They may have forgotten all the formulas, but they know how to learn mathematics best.

How to learn mathematics? Professor Ding Jiu of the Department of Mathematics at the University of Southern Mississippi has a unique view on this. He has been teaching for more than 33 years. Recalling his experience in the Department of Mathematics at Michigan State University, he has a lot of experience and feelings. During the four and a half years of studying, the teachers of the courses he took and audited had their own teaching characteristics and personal styles. The comprehensive course settings and the more discussion classes there are, the more knowledge he has gained.

He believes that Chinese education places too much emphasis on mnemonics, tapping the storage function of students' brains to the extreme, but ignoring the most important function of the brain - thinking. Not being able to think or not wanting to think is one of the consequences of China's thousands of years of rote learning.

Written by Ding Jiu (Professor of Mathematics at the University of Southern Mississippi)

New Year's Day in 1986 was quite memorable for me. At noon that day, I flew from Shanghai to the United States to study for a doctorate, went through customs in San Francisco and stayed overnight, and flew to Michigan State University the next morning. So I spent "half of New Year's Day" in both China and the United States that day.

In the summer of 1990, I received my Ph.D. in applied mathematics from Professor Li Tianyan (1945-2020), and was immediately hired by the Department of Mathematics at the University of Southern Mississippi, where I have been teaching for 33 and a half years. Recalling the details of my teaching, I have a lot of feelings.

A Blot on the Examination History

Professor Jacob Plotkin (1941-), the director of graduate affairs in the Department of Mathematics where I studied, was well aware of the strengths and weaknesses of Chinese graduate students at that time. On the first day I registered, he advised me: Don't panic about mathematics, focus on English first. Although I followed his advice and took two courses in English listening and writing, in addition to officially registering for a mathematics course taught by Professor David H. Yen (1934-2011), I also audited several graduate courses. Two of them were courses for the doctoral qualification exam, one of the two major exams for doctoral students. The purpose was to kill two birds with one stone: it could help improve English listening skills and review and consolidate the mathematics content of the qualification exam. Professor Yen was the academic advisor assigned to me by the department. If I needed guidance on taking courses, I could ask him for advice.

The two one-year courses on which the doctoral qualification exam was based covered content—analysis and algebra—which I had basically learned when I was an undergraduate in the Department of Mathematics at Nanjing University, but I had not reviewed them for many years, so I had to review them. In fact, on the evening of January 2, when I arrived at the school, Shen Yunqiu, a doctoral student who had always been very helpful to me, told me that the department had already taken the first half-year qualification exam that day, and would continue to take the second exam the next day. He suggested that I give it a try, because according to the regulations, if I failed the qualification exam right after entering the school, it would not be considered a failure. So I went to try my luck the next morning, drowsily, but encountered a difficult problem because my memory bank had missed many formulas due to years of disrepair, and I failed the exam. Although this bold and hasty exam did not leave a bad impression on my future doctoral thesis supervisor, Professor Li Tianyan (but he often said that the qualification exam was only used to "test American idiots"), I was really ashamed because it was an indelible stain in my exam history. I heard that before the start of the fall semester, two young masters from Sichuan University came to pursue their doctoral degrees. They asked the department for the qualification exam papers from the past few years, and after preparing for ten days, they passed the two exams in early September. Four months later, when the winter semester started, they passed two more difficult "doctoral preparatory exams." Such a speed made me feel inferior. I have always respected the spirit of these two Chengdu mathematical geniuses - Lü Kening and Zheng Dechao - and their way of doing research and being a person. Over the past few decades, they have become leaders in their respective research fields.

However, I have confidence in my ability. When I first entered Nanjing University, I saw that my classmates who had entered a month earlier seemed to be "working hard" and "martial arts masters". Many people studied ahead of time and seemed to know everything. I was so scared that I lost weight in a week, which surprised my brother who came to Nanjing on a business trip and visited me. But soon I found that as long as I could really understand the key mathematical concepts and proceed step by step, I would be invincible in learning. The self-study ability that I had gradually developed over the years came in handy at this time, and I soon stood out among my classmates who were admitted in the same batch. I still clearly remember what Mr. Wang Yongcheng (1939-), who taught us the course "Analytical Geometry of Space" in the first semester, said to us: "I almost forgot all the formulas when I graduated from college, but I learned how to learn mathematics." Indeed, mastering learning methods is far more important than remembering what you have learned. This is the same as the meaning of the idiom "It is better to teach a man how to fish than to give him fish." Later, Mr. Wang went to teach at Shanghai Jiaotong University and became a computer expert with great achievements in Chinese typesetting. This may have something to do with his learning "how to learn mathematics". His early study experience of "forgetting all the formulas" verified the famous saying of Japanese Fields Medal winner Hironaka Heisuke (1931-): "The purpose of learning is to master wisdom."

