On Undergraduate Science and Engineering Education

   (I)On science and engineering education, the large background we need to clarify is: the talent and thinking ability of students around us can be divided into two basic levels-normal and excellent; for instance, in the mathematics department of Fudan University where I studied before (Fudan is one of the best universities in China), the majority of students belong to the ‘normal’ level and I am in this category, while a small number of them belong to the gifted, bright level, like Weixiao Shen and Wenjun Wu. The difference of thinking ability between these students is huge: the learning efficiency of ‘good’ students is tens of times higher than ‘normal’ ones. (We all know few students who can well learn everything by only studying 3 or 4 hours each day) The big gap of thinking ability embodies in 3 major aspects: the depth of understanding, the proficiency and the creative utilization of the same content. From the perspective of actively solving problems and mastering specific knowledge (these two things are actually the same), the ‘normal’ students only grasp few key points, thus, they can barely solve a small part of after-school exercise, while ‘good’ students can solve almost all the problems, namely, they have grasped most content in undergraduate courses.

   Over my four years of undergraduate life, I studied almost day and night due to the enormous enthusiasm for mathematics, in the daytime, I intensely learned in the 2^nd and 3^rd Teaching Building except class and meal time, and I also worked against time in the 4^th and 5^th Teaching Building at night; from 8 a.m. to 12 p.m., my learning plan is always compactly arranged. During this time, I read many reference books with three aims: deepen the understanding of particular courses, solve the after-school exercise and prepare for the final exam; for instance, I carefully read 2 or 3 reference books of higher algebra and many of mathematical analysis, complex analysis in the library. To work out homework assignments, we had to devote a great deal of time and energy in learning, and I still remember the stressful scenarios of studying functional analysis, algebraic topology, partial differential equation (PDE), differential geometry and other courses in the classroom for handing in homework on time. In the undergraduate stage, we earnestly listened to our teachers in the class and repeatedly read books after class, and learning is the keynote melody throughout our college days, and I think most of scientific students have experienced a similar life condition. However, even with such long-term hardworking, my learning effect was still very bad at graduation; back then, for mathematical analysis, I could just solve about 10%, simplest problems, and I also made little sense of PDE (from today’s higher point of view) since I couldn’t solve most of relevant problems. To sum up, during undergraduate, I was very distressed about the central issue of being unable to actively solve problems. This basic phenomenon arises from many reasons, and among them, the core reason is that my thinking ability was low, which led to a very superficial understanding of particular knowledge, and I just had a vague, shallow impression after learning it once. Broadly speaking, the reason of my poor learning in undergraduate is my low thinking ability, since I was at the ‘normal’ level then. I think my own condition is not an isolated case and the majority of science and engineering students have a similar experience, namely, they do not learn most professional courses well when they graduate.

   In my senior year (due to the special course arrangement of Chinese university, we don’t have new classes to learn except writing senior thesis and interning in that year), I did not relax too much like some students, instead, I still kept learning, and at that time, since I had a lot of free time, I embarked on relearning undergraduate courses, and the courses I relearned include real analysis, partial differential equation, complex analysis, etc, and I listened to the teaching again in the classroom then, thus, my knowledge and independent views still grew, and my thinking ability was still enhancing every day, but the effect was not very ideal, I still couldn’t actively solve problems, and from today’s viewpoint, my understanding then was still very scattered and vague. Over a long period of time, the sense of anxiety stemming from being unable to actively solve problems was the major psychological condition when I faced mathematics.

   Until graduation, I did not master the Taylor expansion of many common functions, and just had a very superficial, disordered and vague impression of many basic contents, like separation variable method in partial differential equation and permutation group, which is rather ashamed, but it is a basic fact.

   In the graduate stage, the situation changed qualitatively, and I did not concentrate my energy in research like many others, instead, I focused on relearning undergraduate courses, and this broad strategy has a very good effect. (The reason for this is that I was very distressed about being unable to solve after-class problems and I think it is an essential issue I can’t evade) In the first one year and a half, to pass the qualifying exam, I repeated functional analysis, numerical PDE and abstract algebra for over 10 times, and from the spring semester of the 2^nd year (January, 2013), I repeated mathematical analysis (calculus and its theoretical foundation), half of higher algebra, 2/3 of abstract algebra, special relativity, 2/3 of partial differential equation, C++ and numerical PDE for about 50 times (I repeated them simultaneously). There are three basic reasons for relearning undergraduate courses: firstly, expansion of knowledge; secondly, improvement of ability; thirdly, accumulation of independent ideas about related courses. These 3 aims are highly unified and intertwined. With the constant repetition for over 3 years, in the summer of 2016, my ability finally improved beyond the ‘normal’ level, which is a qualitative breakthrough. The reason for this breakthrough is that my thinking ability has greatly improved, and this improvement happens day by day; in the short term, the improvement is small, but we can feel it; in the long term, I can clearly feel the giant leap of my thinking ability. For instance, from freshman to junior, my thinking ability was always improving, though it was not ideal then, the process of improvement was clear, and from the 1^st year of graduate to 5^th, my thinking ability was still enhancing, and at the summer of my 5^th year, it finally improved to a good condition.

   For courses I didn’t repeat in the 5 years after undergraduate, my impression is very hazy and vague, like algebraic topology (key points like mapping lifting) and higher algebra (vital content like Jordan canonical form), which clearly proves that my thinking ability then was bad and there are serious problems of my internal understanding about these knowledge. I think this kind of intellectual condition is a basic experience shared by many students: when we learn in undergraduate, our main feeling is fully dim and specious.

   In the summer of 2016 (at the end of August), after 5 years of repeating, I finally felt that these courses, like calculus, were somewhat simple, and it became natural for me to think about their questions and I can actively solve over 2/3 of related problems (in the past, I felt that these questions were very hard and I could not find the clues, but now, I realize that they are all basic problems). This hard but profitable experience gives me a deep enlightenment: if we spend a great deal of time repeating one course, then sooner or later we can master them. In mathematical learning, complex and simple, abstract and concrete are all relative; when we don’t learn these courses well, we will naturally feel that a number of theories and questions are hard and abstract, but if our overall understanding deepens, we will feel them concrete and familiar.

When looking back this long process, I can surely say that for these courses, like abstract algebra, I couldn’t solve most of their problems even I repeated them for 49 times, and until the 50^th time (Jan, 2016), I began to be able to solve a small number of the problems; since then, I still had a great improvement after every repetition, and at the 55^th time ( August, 2016), I felt that my mastery of these courses was already deep, organized and lucid (I could solve over 2/3 of the problems, and moreover, these problems became natural and familiar for me). When I repeated the 55^th time, I not only felt that I could solve most problems, also felt I had brought the major points together, and my line of thinking became much more clear when facing these problems and I also felt that they were much easier, correspondingly, I had a relaxed feeling, and moreover, this is a holistic phenomenon. Therefore, I hear some students in physics say that they suffer from learning theoretical mechanics since they can’t solve most problems in it, and this basic fact is easy to understand because they just repeated this course for 20 times which is far from 50 times, thus, they inevitably could not solve its problems and they suffered. We know that when we read professional literature, we will feel that as if we learned nothing in the first time when we read it again, and I have a similar intellectual experience when relearning undergraduate courses, and even more serious, because I felt that I had learned nothing in the previous 49 times when I read the 50^th time, and until the 51^st time, I finally felt that I really had learned something. In the middle stage, when I repeated these courses, I found that there were so many key points I had completely missed before, I didn’t know when this endless process would finish, and for many times, I suspected whether I could grasp these courses; of course, I finally felt confident after repeating over 50 times. To sum up, I spend over three years in learning just 4 undergraduate courses. ( We need to point out that it is not enough to solve over 2/3 problems, and we have to solve almost all of them, especially some hard questions, and only in this way can we achieve the real goal of relearning undergraduate courses because we can get sufficient depth of understanding only by solving many hard questions.)

The reason of repeating science and engineering courses for over 55 times is that these knowledge has 4 basic characteristics: 1immense information, every course has hundreds of thousands of information and there is much condensed information on every page; 2the points are interrelated, intertwined and interacted, which form an organic thought system; 3the points are delicate, many details are actually crucial; 4knowledge points are rather deep, abstract and hard to master. Due to these 4 characteristics, especially the first one, science and engineering knowledge is hard to learn; thus, scientific study and innovation is always a slow process (with the great improvement of overall thinking ability, I think I can learn the left course within 10 repetitions). In humanity and social sciences, like history, the information is also enormous, but we don’t need so many repetitions; the works of political philosophy (like John Rawls’ A Theory of Justice) may require over 20 repetitions, but they naturally do not need as many as 55 times. The interplay of scientific knowledge determines that for any course (like special relativity), to learn it well, we need to master all the points without any gap. Therefore, scientific study has a high demand for us.

