How to Cultivate Students' Logical Thinking in Primary Mathematics Classroom

First, the infiltration of virtue-to cultivate the aesthetic power of thinking

Knowing the gains and losses of right and wrong, beauty and ugliness is the ideological basis for a person to do something. Education should always pave the way for improving students' ideological understanding. Use positive examples to provide the power of example; Draw negative lessons and enhance the sense of hardship; Show the role of theme content in stimulating interest; Explore the aesthetic factors of subject content and cultivate sentiment; Reveal the philosophical material contained in the subject content and improve the ability to perceive the world and know yourself; Wait a minute. Make students gradually form the aesthetic strength of personality, behavior, appreciation and dialectical materialism. For example, in the teaching design of Pythagorean Theorem, students are arranged to go home to find relevant information about Pythagorean Theorem before class: there are about 322,000 Pythagorean Theorem related contents that can be searched online; There are about 72,500 related contents in the Pythagorean Theorem Proof Method; "Some data show that there are more than 500 ways to prove Pythagorean theorem, and only the mathematician Hua in the late Qing Dynasty provided more than 20 wonderful methods." "This is unmatched by any theorem."; Up to now, the Pythagorean Theorem is the earliest recorded by China, a famous mathematical work in ancient times, The Sutra of Zhouyi, which was written around 1 century BC, more than 500 years earlier than Pythagorean mathematician Pythagoras (Pythagorean Theorem is usually called Pythagorean Theorem in the West). Students will deeply feel the beauty of mathematical graphics, and at the same time, they will learn about the outstanding contributions made by ancient mathematicians in China to the field of mathematics, and further enhance their national pride.

In fact, as far as learning itself is concerned, if a student does not have good aesthetic ability, he will fall into the mud pit of "metaphysics", which will lead to heavier reading burden and little effect, and eventually he will be trapped by books. Or because the simple and complicated problem of who and why to study is not solved, the internal dynamic mechanism is paralyzed, which makes reading, a hard work that needs long-term effort and perseverance, a mere formality, resulting in a waste of financial, material and human resources. Therefore, improving the aesthetic power of thinking is the primary task to effectively develop other thinking abilities and thinking qualities.

Second, timely modeling-cultivate the flexibility of thinking.

Mobility is an important symbol of profundity and flexibility of thinking, and this ability depends on various forms of modeling in teaching activities. There are mainly the following two aspects: first, the transfer between teaching activities and social activities; Second, the transmission of ideas and methods between different disciplines and different contents. Through accurate (such as the transformation between practical problems and mathematical problems) and fuzzy (such as the application of "bridge" in solving problems) modeling, students can constantly gain perceptual knowledge of the ways and methods of communicating with different objects, and gradually rise to rational knowledge, thus forming and developing the liquidity of thinking. For example, in the teaching of equation application problems, students should be trained to start from "problems", establish mathematical models through analysis, association and abstract generalization, solve and test the models, and finally solve the problems. It is beneficial to cultivate students' application consciousness and practical ability, improve students' ability to analyze and solve problems, apply mathematics knowledge to economy, finance and trade, make students truly realize the value of mathematics, improve their interest in mathematics learning, and get better training in the basic quality of mathematics at the same time, laying a good foundation for future social and lifelong learning.

Another example: Why does the equation 2x2+3x+5-2a=0 have a real number solution when a is a real number?

Thinking analysis: The inspiration comes from natural or social phenomenon-"birds of a feather flock together" (1), which can be transferred to the strategy of "concentrating variables and separating variables" in mathematical problem solving (fuzzy modeling obtained through association and analogy). So the original equation is transformed into 2x2+3x+5=2a.

Because functions and equations are expressed in the form of "equality", the communication of this structure provides them with opportunities for communication. Therefore, there are:

Idea 1 (building a function model and transforming it into a function problem):

Considering that 2a is a function about x, the problem is transformed into finding the range of function 2a=2x2+3x+5.

"Number" and "shape" are the two main channels for us to enter the palace of mathematics, and functions and equations are the important ties for their communication. So, there are:

Idea 2 (establishing functional model to enter morphological state):

Let the function y = 2x2+3x+5 (); Constant function y=2a

By studying the relationship between two function images, the problem is solved.

Training of changing questions (further migration): can you construct questions (triangle, geometry, application questions, etc. ) different from the original question mode and answer them?

This kind of open question provides a broad stage for students' imagination.

