The concept itself has its strict logical system. Under certain conditions, the connotation and extension of a concept are fixed, which is the certainty of the concept. Due to the continuous development and change of objective things and the deepening of people's understanding, the concept of reflecting the essential attributes of objective things is also constantly developing and changing. The concept teaching in primary school is often carried out in stages, taking into account the acceptance ability of primary school students. For example, the concept of "number" has different requirements at different stages. At first I only knew 1, 2, 3, ..., and then I gradually knew zero. With the growth of students' age, I introduced fractions (decimals), and then gradually introduced positive numbers and negative numbers, rational numbers and irrational numbers, extending numbers to the range of real numbers and complex numbers. Another example is the understanding of "0". At first, we only knew that it meant no, and later we knew that it could mean no unit in digits, and we also knew that "0" could mean boundary and so on.
Therefore, the systematicness and development of mathematical concepts and the stages of concept teaching have become a pair of contradictions to be solved in teaching. The key to solve this contradiction is to grasp the phased goal of concept teaching.
In order to strengthen concept teaching, teachers must carefully study textbooks, master the system of primary school mathematics concepts, and understand the context of concept development. Concepts are developing step by step, and they are interrelated. Different concepts have different specific requirements, even the same concept has different requirements at different learning stages.
The meanings of many concepts are gradually developed, which are generally given by description and then defined. For example, three leaps in understanding the meaning of fractions. For the first time, before learning decimals, let students have a preliminary understanding of fractions. "As mentioned above,,,,, and so on are all fractions." Through a lot of perceptual and intuitive understanding, combined with specific things to describe what a fraction is, we can initially understand that a fraction is an average score and understand who is who's score. The second leap is from concrete to abstract, and the unit "1" is divided into several parts on average, indicating that one or more of them can be expressed by fractions. Abstractfrom concrete things. Then the definition of score is summarized, but the concept of score is given descriptively. This is a leap of sensibility. The third leap is the understanding and expansion of the unit "1". The unit "1" can not only represent an object, a graph, a unit of measurement, but also a group. Finally, it is abstracted that whoever divides is the unit "1", so that the three levels of units "1" and "1" cannot be achieved overnight. It is necessary to show the development process of knowledge and guide students to understand scores in the process of knowledge development.
Another example is the understanding of cuboids and cubes. Many textbooks are divided into two stages. In the lower grades, the initial understanding of cuboids and cubes first appeared. Students can observe some physical objects and physical drawings, such as cartons containing ink bottles and Rubik's cubes. Accumulate some perceptual knowledge about cuboids and cubes, know what shapes they are and know the names of these shapes. Then through operation and observation, we can know how many faces a cuboid and a cube have and what shape each face is, so as to further deepen our perceptual understanding of cuboid and cube. Then abstract the figures (not perspective views) of cuboids and cubes from the objects. However, in this teaching stage, students are required to know the names of cuboids and cubes, and to be able to identify and distinguish these shapes. Just stay at the level of perceptual knowledge. The second stage is in the senior grade. Teaching should be introduced from examples. When teaching the knowledge of cuboids, let students collect the objects of cuboids first. The teacher first explains what are the faces, edges and vertices of a cuboid. Ask students to count the number of faces, edges and vertices respectively, measure the length of edges, calculate the size of each face, and compare the relationship and difference between up and down, left and right, front and back edges and faces. Then the characteristics of cuboids are summarized. Then abstract the geometry of the cuboid from the instance of the cuboid. Then, students can observe the figure by comparing it with the real thing, and find out how many faces and edges they can see at most without changing the observation direction. What is invisible and how to show it in the picture. Students can also think about it and have a look, gradually understand the geometric shape of a cuboid and form a correct representation.
When grasping the phased objectives, we should pay attention to the following points:
(1) At every teaching stage, the concept should be clear, so as not to cause conceptual confusion. Some concepts are not strictly defined, but according to students' acceptance ability, they should either replace definitions with descriptions or reveal the essential characteristics of concepts in plain language. At the same time, pay attention to strict definition in the future.
