Yesterday, we gathered here (at Tsinghua University) to award the Dongrun-Yau Science Award, and then we held the Forum, which was mainly participated in by teachers and some students from the secondary schools. Therefore, I talked about the mathematics education of secondary schools.
It is needed to explain that I don’t know the mathematics education of the secondary school, with no research in this aspect. Here I just comment on this topic, and I hope you can correct me.
First of all, I would like to say a few words about the role of mathematics. Not that long ago, Vice Chairman Yandong Liu, who is responsible for the national science and education, said that some parents even put forward abolishing the mathematics in the college entrance examination. Of course, Comrade Yandong Liu is very clear and she definitely doesn’t do so. I would like to say a few words. Mathematics is the basis of natural science, and is an important tool for various subjects. Especially in the past half a century, it has a very important role on the traditional sciences like physics, chemistry, biology, astronomy, geography and engineering science, and some emerging scientific fields, such as the information science, aerospace science, energy science, environmental sciences, atmospheric sciences, and high and new technologies.
In addition, the role of mathematics is also growing in economy, finance, investment, insurance, securities, actuarial and management, logistics. Look forward to the future, this trend should develop fast. That is to say, the application and penetration of mathematics to these fields of science will be more. Why? In the rapid development of science, high and new technology, economy and management, there is an important aspect to develop in this direction of informatization, digitization and refinement. Each science should develop in this direction because it wants to apply computer, the extremely powerful tool, which can’t be separated from the use of mathematics. For example, we have to explore the mathematical models of these problems, and use the appropriate calculation methods. We have to estimate the rate of convergence and error.
Now, we have entered into the era of big data, and we must do the large-scale science and engineering calculation. In addition, there is another very important aspect. We have to carry out the modern national defense construction, such as the information security and the anti-missile system. Mathematics play a very basic role in these areas.
Of course, what we just said of these roles was what we could see. In fact, the role of mathematics is more important in the cultivation of personnel quality, which is invisible and cannot be seen. The key to the secondary education or the later high-level education is to improve the quality and abilities of the students. After learning the mathematics, many capabilities will be greatly upgraded, such as the geometric intuition and spatial imagination ability, analysis, conclusion and comprehensive abilities, the rigorous logical reasoning ability, demonstration and calculation. We now emphasize the importance of innovation. In fact, the research, development and innovation are inseparable from the talents, while mathematics play a very important role in the talents’ quality and ability.
On some occasions, I met many academicians in the Chinese Academy of Sciences and other fields of the Chinese Academy of Engineering, including the academicians majoring geoscience. Many of them talked to me about the reason why some good results could be made in this field. Because they once ardently loved the mathematics and laid the foundation, when they were in the middle schools and universities. In the 1950s, there were the very well-known books “Infinitesimal Calculus Course” translated from the Soviet Union in mathematics. Although an academician was studying geoscience, he tried to fulfill the book. When he strove for these books about the Infinitesimal Calculus Course, which composed 3 volumes and 8 books. He said that he was able to make achievements in terms of geography, because he had laid a better mathematical foundation at that time, so as to enhance the attainment and ability.
Now, I talk about the secondary school education. I think there is a misunderstanding that the number of students enrolled into the world-class elite schools or into the Tsinghua University and Peking University is viewed as the goal in many well-known secondary schools. It seems that everything is just fine by achieving this goal. I think this should not be the case. The goal should be the number of students who have become the high-level scholars and talents in all the majors. Why is that? Cultivating talents is a long-term process, just like a marathon in physical education. Our education contents and education pattern make many students from the elementary schools to the secondary schools haven’t been taken the enlightened way in a long stage. Nevertheless, they were forced and instilled to learn. Their original interests in learning have been changed almost. After the study of ten years, they have exhausted all their interests and strengths when they are admitted to the prestigious universities. After entering into the university, they just wanted to lie down and rest. It is like the marathon sportsmen who have just run one or two kilometers with all-out sprint. Although they can temporarily lead, they have no stamina, without reaching the end and can’t win.
I have recently read in the newspaper that 30% of the students from Tsinghua University and Peking University have become the so-called "hollow people". Of course, the report of newspapers did not give a precise definition of "hollow people". My understanding of the "hollow man" is a person, with mild depression to study. He has no thoughts for and no interests in many things. This is inseparable from the growth of our students, because they should conform to the law of education in such a long process from the primary school to the secondary school, which should be like a marathon. However, request on him is not to put the final victory, but to be in the lead in a short distance of 1 to 2 thousand meters ahead. He should have run more than forty thousand meters, but he is asked to be in the first place with more than 1,000 meters, without considering his interestand the allocation of his whole ability.