   In repeating undergraduate courses, I have the following 5 guiding principles: 1improvement of thinking ability every day; 2fewer but better, I don’t want to repeat 10 courses together, instead, I just repeated 4 to 5 courses at the beginning, and I will not learn other courses before completely mastering them, and obviously, we have learned them well only if our understanding is thorough and clear, and the main criterion of fewer but better principle is that we can solve most after-class problems; moreover, due to the precise characteristic of scientific knowledge, to solve a problem, it is not enough if we just get the overall clue, we must get a precise result or a clear proof; 3depth, our understanding of the courses must be deep enough, thus, we should solve some hard problems since only hard problems can train our high level professional qualities; 4keep repeating and proficiency, repeat over 55 times at the beginning; 5the main philosophical idea of Grothendieck: solution of one problem is based on overall foundation and intuition, and our understanding becomes mature only after we feel the problems are natural and trivial, and the solution then naturally emerges, and the process of solution should break up into a series of small and natural steps. (This insight can explain why we can’t solve problems, the answer is we haven’t repeated enough times, thus our understanding is not deep enough and also not proficient enough, thus, the intuition of solution cannot emerge. The essential cause of not being able to solve problems is that our understanding of a block of knowledge is bad and our overall foundation is defective.)

   To sum up, among all the undergraduates worldwide, including students in US, European countries, Japan or Brazil, the reason of their failing to solve problems and fear for exams can be divided into two categories: firstly, they have good talent and strong thinking ability, but their foundation is bad, which can be solved by supplementing high school knowledge in a short time; secondly, their gift is poor and their thinking ability is weak, and the solution of this problem will require a long time and it will probably cost 8 or 9 years. Among them, the second condition is the mainstream in science and engineering education. For me, I thoroughly mastered only 4 courses at the end of the 5th graduate year after learning mathematics over 9 years and devoting huge energy in these courses, like mathematical analysis (if including my thoughtful and independent views which can double the amount of information, my grasped content can be broader), and this cost-benefit is low but it is perhaps the reality of scientific learning.

   For many American undergraduates, it is a basic issue that their calculus is bad, and some of them are just bad at calculus, while others’ issue is a weak high school foundation, and therefore, to learn enough professional skills, I think these students need to repeat high school’s basic courses.

   (II) To summarize our analysis, we can realize that for most Chinese scientific graduates (including those in top Chinese universities), their fundamentals are rather weak and they can’t solve most after-class problems. Therefore, we should pay sufficient attention to these basic courses, and graduates of many majors, including computer science, mechanical engineering, electronic engineering, statistics, aerospace engineering, civil engineering, petroleum engineering, communication engineering, chemical engineering, physics, chemistry, etc, should repeat the undergraduate courses if necessary. In my opinion, many electronic engineering graduates in Chinese Academy of Sciences have two problems about undergraduate courses: firstly, their mastery of concrete knowledge is bad; secondly, they cannot solve the majority of problems in them, and this may seem difficult to believe for the laymen, but in fact, more than half of the students have this basic issue.

   Take professors who have worked for many years as an example, we can see the fundamental importance of this point, whether in China or in America, some young (about 35 years old) professors’ basic skills are somewhat lousy, and they even cannot solve hard problems in mathematical analysis (which is one most basic course); therefore, they must repeat undergraduate courses and learn as many basic skills as possible. In today’s graduate school, many students begin to do independent research after passing the qualifying exam in the 2^ndor 3^rd year, while their fundamentals are not solid enough and they can’t actively solve most problems, and therefore, we think this research method can only get unimportant results.

   For me, though my grade was in the first 15% in undergraduate and I got A in all the dozen courses in graduate (these courses are all concrete courses like real analysis and partial differential equation, and are not courses in the ‘research’ and ‘directed study’ category, and I learned these courses mostly in the first two years of graduate), my mastery of these courses was very bad, for example, about point set topology, I learned it in junior year and also in the 1^st and 2^nd year of graduate, however, even I attended it for 3 times, my learning effect was still very poor, and I almost couldn’t solve any problem, namely, I missed most of its key points (my experience about real analysis was also similar, I attended it for 3 times in undergraduate and graduate, but the effect was also bad), and obviously, this experience is quite universal. Since the sophomore year, I realize two basic issues: firstly, I can’t understand much content of the basic courses; secondly, I can’t solve most problems in them.

   For many PhDs in top universities, like Harvard, Princeton and MIT, the situation is somewhat similar; as we know that good PhDs in statistics have already published 4 or 5 high quality papers and good electronic engineering PhDs also have published many papers, while some statistics and electronic engineering PhDs do not have any paper, which is a clear demonstration of their poor foundations, namely, about basic courses, they are poor in concrete knowledge and they also can’t solve most problems. In a word, I think the basic issue discussed in this essay also applies to them.

  (III)By laying down related foundation, we can better understand subsequent courses; for example, the deep impacts of mathematical analysis[1] to real analysis include: 1 the type of Riemann integrable functions is not resolved in mathematical analysis, which is thoroughly resolved in real analysis; 2 function series has a high demand for uniform convergence, while in real analysis the demand is somewhat lower; 3 exchange of integral is too strict in mathematical analysis which is improved by Fubini theorem in real analysis. The impacts of mathematical analysis to functional analysis include: 1 Baire category theorem can solve problems like discontinuous points of functions, the convergence of Fourier series, etc; therefore, Baire category theorem not only has a far reaching impact on the main theorems in functional analysis, it also has roots in mathematical analysis; 2 the part of Fourier series is generalized in Hilbert space of functional analysis, and the best square convergent property of Fourier series and the Parseval equation are the most important special case of corresponding theory; 3 some central themes in mathematical analysis is deepened in functional analysis, including sequence convergence, topological property, differential operators, integral operators and etc. It is widely known that probability theory has a deep influence over statistics, and in probability theory, we need to compute many probability distributions which are closely related with multiple integral and series. As to the direct enlightenment of concepts and methods of mathematical analysis to point set topology are more familiar to us. Before repeating the undergraduate courses for 55 times, I just had a vague impression of mathematical analysis’ fundamental influence on these courses, however, with the solid foundation in both knowledge and problems, I then get a deep and natural feeling. For instance, I did not have a deep feeling with the importance of Taylor expansion to error analysis of finite difference and the value of residue theorem to abnormal integral computation until I did a lot of problems about Taylor expansion and integral in mathematical analysis, and there are innumerable similar examples. The influence of mathematical analysis to relevant courses can be divided into two levels: direct and indirect, and we can easily list many concrete examples of direct impacts, while indirect impact permeates into aspects like mathematical sense and quality, though hard to describe by language and we use them every day without noticing, they are also very important. As correctly pointed out by Zhiwei Yun, the impact of mathematical analysis and higher algebra can be extended to graduate courses, and those following courses are easy to learn if we completely grasp these two courses. The basic courses are the bedrock of thought system in modern mathematics and if we do not learn them well, our understanding of mathematics will have many original defects. My own feeling is that I feel especially relaxed in learning partial differential equation(PDE) and numerical PDE after I have a solid foundation in calculus, and then I feel very natural and strict about many complex deductions in them.

   A concrete point in basic course, like the fundamental theorem of homomorphism in group theory, has 3 levels of enlightenment for us: firstly, the concrete questions this theorem can solve; secondly, the meaning and value of it to the entire abstract algebra; thirdly, the rich intension derived from it for the whole mathematics. Moreover, all the mathematical points may have these three levels of value. To sum up, laying down the foundation mainly has 3 purposes: firstly, improvement of overall ability (for instance, to enhance our thinking ability from the ‘normal’ level); secondly, form an orderly, original understanding of the holistic thought system in the conceptual level; thirdly, the accumulation of concrete knowledge, which we are familiar with; the whole process is an organic unity of these three aspects.

   We can point out more concrete examples, for instance, the complex integral of the smooth curve in complex analysis has a direct connection with the first type of curve integral in calculus, and if we have a solid foundation in the latter one we will feel especially easy when we learn the former; as another example, there is polynomial related theory in abstract algebra, while there is a very similar theory in higher algebra, and higher algebra discusses the remainder theorem of polynomial, UFD property, greatest common divisor and relevant issues, while in abstract algebra, the relevant discussion has both thought successions and new changes (since we have a larger, more abstract theoretical framework), and if we are quite familiar with the polynomial theory in higher algebra, we will feel very easy to learn the corresponding theory in abstract algebra. There are innumerable similar examples, namely, the related theories and ideas in basic courses has a direct, comprehensive and strong influence over the subsequent ones, and the former has a lot of original roots, which sufficiently testifies the importance of foundation.

