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2.4 Features of the formation of mathematical concepts in grades 5-6 28

Introduction

Chapter 1
Fundamentals of Methods for Studying Mathematical Notions

1.1 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 needs of science and have no analogues in the real world; they have a high degree of abstraction. By virtue of this, it is desirable to show students the emergence of the concept being studied (either from the need of practice, or from the need 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. Consider the relationship between the scope and content of a concept. If the content corresponds to reality and does not include contradictory signs, then the volume is not an empty set, which is important to show to students when introducing a 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.

The content of a concept is identified with its definition, and the volume is revealed through classification. Classification - dividing a set into subsets that satisfy the following requirements:

o must be carried out on one basis;

o classes should be disjoint;

o union of all classes should give the whole set;

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

There are the following types of classification:

1. On a modified basis. The objects to be classified can 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 species concepts, one of which has this feature, and the other does not.

Example .

Let's highlight the objectives of teaching classification:

1) development of logical thinking;

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

1.2 Definition of mathematical concepts, primary concepts, clarifying description

Define an object - choose from its essential properties such and so many that each of them was necessary, and all together sufficient to distinguish this object from others. The result of this action is recorded in the definition.

By definition such a formulation is considered that reduces the new concept to the already known concepts of the same area. Such reduction cannot continue indefinitely, therefore science has primary concepts , which are not defined explicitly, but indirectly (through axioms). The list of primary concepts is ambiguous, in comparison with science, in the school course there are much more primary concepts. The main method for clarifying, introducing primary concepts is the compilation of pedigrees.

In a school course, it is not always advisable to give a strict definition of concepts. Sometimes it is enough to form the right idea. This is accomplished with belt nagging descriptions - sentences available to students that evoke one visual image in them, and help to assimilate the concept. There is no requirement here to reduce a new concept to those previously studied. Assimilation should be brought to such a level that in the future, without remembering the description, the student can recognize the object related to this concept.

1.3 Ways to define concepts

By logical structure definitions are divided into conjunctive (essential features are connected by the union "and") and disjunctive (essential features are connected by the union "or").

The selection of essential features fixed in the definition and the fixed connections between them is called logical-mathematical analysis of the definition .

There is a division of definitions into descriptive and constructive.

Descriptive - descriptive or indirect definitions, which, as a rule, have the form: "an object is called ... if it possesses ...". The fact of the existence of a given object does not follow from such definitions, therefore all such concepts require proof of existence. Among them, the following ways of defining concepts are distinguished:

· Across closest genus and species difference. (A rhombus is called a parallelogram, two adjacent sides of which are equal. The concept of a parallelogram is generic, from which the concept being defined is distinguished by means of one specific difference).

· Definitions-conventions- definitions in which the properties of concepts are expressed using equalities or inequalities.

· Axiomatic definitions. In science itself, mathematics is often used, and in the school course it is rarely used for intuitively clear concepts. (The area of ​​a figure is a quantity whose numerical value satisfies the conditions: S (F) 0; F 1 = F 2 S (F 1) = S (F 2); F = F 1 F 2, F 1 F 2 = S (F ) = S (F 1) + S (F 2); S (E) = 1.)

Definitions via abstraction. They resort to such a definition of a concept when something else is difficult or impossible to implement (for example, a natural number).

· Definition-negation- a definition in which not the presence of a property is recorded, but its absence (for example, parallel lines).

Constructive (or genetic) are definitions that indicate a method for obtaining a new object (for example, a sphere is a surface obtained by rotating a semicircle around its diameter). Such definitions are sometimes distinguished recursive- definitions that indicate some basic element of a class and a rule by which new objects of the same class can be obtained (for example, the definition of a progression).

1.4 Methodological requirements for the definition of the concept

· The requirement of scientific character.

· Accessibility requirement.

· The requirement of commensurability (the volume of the defined concept must be equal to the volume of the defining concept). Violation of this requirement leads to either a very broad or a very narrow definition.

· The definition should not contain a vicious circle.

· Definitions must be clear, precise and free of metaphorical expressions.

· Minimum requirement.

