First, the importance of cultivating junior high school students' mathematical thinking methods
The so-called mathematical thought is the understanding of the essence of mathematical knowledge. It is the guiding ideology of establishing mathematics and solving problems with mathematics, such as modeling, statistics, optimization, reduction, classification, integration, combination of numbers and shapes, transformation, equations and functions. The so-called mathematical method refers to various ways, means and approaches adopted in the process of raising and solving problems (including internal problems and practical problems in mathematics) in mathematics. The mathematical methods that junior middle school students should master are collocation method, method of substitution method, undetermined coefficient method, parameter method, construction method and special value method. Mathematical thinking is closely related to mathematical methods. When emphasizing the guiding ideology, it is called mathematical thought, and when emphasizing the operation process, it is called mathematical method.
According to the requirements of the mathematics syllabus, the nine-year compulsory education syllabus has clearly included the mathematical thinking method into the basic knowledge category, which refers to the mathematical thinking method reflected by the concepts, properties, laws, formulas, axioms and their contents in mathematics. The mathematics content of middle school students includes mathematical knowledge and mathematical thinking methods. Mathematical thinking method produces mathematical knowledge, and mathematical knowledge contains thinking method, which is conducive to revealing the spiritual essence of knowledge and improving students' comprehensive quality and mathematical literacy.
From the perspective of education, mathematical thinking method is more important than mathematical knowledge. This is because: mathematical knowledge is stereotyped and static, while thinking method is developing and dynamic. The memory of knowledge is temporary, and the mastery of thinking methods is permanent. Knowledge can only benefit students for a while, and thinking methods will benefit students for life. Strengthening the cultivation of mathematical thinking methods is more important than imparting knowledge, and mastering mathematical thinking methods is beneficial to solving any practical problems. Therefore, mathematics teaching must attach importance to the teaching of mathematical thinking methods.
Practice has proved that cultivating junior high school students' mathematical thinking methods can effectively stimulate students' interest in learning, fully mobilize their enthusiasm and initiative in learning, constantly improve and develop their cognitive structure, enable students to apply existing thinking methods to the process of learning new knowledge, turn complex problems into simple ones to solve, improve learning efficiency and improve their ability to analyze and solve problems. At present, the combination of numbers and shapes, classified discussion, equations and functions are the focus of examination papers in various places. Therefore, we should also attach importance to the cultivation of junior high school students' mathematical thinking methods, and examining students' mathematical thinking methods is the only way to examine students' ability.
Second, the main mathematical thinking methods in junior high school
There are many mathematical thinking methods in junior high school mathematics, the most basic and main ones are: reduction thinking method, combination of numbers and shapes thinking method, classified discussion thinking method, function and equation thinking method and so on.
1. Corresponding ideas and methods
In the introduction teaching of algebra in grade one, there is a calculation problem of algebra evaluation. Through calculation, it is found that the value of algebraic expression is determined by the value of letters in algebraic expression, and different letter values can get different calculation results. Here, there is a correspondence between the value of letters and the value of algebra, then there is a correspondence between real numbers and points on the number axis, and there is a correspondence between ordered real numbers and points on the coordinate plane ... In this teaching design, attention should be paid to infiltrating corresponding ideas, which not only helps students to see problems from a changing point of view, but also helps to cultivate their functional concepts.
2. The idea and method of combining numbers and shapes.
The idea of combining numbers and shapes refers to a thinking strategy that combines numbers and shapes to analyze, study and solve problems. Mr. Hua, a famous mathematician, said: "Numbers and shapes are interdependent, how can they be divided into two?" If there are fewer numbers, there will be less intuition, and if there are fewer shapes, it will be difficult to be meticulous. The combination of numbers and shapes is good, and everything is separated. " This fully shows the importance of the combination of numbers and shapes in mathematical research and application.
3. Overall ideas and methods
Holistic thinking means that when considering a mathematical problem, we should not focus on its local characteristics, but on the overall structure of the problem, understand the essence of the problem from a macro perspective through comprehensive and profound observation, and treat some independent but closely related quantities as a whole. Holistic thinking is widely used to deal with mathematical problems.
4. The idea and method of classification
There are many examples of classification in the textbook, such as rational number, real number, triangle, quadrilateral, etc., which can not only let students know the importance of classification: First, make related concepts systematic and complete; The second is to make the extension of the concept of classification clearer, deeper and more specific, and also to enable students to grasp the main points of the score: (1) classification is carried out according to certain standards, and the classification results are different with different standards;
(2) It should be noted that the classification results are neither missing nor overlapping;
(3) The classification should be gradual, and it can't be overstepped.
5. Ideas and methods of analogy and association
When considering some problems, mathematics teaching design often puts forward assumptions and conjectures based on the similarity between things, thus extending the attribute analogy of known things to similar new things and promoting the discovery of new conclusions. Teaching provides background materials for thinking, which not only enlivens the classroom atmosphere, but also helps to complete the learning of new knowledge in a harmonious and relaxed atmosphere.