(2) When a teaching stage is completed, it should be pointed out that the concept develops and changes according to the specific situation. For example, after a student knows the cuboid, he thinks that any piece of paper in the textbook is also a cuboid. It shows that students have a further understanding of the concept of cuboid, and teachers should affirm it.
(3) When developing concepts, teachers should not only point out the connections and differences between original concepts and development concepts, so that students can master them, but also guide students to learn related concepts and pay attention to their development and changes. For example, the concept of "multiple" usually means that if the quantity of A is regarded as 1 and there are so many quantities of B, then the quantity of B is several times that of A. After the introduction of fraction, the concept of "multiple" has been developed, which includes the original concept of "multiple". If the quantity of A is regarded as L, the quantity of B can also be a fraction of the quantity of A. ..
Therefore, in the teaching of mathematical concepts, it is necessary to clarify the order of concepts and understand the internal relations between concepts. With the development and change of objective things and the deepening of research, mathematical concepts are constantly evolving. Students' understanding of mathematical concepts also needs to be gradually deepened with the improvement of mathematics learning. When teaching, we should not only pay attention to the teaching stage, but also mention the later requirements to the front, which is beyond the students' cognitive ability; We should also pay attention to the continuity of teaching, leave room for the teaching of the previous concepts, and lay the groundwork for the later teaching. So as to deal with the relationship between the stages and continuity of mastering concepts.
2. Strengthen intuitive teaching and handle the contradiction between concrete and abstract.
Although most of the concepts in the textbook are not strictly defined, they are all based on the actual cases or existing knowledge and experience that students know, and try to help students understand the essential attributes of concepts through intuitive and concrete images. For the concepts that are not easy to understand, we will not give a definition for the time being or adopt the method of gradual infiltration in stages to solve them. But for primary school students, mathematical concepts are abstract. When they form mathematical concepts, they generally need to have corresponding perceptual experience as the basis. It takes some time to put perceptual materials back and forth in their minds, from vague to gradually clear, and from many related materials, through their own operations and thinking activities, they gradually establish the general appearance of things and separate the main essential characteristics or attributes of things, which is the basis for forming concepts. Therefore, in teaching, we must strengthen intuition and solve the contradiction between the abstraction of mathematical concepts and the visualization of students' thinking.
(1) Transform concreteness and abstraction through demonstration and operation.
In teaching, we should try our best to transform some relatively abstract contents into concrete contents through appropriate demonstrations or operations, and then abstract the essential attributes of concepts.
The basic knowledge of geometry, whether it is the concept of line, surface and body, or the concept of characteristics and properties of graphics, is very abstract. Therefore, we should strengthen demonstration and operation in teaching, so that students can understand these concepts through testing, touching, swinging and spelling, and thus abstract them.
For example, the concept of "pi" is very abstract. Some teachers arrange for each student to make a circle with a self-defined radius before class. In class, let each student write three contents in the classroom exercise book: (1) Write the diameter of the circle he made; (2) Roll the circle by yourself, measure the length of the circle once, and write it in the exercise book; (3) Calculate that the circumference of a circle is several times the diameter. At the end of the course, each student is required to report his own calculation results and organize the results into the following table.
The diameter of a circle (cm) The circumference of a circle (cm) is several times the diameter.
A26.23. 1
B39.63.2
C4 12.63. 15
D5 15.73. 14
Then guide the students to analyze and find that no matter the size of a circle, its circumference is always a little more than three times its diameter. At this time, it is revealed that this multiple is a fixed number, which is mathematically called pi. Then ask the students to draw a circle at will and measure the diameter and circumference to verify. In this way, students are guided to analyze, synthesize, abstract and summarize a large number of perceptual materials, and abandon the non-essential attributes of things (such as the size of a circle and the units used in measurement). ), grasp the essential characteristics of things (the circumference is always a little more than three times the diameter), and form a concept.
In this way, with the help of intuitive teaching, teachers make use of students' original basic knowledge, and gradually become abstract, closely linked and clear-cut. Through physical demonstration, students can establish representations, thus solving the contradiction between the abstraction of mathematical knowledge and the visualization of children's thinking.
