How to cultivate students' thinking quality in mathematics classroom teaching Abstract: Mathematics is a major basic subject to cultivate students' thinking ability. The focus of mathematics teaching reform should be to guide students to master learning methods through their own thinking activities. Therefore, the implementation of quality education and the cultivation of thinking ability are the core, while the classroom is the main position of thinking training. In teaching, teachers should take thinking as the core, take training as the main line, follow students' psychological and cognitive laws, adopt flexible and diverse teaching methods, develop students' thinking in time, promote the transformation of students' thinking from unknown to known, from image thinking to abstract thinking, from single centralized thinking to divergent thinking, and improve students' thinking quality. Key words: thinking quality, the training method of mathematics teaching. Thinking quality refers to the external manifestation of the particularity of individual thinking activities, and its essence is the personality characteristics of human thinking. It includes rigor, flexibility, profundity, extensiveness, criticism and agility of thinking. Thinking quality reflects the difference of each individual's intelligence or thinking level. Whenever people encounter problems in their work, study and life, they should always "think about it". This kind of "thinking" is thinking. It transforms perceptual material processing into rational knowledge and solves problems through a series of processes such as analysis, synthesis, generalization, abstraction, comparison, concretization and systematization. We often say that concept, judgment and reasoning are the basic forms of thinking. Both students' learning activities and all human inventions are inseparable from thinking. Thinking ability is the core of learning ability, and cultivating high-quality thinking is one of our most important learning tasks. Quality education, in order to improve students' quality in an all-round way, should inspire students to think in various ways in the teaching process, so that they are good at thinking and diligent in thinking. The development of individual thinking ability not only follows the general regularity, but also reflects the difference of personality, which is reflected in the intellectual characteristics of thinking, that is, the intellectual quality of thinking. On the one hand, this quality is formed in the practice of solving problems, on the other hand, it directly affects the solution of new problems. The purpose of strengthening thinking training in classroom teaching is to let students learn how to master thinking, and to cultivate students' good thinking quality. Below, how to cultivate students' thinking quality in mathematics teaching, I talk about my own views, which are divided into the following six points: First, how to cultivate the agility of thinking. Agile thinking refers to the speed of thinking activities, which reflects the keenness of students' intelligence. Let students think quickly, that is, let students think quickly and finish what they should think in a blink of an eye, which is on the one hand; On the other hand, we should think rationally. These two aspects coexist. Thinking fast, but unreasonable, such "fast" is actually a waste of time, because it has no practical significance; Thinking is reasonable, but it is unusually slow. Obviously, this is a manifestation of low quality of thinking. Therefore, only when these two aspects are achieved can it be called agile thinking. People with quick thinking are good at improvising, thoughtful, able to judge correctly and draw conclusions quickly. Example: As shown in the figure, the side length of the square ABCD is a, and the area of the shadow part formed by drawing a semicircle with each side as the diameter in the square is calculated. If this problem directly finds the graphic area, the visible shadow part consists of eight congruent arches. However, this calculation is obviously complicated. By careful observation and analysis, we can know that the shadow part is divided into the difference between the area of four semicircles and the area of a square. It is easy to draw from the results: s Yin 1 a? Shadow = π () 2× 4-A2 = (-1) A222 The agility of thinking is the efficiency of thinking. In order to improve students' learning efficiency, we must gradually cultivate the agility of students' thinking. First of all, to "seek speed" means that teachers should arrange students' thinking activities and have time requirements so that students' thinking activities can be carried out at a certain speed. Of course, the teacher's speed requirement cannot be divorced from the students' reality, and students should be required to use the speed that students can reach. With the passage of time, the speed requirement for a certain training content can be gradually improved. In this way, students will be trained step by step and their thinking agility will be gradually enhanced. Teachers should put forward requirements for students' calculation speed, especially for assigned homework, and pay attention to improving students' mental arithmetic ability. Secondly, we should learn to "set the situation", that is, teachers use language description or other visual means to show a certain situation, a certain situation and a certain scene, so that students can always be in a certain situation and temporarily become a role in the situation. At this time, thinking must be consistent with the rhythm of the situation and cannot be delayed arbitrarily. In this way, they will become active and active, thus thinking quickly. There is also the need to master basic knowledge, remember relevant theorems and formulas on the basis of understanding, and guide students to master scientific operation methods. It can be seen that the cultivation of thinking agility often requires students to carefully observe the superficial and self-questioning connection of mathematical problems, actively think from the impressions they get, quickly determine the direction of thinking, and find correct and simple methods to solve problems. Second, how to cultivate the profundity of thinking. Deep thinking refers to the abstraction and logical level, depth and difficulty of thinking activities. It is manifested in thinking deeply about problems, being good at generalization and classification, logical