Only by training and cultivating students in these three aspects can each student's mathematical ability develop to its due level.
First, the acquisition of mathematical knowledge and the infiltration of mathematical thinking methods
In mathematics activities, students are most concerned about the method of solving problems, which is often called mathematical method. It refers to the specific thinking direction and operating procedures provided for solving mathematical problems under the guidance of mathematical thought.
Middle school mathematics methods can be divided into three categories:
(1) In terms of cognitive methods, there are "observation and experiment, comparison and classification, induction and analogy, imagination, intuition and epiphany". These mathematical methods are implied in teaching materials, and students must be guided to dig and practice repeatedly in solving problems, from perceptual knowledge to rational knowledge, and finally achieve flexible application.
(2) Logically speaking, there are "complete induction and incomplete induction, synthesis, analysis, deduction, reduction to absurdity, and the same method".
(3) There is also a kind of mathematical method in the textbook, which is completed by several specific operation steps, such as elimination method, collocation method, method of substitution, undetermined coefficient method, equal product method, basic graphic method and so on. Mathematical thought is the basic viewpoint of mathematical activities. In teaching, students should be made aware of its inherent laws and essence. Mathematical thought is a high generalization of the inherent laws and essence of mathematical knowledge and methods, which is of guiding significance for solving mathematical problems. The mathematics thoughts in middle school textbooks include "symbol and argumentation thoughts, set and correspondence thoughts, axiom and structure thoughts, system and statistics thoughts, transformation and dialectical thoughts". How to infiltrate mathematical ideas into students in teaching?
(1) refining mathematical ideas in knowledge learning
Mathematical thoughts are implicit in textbooks, and there are abundant mathematical thoughts at the development point of knowledge and the occurrence point of new knowledge. In teaching, students should be inspired to pay attention to refining mathematical ideas, such as exploring the sum of internal angles of polygons, which can guide students to transform polygons into triangles to deal with and extract ideas from them.
(2) Summing up mathematical ideas in the study of mathematical methods.
While students master knowledge, they should be further guided to sum up mathematical methods to solve the same problem in mathematics. Students are not only required to use these mathematical methods flexibly to solve mathematical problems, but also to link these mathematical methods with existing mathematical methods and summarize their characteristics. And reveal its inherent law and essence, so that students can deeply understand the role of this kind of * * * in solving mathematical problems. For example, equations in algebra, method of substitution in equations, angles, line segments and intermediate ratios in geometry all reflect the idea of variables.
(3) Strengthen mathematical thinking when summing up.
In summary, students should not only arrange knowledge structure and mathematical methods, but also strengthen the leading position of mathematical thought and the role of solving mathematical problems. Especially at the end of the chapter, exercises should be carefully compiled, so that these exercises can not only reflect the important knowledge and mathematical methods of the whole chapter, but also reflect the main mathematical ideas of this chapter, so that students can realize what role the mathematical ideas of this chapter play in solving mathematical problems. For example, in the summary of the chapter of trigonometric function, students should strengthen symbolic thinking, corresponding thinking and structural thinking after finishing the knowledge structure and mathematical methods, and embody them with corresponding exercises, especially structural thinking, so that students can master complex problems or figures and how to establish the structure of right triangles to solve problems.
Second, the cultivation of mathematical ideological quality
Because solving mathematical problems is a transformation process from conditions to conclusions, which has certain directionality. Therefore, in teaching, the training of concentrated thinking and divergent thinking is the main content of cultivating students' thinking quality.
From the formal point of view, concentrated thinking is "directional, hierarchical and convergent" in content, and it is "seeking sameness and concentration".