(2) Combining with the actual life of students, the concrete and abstract transformation is carried out.
Many quantitative relations in teaching are abstracted from concrete life contents. Therefore, in teaching, we should make full use of students' real life and adopt appropriate methods to transform concrete and abstract, that is, to transform abstract content into students' specific life knowledge, and then abstract students' life knowledge into teaching content.
For example, in the teaching of multiplication and method of substitution, students are often asked to answer such exercises first: How much does it cost to buy two boxes of pens for a box of 10 pens and a 3 yuan? Students find that there are two ways to solve this problem. One way is to find out "how much is a box" first, and then find out "how much is two boxes". The formula is (3× 10)×2=60 yuan; The other is to find out how many pens a * * * has first, and then find out how much two boxes are. The formula is 3×(2× 10)=60 yuan. The teaching of Multiplication and Distribution Law also allows students to answer similar questions, such as: a coat, 50 yuan, a pair of pants, 30 yuan. How much does it cost to buy five suits like this? In this way, with the help of students' familiar life scenes, abstract problems become concrete.
The relationship between unit price, total price and quantity in the same common quantity relationship; The relationship between distance, speed and time, workload, work efficiency and working time should be abstracted through specific topics in combination with students' life experience, and then these relationships should be used to analyze and solve problems. This kind of training is conducive to the gradual transition of students' thinking to abstract thinking, and gradually relieves the contradiction between the abstraction of knowledge and the concrete image of students' thinking.
However, using intuition is not an end, it is just a means to arouse students' positive thinking. So concept teaching can't just stay in perceptual knowledge. After students acquire rich perceptual knowledge, they should abstract and summarize the observed things, reveal the essential attributes of concepts, make knowledge leap from perceptual to rational, and form concepts.
3. Follow the characteristics of primary school students' learning concept and organize a reasonable and orderly teaching process.
Although primary school students acquire concepts in two basic forms: concept formation and concept assimilation, and the formation of each concept has its own characteristics, however, how to acquire concepts generally follows the concept formation path of "introduction, understanding, consolidation and deepening". The following is a description of the teaching strategies and problems that should be paid attention to in all aspects of concept teaching.
The introduction of the concept of (1) should pay attention to providing rich and typical perceptual materials.
In the process of introducing concepts, we should pay attention to let students establish clear representations. Because the establishment of clear typical representations that can highlight the essence of things is an important basis for the formation of concepts, no matter how to introduce concepts in the concept teaching of primary school mathematics, we should consider how to let primary school students establish clear representations in their minds. At the beginning of concept teaching, we should provide students with rich and typical perceptual materials by intuitive means, such as objects, models, wall charts or demonstrations, etc., to guide students to observe, and let students operate by themselves in combination with experiments, so that students can get in touch with related objects and enrich their perceptual knowledge.
For example, in a class about the meaning of teaching scores, a teacher provided students with various operating materials in advance in order to break through the teaching difficulties of unit "L": a rope, four apples, six pandas, a rectangular piece of paper, a line segment with a length of L meters, etc. Through comparison, it is concluded that an object, a unit of measurement and a whole can use the unit "1".
However, when introducing concepts, we should pay attention to three points: first, the selection of materials should be accurate. For example, the understanding of angle, the angle in primary school is a plane angle, so that students can observe the angle on the blackboard, writing and other planes. Some teachers ask students to observe the angle between two adjacent walls of the classroom, which is two sides, which is not exact for the teaching requirements of primary schools. Second, the selected materials should highlight the essential characteristics of the knowledge taught. For example, the essential feature of a right triangle is "a triangle with a right angle". As for which angle in the triangle this right angle is, the size and shape of the right triangle are not important. Therefore, different graphics should be presented in teaching, so that students can identify their unchanging essential attributes in different graphics.
(2) To understand the concept, we should pay attention to the discrimination of positive and negative examples and highlight the essential attributes of the concept.
The understanding of concept is the central link of concept teaching. Teachers should take all means to help students gradually understand the connotation and extension of concepts, so that students can master concepts on the basis of understanding. Ways to promote conceptual understanding are:
1) Analyze the true meaning of keywords in concepts.
