The so-called mathematical concept is the essential attribute of things in quantitative relationship and spatial form, which is formed by people extracting their essential attributes from many attributes of the objects studied by mathematics through practice. It refers to those mathematical nouns and terms. Numbers, figures, symbols, nouns, terms, definitions, rules, etc. , reflecting the essential attributes of numbers and shapes in primary school mathematics, are both mathematical concepts. )

Mathematical concepts are the foundation of mathematical reasoning and judgment, the foundation of establishing mathematical theorems, rules and formulas, and the starting point of forming mathematical thinking methods. Therefore, the key to learn mathematics well is the study of mathematical concepts, and the teaching of mathematical concepts is an important part of mathematics teaching.

First, the meaning and definition of mathematical concepts

The formation of mathematical concepts starts from a large number of practical examples, finds out the essential attributes of a class of things through comparison and classification, then tests and corrects the discovered attributes through specific examples, and finally gets the definition through generalization and represents it with symbols. In fact, it should contain two meanings: first, mathematical concepts represent a kind of things, not individual things. For example, "triangle" can be represented by the symbol "△". At this time, any figure with three angles and three sides like "△", regardless of size, is called a triangle, that is to say, the concept of triangle refers to all triangles: equilateral, isosceles, unequal, right angle, acute angle and obtuse angle. Secondly, mathematical concepts reflect the essential attributes of a class object, that is, the intrinsic and inherent attributes of the class object, rather than those superficial non-essential attributes. For example, the concept of "circle" embodies that "the distance between a fixed point on a plane is equal to a set of points with fixed length". According to these properties, we can distinguish "circle" from other concepts.

We call the sum of the * * * and essential attributes of all objects reflected by a concept the connotation of this concept, and the scope of all objects suitable for this concept the extension of this concept. Generally speaking, defining a concept is to imply connotation or extension. Generally speaking, there are several ways to define mathematical concepts:

1. Definition of convention

Because of the development of mathematics itself, terms are sometimes given specific meanings through regulations. For example, "the zeroth power of a number that is not equal to zero is equal to 1", which defines the meaning of zero exponential power, but it should be noted that the conventional formula cannot follow one's inclinations and must conform to objective laws.

2. Descriptive definition

Mathematics is a rigorous science, and every new concept is always defined by some known concepts, which must be described by other known concepts, thus forming a series of concepts. In the concept of series, cycles are not allowed. Therefore, there are always some concepts that cannot be defined by other concepts. This kind of concept, known as the basic concept in mathematics, is also called "original name" (or undefined concept or original concept), and its meaning can only be vividly described by means of other terms and their respective characteristics. Such as: points, lines and surfaces in geometry, sets and elements in algebra.

3. Constructive definition

This definition is given by describing the occurrence and formation process of the concept itself. For example, an ellipse is defined as "the locus of a point, and the sum of its distances to two fixed points on a plane is equal to a fixed length, and the locus of this point is called an ellipse".

4. Definition of genus plus species difference

If a concept is subordinate to another concept, the latter is called the genus concept of the former, while the former is called the species concept of the latter. Real number is the general concept of rational number, and rational number is the concept of real number.

Under the same generic concept, the difference of attributes contained in each concept is called species difference. For example, for the general concept of quadrilateral, parallelogram and trapezoid are its two concepts, and their differences are: "two groups of opposite sides are parallel respectively" and "one group of opposite sides is parallel and the other group is not parallel".

To define a concept by genus plus species difference is to put a concept into another broader concept to describe its meaning. The usual method is to express it by adding species difference to adjacent genera. For example, the definition of parallelogram, its adjacent concept is quadrilateral, the difference is that two groups of opposite sides are parallel respectively, so the definition of parallelogram is expressed as "two groups of parallelograms with parallel opposite sides are called parallelograms".

In addition, in textbooks, you will encounter some methods to define concepts by revealing their extension. Definition of real numbers: "Rational numbers and irrational numbers are collectively called real numbers".

Finally, it needs to be declared that definition is the way of mathematical concepts, and the above analysis is relative and not strict. For example, the definition of "the angle formed by a straight line in a different plane" can be considered as conventional, that is, it is stipulated that "the acute angle or right angle formed by parallel lines of a straight line in a different plane passing through any point in space is called the angle formed by a straight line in a different plane", and it can also be understood as biological, that is, taking points as parallel lines to form two pairs of antipodal angles, and the acute angle or right angle is called the angle formed by a straight line in a different plane. In a word, our understanding of definition is not to distinguish which definition mode it belongs to, but to clarify the extension and connotation of the concept and then use the concept to solve the problem.

Second, how to teach mathematical concepts

Mathematical concepts, even those primitive concepts, cannot be literal. In teaching, we should grasp its connotation, which is the basis of mastering concepts; It is also necessary to understand its extension in order to understand and expand the concept; At the same time, we should understand all the provisions and conditions in the concept one by one and fully understand them, so that we will be more impressed and better grasped.

Generally speaking, around a mathematical concept, we should try to clarify the following aspects:

(1) reveals essential attributes. What is the object of this concept discussion and what is its background? What are the terms and conditions in this concept? How do they relate to what they have learned in the past? What exactly do these terms and conditions mean?

