As early as 1940s, Hua was one of the leading mathematicians in the field of number theory. But he is not satisfied, he will not stop, he would rather start a new stove, leave number theory and learn algebra and complex analysis that he is not familiar with. How much perseverance and courage are needed!
Hua is good at telling profound truth in vivid language. These words are concise, philosophical and unforgettable. As early as the SO era, he proposed that "genius lies in accumulation, and cleverness lies in diligence". Although Hua is brilliant, he never mentions his own talent. Instead, he regards "diligence" and "accumulation", which are much more important than cleverness, as the key to success, and repeatedly educates young people to learn mathematics so that they can exercise themselves all the time. In the mid-1950s, in response to the problem that some young people in the Institute of Mathematics were complacent after making some achievements, or kept writing papers at the same level, Hua put forward in time: "There must be speed and acceleration." The so-called "speed" means producing results, and the so-called "acceleration" means constantly improving the quality of results. Just after the "Cultural Revolution", some people, especially young people, were influenced by bad social atmosphere, and some departments were eager for success, frequently demanding grades, evaluating bonuses and other practices that were not in line with scientific laws, which led to the corruption of the style of study. Performance for shoddy, fame and fortune, wanton boasting. 1978, he earnestly put forward at the Chengdu conference in chinese mathematical society: "Early publication, late evaluation." Later, it was further put forward: "Efforts are in me, and evaluation is in people." This actually puts forward the objective law of scientific development and scientific work evaluation, that is, scientific work can gradually determine its true value after historical test, which is an objective law that is independent of human subjective will. "
Hua never hides his weaknesses. As long as he can learn, he would rather expose them. When he visited Britain at the age of seventy, he changed the idiom "Don't teach others an axe" to "teach others an axe" to encourage himself. In fact, the previous sentence is that people should hide their shortcomings and not expose them. Did Hua go to college to help others with their expertise, or did he turn lectures into formalism because he didn't specialize in others? Hua chose the former, that is, "wait a minute, and you will arrive at the door." As early as the 1950s, Hua compared mathematics to playing chess in the preface of Introduction to Number Theory, calling on everyone to find a master, that is, to compete with great mathematicians. There is a rule in chess in China, that is, "A gentleman does not regret watching chess without saying a word". 198 1 year, in a speech in Huainan coal mine, Hua pointed out: "Watching chess is not a gentleman, help each other; I regret the gentleman and change my shortcomings. " It means that if you see someone else's work problems, you must speak up. On the other hand, when you find something wrong with yourself, you must correct it. These are "gentlemen" and "husbands". In view of the fact that some people retreat when they encounter difficulties and lack the spirit of sticking to the end, Hua wrote on a banner for Jintan Middle School: "People cannot say that the Yellow River will not die, but I say that the Yellow River will be stronger."
When people get old, their energy will decline, which is a natural law. Hua knows that time waits for no man. 1979 when he was in England, he pointed out: "The village is old and easy to empty, and people are old and easy to disperse. The scientific approach is to abstain from empty and scattered. I am willing to stick to it all my life. " This can also be said to be his "determination book" to fight against his aging with the greatest determination, so as to spur himself. The patient with the second myocardial infarction in Hualuosuo still insisted on working in the hospital. He pointed out: "My philosophy is not to prolong life as much as possible, but to do more work during the day." If you are ill, you should listen to the doctor and have a good rest. But his indomitable spirit is still valuable.
In a word, all of Hua's expositions are permeated with a general spirit, that is, constant struggle and continuous progress.
Zu Chongzhi (429-500) had a grandfather named Zuchang, who was an official in charge of royal architecture in Song Dynasty. Zu Chongzhi grew up in such a family and learned a lot from childhood. People all praise him as a knowledgeable young man. He especially likes studying mathematics, and he also likes studying astronomical calendars. He often observes the movements of the sun and planets and makes detailed records.
When Emperor Xiaowu of Song heard of his fame, he sent him to work in a government office specializing in academic research in Hualin Province. He is not interested in being an official, but he can concentrate more on mathematics and astronomy there.
