1 how to develop mathematical thinking
Reasonably plan the classroom content and stimulate students' interest in learning.
Mathematics knowledge is complicated, and theorem after theorem seems puzzling, so it is very important to plan the course content reasonably, stimulate students' interest in learning and help them regain their confidence. Then let's take the chapter "sine theorem of trigonometric function" as an example to analyze it concretely: first of all, we know that sine theorem is the relationship between the three sides of any triangle and the sine value of the corresponding angle. It is related to triangles, so at the beginning of the class, let's ask a question like "A mountain is too high for us to climb, or there is a river blocking it, how to calculate the height of the mountain". The first step is to arouse students' curiosity with practical questions;
Then we introduce the concept of sine theorem. After introducing the concept and formula, let students make a guess, encourage them to question boldly, and let them reason whether this sine theorem is correct or how to draw this conclusion according to the formula. The second step is to change the role of scientists and make their own inferences. In the process of students' own reasoning, there must be different ways of thinking, and what is right will naturally be wrong. At this time, teachers need to participate and help, listen to students' opinions and ideas, and give correct guidance to mathematical thinking in time to prevent them from falling into the misunderstanding of thinking and ignoring the blind spot of knowledge. The third step is to give students timely guidance and attach importance to mathematical thinking. Then at this time, we can do some examples to let everyone feel the application of the theorem, which is convenient for deepening.
Reflection on the loopholes in contemporary education
Since the importance of thinking in the teaching process has been made clear, why is the thinking ability of China people not as good as that of other countries? The survey shows that China people can get good grades in mathematics, but they have made little achievements in academic research of mathematics. This proves that there are still loopholes in our education, which is also a problem that we must find. (1) Most teachers prepare lessons from textbooks and teaching reference materials, while some inexperienced teachers completely copy the contents of books. Whether students can really understand the actual situation or not. Therefore, in teaching, teachers often write a blackboard on the blackboard, but like the order and content in the book, students are too lazy to look up at the blackboard, and their attention gradually declines.
(2) Most students lack the ability to think actively, and they lack the ability to find and ask questions. They rely more on the teacher's direct explanation than on active thinking. (3) Students have not laid a good foundation in primary school and junior high school, and their mathematics learning ability is poor, so there are problems that they can't keep up. There are so many classrooms in high school that it is difficult for teachers to take care of every student, so students who are not good at learning are ignored. As time went on, they lost confidence. (4) Teachers pay too much attention to performance evaluation, and attend classes in a hurry, fast and urgent. Some teachers blindly speed up teaching in order to complete teaching tasks and teaching objectives, and some obscure knowledge points pass by, telling students that this is a theorem, but not explaining why this theorem is, which is very unfavorable for cultivating students.
2 How to cultivate primary school mathematics thinking
Attach importance to practical operation and mobilize the development of thinking
Operation is not a simple physical action, but is closely related to the thinking activities of the brain. The thinking of junior children begins with action, and their thinking has the characteristics of intuitive action, which is in the transitional period from image thinking to abstract logical thinking. In the teaching process, teachers can start with intuition, and let students carry out practical activities such as concrete hands-on operation through observation and imagination, which is conducive to improving students' enthusiasm and initiative in learning mathematics.
For example, teach "33-8=?" At that time, the teacher took out three bundles of sticks (10 bundle 1 stick) and three sticks for the students to set. Students take out eight sticks from these sticks. If one stick is not enough, they open 1 bundle sticks and combine them with three sticks to make 13 sticks. 13 sticks take out eight sticks and leave five. In this way, through hands-on operation, students can clearly understand that when calculating two digits minus one digit, if one digit is not enough, one 10 should be taken from ten digits and combined with one digit for reduction. Practice has proved that guiding students to combine their hands and brains organically in teaching can open up students' thinking and promote its development.
Grasp the starting point of thinking and develop students' thinking
The context of mathematical knowledge is connected and closely linked, and always constitutes the knowledge system of each unit according to the natural law of occurrence-development-extension. The same is true of students' thinking process of acquiring knowledge, either starting from existing experience or introducing old knowledge, which is the beginning of thinking. Starting from the starting point of students' thinking, we should grasp all levels of thinking development and gradually deepen it until the end. If this beginning does not conform to the students' knowledge level or thinking characteristics, students will feel that there is no way to solve the problem and the thinking thread will not develop in an orderly manner.
For example, in the teaching of the word "continuous division" in the ninth volume of the new textbook, the word "continuous division" is divided into two division words related to life, so that students can analyze the quantitative relationship and calculate in parallel. Then it shows the application of continuous division. Read the question, understand the meaning of the question and analyze the quantitative relationship, so that students can understand that this question is different from the above two questions. Then I inspired this question: "Can you work out in one step how many kilograms of milk Niu Yi produces every day?" The students all answered "no", and then I asked them: "Since this question can't be worked out at all, what should be counted first?" Then ask the students to analyze the answers in groups. When exchanging reports, some groups gave two algorithms, and some even gave more than three methods. In this way, gradually deepening understanding from problems can not only solve the problem that students can't start in the process of thinking, but also help to develop students' thinking and cultivate their fluency.
3 How to improve students' mathematical thinking ability
Optimize classroom teaching and open divergent thinking
Classroom teaching is the main position and battlefield to implement quality education, and it is also necessary to start with classroom teaching to cultivate students' divergent thinking ability. The cultivation of thinking ability is one of the core goals of mathematics classroom teaching. In classroom teaching, teachers are like directors and students are actors. Therefore, teachers should organize teaching in various ways to realize diversification of teaching objectives, scientification of teaching contents, optimization of teaching methods and multidirectional information transmission, guide students to put forward new ideas, schemes and methods to solve problems in time, create a positive and harmonious teaching environment, and open the door for students to divergent thinking.