abstraction, grasping the essence and laws, carrying out systematic cognitive activities, and being good at predicting and guessing the development process of problems. The profundity of students' thinking is reflected in the comprehensiveness and depth of thinking, in the use of logical thinking methods, taking into account all the conditions related to the problem, studying and grasping the essence of the problem, and solving the problem correctly and simply, and in the personality differences in forming concepts, making judgments, reasoning and argumentation. People with profound thinking quality can see major problems and reveal the most important laws from others' simple or even dismissive understanding. On the contrary, people with superficial thinking are often confused by some superficial phenomena, fail to see the essence of the problem, are not good at deliberation, and often draw conclusions with a little knowledge. For example, if the radius ⊙O is 13 ㎝, the chord AB‖CD, AB=24 ㎝, CD= 10 ㎝, find the distance between AB and CD. This is a "no map" problem, and students are prone to make the following mistakes. Myth: Students are easily influenced by fixed thinking, and draw a graph as shown in figure (1). Cross O is perpendicular to AB and CD respectively, and cross CD and AB to E and F respectively, connecting OA and OC. At Rt△OCE: OE= OC 2? CE 2 = 132？ 52 in Rt△OAF =12 (), OF= OA2? AF 2 = 132？ 122 = 5 (∴ ef =12+5 =17 (㎝). So the distance between AB and CD is 17 ㎝ Analysis: This solution is incomplete because it misses another case. So the correct answer should be 17 ㎝ or 7 ㎝. My thinking: the circle is not only an axisymmetric figure, but also a centrally symmetric figure, and it also has rotation invariance. These characteristics of the circle determine that some problems about the circle will have multiple solutions. If students don't pay attention to solving problems, it is easy to miss the solution. When answering such questions, we need to discuss them one by one according to certain standards, so as not to miss the solution. The mistake of this problem lies in the uncertainty of the position of two parallel chords and the center of the circle. Paying attention to cultivating and developing the profundity of students' thinking is conducive to students mastering mathematics knowledge and skills more systematically and firmly, and to students learning actively and vividly. In view of this, we should start from our own personality and gradually improve the profundity of thinking. Third, how to cultivate the broadness of thinking. The broadness of thinking means that people are good at looking at problems comprehensively in the process of thinking, focusing on the relationship between things, thinking about problems from many angles and finding out the essence of problems. It embodies the width and breadth of thinking. Because of their young age, students often confine their thinking process to a narrow range. To cultivate the broadness of thinking, we should not only train students to think more comprehensively, but also guide them to fully understand the relationship between things, analyze and study problems from various aspects. The broadness of mathematical thinking is manifested in the open mind, which can not only see the whole problem, but also take into account the details of the problem; We can not only grasp the problem itself, but also take into account other related issues; Be good at induction, summary and classification, and form a knowledge structure. The broadness of mathematical thinking is multi-level and multi-angle three-dimensional thinking Generally speaking, you need to have rich mathematical knowledge and experience to form a broad thinking. Overcome the mindset and cultivate the broadness of thinking. Stereotype is the preparation state of psychological activities caused by the pattern of psychological operation, which is also satisfactory. Due to the influence of previous mathematical experience, students' current psychological activities show a certain tendency, and they always want to follow the mastered rule system in the process of solving mathematical problems. The fixed thinking sometimes leads to negative transfer and negative influence, which is manifested in the rigidity and narrowness of thinking. Under the obstacle of stereotype, students' learning forms are stylized and modular, and they lack adaptability. For example, in the evaluation question: "Given X- 1 1 = 1, find the value of X2+ 2", many students are used to finding the value of x first and then substituting it into the evaluation, which makes the problem-solving complicated. It is because I am not good at discovering the relationship between known conditions and evaluation formulas, and the relationship between known conditions and learned complete square formulas. In order to overcome the psychological obstacle of thinking pattern, in the teaching process, students should be trained to use the "double-base" pattern to consolidate and master mathematical knowledge, and at the same time, they should be trained to be good at breaking the pattern, so that when they encounter unfamiliar mathematical problems, they will neither fall into the "pattern" nor be helpless, and they will think about the problem from multiple channels and angles and cultivate the broadness of thinking. Fourth, how to cultivate the thoroughness of thinking. The thoroughness of thinking refers to the depth of thinking activities, the meticulousness and delicacy of logic. The common mistakes are that influenced by thinking set, he doesn't understand the concept and nature thoroughly, he is careless in examining the questions and ignores the implied conditions, which leads to the mistakes in solving problems. Thorough thinking is the basis of solving problems. In the process of solving problems, we should consider the problems comprehensively and systematically and pay attention to the comprehensive application of various conditions, so as to realize the correctness of solving problems. Therefore, it is necessary to observe the structure of the problem from a holistic perspective in order to achieve the purpose of solving the problem, and then solve the problem with a holistic thinking method. Let me give you an example: Example 1: Ignoring the condition that a quadratic equation with one variable has real roots, we know that the square sum of two real roots of equation 2x2-MX-2m+ 1 = 0 is the wrong solution: we can get X 1+X2= 29 from the meaning of the question, and find the value of m? 4 1 ? 