I buried myself in my studies again, just like when I first entered Nanjing University. Although the jet lag bothered me for several months - I wanted to sleep when I entered the classroom during the day, but I was full of energy in the middle of the night. After one semester, I passed the Michigan English test and no longer had to take listening and writing courses. My "conditional admission" due to my low TOEFL listening score before admission was upgraded to a legitimate "regular admission". Since the spring semester, I have focused my main energy on mathematics. Not only did I register for graduate courses that required tuition, but I also audited other mathematics courses. Some of them were not directly related to the future research direction I envisioned at the time, but I preferred to involve different fields and try to expand my knowledge. Just like when I was at Nanjing University, I insisted on reading humanities books and journals no matter how busy I was studying. Here I absorbed the nutrition of pure mathematics like a sponge. In order to pass the doctoral qualification examination as soon as possible, I continued to audit these two basic courses. Among them, Professor Peter A. Lappen, who had a European gentlemanly demeanor, was serious when he taught real analysis. He went to the blackboard as soon as he entered the classroom and wrote on it in a flamboyant manner. He was probably the fastest writer I have ever seen in my life. In early September of that year, I successfully passed the doctoral qualification examination. In January of the following year, I took two doctoral preparatory examinations. From my diary, I saw that I got first place in at least one of them.

The more the better

The comprehensive curriculum and numerous seminars in the Department of Mathematics at Michigan State University have become an inexhaustible source of knowledge for me. The department also has a library with rich collections and elegant decorations. On the first day of my registration, I was almost shocked when I received the key to the department library, which has only the teaching assistant office and the building, from the secretary of the director of graduate affairs. They actually trusted the foreign graduate students who had just come to study so much. Aren’t they afraid that the books would be stolen? I recalled that the library of my alma mater, Nanjing University, once trusted students and implemented open-shelf borrowing for their convenience. As a result, many books disappeared after a few months because some readers firmly believed in the famous saying created by Lu Xun for the protagonist of the novel "Kong Yiji" that "stealing books is not stealing". But in the years I studied here, I never heard of "stealing by the department library". Everyone fully enjoys the convenience of using the library, especially in the quiet night, sitting on a comfortable single sofa, immersed in the ocean of mathematical thinking, it feels really wonderful.

During the four and a half years of studying, the teachers of the courses I took and audited all had their own teaching characteristics and personal styles. The teacher of my first mathematics course, Professor Yan Xianyao, was originally from Shandong and was a direct descendant of Yan Hui, Confucius' favorite student. He received his doctorate from the Courant Institute of Mathematical Sciences at New York University. Richard Courant (1888-1972), a student and colleague of the great mathematician David Hilbert (1862-1943), established this institute in the once glorious mathematical style of his motherland, the University of Göttingen, after escaping Nazi Germany. Its applied mathematics is the leader in the United States. Professor Yan is eleven years older than Professor Li, and they are close colleagues. I took this course because Professor Li suggested it, even though I had already learned the basic content of this course "Numerical Solutions of Partial Differential Equations" in my junior year. Professor Yan is proficient in mechanics and differential equations. His teaching style is like his life style, steady, solid, and logical. His tone of voice is calm and steady. His blackboard writing is small and smart, which forms an interesting contrast with his tall stature. I learned from him the rigor of academic research and the peace of life.