For example, the unit "1", "average score" and "number representing this share" in the definition of score, students will have a deep understanding of the concept of score only if they understand the true meaning of these keywords. For another example, after teaching the concept of "divisibility", students should be helped to judge from the following three aspects: first, the two numbers to judge whether there is a "divisibility" relationship must be natural numbers; Second, the quotient obtained by dividing these two numbers is an integer; Third, there is no remainder. The analysis of the definition is another improvement to help students understand the concept. Definition of triangle height: "Draw a vertical line from the vertex of the triangle to its opposite side. The line between the vertex and the vertical foot is called the height of the triangle, and this side is called the bottom of the triangle. " Here "a vertex", "vertical line" and "vertical foot" are all key words. In order to let students understand the height of a triangle, in addition to the literal meaning, they often need to experience the whole process of drawing "height" through practical operation. It is pointed out that the key to draw "height" is to draw a vertical line, and pay attention to the restriction condition: "Crossing a vertex of a triangle (which can be any vertex) is the vertical line on the opposite side and the line segment between the vertex and the vertical foot". In this way, the actual operation process and the drawn figure of triangle height are compared with the content described in the definition, so that students can accurately understand the definition of triangle height. This is actually to help students analyze the essential attribute after the establishment of mathematical concepts, which not only separates the essential attribute from the definition again, but also makes it clear.
2) Distinguish the positive examples and counterexamples of concepts.
Students' ability to recite concepts does not mean that they really understand them, but also highlights the main characteristics of concepts through examples to help them deepen their understanding of concepts. Teachers should not only make full use of positive examples to help students understand the connotation of concepts, but also make timely use of negative examples to promote students' discrimination of concepts. After the concept is revealed, it is often necessary to organize students to do some exercises according to the teaching requirements. For example, after teaching the classification of triangles by angle, it can be explained that one triangle is not a right triangle and two angles are acute angles. This triangle must be an acute triangle. Let the students make judgments, and lead them to discuss the classification of consolidating triangles, so as to deepen their understanding of the extension of triangle concepts. For another example, after the nature of decimals is revealed, students can judge the numbers 0.40, 0.030, 20.020, 2.800, 10.404 and 5.0000, which "0s" can be removed and which "0s" cannot be removed. So as to deepen students' understanding of the essence of decimals.
3) Transform the narrative or expression of essential attributes.
One of the characteristics of primary school students' understanding and mastery of a concept is that the connotation of a concept is unclear and incomplete, and non-essential characteristics are regarded as essential characteristics. For example, some students mistakenly think that only a rectangle placed horizontally is a rectangle, and they can't recognize it if they put it sideways. Therefore, it is often necessary to change the narrative or expression of concepts so that students can understand concepts from all aspects. The purpose is to grasp the essential attribute of the concept from the variant and eliminate the interference of the non-essential attribute. Because the essential attributes of things can be expressed in different languages, if students can understand and master various narrative and expression methods, it means that students' understanding of concepts is thorough and flexible, rather than rote learning.
4) Contrastively analyze the concept of time approximation.
In primary school mathematics, some concepts have similar meanings, but their essential attributes are different. Such as number and number, number and number, odd number and prime number, even number and composite number, simplification and ratio, time and moment, prime number, prime factor and prime number, perimeter and area, etc. Students are often confused about such concepts, so we must compare them in time to avoid mutual interference.
If you are studying "divisibility", in order to compare with the previous "divisibility", you can design an exercise: Which of the following equations is divisibility and which is divisibility?
( 1)8÷2=4(2)48÷8=6
(3)30÷7=4……2(4)8÷5= 1.6
(5)6÷0.2=30(6) 1.8÷3=0.6
Guide the students to draw the following conclusions through analysis and comparison: the third question is division with remainder, of course, the dividend cannot be said to be divisible or divisible, and other questions can of course be said to be divisible. Among them, there are only questions (1) and (2), and the dividend, divisor and quotient are natural numbers with no remainder. These two problems can be said that the dividend can be divisible by the dividend, and it can also be divisible by the dividend. From the above analysis, let students understand that divisibility is a special case of division, including divisibility and the case that all vendors are finite decimals.