1.5 Introduction of concepts in the school mathematics course

In the formation of concepts, it is necessary to organize the activities of students to master two basic logical techniques: summing up a concept and deriving consequences from the fact that an object belongs to a concept.

Action summing up has the following structure:

1) Selection of all properties fixed in the definition.

2) Establishing logical connections between them.

3) Checking whether the object has selected properties and their connections.

4) Obtaining a conclusion about the belonging of the object to the scope of the concept.

Derivation of consequences - this is the selection of essential features of an object belonging to this concept.

There are three ways in the technique. introducing concepts :

1) Specific-inductive:

o Consideration of various objects, both belonging to the scope of the concept, and not belonging.

o Identification of essential features of a concept based on a comparison of objects.

o Introduction of the term, wording of the definition.

2) Abstract-deductive:

o Introduction of the definition by the teacher.

o Consideration of special and special cases.

o Formation of the ability to bring an object under the concept and deduce the primary consequences.

When introducing a concept the first way, students better understand the motives for the introduction, learn to build definitions and understand the importance of each word in it. By introducing the concept in the second way, a large amount of time is saved, which is also not unimportant.

3) Combined . Used for more complex concepts in calculus. A definition of the concept is given based on a small number of specific examples. Then, by solving problems in which insignificant features vary, and by comparing this concept with specific examples, the formation of the concept continues.

1.6 The main stages of studying the concept at school

In the literature, there are three main stages in the study of concepts at school:

1. When introduction of the concept one of the three above methods is used. During this stage, you need to consider the following:

· First of all, it is necessary to provide motivation for the introduction of this concept.

· When constructing a system of tasks for summing up a concept, provide the most complete scope of the concept.

· It is important to show that the scope of a concept is not an empty set.

· Expand the content of the concept, work on the essential features, highlighting the insignificant.

· In addition to knowing the definition, it is desirable that students have a visual understanding of the concept.

· Assimilation of terminology and symbols.

The result of this stage is the formulation of a definition, the assimilation of which is the content of the next stage. To master the definition of a concept means to master the actions of recognizing objects belonging to a concept, deriving consequences from an object's belonging to a concept, constructing objects related to the scope of a concept.

2. At the stage assimilation of the definition work continues on memorizing the definition. This can be achieved using the following techniques:

· Writing out definitions in a notebook.

· Pronunciation, underlining or any numbering of essential properties.

· Using counterexamples to fulfill the rules of commensurability.

· Selection of missing words in the definition, finding unnecessary words.

· Learning to provide examples and counterexamples.

· Learning to apply the definition in the simplest, but rather typical situations, since repeated repetition of the definition outside of solving problems is ineffective.

· Indicate the possibility of different definitions, prove their equivalence, but choose only one for memorization.

· To learn to construct a definition, to use the compilation of pedigrees for this, explaining the logical structure; to acquaint with the rules for constructing a definition.

· Give similar pairs of concepts in comparison and comparison.

Thus, at this stage, each essential property of the concept used in the definition becomes a special object of study.

3.Next stage - anchorage ... A concept can be considered formed if students immediately recognize it in a task without any sorting of features, that is, the process of summing up a concept is minimized. This can be achieved in the following ways:

· Application of the definition in more complex situations.

· Inclusion of a new concept in logical connections, relationships with other concepts (for example, comparison of pedigrees, classifications).

· It is desirable to show that the definition is given not for its own sake, but in order for it to "work" in solving problems and building a new theory.

Chapter 2
Psychological and pedagogical features of teaching mathematics in grades 5-6

2.1 Features of cognitive activity

Perception. A student of grades 5-6 has a sufficient level of development of perception. He has a high level of visual acuity, hearing, orientation to the shape and color of the object.

The learning process makes new demands on the student's perception. In the process of perception educational information arbitrariness and meaningfulness of students' activities are necessary. First, the child is attracted by the object itself and, first of all, by its outward bright signs. But children are already able to concentrate and carefully consider all the characteristics of an object, highlight the main, essential in it. This feature manifests itself in the process learning activities... They can analyze groups of figures, arrange objects according to various criteria, and classify figures according to one or two properties of these figures.