I also learned useful knowledge from other professors' classes, witnessed their lecture characteristics, and was able to be influenced by their unique teaching art. In the 1988-89 academic year, I registered for Professor Sheldon Axler's Advanced Functional Analysis. Professor Axler has a typical American style, cheerful personality, and advocates freedom. He graduated from Princeton University with an undergraduate degree. He is the grandson of Paul Halmos (1916-2006), a Hungarian mathematician and a master of mathematical writing and speaking in the American mathematics community. His thesis advisor is Professor Donald E. Sarason of the University of California, Berkeley. Like his master, he is also good at teaching, with clear explanations, fluent blackboard writing and beautiful English complementing each other. I have liked functional analysis since my undergraduate studies, and I almost took the postgraduate examination in this subject before graduation. So I decided to "go to the next level" through Professor Axler's class. The concept of functional analysis is to computational mathematicians what advanced calculus is to engineers. This course has no textbooks, only two reference books are specified - Functional Analysis by Walters Rudin (1921-2010) and A Course in Functional Analysis by John B. Conway (1939-). For three consecutive semesters, there was no exam, and the semester grade was calculated based on the total score of the assigned homework. What is amazing is the professor's grading standard: if the assigned exercises are done correctly, you will get 10 points, if you don't do them, you will get 0 points, but if you get them wrong, you will get minus 5 points. This makes the students who take the course nervous and dare not make mistakes. Because there are no exams, it can be imagined that some exercises are quite difficult. Most of the students who take the course are pure mathematics doctoral students who will engage in research in analysis, equations, geometry, or topology in the future, and at least two of them are the professor's own disciples. But they were also afraid of getting negative marks, so they often discussed with each other and did not dare to act rashly or hand in their papers rashly. There was a student in the class who had just graduated from a prestigious university through a study abroad fund in China. He planned to study for a doctorate with a famous professor in the department, but a few years later he went to an even more famous professor at an even more famous university on the East Coast to get a degree. Later he became a professor at a research university and also served as a distinguished professor at his alma mater. After all, he only had an undergraduate degree at the time, and he was not very confident about this difficult course, so he often asked me questions. Once the homework was handed out, he got 10 points for a certain question, but I was deducted 5 points because my proof of the solution was too refined and was considered wrong. However, he did the question because of the discussion with me, but he wrote a more detailed answer. Of course, I explained my correct answer to Professor Aksla after class. After careful consideration, the professor agreed to increase the score from negative 5 points to positive 10 points, but he did not want to see the "sloppiness" in my homework in the future. From then on, I remembered the English term "sloppiness" he used, which I saw for the first time. Professor Aksla was such a great lecturer that I and others nominated him for the department's teaching award, which he deservedly won. Later, when I applied for a university teaching position, he wrote me an enthusiastic letter of recommendation, praising my "performance" in his unique one-year course.

However, the professor who really impressed me and had the greatest influence on my later research and teaching career was Mr. Li Tianyan, whom I came to join. In early June 1985, when he visited the mainland of China for the first time, I, who had known him through correspondence, flew to Sun Yat-sen University to listen to his lectures for a week. I witnessed his dancing classroom performances and heard for the first time the famous paper Period Three Implies Chaos co-authored by him and his doctoral supervisor James Yorke (1941-), which he translated as "Period Three Implies Chaos". That week, I learned more mathematical ideas in his lectures than I did in my entire master's degree. This feeling may be considered an exaggeration for those who study step by step. In fact, as long as you read what Professor Freeman Dyson (1923-2020) of the Institute for Advanced Study at Princeton has written, you will know how strong this feeling is. Dyson, who was recruited by Dean Oppenheimer (J. Robert Oppenheimer, 1904-1967) to become a full professor at the age of 30, recalled that when he was discussing with the great physicist Enrico Fermi (1901-1954) at the University of Chicago a physical calculation that he thought was feasible, the other party's 20-minute comments made him fully believe the Chinese idiom "Listening to one word from you is better than reading for ten years."

