According to the psychological process and characteristics of mathematical concept learning, the teaching of mathematical concepts is generally divided into three stages: ① introducing concepts to make students perceive concepts and form representations; ② Through analysis, abstraction and generalization, students can understand and clarify the concepts; ③ Make students consolidate and apply concepts through examples and exercises.
(A) the introduction of mathematical concepts
The introduction of mathematical concepts is the first and very important link in the teaching of mathematical concepts. If the concept is introduced properly, it can closely focus on the theme, fully stimulate students' interest and learning motivation, and lay the foundation for students to master the concept smoothly.
The process of introducing new concepts is to reveal the occurrence and formation process of concepts. The occurrence and formation process of each mathematical concept is different, and some are direct reflections of realistic models. Some are obtained after one or more abstractions on the basis of existing concepts; Some come from the needs of the development of mathematical theory; Some are produced to solve practical problems; Some idealize the object of thinking and get it through reasoning; Others arise from theoretical existence or the structure of mathematical objects. Therefore, in teaching, we must choose different ways to introduce concepts according to the background of various concepts and the specific situation of students. Generally speaking, the introduction of mathematical concepts can adopt the following methods.
1. Introduce a new concept based on perceptual materials.
Use the things that students come into contact with in daily life or the practical problems in textbooks, as well as models, graphs and charts as perceptual materials to guide students to acquire concepts through observation, analysis, comparison, induction and generalization.
For example, by learning the concept of "parallel lines", students can identify some familiar examples, such as rails, the upper and lower edges of doorframes, and the upper and lower edges of blackboards. , and then analyze the attributes of each example to find out the essential attributes of * * * *. Rails have properties: they are made of iron and can be regarded as two straight lines. On a plane, the two sides can extend indefinitely and never intersect. You can also analyze the properties of the upper and lower sides of the door frame and blackboard. By comparison, we can find that they have the same properties: they can be abstractly regarded as two straight lines; Two straight lines are on the same plane; The distance between them is equal everywhere; There is nothing in common between the two straight lines. Finally, the essential attributes are abstracted and the definition of parallel lines is obtained.
The introduction of new concepts based on perceptual materials is taught by the way of concept formation. Therefore, in teaching, we should choose those cases that can fully show the characteristics of the introduced concepts and correctly guide students to observe and analyze. Only in this way can students summarize and generalize the essential attributes of * * * from cases and form concepts.
2. Introduce new concepts with the relationship between old and new concepts.
If there is a certain relationship between the old and new concepts, such as compatibility and incompatibility, then the introduction of new concepts can make full use of this relationship.
For example, when learning the meaning of multiplication, we can introduce it from the meaning of addition. For another example, when learning the concept of "divisibility", it can be introduced from "division". For another example, learning "prime factor" can be introduced from the concepts of "factor" and "prime number". For another example, when learning the concepts of prime numbers and composite numbers, the concept of divisor can be introduced: "Please write down all divisors of the numbers 1, 2, 6, 7, 8, 12,1,15. How many divisors do they have? Can you give a classification standard to classify these figures? Can you find a variety of classification methods? Of all the classification methods you have found, which is the latest? "
3. Introduce new concepts in the form of "problems".
Introducing new concepts in the form of "questions" is also a common method in concept teaching. Generally speaking, there are two ways to introduce concepts with "problems": ① to introduce mathematical concepts from real life problems; ② Introduce concepts from the development of mathematical problems or theories themselves.
For example, when studying Average, teachers can first present students with a life situation of "kindergarten children fighting for candy", so that students can think about why some children are happy and others are unhappy. What should I do to make everyone happy? What should we do next? What are the teachers in this kindergarten likely to do?
4. Introduce new concepts from the process of concept emergence.
Some concepts in mathematics are defined by occurrence. In the teaching of this kind of concept, we can use the visual teaching AIDS of demonstration activities or the method of demonstration drawing to reveal the happening process of things. Concepts such as decimals and fractions can be introduced in this way. This method is intuitive and reflects the viewpoint and thought of movement change. At the same time, the introduction process naturally and irrefutably clarified the objective existence of this concept.
(2) Introducing concepts into the formation of mathematical concepts in primary schools is only the first step in concept teaching. In order to get the concept, students must be guided to understand the concept accurately, to make clear the connotation and extension of the concept, and to correctly express the essential attributes of the concept. Therefore, some targeted methods can be adopted in teaching.
1, comparative analogy.
By comparing concepts, we can find out the differences between concepts, and by analogy, we can find out the similarities or similarities between concepts. For example, when learning the concept of division, we can compare it with the concept of division and find the difference between them. To talk about a new concept by comparison or analogy, we must highlight the difference between the old and new concepts, clarify the connotation of the new concept, and prevent the old concept from having a negative transfer effect on learning the new concept.
2. Use counterexamples appropriately.
In concept teaching, besides revealing the connotation of the concept from the front, we should also consider using appropriate counterexamples to highlight the essential attributes of the concept, especially by comparing the differences between positive examples and counterexamples, so that students can reflect on their mistakes, which is more conducive to strengthening their understanding of the essential attributes of the concept.
