1 How to Cultivate Mathematical Abstract Thinking
Attach importance to learning reflection and cultivate the criticism of abstract thinking
The criticism of abstract thinking is the ability to complete objective evaluation and theoretical evaluation based on objective facts and rationality, and will not be influenced by perceptual and unfounded thoughts. Only people with critical abstract thinking can find mistakes in the study of mathematics knowledge in senior high school, consciously resist perceptual thinking, actively and consciously improve and adjust their thinking activities and improve their mathematical thinking ability. Critical abstract thinking is the key element of high school students' creative thinking, and it is also an effective action that every student must improve his thinking through study and practice.
First of all, we should not be afraid, but face up to our ideological loopholes. In the study and practice, we should find the weak links in our own thinking, and make self-diagnosis and self-reflection as a breakthrough, scientifically monitor the process of mathematical thinking, and find the loopholes and mistakes in our own use of abstract thinking. At the same time, high school students should pay attention to what basic mathematical thinking methods and skills are used in thinking and solving problems, what effect they have had when they are used, and whether they can find more effective methods through exploration; What mistakes have been made in solving mathematical problems, what are the root causes of the mistakes, and how to change the wrong thinking in learning practice.
Strengthen knowledge association and cultivate the depth of abstract thinking
The profundity of thinking refers to abstract logic, which is an important embodiment of the characteristics of abstract thinking and a link that must be paid attention to in the cultivation of abstract thinking ability. When people come into contact with perceptual data, by removing the false from the perceptual data, removing the rough and selecting the fine, people's brain thinking will change suddenly in the cognitive process, resulting in summarized and abstract logic, and the depth of thinking will be greatly improved.
In the study of mathematics knowledge in senior high school, senior high school students can understand the essential attributes and internal laws of mathematics knowledge through thinking generalization. By strengthening the connection between knowledge, they can think more deeply about mathematical problems, thus grasping the essential laws of things, strengthening the profundity of abstract thinking and promoting the improvement of mathematical thinking ability. For example, given that |2m6|+|4n-8|=0, what are m and n respectively? Mastering the concept and essence of absolute value, we can know whether absolute value is negative or not. According to this property, we can know that the sum of these two formulas can be zero only when they are both zero, so m=3 and n=2. After mastering this essence and law, the following problems can be solved quickly by using the method of knowledge transfer: |x-4|+3(2y-5)=0, and the values of x and y can be obtained.
2 How to improve the level of mathematical thinking
Strengthen reflection and improve students' application ability
On the one hand, reflection and summary in learning can help students better review their learning process, on the other hand, students can find their own areas that need improvement in reflection. Reflecting on the learning content in the preview stage can enable students to preview more effectively in the future, and also enable students to better understand related issues. Reflection in teaching analysis stage is of great help to improve students' mathematical thinking and logical ability. Reflection in the training stage allows students to review what they have learned in the process of reviewing the answers to a certain type of questions, so that students can find out the answering skills and specific methods of a certain type of questions in the long-term thinking. Therefore, these have an important influence on the cultivation of students' ability and the development of mathematical thinking.
For example, in the teaching of analysis, the relevant content of quadratic function is used to solve examples. In the process of reflection, students will first analyze the relevant conditions involved. "The purchase price of each piece is 8 yuan, and the price is 10 yuan. You can probably sell 1 10 pieces a day. If the unit price of goods is reduced by 0. 1 yuan, the sales volume can be increased by 65438. What do these conditions have to do with the required maximum profit? In the "five steps" of the analysis stage, the relationship between each step is gradual, which is a very meticulous logical thinking. Finally, on the basis of such a step related to reflection, it seems that students are reviewing this topic, which is actually a summary of the specific application of quadratic function. Once students discover this rule, they will find practical problems related to quadratic function. The general steps to solve the problem are: defining the known conditions-determining what the problem needs to solve, whether to seek the maximum value or something else-how the known conditions relate to the problem-what is the potential established range-listing the analytical formulas to solve according to all the excavated conditions.
Strengthen variant training, improve the flexibility of thinking and cultivate the ability to draw inferences from others.
The so-called mathematical variant training refers to the horizontal or vertical extension of concepts, properties, theorems, formulas and problems from different angles, levels and situations in the process of mathematics teaching. Variant training can help students understand problem-solving methods from multiple angles, from "mastering knowledge" to "understanding ideas"; As the saying goes, it is better to teach people to fish than to teach them to fish. Teachers should let students take the initiative to participate, instead of always "changing" teachers and "practicing" students.
We should encourage students to make bold changes, purposefully and consciously guide them to discover the essence of "unchangeable" from the phenomenon of "changeless", explore the law of "changeless" from the essence of "changeless", help students master what they have learned, and cultivate their innovative consciousness and spirit and the ability to draw inferences from others.
How to Cultivate Students' Abstract Thinking Ability in Mathematics Teaching
1, focusing on thinking in images. First of all, teachers should try to use images in teaching. Thinking in images can enrich students' psychological activities and help them to understand the nature and laws of things more deeply.
Secondly, we should guide students to develop the habit of using intuitive strategies to solve problems. For example, if Xiao Ming and Xiao Jun buy the same book, the book with Xiao Ming's money is short of 1.6 yuan, and the book with Xiao Jun's money is short of 1.8 yuan. If they put all their money together to buy a book, they will spend 2 yuan more. What is the unit price of this book? If students adopt drawing strategy, the problem can be easily solved.