When I was studying for my master's degree under Professor He Xuchu (1921-1990) in the Department of Mathematics at Nanjing University, from the first semester, my mentor asked us to hold seminars. He also participated in person and made inspiring insights from time to time. He not only guided us to enter the research field as quickly as possible, but also cultivated the habit of independent thinking and improved our mathematical presentation skills. I remember that before I reported on the section on the duality theory of linear programming in the textbook "Introduction to Linear and Nonlinear Programming" written by Professor David G. Luenberger (1937-) of the Department of Engineering-Economic Systems at Stanford University, in line with the basic view that "mathematics is an organic whole", I used the concept of duality in functional analysis as an introduction and used analogy to implement the duality idea into linear programming. This first report in my life was well received by everyone, and Mr. He, who usually did not smile, also smiled. I laid a relatively solid foundation in analysis at Nanjing University during my undergraduate studies, and I was fortunate to receive training in seminars during my master's degree. In addition, in the first year of graduate school, Professor Zuhe Shen, who taught us the basic course "Iterative Solutions of Nonlinear Equations", asked us students to teach the second half of the English textbook after one semester of teaching, which further strengthened our ability to study independently and give lectures. After coming to the United States, I confidently became a member of Professor Li's discussion class.

Mentor Li Tianyan's "Scholarly Approach"

Professor Li Tianyan is a legendary figure with a strong personality. After he unfortunately passed away at the age of 75, the following year, I published my memoir Out of Chaos: My Mathematical Affinity with Li Tianyan, which was published by Shanghai Science and Technology Education Press. One chapter in the memoir talks about his "way of studying". He had very high requirements for seminar reports. He did not want to hear only the logical reasoning of theorem proofs or "inexplicable ε-δ language". He wanted to hear the basic ideas behind the conditions and conclusions. If you just want to show by heart that a certain proposition is valid for all natural numbers n, then you are wrong. What he wants to see is your specific deduction when n is equal to 3. You may be able to recite the steps of the argument that is valid for any n, but you cannot handle the simpler special case of n=3, because you do not really understand the theorem and are only at the stage of reciting the proof. There are actually many such students. Professor Li told us that a graduate student asked to change the question of "proving that the product of two compact sets is a compact set" to "proving that the product of any compact sets is a compact set" during her doctoral qualification oral examination, because she had already memorized the proof of the latter famous Tikhonov's theorem.

Before I joined the seminar, only three weeks after I entered the school, I experienced Professor Li's strict requirements for academic reports. That day, he handed me a thick article and said, "I don't know what your level is yet. Report its content to me in three weeks." He was giving me an unofficial "doctoral qualification test"! The doctoral supervisor of the author of the article is the famous Stephen Smale (1930-), who became a young professor at Cornell University after graduation and has already made a name for himself in the academic world. This long article of more than 70 pages uses integral geometry to study the computational complexity of a class of algorithms for linear programming, and I am a layman in this field. However, although I am busy taking classes and auditing classes every day, I also have to deal with the attack of jet lag. I cannot retreat in the face of difficulties, but should take practical actions to live up to the teacher's expectations! I borrowed the classic work on integral geometry by Spanish mathematician Luis A. Santaló (1911-2001) from the department library. My self-learning ability, which has been continuously improved over the years, has enabled me to quickly understand the basic concepts of this subject and to be fascinated by the beauty of studying geometry from a probabilistic point of view. Anyone who has heard that the value of pi can be estimated by a probabilistic method by randomly throwing a needle a large number of times on a plane with two straight lines will be amazed at the genius creation of Georges de Buffon (1707-1788), one of the pioneers of integral geometry, a French botanist and writer, in his seventies.

When I stood in front of the blackboard in Professor Li's office, ready to show off my skills, I saw him sitting on a chair with his feet on the desk (this did not show his arrogance, but was a common phenomenon in the United States). He suddenly said something that surprised me: "You think I'm a fool. I don't know anything." A great professor, but he doesn't know anything? In an instant, I understood everything. He was testing my teaching skills and the effectiveness of my speeches, and whether I could really make "fools" understand "advanced mathematics." I know my shortcomings and strengths. My shortcomings are that my Mandarin pronunciation is poor and my hometown accent is quite strong. This is the result of my starting to make a living at the age of 14 and working with workers who only spoke the hometown dialect for several years before entering university. But I hope to "learn from others' strengths and make up for my weaknesses." This small strength is that as long as I have truly understood something, I can basically make it understandable to ordinary people. The allegorical saying "dumplings in a teapot - there is something in the stomach, but it can't be poured out of the mouth" may not be applicable to me. This is partly due to the innate teaching genes of parents, and partly due to the acquired teaching training at Nanyang Technological University.

After the first report, Professor Li said nothing. He rarely praised people in person, but I knew very well that I had made this "idiot" understand. In the second week, I was more confident. After the report, he said, "Go back and write a report for me." I breathed a sigh of relief, and finally passed his unique "qualification test."