   (IV)How to solve problem is clearly a central problem in undergraduate and I suffered from this problem in all the 4 years, and more than this, I still suffered form this issue at the end of 4^th graduate year, and I still could not actively solve problems then, particularly hard problems; I think many chemical engineering and physics PhDs also have a similar beset. Until the end of the 5^th year after graduation, with 9 years of struggle and grope, I gradually overcame this problem; the central reason of not being able to solve problems is a shallow understanding of the courses and a low thinking ability (we have pointed out it for many times above), and due to a shallow understanding of knowledge, we cannot grasp the essence, central spirit and deep context of the courses, thus, we cannot solve problems. In my own experience, after 4 years of graduation, more precisely, before October, 2015, I could not solve most problems in multiple variable implicit functions, the analytical property of parametric variable integral, multiple integral, the first type of surface integral, etc, and I felt that these problems were out of reach for me, but when I repeated these courses for the 51^st time, I finally managed to solve some problems, and I realized that they were just easiest and most basic problems, and I think this psychological experience is quite universal. When I studied the 51^st time, I finally realized that for the theoretical foundation and concrete examples of the second type of surface integral, I did not master them at all before, and in the condition of failing to completely master them at knowledge level, I naturally could not actively solve problems. Problem solving is based on three major aspects: knowledge points (which is often a holistic phenomenon), mindset and depth of understanding, and in many cases, for a concrete problem, if we do not learn relevant points (for example, if we cannot flexibly use concepts like superior limit and infimum) or the depth of idea is not enough, or we do not have relevant mathematical mindset, then we can’t solve it even we think about it for 1 year. When I repeated for 50 times, I could just solve problems scrappily, and when I repeated the 55^th time (August, 2016), I could extensively solve problems then. My ability of problem solving then was based on deepening of foundation and improvement of thinking ability, namely, my thinking ability had enhanced a lot, and I suddenly felt that the holistic line of thought and key details were very clear. In a word, problem solving are based on the solid overall knowledge foundation and high level of thinking ability, and only with these two conditions together can we solve problems.

   Many people have the following two feelings: firstly, in many cases, we cannot solve problems, then some days later after seeing the keys, we still cannot solve them, and a few days later after seeing the keys again, we think that we will be able to solve these problems, but when doing them we find that we still cannot solve them. (When reading books, we will have a similar experience in facing knowledge points) Secondly, for those problems we can solve (or concepts and approaches we have already mastered), we will also have a new feeling when we read them again: firstly, we are more proficient; secondly, our mastery is more deep and solid; thirdly, our particular understanding is integrated into the organic whole thought system. Broadly speaking, we will experience three stages in problem solving: firstly, we cannot solve the problems; secondly, we can solve them, but reluctantly and with a sense of difficulty; finally, we can easily solve the problems. These three stages is a natural process everyone will go through.

   Here, we must answer one basic question: why problem solving has certain importance? I think there are three major reasons: 1 knowledge in courses is often somewhat abstract, while after-class problems are always concrete and they include many examples, these examples can extend our understanding of relevant knowledge, and the aim of learning is utilization, while after-class problems is an ideal place to creatively use knowledge, and they can give us a preliminary feeling of the utilization of knowledge. 2 To solve some problems, it often requires depth of thought, and some students believe that they have grasped some knowledge but they cannot solve relevant problems, which naturally shows that their understanding is not deep enough, namely, if we can just get a vague intuition but without a clear and detailed line of thought and a precise result, this is obviously a direct sign of a bad mastery of knowledge, and in some occasions, we think that we have understood some point, but, in fact, we do not really understand it and we are still far from real understanding, thus, hard problems are a benchmark of our degree of understanding. A proper example is this: at one time, I think my understanding of the concept ‘uniform convergence’ was sufficiently thorough and I think I had proficiently mastered theory and problems in the book, but when tried to solve relevant problems in this part, I found that I could solve some of them but was not able to solve others, in a word, the related content of this concept is more complex and deep than my previous understanding. In brief, to solve various problems, we need to accumulate many concepts, ideas, approaches and techniques, and many problems require an understanding in conceptual level and mastery in technical level, and when our mastery of concepts and techniques is sufficient, we naturally can solve related questions. 3 Hard problems are often related with many points, and these points are often beyond a certain chapter and even related with other courses, moreover, they are very flexible and stubborn, which is fundamentally important to improve our comprehensive ability and deepen our overall understanding of one course. In a word, problem solving (especially hard problems) is a basic step in scientific learning. 

   An easily observed fact is that scientific problems have 5 basic characteristics: 1Stubborn, complex, many problems need complicated computations. 2Deep, a number of after-school problems require depth of thought, and we need to get enough depth of relevant knowledge to solve them. 3Highly flexible, many relevant problems require flexible techniques and are full of changes, and they cannot be solved by routine and standard process. 4Diverse, the after-school problems often test all the important points in one chapter, not merely aimed at some particular information, thus, they often have a rich, diverse character, and the common situation is that nearly every after-school problem has its own feature, and it requires particular concepts or techniques, namely, every two different problems will use different solution methods. 5Delicate, many problems are related with delicate information and subtle analytical skill, and in fact, all experienced science and engineering workers know that nearly every problem in higher education is very delicate. It is understandable that these 5 basic features of science and engineering problems are often intertwined. (The internal connection of these basic characteristics of scientific problems and basic features of scientific knowledge discussed above is pretty interesting.)

   About this crucial point, we can list some concrete, vivid illustrations. For example, for Taylor expansion, it is well known that it includes many proof problems, and when I repeated the 35^th time (January, 2015), I realized that this part has many inequalities using symmetrical ideas, but I felt these problems were disordered, rambling, and lacked inherent law, thus, I could not prove similar new questions; when I repeated the 55^th time, due to proficiency and independent thinking, I had combed the inner context of this part and my understanding was more comprehensive and mature, and I realized that it was not made up of one single proof skill, but a combination of many proof ideas, and I finally grasped all the proof ideas then. A similar example also happens in the separation variable method of partial differential equation, in the past, I was not confident about its after-school problems; and when I really understand all the details later, I finally understand the solution method of this type of problem, then when facing this kind of problem I already have mature confidence, since I have made sense of all the important points. (My specific computation may have some errors, but I have indeed grasped the whole line of thinking and key sectors about it)

  (VI)Many people may ask: why am I able to actively solve problems after repeating for 55 times? The reason is quite simple, because I really master the ideas, concepts, approaches and techniques of related content. For example, at one time I felt that I could not solve problems of field theory in abstract algebra, and later I realize that it is because I did not really understand many ideas, concepts, approaches and techniques of characteristic of field, field extension, algebraic extension and other aspects, and my mastery of them was not deep enough, thus, I naturally cannot solve related problems. The case of residue theorem is also similar, and I couldn’t actively solve related problems in this part in undergraduate, while in the latter part of graduate, I could solve most of its problems, and then I realized that the reason for my being unable to solve such problems was that I did not really understand this part in the past; in this sense, we can say that active problem solving and real understanding is one thing.

  When I repeated mathematical analysis and abstract algebra for the 49^th time, I found that I still could not solve most problems, and later I realize that it is because I did not really understand most ideas, concepts, approaches and techniques in these courses then.

  Meanwhile, we need to point out that these ideas, concepts, approaches and techniques often exist en bloc and are often interrelated, thus, we can solve relevant problems only after mastering a block of concepts, ideas and techniques, and if we just mastered them isolatedly, we still cannot solve problems; usually, only after we get a deep understanding of the whole chapter, or even the whole course, can we naturally solve relevant problems in this chapter. In a word, the accumulation of knowledge and ability of solving problems are often holistic phenomena.

  (VII)Broadly speaking, the process of repeating undergraduate courses for 55 times is a process of gradually mastering concrete content and details (including numerous ideas, concepts, approaches and techniques), and also a process of gradually deepening specific understanding, and these two processes are completely intertwined; roughly speaking, the process of repeating these courses is a complex process of accumulating ideas, concepts, approaches and techniques and also a process of integrating disordered knowledge into a coherent thought system with depth, breadth,  delicacy and organic connections.