2m？ 1 m, X 1X2= so, 2 2 m? 2 m？ 129x12+x22 = (x1+x2) 2-2x1x2 = () 2-2× =, that is, m2+8m-33=0 2 2 4 gives m14. △ = (-m) 2-4× 2× (-2m+1) = m2+16m-8 ≥ 0, and when m=3, △ > 0; δ< 0 when m=- 1 1. So the correct answer is m=3. Looking at one thing in isolation may lead to one-sided or even wrong conclusions; If we relate related things to understand, it is possible to draw a comprehensive and correct conclusion. Therefore, when solving problems, guiding students to use the method of "mutual connection" can cultivate students' thoroughness of thinking. V. How to Cultivate the Flexibility of Thinking Flexibility of thinking refers to the flexibility of thinking activities, that is, the ability to quickly and simply change thinking from one to another. When thinking lacks flexibility, it is manifested as rigid, rigid or slow thinking. It embodies the transfer of wisdom and ability, and is good at guiding students to solve multiple problems, which is an effective way to cultivate thinking flexibility. Through the training of "multiple solutions to one problem", we can communicate the internal relationship between knowledge, improve students' ability to solve practical problems by using the basic knowledge and skills they have learned, and gradually learn the ability to draw inferences from others. a b c a？ 3b？ 2c Example: The value of = = is known. 3 4 5 2a？ b？ The general method of c a b c is: let = = =K, then a=3K, b=4K and c=5K. 3 4 5 3k？ 3 ? 4k？ 2 ? 5k k 1 Substitute algebraic expression: = = 2? 3k？ 4k？ 5k 7k 7？ 3b 2 c a？ 3b？ 2 c a a a b c a？ 3b？ 2c 1 3？ 12 10 ? Option 2: = = 2 a? b c 2 a？ b 1？ C a 3 3 4 5 2a？ b？ C 7？ 6 4 ? 5 ? 7 ? 3 ? Scheme 3: Considering that the examination of this knowledge point usually appears in the form of filling in the blanks or selecting, on the basis of Scheme 1, special values can be used instead of evaluation. Let a = 3, b = 4 and c = 5. Mathematical thoughts and methods are the essence of mathematical knowledge and the link between knowledge and ability. The thinking method in mathematics is the core of understanding the structural form of mathematics through thinking activities, including the representation concept and concept system as knowledge content, and the thinking ability necessary to master the corresponding knowledge content. While teaching mathematics knowledge, teachers should pay more attention to the infiltration and cultivation of mathematical thinking methods, integrate mathematical thinking methods with mathematical knowledge and skills, and constantly improve students' thinking ability, problem-solving ability and ability to combine with practice. Attaching importance to the education of mathematical thoughts such as set thinking, functional thinking, equation thinking, combination of numbers and shapes, and reduction thinking can help students grasp the essence of problems and draw inferences from others, which is of great significance to improving students' problem-solving ability, and will also double students' interest in mathematics learning and achieve the goal of improving mathematics quality with half the effort. When we talk about the flexibility of thinking, we also emphasize diversity and difference. Cultivating the flexibility of students' thinking is an important teaching link for mathematics teachers. It is mainly manifested in enabling students to think flexibly according to the changes of things and change the original plan in time. It is not limited to outdated or inappropriate assumptions, because the objective world is changing all the time, so it is a manifestation of the flexibility of thinking to ask students to understand and solve problems with a changing and developing vision. In this sense, it can also be called divergent thinking. The greater the flexibility, the more developed the divergent thinking, and the more problems can be solved. The more complete the types of multiple solutions, the more important the migration process is. We often say that "drawing inferences from others" is a kind of advanced divergence and a description of the flexibility of thinking to a certain extent. Sixth, how to cultivate the criticality of thinking The criticality of mathematical thinking is a kind of thinking quality, which means that a person is good at testing the correctness of his thinking and its results according to objective facts and viewpoints. People with critical thinking can make a correct evaluation of all the people and things they meet according to certain principles; When dealing with problems, we can objectively consider both positive and negative opinions, stick to correct opinions and give up wrong ideas. In thinking activities, I am good at estimating thinking materials and checking the thinking process, and I am not blindly obedient or credulous. The criticism of thinking comes from students' adjustment and correction of all aspects of thinking activities, that is, self-awareness. This self-monotonous "adjustment" and "correction" comes from students' understanding of the nature of the problem. Only by in-depth understanding and careful thinking can we make a comprehensive and correct judgment. Therefore, the criticism of thinking is a kind of thinking quality with deep foundation. Critical thinking refers to the degree of independent analysis and criticism in thinking activities, whether to follow the advice of others or to think independently and be good at asking questions. Critical thinking is actually an integral part of problem-solving and creative thinking. The quality of students' mathematical thinking is a unified whole, and all the components complement each other, understand each other, promote each other and complement each other. In the process of teaching, teachers combine them organically, strengthen thinking training purposefully and systematically, and cultivate students' good mathematical thinking quality. Only in this way can we really meet the requirements of quality education for mathematics teaching and fully cultivate students' thinking quality in mathematics learning. In a word, how to cultivate students' thinking quality in mathematics teaching in primary and secondary schools, I think, should be a subject of great interest to our educators. I believe that through our continuous exploration, the quality of our next generation will definitely increase!

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