In schoolchildren of this age, observation appears as a special activity, observation develops as a character trait.

The process of forming a concept is a gradual process, in the first stages of which the sensory perception of an object plays an important role.

Memory. A schoolchild in grades 5-6 is able to manage his arbitrary memorization... The ability to memorize (memorize) slowly but gradually increases.

At this age, memory is rebuilt, moving from the dominance of mechanical memorization to semantic memorization. At the same time, the semantic memory itself is rebuilt. It acquires an indirect character, thinking is necessarily included. Therefore, it is necessary to teach students to reason correctly so that the memorization process is based on an understanding of the proposed material.

Along with the form, the content of memorization also changes. It becomes more accessible to memorize abstract material.

Attention. The process of mastering knowledge, skills, and abilities requires constant and effective self-control of students, which is possible only if there is enough high level arbitrary attention.

A student in grades 5-6 can well control his attention. He concentrates well on activities that are meaningful to him. Therefore, it is necessary to maintain the student's interest in learning mathematics. In this case, it is advisable to rely on aids (objects, pictures, tables).

At school in the classroom, attention needs support from the teacher.

Imagination. In the process of educational activity, the student receives a lot of descriptive information. This requires him to constantly recreate images, without which it is impossible to understand and assimilate educational material, i.e. recreating the imagination of students in grades 5-6 from the very beginning of training is included in purposeful activity that contributes to his mental development.

As the child develops the ability to control his mental activity, the imagination becomes more and more controllable.

For schoolchildren in grades 5-6, imagination can turn into an independent internal activities... They can play mental tasks with mathematical signs in their minds, operate with the meanings and meanings of the language, combining two higher mental functions: imagination and thinking.

All of the above features create the basis for the development of the process of creative imagination, in which the special knowledge of students plays an important role. This knowledge forms the basis for the development of creative imagination in the subsequent age periods of the student's life.

Thinking. Theoretical thinking, the ability to establish the maximum number of semantic connections in the surrounding world, is beginning to acquire an increasing importance. The student is psychologically immersed in the reality of the objective world, figurative-sign systems. The material studied at school becomes for him a condition for constructing and testing his hypotheses.

In grades 5-6, the student develops formal thinking. A student of this age can already reason without associating himself with a specific situation.

Scientists have studied the question of the mental capabilities of schoolchildren in grades 5-6. As a result of the research, it was revealed that the mental capabilities of the child are wider than previously assumed, and with the creation of appropriate conditions, i.e. with special methodological organization learning, a student in grades 5-6 can assimilate abstract mathematical material.

As seen from the above, mental processes are characterized by age characteristics, knowledge and consideration of which are necessary for the organization of successful training and mental development students.

2.2 Psychological aspects of concept formation

2.3 Some pedagogical features of teaching mathematics in grades 5-6

The leading idea modern concept school education is the idea of ​​humanization, placing at the center of the learning process of the student with his interests and capabilities, which requires taking into account the characteristics of his personality. The main directions of mathematical education is to strengthen the general cultural sound and increase its importance for the formation of the personality of a growing person. The main ideas underlying the mathematics course in grades 5-6 are the general cultural orientation of the content, the intellectual development of students by means of mathematics on the material that meets the interests and capabilities of children 10-12 years old.

The mathematics course for grades 5-6 is an important link in mathematical education and the development of schoolchildren. At this stage, the training of counting on the set of rational numbers is mainly completed, the concept of a variable is formed and the first knowledge about the methods of solving linear equations is given, training in solving word problems continues, the skills of geometric constructions and measurements are improved and enriched. Serious attention is paid to the formation of the ability to reason, make simple proofs, and provide justifications for the actions performed. In parallel, the foundations are laid for the study of systematic courses in stereometry, physics, chemistry and other related subjects.

The mathematics course of grades 5-6 is an organic part of the whole school mathematics... Therefore, the main requirement for its construction is the structuring of the content on a single ideological basis, which, on the one hand, is the continuation and development of ideas implemented in teaching mathematics in primary school, and, on the other hand, serves the subsequent study of mathematics in high school.