We all know that the man who finishes the marathon finally lies on the ground, whose state was just similar to the "hollow people" somewhat. We have changed the students with the very good interest into a "hollow people". I think this is a very serious problem in education. In addition, examination-oriented education system is very serious in present education. In fact, there are two aspects in education of the middle school. On the one hand, it is to improve his quality, his level and self-cultivation and to enhance the ability, which is the most important aspect. Of course, this point is achieved through the promotion of his knowledge.
However, review and examination are viewed as a major goal. In order to examine, the teachers from high schools generally don’t teach students in the last year. They all use to review and I think this is very abnormal. I have interest in mathematics from the second grade of the junior middle school. I concentrated on the class with my whole heart. After school, I read some reference books, did more exercises and improved my ability, which also enhanced my knowledge. As a result, I did not review mathematics when I got through an entrance examination into the senior middle school and the examination into the university. For example, I believe that the student who won the mathematics Gold Award of Dongrun-Yau Science Award yesterday won't need to review his mathematics, if he takes the university entrance examination.
This means that if you really meet the requirements of learning, you can improve the quality and enhance the knowledge. In fact, you don’t need to review and have the examination. Of course, this is for the most interested student. I think the review and the examination should be the assistant methods for the classmates, but now it is seen as a very serious thing, which has formed the examination-oriented education and the excessive assignments tactic, causing excessive tension. I have heard that the most intense place is the study and life for 3 years in high school. They can’t undress to sleep because it will delay. We can’t accept going to bed in our clothes, which is hard to imagine and extremely excessive.
For the teaching and learning of mathematics in middle school, I felt quite disappointed when I saw the curriculum schedule used in the middle school a few years ago. We must firstly focus on the systematicness for the learning contents, and the basic contents should be properly guaranteed. For example, the education of middle school should emphasize the application and modernization. Someone said that Euclidean geometry was the thing in Greek era, which was outdated. However, we can’t give up the systematicness and the basic contents of learning contents, so as to master it. At the same time, the basic requirements of all students can’t be too high because it is required for all students. Application and modern contents have been added too much to break knowledge system, which makes students probably difficult to understand well. If you lay the foundation later, students can master these contents through self-study in the future.
For example, on the application, we can give a lecture to guide students to broaden their horizons, and the application is not entirely confined to our daily production and the very simple things in our life. I recall that there had been such an example in the 1950s. In 1958, when the application was emphasized, the mathematical application was the configuration of the wheat field and the dispatch of the vehicle, which was too strong in limitation. In fact, the role of mathematics I have said before is very numerous. There are many deep and comprehensive contents in many subjects of science plus of the high and new technologies. In the field of education in middle school, I think we can use lectures to do some introduction in mathematics, but it is not necessary to make into contents of the textbooks.
On the issue of modernization, I think we must emphasize that mathematics is a strict science following on order and advancing step by step. If the previous content is not learned, sometimes it is difficult to learn new things. Of course, I think it is very important to learn calculus for some strong students, because calculus is the basis of contemporary mathematics.
We know that the biggest difference between the elementary mathematics and the higher mathematics is that the elementary mathematics is the constant mathematics, while the higher mathematics is the variable mathematics. There are so few things about concepts in mathematics of middle school. However, there are many things about concepts during the learning in university mathematics. As a result, we should note some important conceptual things in middle school. We should try to ask the students to master them. For example, the concept of change and the concept of infinity and the limit. We think the function definition in middle school like an expression. In fact, it can be divided into a number of segments, and each of them is used with an expression formula.Another example is about the denseness, completeness, matrix and alternation. In the middle school, teachers can explain calculus well to help students master it if there are conditions. Then, teachers should strengthen students grasp of some conceptual things, so it is easy for students to lay the foundation for their further study.
Finally, I talk about Mathematical Olympiad. In fact, I think its original intention is still good. In China, the domestic Mathematical Olympiad is also for a small part of the students who are very interested in mathematics from Grade Two and Grade Three in high middle school, to participate in the Mathematical Olympiad match in this range. But, unfortunately, it becomes the Mathematical Olympiad industrialization, and almost all of the pupils from the primary school and all the students from the middle school have to learn Mathematical Olympiad. Consequently, we are very disgusted with the Mathematical Olympiad. The Mathematical Olympiad has indeed embarked on an unhealthy road.
The Dongrun-Yau Science Award awarded yesterday is to cultivate the initial scientific research capabilities of students to cultivate their innovative spirit. Hence, I think this is what we really need. I put forward such a point of view. Please point out my mistake and criticize it.