 Because when we learn some parts of knowledge, we will constantly think about their problems, techniques, approaches, ideas, theorems, framework ,concepts and other aspects, and our thinking about them will be a mixed and interweaved complex condition (our actual learning process is certainly not by the order of knowledge points and we learn the theorems, problems, concepts, techniques one by one, but a half-orderly process which gradually permeates), and with the mastery of concrete information, including many theorems, problems and skills, we will deepen the overall understanding of particular knowledge points. In the beginning, when we are exposed to some knowledge, since there is much information, and lots of ideas, skills, details, concepts mingle together, we will have a confused and jumbled feeling; with more and more repetitions, we can gradually sort these information out, and our understanding of their inner context will be clear and these knowledge will be ordered and organized.

 (VIII)The breadth of scientific courses is also an important issue we need to analyze. All the good technological practitioners know that every scientific course in higher education includes tens of thousands of ideas, thousands of concepts and innumerable details, techniques and methods, and we can only gradually master these concrete concepts and details, since they are all different from one another. Since every scientific major has dozens of courses, therefore, even for good students, when facing modern mathematics and physics, they will also have a sense of overly vast, but, if we learn the courses well, we can handle these broad and subtle contents.

  Since every scientific course has two features: firstly, its content is very broad; secondly, it is also complex and delicate; the overlapping of these two basic features together determines that we need to spend a great deal of time in learning related knowledge, and we cannot learn it well if just spending 2 hours each day to study. We all know that, for scientific work, we need to learn it over 8 hours each day for over 10 years to learn it well, since it is a slow process and we must devote much time in learning to well grasp one specific course. (From 2007 in undergraduate to today's 2018, I probably spent about 8 hours each day in these 10 years in studying mathematics.)

(IX)Now I want to more delicately analyze the complex psychological and intellectual experience in repeating the undergraduate courses for 55 times. When facing these courses, in January, 2015, I had repeated them for 35 times, but I was still in a vague, specious, directionless and fragmentary overall condition, and only later (after repeating the 55^th time) do I realize that this condition stemmed from two basic reasons: firstly, my depth of understanding was not enough, and my understanding of various knowledge and ideas then was shallow, and my understanding of the whole course was also superficial; secondly, my concrete accumulation was also not enough, and my mastery of the major ideas, concepts, approaches and skills was specious and I did not really grasp these concrete content.[2] When I repeated the 55^th time, my overall foundation was already deep and solid, and I redid problems in multiple integral, function series, multiple variable calculus and other parts, I then had a deep-seated feeling, and evidently felt that I had a solid knowledge and idea foundation; while, in the past stage, for instance, when I repeated for the 30^th time, since my knowledge structure was fragmentary and shallow, thus, when I did these problems, I mainly relied on luck and casual guess, trial; while in August, 2016, when I repeated the 55^th time, since I already had much relevant knowledge, thought and experience accumulation, thus, when I did these problems, I felt that my thoughts, intuitions, concrete details, skills and other aspects were much more clear. To sum up, in problem solving, the psychological conditions generated by solid foundation and shallow one are two quite different intellectual states.

  Take multiple integral as a concrete example, since my freshman year, I was not confident enough when facing double integral, triple integral and n-dimensional integral, and this psychological condition lasted until I embarked on relearning undergraduate courses in 2013. When in January, 2016, I had repeated undergraduate courses for 50 times, and I had studied the theory and problems of multiple integral for many times, but my understanding was still not clear enough; when it was August, 2016, based on solid foundation, I finally could solve most problems in multiple integral, and then I realized that my previous understanding was not deep enough (thus, my psychological condition of unconfidence before was right, and it naturally reflected my immature mastery of concrete knowledge): in fact, the use of cylindrical coordinates and spherical coordinates was more complex than my previous understanding (it is not very obvious and direct on how to select which coordinates, instead, it requires precise analysis), the variable substitution skills are also richer and more flexible than I thought before, and I also had a more clear understanding of sphere, paraboloid and cone.

  Another impressive example is Fourier series’ term by term integration theorem, term by term differential theorem and best square convergence theorem. Since January, 2015, I relearned the relevant theory or rewatched the relevant video every two month (I had repeated these contents for many times before), and for many times I thought that I had completely understood this part of knowledge, but in September, 2016, I realized that I did not really understand these theorems before, since their conclusions and proofs have rich properties, and my understanding of these theorems then was finally mature and reasonable; the reasons behind include three aspects: firstly, my understanding of the block of Fourier series (it naturally includes these theorems) was much deeper; secondly, my holistic understanding of mathematical analysis was deeper; thirdly, my overall mathematical quality also greatly improved. In a word, due to the overlapping of three basic aspects, my understanding of these theorems was proficient and satisfactory.

  My experience in learning multivariable differential calculus is also similar. Since the spring of 2013, I began to repeat this part of knowledge, and at many stages I falsely believed that I had completely understood them, and until August, 2016, I finally mastered unconditional extreme value, multiple variable Taylor expansion, conditional extreme value, the application of partial derivative in geometry, implicit function theorem and etc; at that time, I finally was able to integrate many information fragments into an organic thought system; indeed, these knowledge is not very abstract, thus not very hard, but they are also complex, and relevant theory, problems, theorems, conclusions include much, delicate information.

 When I repeated for the 20^th time, I could solve part of the problems, but they are the easiest, and moreover, even I could solve part of the questions, it was partially accidental, namely it was not based on deep understanding with holistic foundation, thus, I just could fragmentally solve problems, which proves that we can only locally solve problems by petty trick, to globally solve all the problems we must have a solid, delicate foundation. When I repeated for the 55^th time, I was able to globally solve most problems and feel the organic connections between these problems and the concepts, ideas, approaches, techniques in these courses, namely, the isolated knowledge information began to converge into an overall knowledge foundation, and moreover, the problems I mastered began to be integrated into the overall foundation of one chapter. To sum up, only we repeat enough times and are sufficiently proficient can our understanding of concrete knowledge and concrete problems promote from isolation to integrity.

Also at this stage, I eventually can differentiate what contents are hard and what are easy; before, I falsely thought that some knowledge was hard (like barycentric coordinates), and now I realize that they are just simplest concepts; indeed, a course does include some hard contents (like the proof of variable substitution of multiple integral), but only at this highly proficient stage, I am able to tell what contents are really hard. At the initial stage of learning one course, I feel that almost all the points are hard, but after repeating 55 times, I find that much content is actually quite easy (like the normal plane of curve, the tangent plane of surface, the computation of partial derivative, the computation of Euler integral in calculus, they just follow standard solution methods), but some parts of knowledge are really hard. About this point, we can list many other examples, for instance, about “Legendre polynomials are orthogonal polynomials” in calculus, before 2015, I felt that it was hard and complicated, but in August, 2016, I already felt that it was natural and easy. While, about “if A is UFD, then A[x] is also UFD” in abstract algebra, after I completely master it, I find that it is really deep. However, even for hard content, at this time, with the deepening of overall foundation, my mastery of them is more relaxed, for instance, some computations of parametric variable integral are rather complex, but I have clearly mastered their main ideas, technical details and spiritual essence. In a word, no matter for simple contents or hard ones, my understanding has both enhanced a lot.

(X)Another important point we need to add is that in January, 2016, when I repeated the 50^th time, I found that I still could not solve most problems, and then I was pretty depressed; then in May, 2016, when I repeated the 52^nd and 53^rd time, I finally could solve some problems, but my understanding was not thorough; in August, 2016, when I repeated the 55^th time, I finally had a lucid feeling, until then also, I truly felt the wonder and pleasure of these mathematical knowledge, and finally had a mature and confident mentality of truly mastering some knowledge. Namely, to scientific knowledge, we can understand its real intension only when we learn it well, and only then can we feel its marvel, and it often requires some time of accumulation.

In summary, in most time of relearning undergraduate courses, I was rather depressed; until the last 2 or 3  months, since I felt that I had proficiently mastered much precise knowledge and was close to completely master related courses, meanwhile, my comprehensive ability had enhanced a lot, I began to have a delighted state of mind. Moreover, in the last 2or 3 months, I not only mastered most ideas, approaches and problems of the courses I repeated but also felt that they became especially plain and clear, and I felt easier to understand them, and thus I had a light-hearted mentality, which is a strong contrast with my depressed condition in the first 50 times. To sum up, for the whole process of learning these 3 to 4 courses, in the initial long period, due to the weak foundation, I felt rather difficult to think about many knowledge points, techniques and problems, but in the last 2 or 3 months, due to the deepening and refining of holistic foundation and improvement of proficiency, I began to feel quite relaxed to think about most ideas, concepts and problems in them, which is a very real and pleasant feeling, and I believe practitioners who truly master some courses will all have this proficient, simple and stable overall feeling. Namely, in this long journey, after the first cloudy long road, I was finally exposed to beautiful sunshine at the last stage. In brief, for scientific knowledge, we can feel its real pleasure only when we learn it very well, and then we can flexibly and fully use it, and half-digested knowledge is not very meaningful.