Later I learned that the article was submitted to the journal Mathematical Programming, and the editorial department sent it to Professor Li for review. He came up with this trick, killing two birds with one stone, which not only allowed me to see my mathematical skills and presentation skills, but also gave me the opportunity to practice writing review reports. Of course, he also saved time on reviewing, so why not? This was the first time I reviewed a paper in my academic career. Although it was unofficial because I was still a doctoral student, it meant a lot to me.

In the summer of 1986, Professor Li held a seminar to discuss MIT professor Gilbert Strang's (1934-) new book "Introduction to Applied Mathematics". Although the department did not provide financial support for me that summer because I had not passed the doctoral preparatory exam (I had not even passed the qualification exam), I did not want to work outside the school, because the salary I had earned as a teaching assistant in the past six months was enough to cover the expenses for the next three months, so I naturally participated. Professor Li's two doctoral students, Noah Rhee from South Korea and Zhang Hong from Beijing, plus a disciple of Professor Yan, also participated. Each student had to report on a chapter, and because my master's major was optimization theory, I was asked to report on the chapter on mathematical programming.

This was my first seminar presentation in the United States. Despite the training I had received from the NTU seminar, I prepared for it as if it were my first academic presentation in my life. This was indeed my first seminar presentation in English. I carefully read the chapter in Professor Strong's book, sorted out my experience of this subject over the past few years, and decided to use my own language to geometrically introduce the basic ideas of optimization theory. Before I went on stage with confidence, Professor Li said hello to me and apologized for leaving ten minutes early due to something. That day, I did not use the standard method used by the author to prove the Lagrange multiplier rule in equality-constrained optimization. Instead, I used the concept of Chinese derivatives in multivariate calculus and derived it intuitively at once. My oral English expression ability is definitely still at the beginning stage, but my confident demeanor shows that I have a relatively thorough understanding of this law - this is due to the basic skills I laid when I was a graduate student in optimization at NTU and the serious preparation for the speech. I was sure from the facial expressions of the audience that they really understood. The most satisfying thing for a speaker is to feel the resonance of the audience! Suddenly, I saw Professor Li stand up because he was leaving, but he first uttered a few words that I will always remember, and they were in Chinese: "Well said, you have learned Marxism-Leninism well!"

A few days later, Wang Duo, a doctoral student in the department who was about the same age as Professor Li and with whom he often chatted, told me that Professor Li told him that "Ding Jiu has ideas." Although this five-word comment was an exaggeration, it made me happy because I had basically passed the test in the eyes of my future doctoral thesis supervisor. Traditional Chinese education places too much emphasis on "knowledge", which gave rise to an enviable idiom "learned and rich". However, in the eyes of Westerners, scholars who "have ideas" are creative scholars.

"Forget Light Formula"

Wang Duo was admitted to the university before the Cultural Revolution. Not long after I entered Nanjing University, he was admitted to Peking University as a graduate student. After graduation, he came here to visit Professor Li's senior brother, Professor Zhou Xiuyi (1943-2023), and later became his doctoral student. Professor Zhou is a Chinese from Singapore. After graduating from university, he did not serve in the army for a year like Professor Li, but went straight to the United States. In 1970, he received his doctorate from Professor York. Although he is two years older than his junior brother, he looks very young. With his delicate face and bright eyes, you can tell at a glance that he is a capable person who can make achievements in any field. We chatted for a while when I met him for the first time. When I mentioned in Chinese that a mathematics professor at a famous university in China has a strong theoretical team in a certain professional field, and that his theoretical research is unique in China, he answered me in English: "Not many people do that!", meaning that this research topic is too old, which stunned me all of a sudden and made me respect him very much.