There have been officials who studied astronomy in all previous dynasties in our country. They made calendars according to the results of astronomical research. By the Song Dynasty, the calendar had made great progress, but Zu Chongzhi thought it was not accurate enough. Based on his long-term observation, he created a new calendar called "Daming Calendar" ("Daming" is the title of Emperor Xiaowu of Song Dynasty). The number of days in each tropical year measured by this calendar (that is, the time between winter and sun in two years) is only 50 seconds different from that measured by modern science; It takes less than one second to measure the number of days for the moon to make one revolution, which shows its accuracy. In 462 AD, Zu Chongzhi requested Emperor Xiaowu of Song Dynasty to issue a new calendar, and Emperor Xiaowu called ministers to discuss it. At that time, Dai Faxing, one of the emperor's minions, stood out against it and thought that it was deviant for Zu Chongzhi to change the ancient calendar without authorization. Zu Chongzhi refuted Defarge on the spot with his own research data. Relying on the emperor's favor, Dai Faxing said arrogantly: "The calendar was formulated by the ancients and cannot be changed by future generations." Zu Chongzhi is not afraid at all. He said very seriously, "If you have a solid basis, argue it out. Don't scare people with empty talk. " Emperor Xiaowu of Song wanted to help Dai Faxing, and found some people who knew the calendar to argue with Zu Chongzhi, but Zu Chongzhi refuted them one by one. However, Emperor Xiaowu of Song still refused to issue a new calendar. It was not until ten years after Zu Chongzhi's death that his Da Ming Li was put into practice.
Although the society was very turbulent at that time, Zu Chongzhi studied science tirelessly. His greater achievement is in mathematics. He once annotated the ancient mathematics book Nine Chapters Arithmetic and wrote a book Composition. His most outstanding contribution is to get quite accurate pi. After a long and arduous study, he calculated pi between 3. 14 15926 and 3. 14 15927, becoming the first scientist in the world to calculate pi to more than seven digits.
Zu Chongzhi is a generalist in scientific inventions. He built a kind of compass, and the copper man in the car always pointed south. He also built a "Thousand-Li Ship" and tried it in Xinting River (now southwest of Nanjing). It can sail 100 Li a day. He also used hydraulic power to rotate the stone mill, pounding rice and grinding millet, which was called "water hammer mill".
In Zu Chongzhi's later years, Xiao Daocheng, who mastered the Song Guards, wiped out the Song Dynasty.
During the Northern Song Dynasty in China, there was a well-read and extraordinary scientist, and he was Shen Kuo.
Shen Kuo, Chinese character, was born in Qiantang, Zhejiang (now Hangzhou, Zhejiang) in the ninth year of Tiansheng in Song Renzong (A.D. 103 1). His father Shen Zhou worked as a local official in Quanzhou, Kaifeng and Jiangning. Mother Xu Shi is a well-educated woman.
Shen Kuo studied hard since childhood. Under the guidance of his mother, he finished reading at home at the age of fourteen. Later, he followed his father to Quanzhou, Fujian, Runzhou, Jiangsu (now Zhenjiang), Jianzhou, Sichuan (now Jianyang) and Kaifeng, the capital of China. He had the opportunity to get in touch with the society, understand the life and production of the people at that time, increase his knowledge and show his superhuman intelligence.
Shen Kuo is proficient in astronomy, mathematics, physics, chemistry, biology, geography, agriculture and medicine; He is also an outstanding engineer, excellent strategist, diplomat and politician. At the same time, he is knowledgeable, good at writing and proficient in other people's calendars, music, medicine, divination and so on. Meng Qian Bi Tan, written in his later years, recorded in detail the outstanding contributions of working people in science and technology and their own research results, reflecting the brilliant achievements of natural science in ancient China, especially in the Northern Song Dynasty. Meng Qian's pen talk is not only an academic treasure house in ancient China, but also occupies an important position in the history of world culture.
Japanese mathematician Kazuo Sanshi once said: People like Shen Kuo can't be found in the history of mathematics all over the world. Only China has such people. Dr Joseph Needham, a famous British expert on the history of science, said that Shen Kuo's Meng Xi Talk is the coordinate of the history of Chinese science.
Gauss is a German mathematician, astronomer and physicist. He is regarded as one of the greatest mathematicians in history, as well as Archimedes and Newton.