The deeper reason for the above phenomenon is that every section of scientific knowledge includes a lot of complex information, such as ideas, techniques and details, and we can accumulate some concrete information, like ideas and details, for every more repetition, meanwhile our depth of understanding will also deepen; when I repeated the 30^th time, I only mastered 40% of all the related ideas, skills and details, thus, I had a specious feeling and could not solve most problems; when I repeated the 51^st  time, I had mastered 80% of all the ideas, skills and details, thus, I had learned the main essence of some parts of knowledge and could solve some of the problems; when I repeated the 55^th time, I had mastered over 95% of relevant ideas, concepts and skills, and my overall understanding was deep and clear enough, thus I naturally had a lucid holistic feeling. This fully demonstrates that the depth of understanding is based on concrete information, including many ideas, concepts and details, thus, one people is not eligible to say deep understanding if he does not truly grasp enough concrete information.

Correspondingly, this naturally explains the universal condition when we solve problems, and it is well known that we can only solve the easiest questions at the beginning, which is because we haven’t truly mastered most of the approaches, skills and details of one part of knowledge then, and with the further accumulation of ideas, concepts, approaches, details and deepening of understanding, we can gradually solve more difficult and really hard questions.

(XI)If we take the proportion of after-class problems we can actively solve as a clear criterion, in September, 2014, when I repeated the 30^th time, I could just actively figure out 20% of the after-class problems and they are naturally the simplest 20% problems; in January, 2016, when I repeated the 50^th time, I could actively figure out about 60% of the problems; while in September, 2016, when I repeated the 55^th time, I could already figure out over 80% of the problems and they included most hard questions. When I can actively solve 80% after-class problems, this indicates that my understanding of one specific course is stable and mature, while for some students, if they can just actively solve 20% of the problems, this naturally shows that their mastery is shallow, coarse and very unstable. To sum up, the problems we can actively solve (proportion and level of difficulty, etc) is a good indicator of our mastery of certain courses.

(XII)Here, it is somewhat meaningful to articulate the mental and psychological condition when fully mastering some courses. In September, 2016, when I repeated these courses for the 55^th time, I finally got a sense of mental stability. Before, knowledge of some courses, like mathematical analysis, abstract algebra and C++ made me very anxious since I could not address relevant problems and I was not confident about them both in concrete knowledge and holistic understanding, and my overall feeling of these courses was directionless, fragmentary and specious; when in August, 2016, I finally had a relaxed, proficient overall feeling, and finally could control much information included in them, and then, I had a relaxed feeling of overlooking from above, since I truly mastered most of the ideas, concepts, theorems, approaches, skills in these courses, and moreover, I did some hard questions; thus, I know that I had truly understood these courses and no longer had a sense of fog and puzzle. Meanwhile, when we truly master most content in one course, since we have thoroughly mastered such broad information, we will have a sense of fullness, delight and achievement. 

At this time, when I read books of these courses, for every theorem, every problem and every illustration entering my horizon, I have a sense of proficiency and am familiar with numerous details included in them, and I have a mature confidence when facing these courses, and now I realize that there may be some concrete information I have not grasped, but I am familiar with the major part. At this time, I feel proficient in both concrete details and thought essence of every theorem, concept and example, and I profoundly realize that only by being familiar with all details of one knowledge part can we learn it well, and if we are not sufficiently proficient about some problems and theorems, then it indicates that our understanding is not mature; in a word, for one particular knowledge part, we must be proficient with every concept, detail, idea and skill and then we can study it well.

Before, when facing many theorems and hard questions in some courses, I often had a sense of fear and some sense of mystery, I felt that they are hard, and at the same time, they are also out of reach, and it seemed to me that they have profound intension, but when I am truly familiar with them, to my surprise, I find out that they are actually easy, and the ideas and skills included are very clear, I then have a sense of so-so about them, and this feeling is certainly holistic, namely, I achieve this relaxed condition for all the content of one course. (This again demonstrates that a part of knowledge is often en bloc, and it is unlikely to solve some problems isolatedly, even though we want to solve few limited problems, we must be familiar with the relevant whole part) Also at this time, I realize for the first time what psychological condition is when we well learn one particular course in higher education, and the overall mysterious feeling which covers one part is replaced by a sense of simplicity, proficiency and sufficient confidence. At this point, when facing the content of these courses, I have a similar feeling as facing knowledge in high school, like function and sequence, and I feel that they are all both familiar and simple, and these two basic feelings often affect each other: since we are highly familiar with every concept, skill and detail of one part, it naturally creates a sense of simplicity; meanwhile, only by learning one part of knowledge to a simple degree can it demonstrate that our understanding is mature enough.

(XIII)In undergraduate, the basic issue we face is that our study time is limited; in higher education, the content of knowledge is huge, and only one course, like calculus, has broader information than the sum of high school mathematics, and moreover, it is much harder, let alone we have dozen of courses to learn in college; thus, the common condition of most students is to passively follow the curriculum, tired of learning newly instructed classes, and we do not have time to go over the courses; therefore, the overall learning effect of college classes is bad.

Compared with high school, another big difference of university is that in high school, our abstract thinking ability then is not very strong, and we also do not have many independent views then, thus we do not actively solve problems then, while in college, our problem solving is based on intuition and deep understanding of relevant content, and the thoughtfulness also enhances a lot. Accordingly, in terms of problems need to be addressed, since the knowledge points in elementary education are somewhat few, thus, for every concrete point there are often many questions to repeatedly test it, while in higher education, each after-class problems often has its own unique feature. Namely, for all majors of science and engineering, from high school to higher education, both the content we need to learn and the problems we need to creatively address unconsciously go through an enormous change.

An obvious basic fact is that compared with high school, some major basic features of college knowledge, like the complexity, degree of abstract and difficulty, also have greatly risen. (For example, the quadratic form in higher algebra is more complex, abstract and delicate than high school mathematics like sequence) When facing complex knowledge in undergraduate, we often do not well prepare for it mentally and psychologically, and meanwhile, in undergraduate, we begin to seriously think about life, and all kinds of life problems need us to keep thinking. To sum up, in both life and work, compared with high school, undergraduate stage has a qualitative change, which is a basic fact we need to recognize.

In a word, due to the time pressure of study and increase of difficulty in knowledge, we often passively follow the course progress and do not have sufficient time to digest and absorb information in the courses, which is a basic problem many students face in undergraduate. (In fact, due to the low thinking ability, even though we have enough time to learn, we still cannot master these concrete knowledge, and I finally mastered some courses in freshman year, like mathematical analysis, at the 5^th year in graduate.)

(XIV)For those who have graduated from college or graduate school and entered the society, like graduates of electronic engineering and mechanical engineering, we can surely say that the foundation of the majority of them are rather bad; thus, no matter how disordered and turbulent life is in actual work, they all probably need to repeat undergraduate courses for over 55 times to improve their thinking ability, professional quality and the ability to solve actual problems, and only by doing this can they better qualify for their own work. Since the curriculum in undergraduate is very crowded, and we are constantly preparing for the new exams, thus, we do not have enough time to repeat basic courses for 55 times; therefore, this crucial learning process needs to be taken up after graduation. In the meantime, though our thinking ability is low in undergraduate, and we will feel hard and painful when facing some concrete knowledge, the great deal of time devoted in this stage is also indispensible, since the accumulation in this low thinking ability stage is the necessary premise for the latter part. Therefore, in undergraduate, we must devote much energy in learning, and without  this stage's sustained accumulation, the subsequent learning cannot achieve the improvement of thinking ability.

  (XV)In the complex process of scientific learning, we should pay attention to look at problems from the independent thinking perspective, which is all important for science and engineering workers, and if we master these concrete information thoughtfully and artistically in our own way, we can doubly enrich our understanding; in this process, our independent thinking is tremendously vital, and we need to understand particular knowledge in our own mindset, since everyone has different mode of thinking, thus, we need to brand the knowledge with independent notions, and only this kind of knowledge can have a long life in our mind, meanwhile, only by much independent thinking can we form vivid understanding of particular knowledge and can we innovate in the future, namely, one course has two different levels: knowledge and thought, and the latter basic aspect is absolutely indispensable.

In the process of relearning undergraduate courses, only by forming many independent views can we prepare for our independent research in the future, and independent views are the main bedrock for the future innovation; if we just master concrete knowledge without independent ideas, then our independent research's quality has no guarantee. Only by mastering knowledge in our own way, these thoughts and information can be full of life in our mind, and it can lay a solid foundation for our future creative work, which is an obvious basic fact. Conversely, in scientific field, if we do not have enough independent ideas, then it is very hard to make significant contributions in the future, since the amount of information is not rich and deep enough.

