1 How to improve junior high school mathematics thinking
Training of correct thinking direction
First, logical thinking is multi-directional and guides students to know the direction of thinking. Positive thinking is a thinking method that directly uses the existing conditions and draws correct conclusions through generalization and reasoning. Reverse thinking is a kind of thinking method that starts from the problem and looks for the conditions related to the problem, and changes the one-way association that only works from one aspect into the two-way association that works from two aspects. Lateral thinking is to explore the given knowledge from the local or side, turn the problem into another situation, arouse students' memory of the existing knowledge, communicate the internal connection of knowledge, and thus broaden their thinking. Divergent thinking is the opposite of concentrated thinking. It thinks from different angles, directions and sides, thus producing all kinds of novel ideas and answers. In teaching, we should pay attention to cultivating students' good habits of thinking in many ways, so that students can face all kinds of problems calmly. We should "teach people to fish, it is better to teach them to fish!" Students should be taught how to think, not just a problem.
Second, guide students to find the correct way of thinking. To cultivate logical thinking ability, we should not only let students know the direction of thinking, but also guide students to seek scientific methods in the correct direction of thinking. In order to make students be good at seeking the correct thinking direction, we should pay attention to the following points in teaching: (1) Carefully design thinking sensory materials. To cultivate students' thinking ability, teachers need not only to provide students with rich sensory materials, but also to carefully design and skillfully arrange a large number of perceptual materials, so that students can smoothly realize the transformation from perception to abstraction. (2) Thinking activities based on basic knowledge. The basic knowledge of middle school mathematics includes concepts, formulas, definitions, rules, theorems, axioms and inferences. Students can think about problems according to the above knowledge and find the correct thinking direction. (3) Associating and analogizing with old knowledge. Old knowledge is the foundation of thinking, and thinking is the bridge to new knowledge. Associating analogy from old knowledge is also an effective way to seek the correct thinking direction. Association and analogy compare two similar or similar knowledge or problems, find the connection and difference between them, and then find the correct answer to the question. (4) Repeated training to cultivate multi-directional thinking. The cultivation of students' thinking ability can not be achieved by one or two exercises and training, but needs repeated training and practice. Because students' thinking direction is often single, there is a certain mindset, so we need not only repeated training, but also pay attention to guiding students to think from different directions and cultivate multi-directional thinking.
Attach importance to the cultivation of good thinking quality
To cultivate students' logical thinking ability, we must attach importance to the cultivation of good thinking quality, because the quality of thinking will directly affect the strength of thinking ability. (1) Cultivate the agility and flexibility of thinking. In teaching, we should pay full attention to the examples in textbooks and other solutions in exercises, compare which one is the best, how to analyze it and whether there are any shortcomings, guide students to broaden their thinking, choose the best thinking through association and analogy, and cultivate students' agility and flexibility in thinking. (2) Cultivate the breadth and depth of thinking. Paying attention to the communication of knowledge in teaching can cultivate the breadth and depth of thinking. (3) Cultivate the independence and creativity of thinking. In teaching, we should creatively use teaching materials, participate in thinking in images and cultivate students' independence and creativity in thinking. In textbook examples, the first part is mostly to pave the way for learning new knowledge, and the second part is to consolidate and deepen what has been learned.
Therefore, the teaching focus of the first few examples is to let students clearly understand the principle, while the teaching focus of the last few examples is practice. The following exercises should be further deepened, expanded and divergent. Mathematical thinking method is the essence of mathematics. If you master the mathematical thinking method, you will learn to think. Curriculum standards require the cultivation of social members with mathematical literacy. Mastering mathematical thinking methods is also an important standard for possessing mathematical literacy. In the process of exploring science and developing economy, we need to have certain mathematical knowledge and use mathematical thinking methods. People with mathematical literacy are often good at analysis, comprehensive comparison, general judgment, reasoning and induction. These scientific thinking methods are all cultivated in the infiltration and training of mathematical thinking methods. The mathematical thinking methods in middle schools include equation function thinking, combination of numbers and shapes, reduction thinking, experimental and inductive reasoning thinking, overall consideration of problems, classified discussion thinking and mutual transformation among mathematical models. Teachers should train students to be good at theorizing practical problems, and make solutions to problems through the theoretical knowledge they have mastered, so that students can learn to observe and analyze the real society with mathematical ideas and improve their ability to analyze and solve problems.