The development of all content-methodological lines of the course of elementary mathematics continues: numerical, algebraic, functional, geometric, logical, data analysis. They are implemented in numerical, algebraic, geometric material.

IN recent times the study of geometry has been substantially revised. The purpose of the study geometry in grades 5-6, cognition of the surrounding world is the language and means of mathematics. With the help of constructions and measurements, students identify various geometric patterns, which they formulate as a proposal, a hypothesis. The demonstrative aspect of geometry is considered in a problematic plan - the students are taught the idea that many geometric facts can be discovered experimentally, but these facts become mathematical truths only when they are established by means adopted in mathematics.

Thus, geometric material in this course can be characterized as visual-activity geometry. Education is organized as a process of intellectual and practical activity aimed at developing spatial representations, visual skills, expanding the geometric horizons, during which the most important properties of geometric figures are obtained through experience and common sense.

The content line “ Data analysis », Which combines three areas: elements of mathematical statistics, combinatorics, probability theory. The introduction of this material is dictated by life itself. Its study is aimed at developing in schoolchildren both a general probabilistic intuition and specific methods of evaluating data. The main task in this link is the formation of an appropriate vocabulary, teaching the simplest techniques for collecting, presenting and analyzing information, learning how to solve combinatorial problems by enumerating possible options, creation of elementary ideas about the frequency and probability of random events.

However, this line is not present in all modern school textbooks for grades 5-6. This line is especially detailed and vividly presented in textbooks.

Algebraic the material included in the mathematics course for grades 5-6 is the basis for the systematic study of algebra in high school. The following features of the study of this algebraic material can be noted:

1. The study of algebraic material is based on a scientific basis, taking into account the age characteristics and capabilities of students.

Among the skills that math teaches and that you all need to learn, great importance has skill classify concepts.

The fact is that mathematics, like many other sciences, studies not single objects or phenomena, but massive... So, when you study triangles, you study the properties of any triangles, and there are an infinite number of them. In general, the scope of any mathematical concept is, as a rule, infinite.

In order to distinguish objects of mathematical concepts, to study their properties, these concepts are usually divided into types, classes. Indeed, in addition to general properties, any mathematical concept has many more important properties inherent not to all objects of this concept, but only to objects of some kind. So, right-angled triangles, in addition to the general properties of any triangles, they have many properties that are very important for practice, for example the Pythagorean theorem, ratios between angles and sides, etc.

In the process of centuries-old study of mathematical concepts, in the process of their numerous applications in life, in other sciences, from their scope, some special types were distinguished from their volume, having the most interesting properties, which are most often found and used in practice. So, there are infinitely many different quadrangles, but in practice, in technology, only certain types of them are most used: squares, rectangles, parallelograms, rhombuses, trapezoids.

Dividing the volume of a concept into parts is the classification of this concept. More precisely, classification is understood as the distribution of objects of any concept into interrelated classes (types, types) according to the most essential features (properties). The attribute (property) by which the classification (division) of the concept into types (classes) is made is called basis classification.

A correctly constructed classification of a concept reflects the most essential properties and connections between the objects of the concept, helps to better navigate in the set of these objects, makes it possible to establish such properties of these objects that are most important for the application of this concept in other sciences and everyday practice.

The classification of the concept is made on one or more of the most significant grounds.

So, triangles can be classified according to the magnitude of the angles. We get the following types: acute-angled (all angles are acute), rectangular (one corner is straight, the rest are acute), obtuse-angular (one corner is obtuse, the rest are sharp). If we take the relationship between the sides as the basis for dividing the triangles, then we get the following types: versatile, isosceles and regular (equilateral).

It is more difficult when you have to classify a concept on several grounds. So, if convex quadrangles are classified according to the parallelism of the sides, then in essence we need to divide all convex quadrangles simultaneously according to two criteria: 1) one pair of opposite sides is parallel or not; 2) the second pair of opposite sides is parallel or not. As a result, we get three types of convex quadrangles: 1) quadrangles with non-parallel sides; 2) quadrangles with one pair parallel sides- trapezoid; 3) quadrangles with two pairs of parallel sides - parallelograms.