In particular, independent thinking mainly includes two aspects: speculative level and artistic level. The speculative thinking can enable us to understand particular knowledge from speculative perspective, and artistic thinking can enable us to master concrete knowledge creatively and full of novel vitality, since art requires creativity and creativity also requires art. For instance, Men of Mathematics by ET Bell is well known to us, Bell’s writing is definitely elegant, but his writing lacks depth of speculation, thus, he did not make the most outstanding contributions (As the central fact we have pointed out elsewhere[3] : in the mathematical and physical world, only the masters’ writings have both true deep thought and high artistry, like Laplace, Dirac and Heisenberg, and most of the Nobel Prize winners’ writings also do not achieve this level, after all, the number of masters in the mathematical and physical world is very few). We need to add that workers in engineering majors, like electronic engineering, statistics and computer science, also need to store their own professional knowledge thoughtfully and artistically to some degree, though not as strict as mathematics and physics, and what these majors need is other intellectual capabilities.

(XVI)As we have repeatedly emphasized, in the process of repeating undergraduate courses, the mastery of knowledge is the first aspect, and in this process, our accumulated independent ideas are the second aspect, meanwhile, in this process, the improvement of our thinking ability is the very important third aspect. The huge improvement of thinking ability is also a beautiful gift which repeating undergraduate courses brings to us, for instance, if we repeat basic courses, like mathematical analysis, after we repeat dozens of times, our overall thinking ability will greatly improve, thus, when facing knowledge in computer science (like algorithm and data structure), our absorbing efficiency and depth of understanding will greatly enhance, and then we will feel these particular knowledge is especially easy. Namely, repeating courses like mathematical analysis will lead to an essential improvement of our thinking ability, and will also lead to a great improvement in efficiency when we learn other fields. While if we just superficially read recent papers, our scientific thinking ability will also improve, but not too much, moreover, this kind of superficially learning will not give us valuable experience of completely mastering one particular course, and it will not give us intellectual experience of deeply grasping one subject. Thus, our study of one concrete course in computer science also won’t be deep enough.

Here, we can give one concrete example. Before the summer of 2014, though I had repeated bubble sorting in C language for about 10 times, I felt that it was hard and abstract, while, in the spring of 2016, with the constant improvement of thinking ability, I then felt that bubble sorting was simple, and I easily mastered it with just one repetition. This example actually belongs to a broader illustration, namely, for the entire C language course, in my sophomore year (the autumn of 2008), I once learned this basic course, but my learning then was painful and specious, and I felt that all kinds of knowledge details were disorderly, however, through the improvement of thinking ability by repeating undergraduate courses, in April, 2016, it took me just 2 or 3 hours to master much knowledge which I failed to grasp with hundreds of hours’ study in undergraduate, including character array, two-dimensional array, pointer, structure, linked list, macro (parametric macro definition, nonparametric macro definition), string manipulation function, array initialization, etc, and then, my understanding of these much concrete knowledge was more clear. In summary, the improvement of thinking ability will improve our absorbing and understanding speed of local information, and it will also enhance our mastering efficiency of the large-scale knowledge, and these two aspects are both very valuable for our work and life. Here we also explain the basic phenomenon why talented students around us can learn all sorts of knowledge well by only working 3 to 4 hours each day, and the reason behind is that their thinking ability is strong.

Similarly, as to the statements of “the function series of convergence in measure has almost everywhere convergent subsequence” in real analysis and “n order symmetric matrix can be diagonalized through congruent transformation” in higher algebra, I devoted a great deal of energy to carefully learning them in undergraduate, but I did not understand them at that time; while in June, 2016, I easily grasped them by only one repetition in a short time.

In summary, the enhancement of thinking ability, the improvement of knowledge basis and the accumulation of independent ideas are the three major thought treasures brought by systematically repeating undergraduate courses. As most scientific workers can feel, different scientific workers have huge differences in talent, and one of the basic conclusions of this paper is: this gap in thinking ability can be overcome (the way is to repeat undergraduate courses in suitable situations), but it requires great effort.

As mentioned above, the basic starting point of this paper is to well learn undergraduate courses, and some people may argue that some prominent scientific workers’ foundation is somewhat weak, like Grothendieck and Smale, but they also make first-class contributions; here, we should not overlook one basic fact: indeed, these brilliant mathematicians may be bad at fundamentals, but their thinking ability is very strong and in the ‘good’ level, thus they can quickly learn much new knowledge and integrate them, while the majority of people whose fundamentals are weak also do not have strong thinking ability, thus, they need to repeat undergraduate courses to enhance their ability, otherwise if they just superficially read recent papers, their thinking ability will not have a basic change even in their 40s, moreover, even in 40s, they do not well learn any undergraduate course.

(XVII)In the previous parts, ‘thinking ability’ is an overall concept, and it is used to describe the velocity and efficiency when we absorb concrete knowledge, and since this concept is somewhat too general, we want to analyze its rich intension in this part.

With the improvement of thinking ability by repeating undergraduate courses, our creativity will evidently enhance. For myself, on the weak knowledge foundation before, what I had was just very little creativity, whether it’s about creativity for problem solving, or creativity for extracting new theories, but with basic changes in many aspects, including the deepening of knowledge foundation, the enhancement of intuition and richness of techniques, my creativity has evidently improved, and I get many kinds of creativity (like creative problem solving, creatively think about the concepts, summarize rules, generalize the known ideas, etc), moreover, my creativity has a solid foundation now.

Meanwhile, my abstract thinking ability has greatly improved. A good example is this: in the past, I thought the proof of Baire category theorem and its application were both very abstract, and after repeating it for 30 times, I still had a sense of fear and thought it was too abstract for me, and I even doubted whether my abstract thinking ability was strong enough; later, when I read the 50^th time, after proficient with all its details, I found that it was actually unadorned and familiar. To sum up, in undergraduate, the abstract thinking ability of good students are indeed much stronger than normal students, while if we relearn undergraduate courses, since we can think and master more and more abstract knowledge, our abstract thinking ability will keep growing.

In the meanwhile, our abstract generalization ability will also gradually improve. For many similar questions, similar approaches and similar concepts, we will gradually discover general rules hidden behind, and our capability of using various ideas, knowledge and techniques together will clearly improve, and we will have more and more holistic generalization to the basic features of some courses. At this time, we will gradually digest a part of knowledge and will naturally find more and more knowledge linkages, and moreover, we will gradually form an organic understanding of one whole course. In a word, we will get enormous pleasure in many kinds of theoretical generalization.

With this complex process, our capability of capturing key information and concretely thinking about particular contents will also enhance. When I spent 3 years and 7 months in repeating and thoroughly mastered 4 to 5 undergraduate courses, I realized: in undergraduate, ‘good’ students absorb much concrete information, while ‘normal’ students only absorb some superficial, simple knowledge; here, by relearning undergraduate courses we can solve this basic question. To begin with, through the immense accumulation of concrete knowledge, we will gradually realize which techniques, ideas and concepts play a central role in one proof or question. Meanwhile, our particular understanding to one course will become more concrete, and every course is actually made up of many concrete details, but when our thinking ability is low and our understanding of one course is shallow, we cannot notice much important concrete information at all, only after our thinking ability improves and knowledge foundation deepens can they naturally come into our horizon. To sum up, with repeating undergraduate courses, our mathematical thinking ability will become both more abstract and more concrete, and they will mutually reinforce and promote-the improvement of concrete thinking ability will give more further detailed and further elaborate materials to our abstract generalization, while the improvement of abstract thinking ability will enable us to more easily find huge, diverse concrete information.

In this process of improvement in thinking ability, our computation ability will also promote a lot. In the past, when our foundation is weak and ability is poor, we will feel hard to even do some simple computations, and moreover, we cannot judge whether our result is correct (the reason is that our understanding of a block of knowledge is dim) ; with the deepening of our foundation and the improvement of ability, we will be more flexible in doing simple computations; meanwhile, since we are extensively exposed to numerous complex proofs and problems and are also proficient with them, we can address more and more complicated computations (our computation now is mixed with lots of experience, intuitions, concepts and techniques accumulated over a long time).

Meanwhile, our independent thinking ability will also enhance and deepen. When our knowledge foundation is weak and superficial, we will also have some independent thinking, but at that time, our independent views are hollow and do not have valuable thought essence; when our knowledge foundation deepens and ability promotes, our independent ideas will have a solid foundation, and at this time (if we have a strong desire for independent thinking) we will have more and more our own ideas; moreover, since we have rich and deep learning experience, these independent ideas will be more reasonable, novel and mature.

