How to Cultivate Students' Rational Thinking in Mathematics Learning

As soon as the math exam results come out, students often sigh: "What's wrong with this topic?" And "I will do it, but I am careless." Hearing this, parents are often relieved and told not to be careless in the future, as if the problem had been solved.

In fact, no one wants to be sloppy in the exam, and everyone wants to complete it with high quality, but they can't avoid all kinds of mistakes. This is because the essence is not carelessness, but ability. The word carelessness masks many substantive problems.

I think carelessness is an improper classification of a large number of substantive problems. The so-called carelessness, its subordinates are the defects of students' various abilities in learning. The operation is wrong because there is something wrong with the operation ability; There is deviation in understanding, which is a defect in understanding ability; Considering the problem is not comprehensive and the logic is not strict; A mistake in expression is the problem of expression ability and so on.

There are so-called carelessness in many links, but I don't think we can simply use the word "carelessness" to brush it off. We should realize that this is a question of ability.

We should attach importance to the ability training in mathematics learning, the core of which is good learning hobbies. Let's take surgery for example. Why is the operation wrong? High school mathematics, every step of the operation, mostly the content of primary school mathematics, should not be wrong in theory. Many times mistakes are caused by students' inattention in operation, which leads to all kinds of low-level mistakes.

Of course, like carelessness, the problem of concentration is easier said than solved. People's concentration is often independent of their own will. Maintaining strong concentration in math learning and problem solving is a kind of ability, which needs to be cultivated in daily training, and its foundation is good math learning habits.

In mathematics learning, we should ask ourselves to have a serious attitude and a dedicated spirit in everything we do. This kind of high attention and all-out effort is a very important habit, which is the basis of improving ability and can form a virtuous circle of study, work and life.

Although I used to be an Olympic math coach, I don't agree that everyone should do it, those who have the ability and those who like it should do it. One consequence of blindly studying the Olympic Mathematics is "muddling along". You obviously don't like it and don't want to do it, but it's impossible to concentrate if you force it. It may take a long time, but the efficiency is not high and the quality is not good. Instead, I developed bad habits.

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Simulating the History of Mathematics: Stimulating Interest in Mathematics Learning

Einstein said: "For everything, only love is the best teacher, far exceeding the sense of responsibility". I think, if you are not interested, you will never talk about "love". It seems that we generally believe that mathematics education in primary and secondary schools in the United States is not as good as ours.

Why, on the one hand, we think that the level of basic mathematics education in China is much higher than that in the United States, but on the other hand, there are still many people who question the role of mathematics education and hope that mathematics can "walk out of the college entrance examination"? The answer is actually very simple. If the purpose of mathematics education is to test and the process of mathematics learning is only to solve problems, such mathematics education is of course boring.

The interest of high school students is not simply based on fun, but more importantly, it makes students feel rewarding and enlightening. So what is the harvest? One view is to explain that mathematics comes from practice and can also be applied to practice, and it can be useful in life after learning.

However, there are only a handful of high school mathematics knowledge that can be directly applied to life practice. The application of "artificial" is not only unconvincing, but also a reason for the formation of the theory of uselessness in learning mathematics. In fact, mathematics is the foundation of natural science, which is an accepted fact.

In the process of mathematics learning, especially in senior high school, I object to one-sided emphasis on the relationship between mathematics and practical application, and expect to pay more attention to the useless use of mathematics. Thinking about mathematics education from the perspective of culture and human growth.

The problem of mathematics education now lies in the organic connection between mathematics knowledge and the ideological culture behind it. There are only isolated knowledge points and topics, and there is no vivid process and experience. The emergence of any mathematical concept did not fall from the sky. The development of mathematics has both internal needs and external strength, which are very valuable mathematical resources.

Therefore, the charm of mathematics lies in letting students know the necessity and possibility of the emergence of mathematical concepts in textbooks, guiding them to relive or simulate the occurrence and development of these problems, allowing students to experience the joy of exploration and innovation while accumulating knowledge, and to be inspired by the background and corresponding methods of previous people's research on problems, thus realizing mathematical culture.

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Questioning and improving mathematical ability: what, why and what else?

Mr. Zeng Rong, a middle school affiliated to Fudan University, summed up three things about mathematics learning, namely: what, why and what else?