Try to deal with teaching materials flexibly. Stimulate the imagination of the brain by changing the angle and conditions of thinking. For example, after teaching that a balance is a tool for measuring the mass of an object, the teacher can ask: How can you measure the mass of an object without a balance? At this time, it is necessary to broaden the thinking space according to the students' answers and the situation, so as to improve learning efficiency and cultivate divergent thinking ability. In terms of teaching methods, teachers try their best to use teaching methods such as watching, listening, reading, thinking and practicing to keep students' brains in a positive and exciting state and create favorable conditions for students' thinking development. For another example, teaching can be assisted by practical activities and knowledge contests, which can stimulate and induce students to open their minds, tap their potential, truly realize the coordinated use of eyes, ears, mouth and brain, and achieve the purpose of cultivating students' divergent thinking.
Diversification to cultivate students' thinking quality
(A) the use of open questions to cultivate students' profound, broad and creative thinking.
First of all, the conclusions of open questions are different or the problem-solving strategies are diversified, but there is an internal connection between these different conclusions or diversified problem-solving strategies, that is, "the form is scattered but the spirit is not scattered." [Case] When I was talking about the section of the vertical diameter theorem, I designed such a set of questions: (1) In ⊙O, the chord AB=8cm, the distance from point O to chord AB is 3cm, and the radius is. (2) For 5cm, chord AB=8cm, find the distance from O to chord AB. (3) If the radius ⊙O is 5cm and OP=3cm, what is the shortest chord length among the chords passing through point P? (4) If P is the midpoint of the arc AB, the distance from P is 2cm, and the chord AB=8cm, find the radius ⊙ O. Students can get the auxiliary line of this kind of problem through practice: that is, it constitutes Rt△, and its three sides are half of the chord length, radius and chord center distance respectively. So as to effectively cultivate the profundity of students' thinking. Secondly, students are also broad in solving problems, that is, they do not use the knowledge learned in this unit or this textbook to solve problems.
(B) The use of conjecture is a means to cultivate students' creative thinking.
Regarding conjecture, Paulia has a wonderful exposition: "I want to make a small suggestion. Can students guess the result or part of the result of the question before doing it?" . Once the student expresses his basic idea, he connects himself with the question and is eager to know whether his guess is correct. Therefore, he will take the initiative to care about this problem, care about the progress of the class, and will not take a nap or make small moves. "From Paulia's exposition, we can feel that students don't need to have guesses like scientists, but all guesses that can promote students' learning and cultivate students' creative thinking are very meaningful. Guide students to guess, let students acquire knowledge better, show their innovative ability and improve their self-confidence in learning.
How to cultivate students' thinking ability in primary school mathematics teaching
Find a breakthrough in cultivating mathematical thinking ability
Psychologists believe that cultivating students' mathematical thinking quality is a breakthrough in cultivating and developing mathematical ability. Thinking quality includes profundity, agility, flexibility, criticism and creativity, which reflects the characteristics of different aspects of thinking, so there should be different training methods in the teaching process. The profundity of thinking is the essence of mathematics, which determines that mathematics teaching should be student-oriented and cultivate students' profundity of thinking. The difference of mathematical thinking depth reflects the difference of students' mathematical ability. To cultivate the profundity of students' mathematical thinking in teaching is actually to cultivate students' mathematical ability. In mathematics teaching, students should be educated to look at the essence through phenomena, think about problems comprehensively, and form the habit of asking questions.
The agility of mathematical thinking is mainly reflected in the speed problem under the correct premise. Therefore, in mathematics teaching, on the one hand, we can consider training students' operation speed, on the other hand, we should try our best to let students master the essence of mathematical concepts and principles and improve the abstraction of the mathematical knowledge they have mastered. Because the more essential and abstract knowledge is, the wider its scope of application and the faster its retrieval speed will be. In addition, the operation speed is not only the difference in understanding mathematical knowledge, but also the difference in operation habits and thinking generalization ability. Therefore, in mathematics teaching, students should always be asked about speed, so that they can master the essentials of quick calculation. In order to cultivate students' thinking flexibility, we should strengthen the variability of mathematics teaching, provide students with a wide range of thinking association space, enable students to consider problems from various angles, quickly establish their own ideas, and truly "draw inferences from others." Teaching practice shows that variant teaching plays a great role in cultivating the flexibility of students' thinking. For example, in concept teaching, let students describe concepts in equivalent language; In the teaching of mathematical formulas, students are required to master all kinds of variations of formulas, which is conducive to cultivating the flexibility of thinking.
Using audio-visual media to optimize classroom teaching and develop students' thinking ability
With the deepening of teaching reform, audio-visual media has become an important means to improve teaching quality. In teaching, vivid and sensible audio-visual media has the advantage of displaying three-dimensional graphics with intuitive images, which can create situations, stimulate interest, develop students' thinking ability and enable students to acquire knowledge in a relaxed and happy learning atmosphere. Therefore, the proper use of audio-visual media can make the boring teaching content vivid and the perceptual process vivid, thus mobilizing students' subjective initiative and producing twice the result with half the effort.
It is impossible to cultivate students' thinking ability overnight, it is a systematic process. In teaching, it is necessary to achieve clear teaching objectives, prominent teaching priorities, reasonable teaching methods, step by step and long-term persistence; Only by constantly summing up experience and lessons in teaching and learning from each other's strengths can we achieve the expected results. In a word, the goal of education is to cultivate students' high-level mathematical thinking ability, innovative spirit and ability to solve practical problems. We believe that mathematics education should pay attention to the development of students' innovative consciousness and ability, and make students' thinking modes diversified. This kind of teaching activity can make students think and learn.