With the increase of repetitions, our depth of thought will hugely promote. It is easy to understand that in undergraduate, ‘good’ students’ understanding of one specific course is much deeper than ‘normal’ ones, and when our knowledge foundation is weak and hollow and our ability is bad, our understanding of one particular course is doomed to be superficial, while after our knowledge foundation broadens and deepens, we will begin to be able to understand some profound ideas in the books, meanwhile, we can gradually understand some deep experience of previous people and good students. At this time, if we read relevant books, we will have a deep understanding, not a superficial impression like the past; since we can tell the difference between superficial and deep ideas in one course, we will form a deep mindset and can independently solve many hard problems, and in the meantime, we can lay a good thought and knowledge foundation for the future meaningful innovations.

In this process, our capability of analyzing and comprehensively solving problems will also greatly promote. Due to the enhancement of foundation and improvement of capability, we can more deeply analyze lots of theorems and knowledge, and when facing new problems, we can creatively find some approaches and paths to solve them by a number of methods, like analysis and synthesization. At this time, the depth and delicacy of our analysis will obviously enhance, and we can more flexibly use important ideas in other distant parts. In summary, our abilities of concretely analyzing particular knowledge and problems and comprehensive utilization of many methods and skills will both apparently promote.

To sum up, the concept ‘thinking ability’ has two levels of meaning: firstly, as an overall concept, it is meaningful, and an appropriate example is, in undergraduate, hundreds of my classmates sat in the same classroom and listened to the same courses like complex analysis, real analysis, functional analysis, algebraic topology at the same time, but the difference of understanding between ‘normal’ students and ‘good’ ones was tremendous (I realize this point at 5 years after graduation, after proficiently mastering 4 to 5 undergraduate courses), and the latter ones’ understanding was much deeper and they also mastered much richer information, meanwhile, their abstract generalization ability enabled them to make rich connections of these knowledge, and moreover, their mastery was much more proficient, thus, their creativity of solving problems was much stronger (for those knowledge the ‘good’ students mastered in one hour, we would perhaps spend 50 hours to get the same level of understanding, however, this comparison is not appropriate, because even we spent that time, we can truly understand them only after 8 or 9 years since we enter the college). In summary, this concept can be used to well describe our velocity and efficiency in absorbing knowledge. Secondly, ‘thinking ability’ is a general term for the 8 to 9 capabilities discussed above, and these capabilities represent most basic abilities in science and engineering, and each kind of them is quite meaningful (meanwhile, they are also closely related and mutually reinforce); in fact, good students have these 8 to 9 abilities since freshman year (actually, they gradually have these good capabilities since high school), thus, their learning efficiency is dozens of times to ‘normal’ ones. Only by combining these two levels together can we get a reasonable understanding of ‘thinking ability’.

Finally, we must say that the aims of relearning undergraduate courses are not merely for the promotion of our thinking ability, in this process, the concrete knowledge we master is also hugely important, and the popular notion that education is mainly for the cultivation of ability is very wrong, because: firstly, knowledge and ability is unified, and we can hardly achieve the overall improvement of thinking ability without much concrete knowledge accumulation; secondly, concrete knowledge is very important for students of every major, for instance, calculus is very important and many fields of business need it, and finite element and finite difference methods are also important for many engineering majors; if we just have the empty so-called ability without these concrete knowledge accumulation, how can we creatively treat the concrete problems in our actual work? In a word, accumulation of knowledge is a basic part of education, and its importance is not under the promotion of students’ thinking ability and comprehensive ability.[4]

(XVIII)One basic problem we often encounter is that many workers think the mistakes they make when they compute certain problems (like the second type of surface integral) stem from carelessness, but this view is wrong in many cases, and mostly, our false computations arise not from carelessness, but from lack of depth of understanding, namely, our understanding of one block of knowledge is not deep enough. Many experienced teachers all know that some students’ real ability is lower than they view themselves, and for many problems, they just have coarse thinking and cannot get precise results, and these teachers often think that this problem is due to lack of rigorous attitude and carefulness, however, the reality is that due to a serious problem with the understanding of the whole block of knowledge, we not merely cannot get correct results, our whole line of thinking is also disordered and vague.

For example, in March, 2014, I thought I had completely grasped the analytical properties of function series, while in July, 2016, with two more years’ accumulation, I realized that I did not really master these content in 2014, and my previous understanding had two basic defects: firstly, it was one-sided, namely, I didn’t organically integrate these contents into the holistic framework of mathematical analysis; secondly, it was also superficial, my then understanding lacked enough depth. Here, I can put forward another two experiences from my own learning process: the first experience was in August, 2012, I encountered one problem about maximal ideal and ring in abstract algebra, I could not solve it then, and I was distressed for a period of time, but four years later, I realized that even I could solve that problem in 2012, my understanding of ring and the whole abstract algebra still had big problems, and therefore, my fundamental problem was that there were serious problems about my overall understanding of this part of knowledge, not merely an isolated particular problem. The second thing happened in the autumn of 2015, I felt reluctant in solving one problem which involves energy, kinetic energy, velocity and momentum in special relativity, and that problem was a little complicated for me; later, after repeating special relativity for several more times (August, 2016), I realized that my overall understanding of special relativity before was not proficient enough, and my understanding of much concrete knowledge had a big problem, and when my overall mastery of special relativity further enhanced, I felt especially easy in solving that problem I was unconfident about in 2015. (This experience sufficiently proves that if we want to master one part of knowledge, we must be able to solve almost all the problems in it) In summary, these cases can help us to clarify one misunderstanding: it is just accidental if we cannot solve some problems, and it is also accidental when we falsely compute some problems, in fact, they are all holistic phenomena, and there exists a hard process with dozens of repetitions between a specious understanding and a really thorough one.

Before I relearned undergraduate courses (in 2012), sometimes I felt that my mathematical analysis was not very bad, and it seems that I had mastered its main parts; later, after I spent 3 years and 7 months in repeating it, I began to realize that my mastery then was very bad, and I nearly missed all its essence. This case fully proves that it is indeed very easy to have specious understandings in scientific learning.

(IXX)As described above, we emphasize the enormous importance of creatively independent thinking in the process of laying a solid foundation in undergraduate courses all the time, namely, in the process of laying a good foundation, we need to develop our initiative spirit and absorb broad undergraduate knowledge through our own imagination, and we should brand these knowledge with our own mindset. Meanwhile, we should also view the rich values of laying a good undergraduate foundation imaginatively: firstly, after laying a solid foundation, we can more efficiently control our work and life and can better develop our creative spirit in technology and make real contributions to the development of society, however, in a broader intellectual horizon, we should not be too utilitarian about laying a good knowledge foundation, and its ultimate goal is not for better wages or higher social status, but with broader value of life, namely, laying a good foundation can enhance our intense interest to a particular field and can enable us to enjoy the rich tastes of intellectual food in professional knowledge; meanwhile, solid foundation can broaden our horizon of life, thus we can better develop our imagination and originality, which can make our life more full, rich and profound, to sum up, promoting our interest, imagination and creativity in our own major and moreover, promoting the imagination, exploration enthusiasm and creative attitude towards our whole life is the essential goal of laying a solid undergraduate foundation. To sum up, no matter in the process of laying a solid foundation or after that, we need to see beyond the narrow goals of realistic level, and we should move into broader knowledge and intellectual fields, and should always view our professions and lives with deep curiosity, while laying a solid knowledge foundation can help us better follow our own enthusiasm and interest and can enable us to view life, society and universe with broader and better imagination.[5] In a word, realistic and utilitarian goals will greatly narrow the rich values of laying a good undergraduate foundation. A famous western proverb “All work and no play made Jack a dull boy” is partially consistent with the basic spirit here.