2 Mathematics teaching methods
Establish diversified teaching objectives
"The mathematics curriculum in compulsory education emphasizes starting from students' existing life experience, allowing students to personally experience the process of abstracting practical problems into mathematical models and explaining and applying them, so that students can gain an understanding of mathematics, and at the same time have thinking ability, emotional attitude and values, etc. "
Based on this concept, mathematics curriculum has set diversified teaching objectives from four aspects: knowledge and skills, mathematical thinking, problem solving, emotion and attitude. Mathematics teaching should not only attach importance to knowledge and skills, but also attach importance to emotional attitude. Mathematics teaching should not only pay attention to problem solving, but also pay attention to mathematical thinking process, and put the result and process in the same important position.
Cultivating students' innovative ability in mathematics teaching
Innovative ability is mainly manifested in seeking new methods to solve problems in mathematics teaching. "Learning begins with thinking, and thinking begins with doubt". Students' thinking process of exploring knowledge always starts with problems, and develops and innovates in solving problems. In the teaching process, under the situation created by teachers, students can operate by themselves, think and express with their brains, explore unknown fields, seek objective truth and become discoverers. Students should participate in this exploration process from beginning to end and cultivate their innovative ability. For example, in the teaching of ball volume, I divided the students into three groups in my spare time and asked everyone in the first group to make a hemisphere with a radius of 10 cm; In the second group, each person made a cone with a radius of 10 cm and a height of 10 cm; In the third group, each person made a cylinder with a radius of 10 cm and a height of 10 cm.
One person in each group forms multiple groups. Each group puts the cone into the cylinder, and then fills the cylinder with soil in the hemisphere. The students found the relationship between them. The volume of a hemisphere is equal to the difference between the volumes of a cylinder and a cone. The derivation of the volume formula of a sphere is a perfect example of the flexible application of these thinking methods, which integrates axiomatic thinking, reduction thinking, equal product analogy thinking and cut-and-complement transformation method. Thirdly, through the analysis of the thinking of solving the volume problem in teaching, a systematic and coherent deduction clue of the volume formula is formed, and these thinking methods are clearly presented to students. Students can understand the creative thinking process of mathematicians and stimulate their creative thinking and innovation ability.
3 math classroom interest
Show mathematical culture and cultivate students' interest in mathematics.
In China, mathematics culture has inherited China's long history and profound traditional culture, and mathematics curriculum also shows humanistic consciousness and feelings. Specifically, we can talk about the history of mathematics in the teaching content, and the teacher can talk about a brief history of the history of mathematics in the world; At the same time, we can tell you that Zu Chongzhi, a great mathematician in China, calculated pi to seven decimal places in the Northern and Southern Dynasties, and his density value was the first in the world. It can also tell how Pythagorean Theorem was first created in the world in "Computational Classics" by the director of education in China, and how to apply it. Through these great achievements in the history of mathematics, teachers can enhance students' patriotism, encourage students to learn mathematics well and stimulate their interest in learning mathematics.