6. The method of reverse thinking
The so-called reverse thinking is to turn the problem upside down or think from the opposite side of the problem or use some mathematical formulas and laws to solve the problem. Strengthening the training of reverse thinking can cultivate the flexibility and divergence of students' thinking and effectively transfer the mathematical knowledge they have mastered.
7. Ideas and methods of transformation and transformation
Transformation consciousness refers to the transformation of problems in the process of solving problems, making them simple and familiar with the basic problem-solving mode. It is the idea and method of transforming one mathematical object into another under certain conditions. Its core is to turn the solved problem into a problem with a clear solution program, so as to use the existing theory and technology to deal with it, and cultivate students to observe things and understand problems with the viewpoint of connection, development and movement change.
Third, how to cultivate junior high school students' mathematical thinking methods
(A) the cultivation of mathematical thinking methods should follow the principles
1, permeability principle
The arrangement of textbooks for nine-year compulsory education is carried out vertically according to the logic of knowledge. Mathematical knowledge contains a lot of mathematical ideas and methods. Therefore, in the teaching of specific knowledge, carefully design the learning situation and teaching process, consciously guide students to understand the mathematical ideas and methods contained in them, so that they can understand and master them imperceptibly.
2. The principle of hierarchy
To enable students to master mathematical methods, first of all, teachers should accurately and clearly grasp the level of mathematical thinking methods in junior high school mathematics textbooks. First, we should grasp the level of students' cognitive mathematical thinking methods; Junior high school mathematics methods can be divided into three levels: understanding, understanding and mastering. Understanding: Have a perceptual preliminary understanding of the meaning of mathematical thinking methods and be able to identify them in related problems. Understanding: I have reached a rational understanding of mathematical thinking methods, and I can not only tell what it is, but also know the basic viewpoints and uses. Mastery: On the basis of understanding, master its essence through training and use it to solve some problems. Second, we should grasp the level of a certain mathematical method in different textbooks and different stages. The same mathematical thinking method requires different levels in different grades (or different chapters).
3. Repeatability principle
From a long learning process, students' understanding of various mathematical thinking methods is formed through repeated understanding and application, and there is a spiral rising process from low to high. For example, we should pay attention to the reappearance of the same mathematical thinking method in different stages of knowledge and strengthen our understanding of mathematical thinking methods.
The learning of mathematical thinking method is generally divided into three stages: imitation stage, initial application stage and conscious application stage. The task of teaching is to promote the formation of the first two stages and reach the third stage as soon as possible. In teaching, we should set the hierarchical goal of mathematical thinking methods, and focus on infiltrating thinking methods in different grades and chapters. The whole idea is only to ask students to imitate the teacher to solve problems when they are in Grade One.
After students are exposed to more mathematical problems, the learning of mathematical thinking methods gradually transitions to the initial application stage, and they begin to understand the exploration methods and strategies used in the process of solving problems, and can also summarize them.
(B) in the whole process of knowledge transfer, pay attention to cultivating students' mathematical thinking.
Mathematical thinking is a bridge to form mathematical ability and consciousness, and it is the soul of skills and methods to use mathematical knowledge flexibly. Therefore, in the whole process of applying knowledge, we should pay attention to the application of mathematical thought, from analyzing and exploring ideas to optimizing and implementing solutions, and finally reflecting and verifying conclusions.
1. Infiltrate mathematical ideas in the process of concept formation
Basic mathematical thinking methods are permeated in middle school mathematics textbooks. Mathematical concepts, formulas, rules and other knowledge are "stereotyped" in textbooks, but the basic mathematical thinking methods are not "stereotyped" in textbooks. It exists between the lines in a hidden form and scattered in various chapters of the textbook in an unsystematic way. It needs the guidance of teachers so that students can understand and master it.
2. Infiltrate mathematical ideas in the process of proving formula theorems.
3. Infiltrate mathematical ideas into the teaching of examples.
The cultivation of classified thinking depends on students' handling of specific mathematical problems. Therefore, in example teaching, students should be guided to explore the solutions to some problems by using the idea of classification, and their classification skills should be trained, and corresponding questions should be arranged for training.
4. Infiltrate mathematical ideas in practice.
In the process of consolidating exercises, the idea of classification is further infiltrated.
(C) to cultivate students' ability to consciously apply mathematical thinking methods to solve practical problems
In teaching, we should pay attention to cultivating students' application consciousness and combine science and technology or life problems with mathematics teaching. With the advent of knowledge economy, we should learn a lot of mathematics knowledge in the whole process of receiving education. This is not to use this knowledge to solve specific mathematical problems in the future, but to absorb mathematical thinking methods contained in mathematical knowledge, and to learn profound scientific thinking methods while learning mathematical knowledge. Therefore, in my teaching, I not only impart knowledge, but also teach students learning methods and guide them to turn practical problems to mathematical problems.
Mathematical thinking method is the most widely used knowledge in people's lives. We should seize the opportunity to let students understand and gradually learn to use these thinking methods to solve practical problems. Through the experiment of cultivating junior high school mathematics thinking method, students can learn to learn to learn and innovate, and cultivate their awareness and ability of lifelong learning and development.