Using counterexamples to highlight the essential attributes of concepts, its essence is to let students know the extension of concepts and deepen their understanding of the connotation of concepts. Any object with the essential attributes reflected by a concept must belong to the extension set of the concept, and the construction of counterexample is to let students find out the objects that do not belong to the extension set of the concept. Obviously, this is an important means in concept teaching. However, it must be noted that the counterexample should be appropriate to prevent students from being too difficult, too biased, distracting and unable to highlight the essential attributes of the concept.
3. Rational use of variants.
Understanding concepts by relying on perceptual materials is often because the perceptual materials provided are one-sided and limited, or the non-essential attributes of perceptual materials have obvious outstanding characteristics, which is easy to form interference information, thus weakening students' correct understanding of the essential attributes of concepts. Therefore, we should pay attention to the use of variants in teaching to reflect and describe the essential attributes of concepts from different angles and aspects. Generally speaking, variants include graphic variants, formula variants and letter variants.
For example, when teaching the concept of "isosceles triangle", teachers should not only use common graphics (Figure 6- 1( 1)), but also use variant graphics (Figure 6- 1(2), (3) and (4)) to strengthen this concept, because it is used.
(C) to consolidate the concept of primary school mathematics
In order for students to master the concepts they have learned, there must be a process of concept consolidation and application. Attention should be paid to the following aspects in teaching.
1, pay attention to review in time.
The consolidation of concepts is completed and realized in the understanding and application of concepts, and at the same time, it should be reviewed in time. Consolidation cannot be separated from necessary review. The way to review can be to repeat individual concepts, to review concepts by solving problems, and more to review concepts in the concept system. When concept teaching reaches a certain stage, especially at the end of chapter review, final review and graduation review, we should pay attention to sorting out and systematizing the concepts we have learned, find out the vertical and horizontal relations between concepts, and form a concept system.
2. Pay attention to application
In concept teaching, we should not only guide students to form concepts from concrete to abstract, but also let them use concepts from abstract to concrete. Whether students can grasp a concept firmly depends not only on whether they can say the name of the concept, but also on whether they can use it correctly and flexibly. Through application, they can deepen their understanding, enhance their memory and improve their awareness of mathematics application.
The application of concept can be carried out from the connotation and extension of concept.
Application of the concept of (1)
① Retell the definition of the concept or fill in the blanks according to the definition.
(2) According to the definition, judge right or wrong or correct mistakes.
③ Reasoning according to the definition.
④ Calculated according to the definition.
Example 4( 1) What is a prime number? This is a prime number.
(2) True or false:
27 and 20 are prime numbers ()
34 and 85 are prime numbers ()
Two numbers with common divisor 1 are prime numbers ()
Two composite numbers cannot be prime numbers ()
(3) One angle of an obtuse triangle is 82o, and the degrees of the other two angles are prime numbers. How many degrees can these two angles be?
(4) If P is a prime number, then all natural numbers less than P are coprime with P ... Is this correct? Please explain why?
2. Application of concept extension
(1) example
(2) Identify positive or negative examples. And explain why.
(3) Select cases from the extension of the concept according to the specified conditions.
(4) Classify concepts according to different standards.
Example 5( 1) lists the cylindrical objects you have seen.
(2) Which shaded parts in the picture below are fan-shaped? (Figure 6-2)
Figure 6-2
(3) The simplest true fraction with denominator of 9 has a false fraction with numerator of 9, and the smallest is
(4) Natural numbers 2- 19 are divided into two categories according to different standards (at least three different classification methods are proposed).
The application of concepts can be divided into simple application and comprehensive application. After a new concept is initially formed, simple application can promote the understanding of the new concept. Comprehensive application is generally after learning a series of concepts, combining these concepts can cultivate students' comprehensive application ability.
Pay attention to discrimination
With the deepening of learning, students have more and more concepts, some of which are expressed in the same words, some of which are similar in connotation, which is easy to confuse students, such as prime number and prime number, divisibility and division, volume and volume. Therefore, in the consolidation stage of concepts, we should pay attention to organizing students to use comparative methods to understand the differences and connections of confusing concepts, so as to promote the accurate distinction of concepts.
Regarding the area and perimeter, students can be organized to meet in the following aspects.
(1) What is the circumference of a rectangle? What is the area of a rectangle?
(2) What are the commonly used units of measurement for perimeter and area?
(3) In Figure 6-3, are the perimeters of two figures A and B equal? Are the areas equal?
Figure 6-4
Figure 6-3
(4) Each small square in Figure 6-4 represents a square centimeter. The area of this figure is, the perimeter is, cut a knife, and then put it into a square. The perimeter of this square is, and the area is.
Mathematical concepts are expressed by words or phrases, but some words are influenced by daily language, which will cause illusions and obstacles to students' understanding. For example, the concepts of height, bottom and waist in geometry knowledge can easily make students have the illusion of "vertical direction" and literally "below" and "side". "Reciprocal" strengthens the intuitive understanding of the inversion of numerator and denominator, and weakens the essential attribute of "the product of two numbers equals 1". Therefore, in teaching, students should be helped to distinguish the daily meaning of some words from the professional mathematics meaning, correctly understand the words representing concepts, and thus accurately grasp the concepts.