For example, in teaching, the author asked students to answer the famous "will question": there was an old man who had three sons and seventeen horses. On his deathbed, he asked his sons to divide the horses according to his will. He said: I will leave all seventeen horses to my three sons, half for the eldest son, one third for the second son and one ninth for the younger son. It is not allowed to kill horses and dismember them. As soon as the topic came out, some students said it was too simple and couldn't wait to start writing, but soon they said the number of the topic was wrong. When the students had a heated discussion, the author suggested borrowing, and some students came up with a solution to the problem: lend a horse to three people. The old man has 17 horse, plus a borrowed horse and a *** 18 horse. So the three brothers got nine, six and two horses respectively according to half, one third and one ninth of 18 horses. 9+6+2= 17 (horse). There is one horse left, and it is returned to the borrower. After the author affirmed it in time, he guided the students to discuss. By using the method of proportional distribution, simplifying1/2:1/3:1/9 can get 9: 6: 2, which is exactly 9+6+2= 17. It can be seen that 9 horses were given to the eldest son, 6 horses to the second son and 2 horses to the younger son, which not only exactly divided all the horses 17, but also met the ratio of1/2:1/3:1/9. While students feel the ingenious method of "borrowing", they also realize the fun of thinking brought by mathematics learning, thus stimulating students' interest in learning.
Create interesting teaching situations to improve students' interest in learning mathematics.
Interest is the best tutor in learning and the driving force to promote students' learning. Creating a good teaching situation can make learning have a good start. Teachers create interesting teaching situations, which can make students concentrate on their studies. As long as students truly realize the "infinite mystery and fun" of learning mathematics, they will be willing to learn mathematics and accept it.
For example, when learning "the positional relationship between a straight line and a circle", teachers can deeply study the contents of the textbook and carefully create the following problem teaching situations in combination with the actual situation of students' learning. Teacher: Have the students ever seen the sunrise rising slowly from the sea level in the morning? The students answered one after another: I have seen the rising sun. Teacher: If we regard the sea level as an infinite straight line and the sun as a super-large circle, what is the positional relationship between the straight line and the circle when the sun rises above the sea level in the morning? Can students draw it graphically? In classroom teaching, teachers introduce examples that students are familiar with in real life, so that students can experience "life-oriented" math problems, make students feel cordial, better introduce the content of new courses, and make the learning atmosphere relaxed and happy.
4. Cultivate divergent thinking in mathematics
Multiple solutions to one problem
When adopting "multiple solutions to one problem", students should be guided to observe and think from different angles in order to find different methods to solve problems. At the same time, we should guide students to compare various methods, optimize problem-solving methods, pay attention to find out the conditions and reasons for the existence of multiple solutions to the same problem, and explore its internal laws. Cultivate students' divergent thinking of seeking difference and innovation, and realize and improve the fluency of thinking. Through the training of multiple solutions to one problem, students can seek solutions to problems from multiple angles and ways and open up new ideas to solve problems. So as to comprehensively use different knowledge, select the best scheme from the comparison of various schemes, summarize the law of solving problems, improve the ability of analyzing and solving problems, and enhance the divergence and creativity of thinking.
For example, the ratio of A to B is 3: 1, A is 45, and B is what? There are several algorithms for this problem: ① 45; ②45? ; ③45? 3? 1; ④45? 3; ⑤ = ; ⑥ = etc. After the calculation, guide them to discuss one by one, let the students speak their own ideas, explain the truth, and find clever and simple algorithms. Regular training with multiple solutions to one problem is conducive to developing problem-solving thinking, cultivating students' divergent thinking ability and integrating what they have learned.
A theme is changeable.
"One topic is changeable" is a variant of the topic structure, which turns a topic into multiple topics, while the nature of the topic remains unchanged. Ask students to answer these questions and think about them at any time according to the changing situation, and find out the differences and connections between them, as well as the special and general relations. It can not only enable students to review, review and comprehensively apply what they have learned, but also enable students to learn the knowledge, skills, methods and techniques firmly and learn to live, thus cultivating the flexibility of thinking and the adaptability of solving problems.
Cultivate students' turning wit and flexibility of thinking, and realize the flexibility of divergent thinking. By changing the conditions, conclusions and propositions, the exercises will become more valuable and innovative new problems, so as to apply more knowledge to solve the problems and achieve the effect of "practicing more on one question" and "getting more from one question". Make students' thinking ability improve with the constant change and solution of problems, effectively enhance the agility and adaptability of thinking, and cultivate and develop creative thinking.
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