Quite often, a concept is classified in stages: first, on one basis, then some species are divided into subspecies on a different basis, etc. An example is the classification of quadrangles. At the first stage, they are divided on the basis of convexity. Then the convex quadrangles are divided according to the parallelism of the opposite sides. In turn, parallelograms are divided according to the presence of right angles, etc.

When carrying out the classification, certain rules must be followed. Let us indicate the main ones.

1. As a basis for classification, one can take only a common feature of all objects of a given concept. So, for example, it is impossible as a basis for classification algebraic expressions to take the sign of the arrangement of terms by degrees of some variable. This feature is not common to all algebraic expressions, for example, it does not make sense for fractional expressions or monomials. This feature is possessed only by polynomials; therefore, polynomials can be classified by the highest degree the main variable.
2. The basis for the classification should be taken the essential properties (attributes) of concepts. Consider again the concept of an algebraic expression. One of the properties of this concept is that the variables included in an algebraic expression are denoted by some letters. This property is general, but not essential, because the character of the expression does not depend on what letter this or that variable is designated. Thus, algebraic expressions x + y and a + b is essentially the same expression. Therefore, you should not classify expressions on the basis of the designation of variables by letters. It is another matter if we take as the basis for the classification of algebraic expressions the attribute of the type of actions by which the variables are connected, that is, the actions that are performed on the variables. This common feature is very essential, and a classification based on this feature will be correct and useful.
3. At each stage of the classification, only one kind of basis can be applied. You cannot simultaneously classify a concept on two different grounds. For example, it is impossible to classify triangles at once both in size and in the ratio between the sides, because as a result we get classes of triangles that have common elements(for example, acute-angled and isosceles or obtuse and isosceles, etc.). The following classification requirement is violated here: as a result of classification at each stage, the resulting classes (types) should not overlap.
4. In the same time classification for any reason must be exhaustive and each object of the concept must fall as a result of classification into one and only one class.

Therefore, the division of all integers into positive and negative is incorrect, because the integer zero did not fall into any of the classes. We should say this: integers are divided into three classes - positive, negative and the number zero.

Often, when classifying concepts, only some classes are clearly distinguished, and the rest are only implied. So, for example, when studying algebraic expressions, only such types of them are usually distinguished: monomials, polynomials, fractional expressions, irrational. But these types do not exhaust all types of algebraic expressions, therefore such a classification is incomplete.

Complete correct classification of algebraic expressions can be done as follows.

At the first stage of the classification of algebraic expressions, they are divided into two classes: rational and irrational. At the second stage, rational expressions are divided into whole and fractional ones. In the third step, whole expressions are divided into monomials, polynomials, and complex whole expressions.

This classification can be presented as follows

Assignment 7

7.1. Why can't rational numbers be classified according to their parity?

7.2. Establish whether the division of the concept is correct:

a) The values ​​can be equal or unequal.

b) Functions are increasing and decreasing.

c) Isosceles triangles can be acute-angled, rectangular and obtuse-angled.

d) Rectangles are squares and rhombuses.

7.3. Divide the concept " geometric figure"by property to occupy a part of the plane and give examples of each type.

7.4. Build possible classification schemes for rational numbers.

7.5. Build a classification scheme for the following concepts:

a) a quadrangle;

b) two corners.

7.6. Classify the following concepts:

a) triangle and circle;

b) corners in a circle;

c) two circles;

d) line and circle;

e) quadratic equations;

f) a system of two equations of the first degree with two unknowns.

Ostensive definitions are such definitions that introduce a concept by demonstrating, showing objects that are designated by this term.

Mathematics, unlike other sciences, studies the world around us from a special perspective. Any mathematical objects this is the result of the separation from objects and phenomena of quantitative and spatial properties and relationships. That. mathematical objects don't really exist. These are ideal concepts, they exist only in the thinking of a person and in those signs and symbols that form a mathematical language. Moreover, in the formation of mathematical concepts, in addition to abstraction, they are credited with such saints that no real object possesses.

Basic mathematical concepts: point, line, plane, plurality, number, value, arithmetic operation.