Although the number of extracurricular mathematics books I read in college may be no less than the textbooks, the training mechanism of graduate students in China restricts us from taking courses in other fields. After coming to the United States, I basically only took those subjects that I had not systematically studied in China. In the second year, when I saw that Professor Zhou was going to teach a one-year graduate course "Ordinary Differential Equations and Dynamical Systems", I signed up for the course without hesitation. Professor Zhou was only in his forties at the time, had several research grants, had the highest salary in the department, and was in the most productive period of his research career. The Theory of Bifurcation Method, co-authored by him and Professor Jack K. Hale (1928-2009) of Brown University, published in the early 1980s, is one of the main reference books in this field. His teaching style is very different from that of Professor Yan or Professor Aksla, and each has its own merits. Professor Yan's teaching is like a gurgling stream, nourishing the heart, but it is difficult to see a magnificent move; Professor Aksla's classroom proofs are clear, logical, and watertight; Professor Zhou's performance on the podium is unrestrained, free-spirited, thinking from the big picture and not being particular about details, and has a strong research flavor. In the process of deduction, he often asked us to help because he forgot the trigonometric function identity that should be used in the next step. This proves from a certain aspect that the aforementioned teacher Wang Yongcheng's frank "forgetting the light formula" is indeed a kindred spirit. What does it matter if you can't remember it? What we students need most is ideas, and Professor Zhou taught us the idea of ​​doing research!

When the theory of relativity was resounding throughout the world, there was an ignorant and fearless young man who wanted to test the great Albert Einstein (1879-1955) by asking him how fast sound traveled in water. Einstein replied calmly: Sir, I can't remember, but you can find it in any physics textbook. I was once on a plane returning to China, chatting with an American sitting next to me about Chinese education. This graduate of the Department of History at the University of North Carolina at Chapel Hill has been guiding the production of export furniture in Guangdong for many years. He told me that Chinese engineers are very obedient in technology, following foreign design plans step by step, and rarely taking a different approach. When talking about the lack of innovation that is prevalent among Chinese students, he quoted a witty remark made by a compatriot who taught physics in Beijing: If a teacher teaches the equation 2+3=5 in class, American students will ask why 2 + 3 is not equal to 4 or 6, while Chinese students will silently memorize the formula in their hearts. Not being able to think or not wanting to think is one of the consequences of China’s thousands of years of rote learning education.

Chinese education places too much emphasis on mnemonics, tapping the storage function of students' brains to the extreme, but ignoring the most important function of the brain - thinking. Therefore, this kind of memory cannot complement and keep pace with understanding. The mechanical memorization that is encouraged everywhere in schools is, in a word, rote memorization. Animals also have memory. Domestic dogs wag their tails to acquaintances and bark at strangers. Carrier pigeons can fly back to their homes thousands of miles away. But the difference between humans and animals is that humans can think, so humans can be defined as "thinking animals." Humans can transform nature and change the world, while animals cannot, which is attributed to the thinking characteristics of humans, not the memory characteristics. Otherwise, our pet dogs and cats can also rule the world. Unfortunately, our current popular test-oriented education can only drive our students to become unthinking people who can only recite formulas. The great American physicist Richard Feynman (1918-1988) had a good father. He told his son from an early age that knowing only the names of things without knowing the essence of things is equivalent to knowing nothing. He gave an example, saying that there is a bird that has different names in different languages. Even if you can memorize a hundred names, but don't know the habits of the bird, you still know nothing about the bird. This is indeed a wonderful metaphor for our education method of "knowing what it is but not why it is so."

No one will take a nap in his class

If Professor Aksla's course grades are not based on exams, but only on regular homework, and his unique grading standards are a bit different, then Professor Zhou's simple method of grading the subjects of students who registered for his class was even more unexpected. I heard later that those who submitted class assignments basically got A, and those who did not submit them also got B. In other words, as long as a baby can register for this course, he can get a B, because babies certainly can't do homework. Regardless of whether this grading method is reasonable or too loose, students at least appreciate the art of doing research from this course. Some people can even start from here and rush to the forefront of popular research. Professors who really do research are often particularly generous to students in terms of grades. However, some professors who have long abandoned their guns and armor in the research field regard grades as the last signboard to maintain their professorial authority and are unwilling to relax easily. When the average score of students is 89.9, they can only get the unlucky B and have no chance of A.