Gauss 1977 was born in a craftsman's family in Brunswick on April 30th, and 1955 died in G? ttingen on February 23rd. When I was a child, my family was poor, but I was extremely smart. I was educated by a noble. From 1795 to 1798, I studied at the University of G? ttingen, and 1798 transferred to Helmstadter University. The following year, he received his doctorate for proving the basic theorem of algebra. From 1807, he served as a professor at the University of G? ttingen and director of the G? ttingen Observatory until his death.
Gauss's achievements cover all fields of mathematics, and he has made pioneering contributions in number theory, non-Euclidean geometry, differential geometry, hypergeometric series, complex variable function theory, elliptic function theory and so on. He attached great importance to the application of mathematics, and emphasized the use of mathematical methods in the research of astronomy, geodesy and magnetism.
A Brief History of Mathematical Competition
Mathematics competition is similar to sports competition. It is an intelligence competition for teenagers, so the Soviet Union pioneered the term "Mathematical Olympics". In similar intelligence contests with basic science as the competition content, the mathematics contest has the longest history, the largest number of participating countries and the greatest influence. The more formal math competition began in Hungary in 1894, and has been held for more than 90 times up to now, except for seven sessions suspended due to two world wars and the incident of 1956. The mathematics competition in the Soviet Union began at 1934, while the mathematics competition in the United States began at 1938. Apart from the three years during World War II, these two countries have held more than 50 competitions. Other countries with a long history of mathematics competitions include Romania (1902), Bulgaria (1949) and China (1956).
1956, the eastern European countries and the Soviet union formally defined the plan of the international mathematical olympiad. 1959, the first international mathematical olympiad (1MO) was held in brasov. It will be held once a year. Except for 1980, which was suspended due to the economic difficulties of Mongolia, the host country, it has been held for 40 times so far. More and more countries are participating. Only 7 countries participated in the first session, and by 1980, there were 23 countries; To 1990, there are 54.
It must be pointed out that there have been some math competitions before the above history. For example, the Soviets said that a math contest was held in the era of 1886. Another example is 1926. A abacus competition was held in China and Shanghai, with students, banks and bank employees. Hua, a first-year student in China Vocational School, won the championship with his wisdom. These are all stories about math competitions, not in the official history.
Second, the development of mathematical competitions
Mathematics competitions have gradually developed from a single city to the whole country and then to the whole world. For example, the math competition in the Soviet Union started in Leningrad and Moscow, and 1962 was extended to the whole country. In the United States, there was no national mathematics competition until 1957.
Mathematical competition activities are also gradually developed from shallow to deep. In almost every country, mathematical competitions are organized by some famous mathematicians. The test questions are very close to the exercises in the middle school textbooks, and then gradually deepen. Some mathematicians spend more energy on selecting topics and organizing competitions. At this time, the test questions are gradually out of the scope of middle school textbooks. Of course, it is still required to state the questions in elementary mathematical language and solve them by elementary mathematical methods. For example, at the beginning of the Soviet Mathematical Competition, famous mathematicians such as André Andrey Kolmogorov, Aleksandrov and Tommy Tam Nie all took part in this work. In the United States, famous mathematicians such as boekhoff and his sons Paulia and Kaplanski participated in this work.
After the start of the International Mathematical Olympiad, the preparation work of the participating countries is often mainly an intensive training for athletes to broaden their knowledge and improve their ability to solve problems. This kind of training class is very difficult, much deeper than middle school mathematics. At this time, several mathematicians are needed to specialize in this activity. In countries with good math competitions, competitions often take the form of pyramid competitions and are selected at different levels. For example. The Soviet Union is divided into five levels of competitions, namely, school-level, municipal, provincial, national and all-Soviet competitions. The number of participants in each level is about110 of the previous level, and eight special math schools (or math Olympic schools) have been established to train students with good math quality.
Although the mathematics competition has a long history, it has developed and changed greatly in the recent 10 years, and the related work has become more and more specialized. We should pay close attention to its development and understand its laws.