Give the definition, name and symbol of the concept, and reveal the essential attributes of the concept. For example, to learn the concept of quadratic function, first learn its definition: "Y = AX2+BX+C (A, B, C, is a constant. A≠0) then y is called the quadratic function of x ". Another example is that when a teacher is teaching "Understanding of Cuboid and Cube", after guiding students to introduce the concepts of Cuboid and Cube into the classification of objects with different shapes, he promptly guides students to draw all sides of Cuboid or Cube on paper, and carefully observes the characteristics of each side, so as to understand what is called "edge" and what is called "vertex". Make a model of "cuboid" or "cube", discuss the characteristics of vertices and edges of cuboid and cube while observing, and finally guide students to summarize and summarize the characteristics of "cuboid" and "cube" so that students can fully understand the connotation and extension of these two concepts.

② Discuss counterexamples and special cases. Classify concepts in a special way, discuss various special situations, and highlight the essential attributes of concepts. For example, the special cases of quadratic function are: y=ax2, y = ax2+c, y = ax2+bx and so on.

③ Contact between old and new knowledge. What are the terms and conditions in this concept? How do they relate to what they have learned in the past? Linking new concepts with related concepts in the original cognitive structure, incorporating new concepts into the corresponding concept system and assimilating new concepts. For example, link quadratic function with linear function and function, and put it into the system of function concept.

④ Instance confirmation. Identify positive examples and counterexamples, confirm the essential attributes of new concepts, and accurately distinguish new concepts from related concepts in the original cognitive structure. For example, give examples of Y = 2x+3, Y = 3x2-x+5 and Y =-5x2-6 for students to identify.

⑤ Specific application. According to the conditions and regulations in the concept, what basic properties can be summarized? What are the functions of these attributes in the application? Through various forms of concept application, we can deepen our understanding of new concepts and integrate related concepts into a whole structure.

Above, we just introduced the general model of concept teaching process. The whole process can be summarized into three stages:

(A) the way to introduce concepts

The concept of mathematics itself is abstract. Therefore, the introduction of new concepts must be based on students' understanding level and closely linked with production and life practice. Different concepts are introduced in different ways. For some primitive concepts and some abstract concepts, teachers should introduce them through a certain amount of perceptual materials and keep close contact with real life, so that students can "see and feel". When citing examples, we must grasp the original characteristics of the concept and pay attention to revealing the true meaning of the concept. For example, the concept of "plane" allows students to observe some things in life, such as desktop and calm water, and draw conclusions through their own exploration and communication with classmates. However, teachers must find ways to let students obtain the essential characteristics of "infinite extension and no thickness".

(B) the method of forming the concept

Understanding a special psychological process, because there are some differences between each student, the time required to complete this process is not necessarily the same. But as far as the cognitive process is concerned, it can't jump. In teaching, the introduction of concepts, so that students can initially grasp the definition of concepts, does not mean the formation of concepts. There must also be transformation and manufacturing from the rough to the fine, from the false to the true, from this to that, from the outside to the inside. We must dialectically analyze concepts on the basis of perceptual knowledge, and further remind the essential attributes of different concepts in different ways.

1. After mastering the essential attributes of the concept, students should be guided to do some exercises. For example, after introducing the concept of factorization, students can choose one of the following exercises to answer.

Which of the following transformations from left to right belongs to the decomposition factor? Which ones are not? Why?

①(x+2)(x-2)= x2-4；

②(a2-9)=(a+3)(a-3)；

③a3-9a = a(a2-9)；

④x2-y2+ 1 =(x+y)(x-y)+ 1；

⑤x2y+x=x2(y+ 1)

By answering questions, especially explaining the reasons, students' ability to make simple judgments by using concepts can be initially cultivated. At the same time, every time you make a judgment, the essential attributes of the concept will reappear in your mind. Therefore, it is effective to promote the formation of concepts.

2. Deepen the understanding of concepts through variants or graphics. Another example is learning the concept of trapezoid, which can provide the following graphics for students to observe:

Here we should pay attention to three points: first, the perceptual materials (trapezoid) should be sufficient, not too little, and there is no need to be too much. Too little is not conducive to students' understanding of the law and forming appearances; Too much will waste time and energy. Second, we should guide students to analyze and synthesize each material. sequence

Third, we should pay attention to variations, and all materials should reflect all the essential attributes of this point.

3. By comparing the old and new concepts, grasp the internal relationship between concepts and form a correct concept. Another example is teaching the concepts of divisor and multiple. We can start with the concept of "divisibility" and introduce the concept.

(C) the development of the concept

After students master a certain concept, it does not mean the end of concept teaching. They should teach concepts from a developmental point of view.

1. Lose no time to expand the meaning of the extended concept. A concept is always embedded in a group of concepts. There are criss-crossing internal relations between them, which must be revealed clearly. For example, after learning the meaning of comparison, we should link "comparison", "fraction" and "division" in time to find out their connections and differences, so that students can condescend and examine the concept of "comparison" in a broad background and deepen their understanding of the concept.

2. Form a certain understanding at a certain stage. Abstract concepts cannot exceed the requirements of textbooks, otherwise it will exceed the students' tolerance. For example, when students learn addition in senior one, they only realize that addition means "together" and "two numbers together", but they can't tell them the exact definition: "The operation of combining two numbers into one number is called addition".

In short, to improve the teaching level of mathematical concepts in primary and secondary schools, teachers should consciously cultivate students' mathematical thinking mode, quality, ability and methods in the practice of concept teaching. Deepening students' understanding of mathematical concepts is the premise and key for students to master mathematical knowledge and enhance their ability, and is the only way to learn knowledge well.