Third, simulated discovery-cultivating the inquiry ability of thinking

"Innovation is the soul of a nation's progress and an inexhaustible motive force for a country's prosperity." It is an important task of quality education to let students learn to discover and innovate. Constructing the scene of knowledge discovery and formation, exposing the process of teachers' learning, research and cognition, reducing the possibility of knowledge and ability formation as much as possible and increasing the inevitability; It is not only the requirement of the law of thinking development, but also the need of effectively forming and developing students' cognitive structure, so as to create a good environment that is predictable, accessible and conducive to active construction, and make students' thinking naturally extend; At the same time, it can stimulate students' desire for discovery and innovation, drive inquiry behavior, train their inquiry ability, and lay a good thinking foundation for future discovery and innovation. For example, in the teaching design of Pythagorean Theorem, students can draw two right-angled sides of a right-angled triangle in groups of 3 and 4 respectively, and another group of students can draw two right-angled sides of 6 and 8 respectively. Then measure the length of the hypotenuse, square the three sides respectively, find out the relationship between them, and guess the Pythagorean theorem. (Operation-Observation-Guess) Cultivate students' ability to explore problems.

Fourth, the idea of inspiration-to cultivate the generalization ability of thinking.

The basic idea of discipline is the soul of discipline knowledge, the basic viewpoint of dealing with problems and the rational understanding of discipline content. Its concentrated expression is the abstract generalization power of thinking. For example, mathematical ideas (transformation ideas, functions and equations, combination of numbers and shapes, classified discussion ideas, etc.). It is empty when it is not perceived, so it is difficult to be perceived, but once understood, it has great power to guide the solution of problems, which can act on people's thinking for a long time and play a role in different fields. The teaching of mathematical thought can be divided into two links: one is to teach, abstract and summarize through problem-solving reflection; Second, demonstration, through the guidance of ideas to find a solution to the problem, especially when the problem-solving ideas are blocked. In order to let students gradually feel the existence, acquisition and application of "thinking" and improve the generalization ability of thinking in the process of understanding thinking.

Fifth, encourage speculation-cultivate the intuition of thinking.

Intuition is a creative thinking ability. The development of this ability depends on the continuous role of guessing consciousness. Of course, guessing should be based on certain knowledge, so as not to guess at random; We should form a rigorous and responsible scientific attitude with strict argumentation as the backing. Reasonable guessing is to establish a scientific goal, which can not only optimize the channels of solving problems, but also train the intuition of thinking well.

In the process of learning new concepts, propositions and theorems in mathematics teaching, we should try our best to let students perceive the whole process of the occurrence and development of new knowledge through their own initiative, so that students can gradually acquire the abilities of collecting and processing information, analyzing and solving problems, expressing words, practicing and cooperating, form innovative consciousness, carry out cognitive activities and open innovative thinking. Regarding the teaching mode of concepts, propositions and theorems, I have made the following "program" design: step one, create situations, and students ask questions and guess; The second step is the exploration of students' various forms of thinking; The third step is the teacher's guidance; The fourth step is to combine students' autonomy and cooperation; The fifth step is to combine students' language and thinking to form new concepts, propositions and theorems. The sixth step, teachers and students * * * cooperate to evaluate and supplement optimization. Through the operation of the above procedures, while exercising intuition, it often brings "bright prospects" for solving problems. Teachers should change from autocracy to democracy, students should change from passive acceptance to active exploration, and classroom teaching should change from closed singleness to open pluralism. These changes will certainly be conducive to the cultivation of students' scientific spirit and the formation of innovative thinking. Students also cultivate the intuition of thinking through careful observation. First of all, Whistler said: "The essence of teaching art lies not in imparting, but in inspiring, awakening and encouraging." Appropriate and timely evaluation can effectively help students adjust their learning progress, learning attitude and learning methods, which will become a powerful driving force and greatly encourage students to participate independently, be diligent in exploration and be brave in innovation. In mathematics classroom teaching, students often have different ideas and solutions from the standard answers prepared by textbooks or teachers. Teachers should fully affirm and praise in the process of classroom teaching, and never turn a blind eye or turn a deaf ear. This kind of praise can greatly stimulate students' innovative consciousness, and also encourage other students to dare to think and do. This is the bud of innovation and should be taken care of.

Six, cited without hair-cultivate exploratory thinking.

The main task of teachers is to "organize and guide students' study life, so that they can' learn with inner experience and creation'". Therefore, if possible, we should return the rich exploration process and sufficient exploration time to students. As Comrade Mao Zedong said, "If you want to know the taste of pears, you should taste them yourself. "Let students experience the sour, sweet, bitter and spicy in the process of cognition, so as to obtain full perceptual knowledge, which not only lays a solid foundation for rational understanding, but also helps to establish self-confidence, form the independence of thinking, and then induce the exploration of thinking. It is a good way to cultivate students' exploratory thinking habits and a necessary means to cultivate excellent thinking quality. For example, when learning the internal angles and formulas of polygons, I will ask students to try how the formula (n-2) × 180 is derived. Students can be instructed how to divide a quadrilateral into triangles first. According to the sum of triangle internal angles is 180, and the sum of quadrilateral internal angles is 360. Then think about the sum of the internal angles of pentagons and hexagons. The formulas of internal angle and internal angle of n polygon are obtained by analogy. There are many ways to divide a polygon into triangles during this query. Let students think boldly and do it. It can activate students' enthusiasm for exploration, enable them to solve problems in exploration, and at the same time experience the connotation of assimilation, concretization, specialization and other strategies, as well as the role of the thinking form of "association". Only by leaving space for students to explore and letting them know the methods of exploration can students really enter the role of exploration, which requires a scientific grasp of the degree of "introduction"