Third, the role of mathematics competition
1. Choose young people with mathematical talent. Because the winners are selected on the basis of step-by-step competition and gradual deepening of assessment, the winners should not only have a solid and extensive mathematical foundation, but also have a flexible and witty mind and creative ability, so they are often diligent and intelligent teenagers. These people have a great chance of becoming talents in the future. More and more countries attach importance to mathematics competition, which is one of the important reasons for its rapid development in the world. In Hungary, famous mathematicians Fay, Ritz, Xie Gui, Koenig, Hal, Rado, etc. are the winners of the mathematics competition. In Poland, the famous number theory expert Zinzel was the winner of a math contest. In the United States, Milnor, Manford and Quinlan won the Fields Prize in Mathematics, and many outstanding young people became famous physicists or engineers, such as the famous mechanic Feng? Carmen.
2. Stimulate teenagers' interest in learning mathematics. Mathematics is becoming more and more important and essential in all natural sciences, social sciences and modern management. Due to the development of electronic computers, various sciences tend to be more in-depth and mature, from qualitative research to quantitative research. Therefore, it is almost necessary for teenagers to learn mathematics well for them to learn all sciences well in the future. Mathematics competition introduces healthy competition mechanism into teenagers' mathematics learning and stimulates their self-motivated and creative thinking. Because the mathematics competition is conducted at different levels, the questions before the national competition basically did not jump out of the scope of middle school mathematics textbooks, which is suitable for the majority of teenagers to participate. However, it should also be admitted that there are differences, even great differences, between people's talent and mathematical quality. National competitions and subsequent competitions and training can only be carried out among a small number of people, and a small number of teenagers with good math quality can afford it. For example, Australian teenager Tory? Tao won the bronze medal, silver medal and gold medal in the 27th, 28th and 29th International Mathematical Olympiad at the age of 10, 12 and12 respectively. Of course, some university teachers and math researchers need to take part in high-level math competitions.
3. Promoted the reform of mathematics teaching. After the mathematics competition enters a high level, the content of the test questions is often the elementary of higher mathematics. This not only adds fresh content to middle school mathematics, but also may prompt middle school mathematics teaching to reflect on a new basis and change from quantitative change to qualitative change in the process of gradual accumulation. Middle school teachers can also learn new knowledge, improve their level and broaden their horizons in the process of participating in math competitions. In fact, some math teachers have gradually tasted the sweetness in this activity. Therefore, mathematics competition may also be one of the "catalysts" for the reform of middle school mathematics curriculum, which seems to be better than the top-down "indoctrination" method. In the early 1960s, the so-called modernization movement of middle school mathematics teaching in the West tried to replace the old middle school mathematics content with some modern mathematics, but it adopted the method of top-down indoctrination, resulting in the intuitive thinking process that was divorced from the teacher's level and students' sequential learning. Now it is basically blown by the wind and declared a failure. On the contrary, math competition may be a way. In China, middle school students are under great pressure to take the college entrance examination, and middle school teachers are rushing about for it, which makes them feel that the road is getting narrower and narrower. Mathematics competition may lead to the reform of mathematics teaching in middle schools.
Fourth, competition mathematics-Olympic mathematics
With the development of mathematics competition, a special mathematics discipline-competition mathematics, also known as Olympic mathematics, has gradually formed. The task of competition mathematics is to put advanced mathematics into elementary mathematics, express the problems of advanced mathematics in the language of elementary mathematics, and solve these problems by elementary mathematics. The background of problems and even solutions here often comes from some advanced mathematics. Mathematics can be divided into analysis and algebra according to its methods, that is, continuous mathematics and discrete mathematics. At present, calculus does not belong to the category of international mathematical Olympics, so decentralized discrete mathematics is the main body of competitive mathematics. Many topics of the International Mathematical Olympiad come from mathematical theory, combinatorial analysis, modern algebra, combinatorial geometry, functional equations and so on. Of course, it also includes plane geometry in middle school curriculum.
Competition mathematics is different from these fields of mathematics. Usually, mathematics often seeks to prove some generalized theorems, while competition mathematics only seeks some special problems. Generally, mathematics pursues to establish general theories and methods, while competition mathematics pursues to solve special problems with special methods. Once a problem comes out, it becomes an old problem and needs to continue to create new ones. Competition mathematics belongs to the category of "hard" mathematics, and usually, like pure mathematics, it takes its inherent beauty, including the simplicity of questions and the cleverness of answers, as an important criterion to measure its value.
Competition mathematics can't develop independently without the existing branch of mathematics, otherwise it will become passive water, so it is often run by experts in some fields. For example, Shan Zun, an excellent coach of China delegation who participated in the International Mathematical Olympics, is an expert in number theory.
