"The movement against the Tao" is one of the main arguments of Laozi's philosophy. The general idea is that everything in nature and human society goes to one extreme and the other. To borrow Hegel's words, everything contains its own negation (1).

1. 1 is relevant.

Laozi said, "Digging doors and windows to build a house can be used as a room." If you dig doors and windows to build a house, you will have empty parts in the doors and windows walls, and then you will have the function of a house. Therefore, "you" brings convenience to people and "you" plays a role. )"

Laozi's view of existence can be explained by a popular example. If a cup is empty, then there can be "existence" in this "nothing"-a cup can be filled; If it is full of things, nothing can be born from this "existence"-it can't accommodate anything else. Therefore, existence and nothingness are mutually causal and mutually changing.

If a teacher talks endlessly, this kind of "being" will produce students' interest in learning and thinking ability. Students should be given enough time to think, create, supplement, recall and evaluate. The correct use of this "existence" will produce the "existence" of students' learning interest and thinking ability.

Draw a circle, leave a small gap at the end, and look at it again, and your mind will tend to complete the circle. Gestalt psychology school's "Gestalt Pressure" theory holds that when people watch an irregular and imperfect shape, they will have an internal sense of tension, forcing the cerebral cortex to move nervously to fill the "defects" and make it "gestalt", thus achieving inner balance.

This kind of "gestalt" psychology, the author has a deep understanding. A class, the topic is "the quantitative product of plane vectors (first class)"

After listening to the lesson, the author has been wondering why the textbook only studies the sum, difference and product of plane vectors, but not the quotient. I asked several teachers, but I couldn't say clearly. This question lingered until one day, when I got an explanation from a university teacher. For such problems, we can at least tell students and study them later.

Another example is blackboard writing. Some classes use multimedia, and a large piece of information comes out on the screen at once, and the amount of information doubles. Man's "irrigation" has evolved into machine irrigation. In fact, sometimes the contents of blackboard writing and courseware are not necessarily complete. What we want to express should not be confined to the inside of the picture, but should let the image factors inside the picture echo the things outside the picture, that is, what is stored in textbooks or students' brains. Or let the students think and follow the incomplete images displayed. Make the incomplete become complete and the defective become a gestalt, so that students' aesthetic needs can be accepted statically from a single direct narrative, and tend to multi-level philosophical trends thinking, so that the defective things can gain a complete concept in students' eyes, thus generating psychological satisfaction.

1.2 Relationship between Majority and Minority

Laozi said: "Take less, get more." Take less, get more, and be greedy, and you will be confused. ) Some activities have their limits relative to the objective environment. If a person eats too much, what is good for his health will become harmful and he will get sick. A person should only eat the right amount of food. This is the law of things changing. (3) Lao Tzu called them "Chang".

1.2. 1 How much have you taught and explored?

The new mathematics curriculum standard requires teachers to "help them truly understand and master basic mathematics knowledge and skills, mathematics thoughts and methods, and gain rich experience in mathematics activities in the process of independent exploration and cooperative communication." "Effective mathematics learning activities cannot rely solely on imitation and memory. Hands-on practice, independent exploration and cooperative communication are important ways for students to learn mathematics." "Full-time irrigation" is not feasible, and students' independent exploration must be paid attention to.

But independent exploration does not rule out teaching. How to deal with the relationship between teaching and inquiry? Appropriate teaching and exploration should be carried out according to the actual teaching content, and it should not be biased. Generally speaking, the following four situations should be said less: 1. Students can basically understand the contents in the textbook; 2. Not accepted by most students; 3. Only short-term effect; It is more complicated now, and it will be obvious in the future. However, not everything can be explored. Some conclusions were reached hundreds or even thousands of years later, so it is not necessary and impossible for students to explore through a class. Professor Zhang Dianzhou believes that there are three kinds of knowledge that should not be "explored": 1. Prior knowledge; 2. incomprehensible knowledge; 3. Procedural knowledge (4).

1.2.2 Ask a few questions in class

The author once listened to a municipal public class for 5 minutes, and was shocked by the students' rapid-fire questions, so he began to count "positive". Rhetorical questions don't count. In the next 40 minutes, * * * asked 1 14 questions! About 20 seconds on average. Most of these problems are cut out, lacking in thinking value, or arbitrary and lacking in purpose; Or the content is simple, the pursuit of lively atmosphere.

If there are too few questions in class, it will become "full house irrigation". There are too many problems, and the classroom atmosphere is vigorous on the surface, which will actually create a false scene of prosperity and lack effective thinking training.

Classroom questioning should at least meet the following conditions: 1, purposeful and not arbitrary; 2. It is enlightening and can activate students' thinking; 3. Moderate depth, in the recent development area of students' knowledge and ability; 4. Facing all, let everyone think and gain as much as possible.

According to this standard, the author calculated that at least 80 questions of 1 14 can be cut off.

1.2.3 How much is the problem-solving training?

There is a teacher whose "teaching effect" is surprisingly good. She comes first in every exam. She is a "tumbler". Everyone was surprised and couldn't understand. One day, the author went to learn from the scriptures and talked about the solution of "a unitary fractional equation that can be transformed into a quadratic equation". Three kinds of questions should be taught in three classes. The teacher finished his "intensive lecture" in 15 minutes without speaking. The rest of the time is "practice more". The quiz before class shows a high correct rate. However, the author is full of doubts. After class, he found a student and asked him to solve the equation:, which was solved soon. Can he not check? He said firmly: no, ask him why, "this is a fractional equation, and both fractional equations have to be tested." Then ask: "Why do you have to take the fractional equation test?" "It is required by teachers and books." In fact, in the whole process of solving the problem, there is no case of "multiplying the formulas that may be zero on both sides of the equation", and the scope of each step of the solution has not changed, so there is no need to test.

Does practice make perfect? Yes! But it depends on what it is. Ouyang Xiu mentioned in the article "Selling Oil Weng" that Chen Kang Su Gong's skillful shooting and the oil Weng's stunt of "putting a gourd on the ground, covering his mouth with money and draining it with a spoon" are the results of "practice makes perfect", but they belong to the skill level and cannot be applied to learning. Psychobehaviorism stimulus-reflex theory. The more standardized the behavior, the basis is a series of animal experiments and human psychometric experiments. The problem is that these experiments can't explain the slightly complicated mathematical learning phenomenon. Behaviorism locates knowledge understanding at the level of knowledge memory, and does not distinguish between "mechanical memory" and "memory based on understanding". In fact, behaviorism only pays attention to people's external behavior, but does not study people's internal thinking process, so it is impossible to discuss "knowledge understanding" in depth.

Laozi pointed out, "A sage teaches no words without doing anything." The sage treats all problems with a "inaction" attitude and teaches by example. In fact, the meaning of "inaction" does not mean inaction at all, but it needs to be done less, so don't do it against nature. If a person does too much, it will become harmful. Moreover, it will be harmful.