From the analysis of the logical structure of the textbook, the directionality, hierarchy and cohesion are obvious, but the process of guiding students to explore every knowledge point is implicit in it. Therefore, students should be guided to read the textbook carefully and systematically after learning a unit or a chapter. Combine the form and content of concentrated thinking, write the thoughts after reading or make the thinking chart of teaching materials, so that students can understand the connotation of concentrated thinking. By analyzing the process of solving mathematical problems, create a situation of concentrated thinking, guide students to comprehensively analyze the existing information in the conditions, and divide the problems into several small problems in turn according to the conclusion direction. Every time a small problem is solved, students can understand that its conclusion directly affects the thinking direction of the next small problem, and the scope of thinking search will be narrowed and the problem will be gradually solved. Obviously, in the process of solving problems, the quality of students' concentrated thinking is cultivated.
The teaching of concepts, properties and theorems can also provide students with a divergent thinking situation to explore problem-solving methods. This kind of thinking looks from the direction. It is reverse, horizontal and multi-directional "; In content, it is flexible and open. People often talk about reverse thinking and seeking difference thinking, but in the process of solving mathematical problems, the starting point of analyzing problems is different, and the purpose is to try to transform conditions into conclusions. Therefore, different divergent situations should be created according to different teaching contents in teaching. Make students use the existing mathematical knowledge and thinking methods to put forward their own views from different angles, so that students' divergent ideological quality can be fully tempered.
In teaching, there are several ways to cultivate divergent thinking:
(1) divergence condition, the conclusion remains unchanged. Inspire students to use known mathematical knowledge and thinking methods to explore problems from different angles as much as possible, and find out all possible quantitative relations or graphic position relations of holding conclusions.
(2) The conclusion is divergent and the conditions are complete. Inspire students to explore all qualified conclusions through imagination, guess, try and intuition in the process of exploration.
(3) The process of solving mathematical problems is divergent, that is, the conditions are complete and the conclusions are certain. Guide students to explore various ways to solve problems by using known mathematical knowledge and thinking methods based on conditions and conclusions and different information.
Third, the cultivation of innovative consciousness.
The so-called innovative consciousness refers to the uniqueness, flexibility, flexibility and pioneering in solving mathematical problems, and then forms the tendency of individual initiative. This tendency of individual initiative is not only related to students' innate conditions, but also to teachers' careful training and correct inspiration, guidance and encouragement. Therefore, students' curiosity should be used in teaching to inspire them to find problems independently, guide them to use existing mathematical knowledge and thinking methods, explore the unknown flexibly, encourage them to explore, and gradually form a tendency of personal initiative.
As can be seen from the textbook, the occurrence and development of mathematical knowledge is a dynamic process, so we should create a dynamic thinking situation for students in teaching. Create various situations from simple to complex, from special to general or from general to special. In this dynamic process, students are inspired to find out which practical problems are related to the mathematics content in real life, so as to cultivate students' unique, flexible and mobile personal initiative tendency in dynamic exploration. Teaching not only inspires students to explore problems with divergent thinking, but also guides students to summarize some special conditions (or conclusions) in conditions and conclusions, and to specialize general conditions (or conclusions), so as to guide students to temper their original and flexible personal initiative tendency from the dynamic changes of quantitative relations and graphic position relations.
How to cultivate students' habit of developing mathematical thoughts?
(1) Explore the properties of existing mathematical models.
Some mathematical model properties are formed by some special mathematical elements, and students can be guided to use these special mathematical elements to discover "new properties" in teaching. For example, review plane geometry and know three sides of a triangle. You can find the relationship between the height of a triangle and its three sides. Then, if you know three sides, the median line of one side and the bisector of an angle, can you find it?
(2) Develop the application of the learned mathematical knowledge.
When students have learned a certain knowledge point, they can be guided to do their own questions, answer with the concepts and properties they have just learned, and sometimes they can be guided to appropriately broaden the scope of their own questions. For example, algebraic problems are extended to geometric problems, and geometric problems are extended to algebraic problems. Let students spread their thinking wings, apply what they have learned and correct the phenomenon that goes against scientific common sense.
(3) Use the examples in the textbook.
The examples in the textbook are typical and profound, so as to guide students to make full use of examples, reveal their profundity and understand their typicality. Make students' learning achieve the effect of giving inferences by analogy.