The spirit of the International Mathematical Olympiad encourages the use of ingenious elementary mathematical methods to solve problems, but it does not exclude the use of advanced mathematical methods and theorems. For example, in the 3rd 1 international mathematical olympiad, some students used Bertrand hypothesis, that is, Chebyshev theorem, that is, when n is greater than1,there must be a prime number between n and 2n, and some students used Scherbinssey theorem when solving problems, that is, a square table becomes the general solution of the sum of s squares. These theorems can only be found in "Introduction to Number Theory" written by Hua (a textbook for graduate students of university mathematics department) or in more professional books. This is not only "killing the chicken with an ox knife", but also according to a foreign coach, "They are bombing mosquitoes with atomic bombs, but mosquitoes are killed!" This is allowed, but it is not encouraged by the International Mathematical Olympiad.
A difficult problem in the international mathematical Olympics needs to be written in three or four pages after simplification, which not only greatly exceeds the depth of middle school textbooks, but also is not lower than the depth of general courses in university mathematics departments, and certainly does not include the breadth of university courses. In fact, in the course of university mathematics department, not many people prove a theorem with three pages. A good test answer is roughly equivalent to an interesting essay. Therefore, it is quite scientific to use these questions to evaluate the mathematical quality of teenagers. Their solution requires participants to have extensive basic knowledge of mathematics, plus wit and creativity. This is completely different from a simple intelligence test. International mathematics competitions generally range from the fourth grade of primary school to the second grade of college. Because primary school students have little basic knowledge, the so-called math contest during this period is actually a quiz. For college students, systematic learning should be emphasized, and an overall understanding of mathematics is needed. Therefore, the focus of mathematics competition should be middle school, especially high school.
Now, we have accumulated a wealth of math contest question banks for middle school teachers, students and math enthusiasts to practice. There is also a special competition mathematics magazine in the world.
5. China Mathematics Competition
China's math competition began at 1956, when senior high school math competitions were held in Beijing, Shanghai, Wuhan and Tianjin. A number of famous mathematicians, such as Hua, Su, and so on, actively led and participated in this work. However, due to the "Left" influence, 1965 only opened six times sporadically. After the "Cultural Revolution" began, the mathematics competition was regarded as a set of "customs, capital and repair" and was forced to cancel it all. It was not until the downfall of the Gang of Four and the resumption of the China Mathematics Competition in 1978 that it embarked on the road of rapid development. The math contest before 1980 belongs to the primary stage, that is, the test questions are not divorced from the middle school textbooks. After 1980, it gradually entered the advanced stage. China participated in the International Mathematical Olympiad for the first time in 1985, and was among the best in 1986. 1989 and 1990 won the first place in the total score of the team for two consecutive years.
China successfully held the 3 1 International Mathematical Olympiad, which indicated that the level of China's mathematical competition reached the international leading level. First, China won the first place in the total score of the team, which shows that China's pyramid-shaped competition selection system at all levels, Olympic Mathematics schools and centralized training systems are all perfect. Second, mathematicians in China simplified and improved more than 100 test questions provided by 35 countries, and recommended 28 questions for leaders of various countries to choose from. As a result, five questions were selected (* * * needs six questions), which shows that the level of competition mathematics in China is quite high. Third, the test papers of students from all countries are first corrected by leaders of various countries, and then coordinated and recognized by the host country. We organized nearly 50 mathematicians as coordinators, scored accurately and fairly, and completed the coordination task half a day ahead of schedule, which shows that China's mathematics has considerable strength. Fourthly, this is the first time that the International Mathematical Olympiad has been held in Asia. China's outstanding achievements have inspired developing countries, especially Asian countries. In addition, the organization of this competition is also quite good.
