horizontal location of a line, the base of a triangle, etc. )
Understanding the definitions is helped by analyzing the logical structure of the definition. For this purpose, concept recognition algorithms, mathematical dictations and tests are compiled.
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