After learning ratio, you can use the list method to design exercises about the relationship between ratio, division and fraction, and make clear the difference between "division is operation, fraction is number and ratio is relationship".
3) Pay attention to the application of concepts and give full play to the role of concepts.
The correct and flexible use of concepts requires students to use concepts correctly and flexibly to judge, reason, calculate and draw. And use concepts to analyze and solve practical problems. The purpose of understanding the concept is to use the concept, and the methods of using the concept are:
1) Boot instance
This requires students to simply apply the concepts obtained to practice, illustrate the concepts through examples, and deepen their understanding of the concepts. Experienced teachers, according to the characteristic that primary school students' understanding of concepts is usually concrete, always let students concretize concepts with examples after analysis, synthesis, abstraction and generalization. From concrete to abstract, and then back to concrete, it conforms to the cognitive law of primary school students and enables students to grasp the connotation and extension of concepts more accurately.
For example, after students get the concepts of true score and false score, they can give some examples of true score and false score respectively. After understanding the characteristics of cylinders, let the students talk about what objects are cylindrical in daily life.
2) Used for calculation and drawing.
For example, students can easily calculate the following questions after learning the algorithm of multiplication.
104×2548×25 10 1×35×2
14×99+ 1425×32 146+9× 146
(80+8)×258×( 125+50)34×5×2
After mastering the basic nature of scores, students are required to skillfully make general scores and converted scores, and explain the basis of general scores and converted scores. After learning the nature of decimals, students can simplify or rewrite decimals as needed. After learning isosceles triangle, you can design a set of operation problems; Draw an isosceles triangle; Draw an isosceles triangle with a vertex angle of 60 degrees; Draw an isosceles right triangle with a waist length of 2 cm.
3) Apply it to life practice
The concept of mathematics comes from life, so it must return to the reality of life. Teachers guide students to use concepts to solve mathematical problems, which is a process of cultivating students' thinking and developing various mathematical abilities. Moreover, only by letting students apply the mathematical concepts they have learned to real life can they consolidate the concepts they have learned and improve their skills in using mathematical concepts. Therefore, teachers should consciously deepen and develop students' mathematical concepts on the basis of mastering the logical system of primary school mathematics textbooks according to the content of textbooks and the reality of students.
For example, after learning the area of a circle, a teacher designed such a question: "We learned the area formula of a circle. Who can find a way to calculate the cross-sectional area of poplar trunks on the school playground? " The students discussed it, and some said that the radius must be known first to calculate the circular area, and the radius can only be calculated if the tree is cut down; Some people don't agree with this, thinking that trees will die as soon as they are cut down. At this time, the teacher further guided and said, "Then can you come up with a method to calculate the cross-sectional area without cutting down trees?" Let's discuss it again. "Students are eager to get the right answer. Through active thinking and argumentation, they finally found a good way, that is, first measure the circumference of the trunk, then calculate the radius, and then use the area formula to calculate the cross-sectional area of the tree. After class, many students went to the playground to actually measure the circumference of the trunk and calculate the cross-sectional area. For another example, when teaching proportional application problems, students can be inspired to use the relationship between the height of flagpole and the length of shadow to calculate the height of flagpole skillfully. In this way, by creating effective teaching situations and teachers' timely guidance, students' thinking is not only inspired, but also their interest and ability to apply what they have learned are cultivated and their understanding of the concepts they have learned is deepened.
(4) Pay attention to the comparative classification between concepts and deepen the concepts.
The characteristics of primary school mathematics knowledge are strong systematicness and close connection. However, due to the limitation of primary school students' thinking development level and acceptance ability, the teaching of some knowledge is often completed in several classes or semesters, which inevitably weakens the connection between knowledge to varying degrees. At a certain stage, we should systematically sort out some related concepts or laws, so that students can establish a network of knowledge in their minds and form a good cognitive structure. Especially in the middle and senior grades, students can be guided to classify concepts, clarify the connections and differences between concepts, and form a concept system.