In China, from the mathematicians of the older generation, young and middle-aged mathematicians to primary and secondary school teachers, thousands of people joined forces with Qi Xin to achieve today's achievements in the mathematics competition. Special mention should be made to Hua here. He not only advocated China's math contest, but also wrote five pamphlets, from Yang Hui's triangle, from Zu Chongzhi's pi, from Sun Tzu's "magic calculation", mathematical induction and mathematical problems related to the honeycomb structure. These are his competition math works. 1978 After China resumed the math competition, he personally presided over the test questions and wrote comments for the answers. China's other outstanding competition mathematics works include Duan Xuefu's Symmetry, Min Sihe's Lattice and Area, and Jiang Boju's A Pen and the Post Road. Wang Shouren should also be mentioned here. Since he cooperated with China, he has been leading and participating in mathematics competitions. Led the China team to participate in the International Mathematical Olympiad for three times, and led the work of the 3rd1International Mathematical Olympiad. After 1980, young and middle-aged mathematicians basically took over the math competition of the older generation in China, and they made positive efforts to push the level of math competition in China to a new height. Qiu Zonghu is one of the outstanding representatives. He has made outstanding contributions from cultivating students to organizing and leading mathematical competitions, from leading the China team to participate in the International Mathematical Olympiad for three times to holding the 3 1 International Mathematical Olympiad.
Six, some questions about the China Mathematics Competition.
1. We should sum up our experience seriously. We should sum up the experience of success and the lessons of failure. Especially in the 22 years from 1956 to 1977, only six mathematical competitions were held on a small scale, and they stopped completely in 16, which was more than twice as long as that in Hungary due to the two world wars, which also reflected the harm of "Left" from one side. It is necessary to allow and even encourage the expression of different views on mathematics competitions, and avoid big bang, ups and downs and "one size fits all". When there are shortcomings, we should calmly analyze and draw a clear line between the irrationality contained in the math competition and the shortcomings in our work.
2. Improve the leadership system. Can we imagine that the State Education Commission and the China Association for Science and Technology, through the chinese mathematical society Mathematical Olympic Committee (or other forms of unified leadership), will lead and coordinate the participation and training of the national mathematical competitions at all levels and the international mathematical Olympics? Set up the Mathematical Olympic Foundation to fund some mathematical competitions and reward winners of mathematical competitions and leaders, coaches and primary and secondary school teachers who have made contributions.
3. Publicity to the society. Publicize the significance and function of math contest, and eliminate misunderstandings, such as "math contest is an intellectual quiz for primary and secondary school students", "talent selection impacts normal teaching", "teachers, especially university teachers, are not in place" and so on. Facts should be used to illustrate the achievements of mathematics competition activities. For example, just before the "Cultural Revolution", in several low-level math competitions, some winners have become talents. For example, Wang Jiagang and Chen Zhihua in Shanghai, Tang Shouwen and Shihe in Beijing, are now famous middle-aged mathematicians in China, and some of them have obtained doctoral tutor qualifications. It was the Cultural Revolution that delayed 10 years, otherwise the achievements would be even greater.
4. Handle the relationship between popularization and improvement. Mathematics competitions need to be held in schools, cities, provinces, the whole country, winter camps and training courses in the form of pyramids. The first three levels are universal, and the test questions should not be divorced from the scope of middle school mathematics textbooks, but should be oriented to the majority of students and teachers. National competitions and follow-up activities are all improved, and the number of participants should decrease rapidly. As for the winter camp and training team, only dozens of students can participate. Mathematical olympiad schools should pay attention to quality, and do it with less and more precision. Students attending math schools should be strictly selected, so as not to hinder their all-round development in morality, intelligence and physique. In addition to winter camps and training courses, a few mathematicians need to concentrate on problem-setting and training. We might as well encourage mathematicians and primary and secondary school teachers to engage in math competitions in their spare time and not interfere with everyone's normal work. In a word, the popularization part and the improvement part of mathematics competition should not be antagonistic, but should be organically combined.
5. Continuing education and training the winners of mathematics competitions. On the one hand, we should fully affirm and encourage the winners' achievements, on the other hand, we should tell the winners of the competition that they must be cautious, modest and prudent, and only after a long and unremitting hoe can they become an excellent mathematician or an expert in other fields. Don't take winning the game as the sole purpose, but as a spur to encourage progress. We should also create better opportunities for the winners of mathematics competitions to study deeply and let them grow rapidly. For example, we can consider letting some universities of science and engineering select some students from the winners of the national high school mathematics competition and not take the exam.
6. Those who have made contributions to the mathematics competition activities, including organization leaders, coaches and primary and secondary school teachers, should fully affirm and reward their work achievements. In their job evaluation, as one of the basis for promotion.