The content of high school mathematics is more abstract, and the concentration of knowledge is greatly improved. Many students suddenly find it difficult to learn high school mathematics, because they don't adjust their learning methods in time with the improvement of their ability requirements, and use the learning mode of junior high school to learn high school mathematics. This is one aspect of the problem.

At the same time, we often pay too much attention to the study or teaching of specific knowledge, but neglect to reveal the truth behind it. In some mathematics teaching, there is often no thinking process, only conclusions, from conditions to conclusions, and there is a lack of reasoning. Compress the mathematical thinking process into a quick answer to the conclusion.

Only pursue the speed of solving problems, not pay attention to the improvement of thinking quality. In this way, students' abilities of inquiry, induction and logical reasoning have not been fully cultivated, and the most effective opportunity to cultivate students' abilities of inquiry, induction and logical reasoning has been lost.

Mathematics learning should take students as the main body, and students can't learn passively. There are a lot of predecessors' creative work in high school mathematics, so I think we need to pay attention to the formation process of mathematical knowledge and concepts. The concepts of mathematical knowledge are all created by predecessors. Students can simulate the process of discovery and inquiry under the guidance of teachers, which is the most real innovation.

For example, in solid geometry, why is the concept of angular size formed by straight lines on different planes defined like this? What kind of mathematical truth is involved? What kind of foundation has been laid for solving the problem? It seems that it takes more time to explore these problems carefully, but it allows students to grasp the important information contained in the concept, excavate the connotation of mathematical concepts, and experience the simplicity and efficiency of mathematics.

You should have the spirit of questioning when learning anything. The key to questioning in mathematics is to question what the essence of mathematical knowledge is, why it is like this, and what else. Only in this way can we finally promote mathematics learning and improve learning ability and thinking quality.

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Mathematics is a culture: Why should we study mathematics?

"Mathematics is useless" and "the topic is difficult" are many people's impressions of high school mathematics, and some people even send out the voice of "Mathematics is out of the college entrance examination" on the Internet. So why does everyone have to learn mathematics that seems to have nothing to do with daily life from primary school, and many people forget it after the exam?

I think we might as well think about this problem from another angle. Probably few people in China ask: What's the use of reading Tang poetry? This is because people all agree that Tang poetry is the traditional culture of China, which contains important educational value and cultural details. The so-called "useless"

We read Tang poetry from childhood, learning and feeling the culture of the motherland. Similarly, learning mathematics is also learning a culture. Mathematics is a kind of world culture, and mathematics education also has educational value and cultural inheritance.

Mathematics stands for reason. The basic laws of nature can be described by mathematics, so the process of learning mathematics is the process of learning how to understand our world. Learning mathematics not only cultivates people's logical ability, but also cultivates scientific attitude and rational spirit.

For example, mathematics is a deductive reasoning system based on axiomatic system. Whether a question is true or not should be demonstrated by strict reasoning, not by intuitive imagination, which sets up a scientific view of truth for students.

Similarly, from primary school to high school, they have experienced several episodes of expansion. We can see that the expansion of numbers is not simply denying the past, but the development on the basis of retaining the core properties of the original number set, which is the development of inheritance. We can feel the harmony and unity of inheritance and development, which is consistent with the development of our society.

A student came back to see me after working for many years and said, "Miss Li, I can't do these math problems." I asked jokingly, "Do you think all your previous math studies were in vain?" The classmate replied: "It's not for nothing, it's the way of thinking." Just like reading Wang Wei's works when he was young. When he grows up and sees the desert again, he will think of "the desert is lonely and straight" and have a poetic life experience.

The same is true of mathematics, which has been inseparable from philosophy since Heraclitus and Pythagoras. For example, the concept of irrational numbers such as root number 2 is impossible to produce intuitively. It comes from people's thinking, but it reveals the reality better. It is not Euclidean geometry, beyond people's daily understanding, but provides a mathematical model for Einstein's theory of relativity.

In the history of mathematics development, there are countless similar examples. Therefore, the study of mathematics can help us think about how to look at the world, understand the world, better understand the world and form rational thinking. This is the culture of mathematics, which is why everyone should learn mathematics from the first grade of primary school.