Seven, provide setbacks-cultivate the tenacity of thinking.

The tenacity of thinking is gradually formed and embodied by constantly overcoming difficulties in the face of setbacks. Without the baptism of setbacks, there will be no indomitable thinking and will quality, and it is difficult for people who lack this will quality to succeed. Therefore, in teaching activities, it is very necessary to provide students with moderate opportunities for frustration exercise, and it is also the professional responsibility of teachers. When students encounter difficulties, teachers should give appropriate guidance, not enthusiastic answers! Otherwise, it not only reduces the difficulty of students' thinking, but also cultivates students' randomness, which will inevitably lead to the rigidity of students' thinking and the fragility of their will. We believe that timely and appropriate advocacy of "don't answer, don't ask, don't answer, don't discuss and don't answer", supplemented by appropriate monitoring, is very beneficial to hone students' will and cultivate their learning ability.

Eight, set a trap and then pull it out-cultivate the profundity of thinking

Both cognitive psychology and classroom teaching practice show that it is often better to attack concepts that are easily influenced by negative transfer and theories that are easily superficial in understanding than to explain them one by one and guide them positively (that is, "if it is difficult, it will be reversed"). Designing traps will make students unconsciously fall into them, and then let them "break free in pain", which will promote the profound development of thinking. For example, in the teaching of fractional equations, in order to let students deeply understand the problem of increasing roots, I designed such a topic for students:

When m is what value, the fractional equation has real roots. Most students follow the basic method of solving fractional equation: remove the denominator, then solve it, and then mistakenly think that m should be any real number. However, this fractional equation is not considered. When m=5 or m=-3, there will be roots.

But the trap should be "suspicious where there is no doubt" and the difficulty should be moderate, so that we can think with doubt.

Nine, multi-directional induction-cultivate the flexibility of thinking

The flexibility of thinking lies in whether you can examine and analyze problems from different angles, or choose a way that suits you to understand and study problems. In particular, the teaching of difficult textbooks can induce students to discuss and express from different angles, form a broad thinking space and provide flexible thinking options, which can not only cultivate the flexibility of thinking, but also help to connect with the learning experience of students at different levels. This is also one of the "teaching students in accordance with their aptitude" in classroom teaching. For example, in the application teaching of quadratic equation of one variable, the problem of designing flower beds is put forward to students: there is a rectangular garden with a length of 4cm and a width of 3cm, and now a flower bed is to be opened in the garden, so that the area of the flower bed is half that of the garden. This is an open topic, there is no fixed answer, and the topic is very participatory, which is suitable for people with different knowledge bases and intelligence levels. At that time, 38 people in the class designed more than ten kinds of schemes and developed their creative ability, which was of great help to form the scientific spirit of bold exploration and innovation. More significantly, it can provide a paradigm for students' divergent thinking.

Ten, advocate questioning-cultivate critical thinking.

Not blindly following, not superstitious, not opinionated, not opinionated, is a reflection of a person's good self-confidence. The formation of this independent personality is synchronized with the critical maturity of thinking. Correct questioning is the key external expression of thinking. Students should be encouraged to be suspicious, have doubts and boldly put forward different opinions; Even if it is wrong or even ridiculous in the eyes of teachers, it is very natural and precious in the process of students' cognitive activities! It embodies the true nature of the cognitive process, is the result of contradictions and conflicts in cognitive activities, and is a "bridge" for the in-depth development of thinking. Therefore, questioning should be regarded as an important form of teaching activities. Let students improve their cognitive structure in questioning; "learn" in questioning, gradually form learning ability and develop creative thinking ability; Learn to criticize and self-criticize in questioning, enhance the awareness of error correction and improve the ability of error correction; Make students gradually form a personality quality that is both modest and prudent and innovative.

The development of thinking ability and thinking quality complement each other, and the roles of different teaching strategies in developing thinking are also complementary and interdependent; Under the school organization form with classroom teaching as the framework, it is particularly important to optimize classroom teaching strategies, which is a necessary condition for the good development of students' thinking and the first link in implementing quality education.