Any mathematical concept is characterized by a term, volume and content.

A term is a word or group of words, which are called elements of a set. The scope of a concept is many of all objects designated by the same term. Distinguish between essential and non-essential properties of objects. Sv-in will be essential if it is inherent in the object, and without it the object cannot exist. Insignificant - the absence of which does not affect essential objects.

a-concept of parallelogram; in-the concept of a rectangle; √вс√а and generic for в; in-specific for a; c-concept of a quadrangle. √а с√с

One and the same concept, for example, a parallelogram, can be generic for the concept of a rectangle or specific for the concept of a quadrangle.

The concepts of isosceles treug. And a rectangular triangle. They are not in a genus-specific relationship. There is a relationship between concepts as part and whole.

For example, a ray is a part of a straight line, a segment is a part of a straight line, an arc is a part of a circle.

If concepts are in genus-specific relations, then there is such a connection between the scope of the concept and its content: the larger the volume, the less its content, and vice versa.

Definition of concepts is a logical operation that reveals the content of a concept. It indicates those essential sv-va, which are sufficient for its recognition. Definitions are divided into explicit and implicit (indirect). Explicit definitions take the form of equality, the coincidence of two concepts.

Example: Parallelogram is called. a quadrilateral whose sides are pairwise parallel. but there is in; a is a parallelogram (a defined concept; b is a quadrilateral, the sides of which are pairwise parallel (a defining concept; a = r + v

Defined concept = generic concept + species difference

Generic: The bisector of the angle is called. ray emerging from the apex of the angle and bisecting the angle / r-generic concept: ray; v-species concept: outgoing from the apex of the angle and bisecting the angle. In elementary school, an explicit definition of genus and species distinction is rarely used. Example: Definition of an even number, rectangle, square, multiplication.

Explicit definitions can have a different structure: a) genetic definitions. A triangle is a figure consisting of 3 points that do not lie on one straight line, and 3 segments connecting them in series. Generic concept and method of construction.

b) recurrent (recursion-return) Arithmetic progression is called a numerical sequence, each term of which, starting from the second, is equal to the previous one, added to the number d constant for a given sequence (difference).

In elementary school, implicit definitions prevail. Implicit definitions are contextual and ostensive. Contextual definitions - in these definitions, the content of new concepts is revealed through the context, analysis of a specific situation that describes the meaning of the concept being introduced. Example: 2 + x = 5

2. Students of primary grades are offered tasks:

1) Which figure is superfluous? Explain the answer.

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, 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 encounter the concepts: point, line, angle, and then - with a whole system of concepts related to 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 the goals of the general education school require.

"The main drawback of the school assimilation of concepts is formalism," says the psychologist NF Talyzina. The essence of formalism is that students, 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. Consequently, the formation of concepts is important, actual problem.

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

Purpose of work: develop guidelines for the study of mathematical concepts in grades 5-6.

Work tasks:

1. To study mathematical, methodological, pedagogical literature on this topic.

2. To identify the main ways of defining concepts in the 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 for the most part are determined with the help of construction, and often the formation of the correct idea of ​​the concept among students is achieved with the help of explanatory descriptions;

· Concepts are introduced in a concrete-inductive way;

· Throughout the process of forming a concept, 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 textbooks on mathematics;

· Experienced teaching.

Fundamentals of Methods for Studying Mathematical Notions

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 needs of science and have no analogues in real world; they have a high degree of abstraction. By virtue of this, it is desirable to show students the emergence of the concept being studied (either from the need of practice, or from the need 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. Consider the relationship between the scope and content of a concept. If the content corresponds to reality and does not include contradictory signs, then the volume is not an empty set, which is important to show to students when introducing a 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 must be carried out on one basis;

o classes should be disjoint;

o union of all classes should give the whole set;

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

There are the following types of classification:

1. On a modified basis. The objects to be classified can 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 species concepts, one of which has this feature, and the other does not.

Example .

Let's highlight the objectives of teaching classification:

1) development of logical thinking;

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

Both types of classification are used in the school. As a rule, at first it is dichotomous, and then according to a modified character.