2. Guide students to learn abstraction step by step. First of all, teachers should pay attention to cultivating students' abstract thinking ability in teaching. Only by getting rid of the concrete image can abstraction make thinking get new results in the form of algorithm. Secondly, abstraction can not only make thinking general, simple and profound, but also have the function of discovering truth. Therefore, teachers should also guide students to solve problems in an abstract way.
3. Pay attention to the role of appearance. Representation is the reflection of the image of things that the human brain has previously perceived that do not directly act on the sensory organs at present. It is not only concrete, but also general. It not only reflects the main characteristics and outlines of individual things, but also reflects the surface characteristics of a class of things. The basis of representation is perception, and teachers should enrich students' perception as much as possible, and mobilize students' multiple senses to participate in perception by observing, operating and experimenting. In the above-mentioned teaching cases, with the help of image thinking, the addition calculation within 10 and the calculation of adding two digits to integer ten and one digit are carried out. The premise is that students must have rich perception and related graphic representations in their minds, otherwise it is difficult to carry out. Thinking in images is a bridge between perceptual knowledge and rational knowledge. In the process of rising from thinking in images to abstract thinking, teachers should attach importance to the role of thinking in images.
4. Formal operation-a good way to train abstract thinking. There is a problem: "A cube is cut into the largest cylinder. What is the volume of this cylinder? " The solution of student 1 is: suppose the side length of a cube is 6 cm, then the diameter and height of the bottom of the cylinder are 6 cm. π×(6÷2)2×6=54π (cubic centimeter), 6×6×6=2 16 (cubic centimeter), 54π÷2 16=π÷4=78.5%. Student 2' s solution is: regard the side lengths of all cubes as a. π×(a÷2)2×a=πa2/4×a=πa3/4 (cubic centimeter), A× a = a3 (cubic centimeter), π a3/4÷ a3 = π/4 = 78.5. Both methods get positive answers, but the first one is to refer to specific data for operation, and the second one is to use letters instead of numbers for operation, that is, parameter method. Obviously, the second method is more abstract and more general. But only six or seven students can think of the second method.
4 How to cultivate junior high school students' mathematical thinking ability
Attach importance to the teaching of mathematical thinking methods and problem solving, strengthen the guidance of learning methods, and improve students' learning ability
Mathematical thinking method is a high generalization of the basic theory of mathematics and the "soul" of mathematical knowledge. Paying attention to the teaching of mathematical thinking methods plays an important role in mathematics teaching. It not only helps students to deeply understand and master mathematical knowledge, but also helps students to apply what they have learned to practice, and can also improve their mathematical ability, especially their mathematical creativity. Modern cognitive psychologists believe that the improvement of students' creative ability mainly comes from the enhancement of knowledge transfer ability, which is closely related to whether individuals can highly summarize what they have learned. The higher the generalization of knowledge, the stronger the migration ability. Therefore, attaching importance to the teaching of mathematical thinking methods is a necessary measure to implement quality education.
Problem-solving teaching is a new teaching method introduced from the west. This teaching method takes mathematical problems as the main teaching content and students' autonomous learning and research and exploration as the main operation mode. The purpose is to let students apply the theory and knowledge they have learned in class to practice, deepen their understanding and mastery of knowledge, experience the process of solving practical problems, accumulate experience and try to create, and enhance the unity and cooperation among students in this teaching process. Its basic model is: studying practical problems to find out difficulties → studying the nature and causes of difficulties → seeking solutions to difficulties → proving solutions to difficulties in theory → testing solutions to problems in practice. It can be seen that paying attention to problem-solving teaching in mathematics teaching can enhance students' learning and application ability, improve students' thinking quality and other qualities, especially improve students' ability to analyze and solve problems and their awareness of mathematics application. Therefore, attaching importance to problem-solving teaching is of great significance to the implementation of quality education.
Mathematical ability is strengthened in reflection.
The process of rethinking and understanding the meaning of the question is to rethink how to obtain information in the process of understanding the meaning of the question at first after the completion of the problem-solving activity. The formation process of reflective thinking is to review how to deal with, reorganize and regenerate information after solving problems. Specifically, it is to recall every step of thinking activities from the beginning of solving problems to the end of solving problems. One task after solving the problem is to reflect on the expression of solving the problem. Reflect on whether the operation is correct, whether the reasoning is rigorous and whether there are loopholes; Reflect on whether the language expression is concise, accurate and complete; Reflect on whether the language expression is concise, accurate and complete; Reflect on whether the solution process can be optimized; Timely improve or correct the problems found.
Reflection is a beneficial thinking activity and conscious impulse to re-learn, constantly reflecting and discovering confusion. Reflection is only a means, and its essence lies in "finding problems" and "solving problems". In this sense, reflection is not as much as possible, but just right. The cultivation of mathematical reflection ability should be organically combined with the cultivation of mathematical ability (thinking ability, spatial imagination ability, ability to solve practical problems, etc.). ), the two can cooperate with each other and develop harmoniously, thus improving the efficiency of mathematics learning and achieving good results.