Of course, the most amazing lecture style was that of Professor Li Tianyan. In the 1987-88 school year, he was invited to be a lecturer at the Institute of Mathematical Analysis of Kyoto University in Japan for one year. It is said that his salary was one and a half times that of a full professor at a Japanese university. The institute has a total of ten lecturers, nine of which are permanent positions and belong to domestic scholars, and only one lecture is invited from abroad. Professor Li gave a series of lectures to the Department of Mathematics of Kyoto University: Ergodic Theory on [0, 1]. During this year, we, the disciples he recruited from the mainland of China, could not see the majestic face of our mentor on a regular basis. We lived freely and lively every day, like mice that could not see a cat, but everyone was very self-conscious and generally worked hard. I have passed two doctoral qualification exams and two preparatory exams, as well as two second foreign language exams. While continuing to take courses, I began to read the latest papers published by others, started to think about problems, and wrote my first academic article after coming to the United States. When Professor Li, who was far away in the East, received the first draft of my article and wrote back to me, he not only gave me academic advice, but also greatly praised my subjective initiative in finding problems and doing research on my own.

In the fall semester of 1988, Professor Li Tianyan, who had just returned from a visit to Japan, decided to teach a one-year ergodic theory course. The material he taught was mainly based on his simple speech manuscripts in Japan. The purpose was to teach formally in class and prepare for the possibility of writing a book in the future. This was the first time I officially registered for his class. The course was called "Selected Lectures on Advanced Applied Mathematics". In addition to almost all of his Chinese doctoral students, there was also an American postdoctoral couple, which forced him to teach in English. In order to improve the English level of visiting scholars from China, the outstanding experimental physicist Ting Zhaozhong (1936-) deliberately did not speak Chinese with them. But Professor Li was just the opposite. As long as there were no foreigners present, he would always speak Chinese. Even in a class where most of the students were American, he would occasionally play some Chinese tricks. Once, he was not very satisfied with the homework performance of several Chinese students who registered for his class. As soon as he entered the classroom that day, he went straight to the blackboard and wrote four big Chinese characters: "High-minded but low-skilled", which made the Americans who did not know the meaning laugh and embarrassed the students who were only about ten years younger than him.

Morris Kline (1908-1992), an American applied mathematician and historian of mathematics who wrote the monumental work Mathematical Thought, Ancient and Modern, once gave the following earnest advice to teachers regarding classroom teaching:

"I would urge every teacher to be an actor. He must use every instrument of the drama to bring life and energy to his teaching. He must use the dramatic effect when it is appropriate, and show passion while telling the facts. The unusual and eccentric behavior will stimulate people's interest, and the humor that is freely used will greatly enliven the class, even if it seems to have nothing to do with the subject."

Sitting in Professor Li Tianyan's class, you probably understand Klein's words better, because he is the model for interpreting this passage. In his class, even people with sleepyheads will not fall asleep. If I had taken his class in my first semester in the United States, my jet lag would not have lasted so long. He is definitely not the kind of old pedant who reads from a textbook, nor is he a standard teacher who narrates proofs step by step, nor is he a teacher whose speed, frequency, and amplitude hardly change over time. When he lectures, his facial expressions change frequently, his body language is exaggerated, and before he spits out new mathematical concepts, he always uses vivid examples to lead the way so that we can enter the classroom calmly. The ups and downs of the tone were sometimes like the gurgling of water, sometimes like the raging waves, and sometimes like the American civil rights leader Martin Luther King (1929-1968) repeating the sentence "I have a dream!" in his famous speech. The continuous emphasis on the same mathematical term, the decibel of the sound became louder and louder, fully arousing our passion.

In the second semester, the American couple did not come to audit the class again. So every class was filled with standard Taiwanese Mandarin. Professor Li's dynamic language became more colorful, and we listened more enthusiastically. Originally, we knew almost nothing about the basic concepts of ergodic theory, but this academic year, we followed him to immerse ourselves in the ocean of theory that combines several disciplines of pure mathematics. "Ergodic" absorbed the fascinating new nutrients. As for me, just like the "butterfly effect" revealed by "chaos", I never predicted that this course would lead to my doctoral thesis.

What this dissertation should be about was something that neither my supervisor nor I had anticipated. Next time I write about the whole process, it may provide some "experience" for young students who are troubled by the topic selection of their master's or doctoral dissertations.

Note:

This article is based on the revised Chapter 4, "Reading Days," of "Personal Experience of American Education: Thirty Years of Experience and Reflection," published by the Commercial Press in 2016.

This article is supported by the Science Popularization China Starry Sky Project

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.

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