(XX)In the meanwhile, we should be clear about the broad goal of well learning undergraduate courses, namely, relearning undergraduate courses should aim at relearning all the basic courses relevant to our own research; since all the basic courses of every scientific major form an organic whole, and these courses have strong internal connections; therefore, we need to lay a solid foundation in all the basic courses related to our research, and only by doing this can we have a broad professional foundation for our future independent work. The interconnection of various mathematical courses is a well known basic fact, for instance: in functional analysis, we can use Banach contractive mapping theorem to solve the existence and uniqueness of the solution of ordinary differential equations, which is a central problem in ODE, and if we are not familiar with ODE, our understanding of the internal value of Banach contractive mapping theorem will relatively narrow, namely, functional analysis and differential equations are closely related; similarly, in abstract algebra, when we solve the impossibility of the trisection of an angle by using field extension, we also use knowledge in complex analysis; while in higher algebra, the proof of the fundamental theorem of algebra can be based on ideas from multivariable calculus and can also be solved by using ideas of algebraic topology and complex function, while the fundamental theorem of algebra is naturally fundamentally important in algebra, namely, algebra have deep connections with many courses, like mathematical analysis and topology. To sum up, the integrity of mathematics is a basic characteristic of modern mathematics, and the interplay of its various courses is deep and extensive and it also touches the central part of these courses, and I think many other scientific majors also have such a characteristic.[6] Thus, if we want to stand out in any scientific major, we have to lay a solid foundation in all the basic courses related to our own research in our major; here, another major aim in relearning undergraduate courses is already clear: due to the central feature of integrity of every scientific major, thus, we need to learn all the undergraduate basic courses related to research well, and the overall process of relearning undergraduate courses mainly is: when we repeat courses in freshman and sophomore years (study should follow a step-by-step approach, thus, we should first relearn courses in freshman and sophomore years), due to our low thinking ability then, our learning rate will be slow and it will cost a long time, but after we finish repeating basic courses in freshman year, due to the great improvement of our thinking ability, the time we spend in relearning courses of junior and senior years will be relatively less.

In the meantime, though this paper insists on the basic view that we should learn most undergraduate courses well, considering the complexity of reality, some people may think that we do not need to pay special attention to those courses irrelevant to our own research, which is naturally understandable.[7] However, we should be familiar with basic courses closely related to our research; for instance, for the students studying PDE, they must be highly proficient with mathematical analysis, functional analysis, differential geometry and ODE, and they need to solve almost all the after-class problems.

(XXI)Finally, we also need to point out that, so far, I have mastered only 4 to 5 courses and there is still a long way to grasp all the basic courses related to my research, but my thinking ability has improved a lot, thus, I think the left part will be relatively easier.

In the autumn of 2016, when I thoroughly mastered some courses, like mathematical analysis, I found that they were not so hard, but I had gone through such a long and complicated intellectual journey, this gave me a complex feeling, and I was doubting : is it necessary for me to spend so much energy in them? After all, these knowledge is very natural, but I clearly know that until 2015, these knowledge was still obscure and deep for me, thus, this intellectual journey is definitely a real experience, and moreover, I think this long learning experience is quite universal in all science and engineering fields. In the autumn of 2016, when the internal context of these courses naturally emerged, and when many parts which I thought were highly tricky before showed an unadorned appearance, I had a plain and natural feeling about this tortuous, interesting and continuously deepening learning experience.

Later, I realize that the reason for which I spent 3 years and 7 months in learning 4 to 5 undergraduate courses well is simple: firstly ,the range of these courses is very broad, and they all include tens of thousands of ideas, thousands of techniques and lots of concepts and approaches, moreover, these concepts and techniques are often interrelated, thus, mastering such a huge amount of information naturally requires a long time; secondly, these courses all have certain depth, and lots of problems and content have considerable difficulty, therefore, our understanding of them will deepen from shallow to profound, and this also takes a long time. To sum up, due to the superposition of two basic reasons-breadth and depth, I spent about 3 years and 7 months in total to learn these courses well.

(XXII) Here, we need to detailedly analyze the basic reasons which lead to superficially read recent papers (this shallow research approach happens among some professional workers at 37 or 38 years old, and also among lots of PhDs in many majors). Many workers who study PDE and numerical PDE master very superficially in basic content of related courses, like PDE and finite element; about PDE, they don’t truly master much content, including the deduction process of D’Alembert formula, separation of variable method, Duhamel’s principle (Duhamel’s principle emerges in many different occasions, like one dimensional nonhomogeneous equation, n dimensional nonhomogeneous equation, its combination with separation of variable method, etc) and energy method (like the deduction of energy inequality) in wave equation, the Dirichlet’s principle in harmonic equation, and they cannot solve relevant questions and also do not really master the complex details of these contents. (Indeed, after we truly master these contents, we will find that they are not really hard, but the superficial research way of these workers leads to their bad learning effect which is far from true understanding). About numerical PDE, they also have a shallow and disordered understanding of much basic content, like the discretization of equation, the deduction process of stiffness matrix and mass matrix of multidimensional equation, the deduction of error analysis of multidimensional finite element, the essence of stability of finite difference equation, the error analysis of finite difference method, etc.

The reason for the basic phenomenon that scientific courses are easy to be superficially learned is that knowledge points of scientific courses have some deeper basic features; take PDE as an example, many knowledge points, like the separation of variable method, the deduction of membrane vibration equation, spherical mean of wave equation and the Green function of harmonic function all have three important basic features: 1complex, 2delicate, 3deep. If we roughly read these knowledge points we often think that we have completely mastered them, but, if we deeply study them, we will find: firstly, they actually include numerous details, and these details all require careful deductions; secondly, they use many complex ideas and have intricate connections with other points in basic courses, this course and following courses; thirdly, they all have certain depth; obviously, nearly all the knowledge points in scientific courses have the above three important characteristics. To sum up, these three basic features of knowledge points in scientific courses lead to a huge difference between a real understanding and a plausible one, which also makes it easy for many workers to learn roughly and superficially.

From the above analysis, we can see that, in science and engineering fields, if we just superficially learn all the courses, we can just make some 2^nd or 3^rd class contributions and can never make really significant contributions, which is undoubted. (Because it is hard to really master one course, and only by really mastering some courses can we make a good contribution)

   You may ask one question: why am I so familiar with the intellectual condition of these students who superficially read recent papers? The answer is simple, because before I relearn undergraduate courses, my understanding of PDE and numerical PDE is similar as theirs; about these courses, my learning then also lacked enough depth and concrete, rich understanding. In a word, even for individuals who specialize in one particular direction (such as the researchers who specialize in differential geometry), their learning about basic courses in their own field is not deep and delicate enough, then it will be very hard for them to get some valuable results in their research with such a weak foundation; thus, our view that we should reinforce our own foundations is somewhat meaningful.

   (XXIII) Below, we want to analyze some negative impacts brought by the research method of superficially reading recent papers. In today’s graduate school, over the five years of PhD study, most PhD students often embark on independent research directly after passing the qualifying exams, namely, they begin the process of reading recent papers and books and doing independent research at this moment. This research method is so universal and popular that almost everyone is used to it, but the negative effects of such a research method also requires our deep thinking, at this time, many students whose thinking ability is ‘normal’ still have serious problems about their elementary courses, which usually leads to a somewhat serious internal defect of their understanding with recent papers, and therefore, the research quality of them almost inevitably has serious problems. Broadly speaking, this research method of superficially reading recent papers stems from a shallow understanding of the basic features of scientific knowledge, learning and innovation.

   It is easy to understand that superficially reading recent papers and superficially learning basic courses are interrelated and interactive. Due to a superficial learning of basic courses, these students’ foundation is weak, thus, their understanding of recent papers has big problems in both width and depth. Conversely, the major reasons of superficially reading recent papers are eagerness to get some original results due to the consideration in job hunting, work, etc; in college, due to the pressure of publishing papers, some workers are eager to publish a certain number of papers, while, in companies, due to the urgent demand of job assignment, they have to keep learning some new skills; thus, they all do not want to spend enough time in elementary courses and naturally lack comprehensive and solid foundation; namely, superficially reading recent papers also leads to superficially learning basic courses. To sum up, we need to have enough reflection about this basic phenomenon.

   Broadly speaking, superficially learning elementary courses will create two basic outcomes: 1 An insufficient mastery of information breadth of all courses, take real analysis as an example, it includes lots of information and details, while a mathematical worker who superficially learns probably just master 20% of the ideas, techniques, information and details included in real analysis, and moreover, they are also the easiest 20%, and it is easy to understand that a worker who superficially learns elementary courses has just mastered 20% easiest content of all courses. 2 Due to the mastered information is not rich and delicate, these workers just have a very shallow understanding of relevant courses, and they can only understand some shallow ideas and can just solve some easy problems (for example, for real analysis, they can just solve about 20% easiest problems), and it is hard for them to understand a lot of deeper ideas and intellectual essence in these courses, and a natural phenomenon is that these workers’ understanding of all courses is very shallow. The deficiency of information breadth and shallowness of understanding will lead to two basic phenomena in research: 1 as to theory and problems, due to a weak foundation, a worker who superficially reads recent books often cannot realize which theory and problems are important, central and meaningful in their own research; 2 as to tools like ideas, techniques and concepts which can build a new theory and solve problems, due to the weak foundation brought by superficially reading basic courses, these students may see some theories and problems are meaningful new directions, but they are not able to solve these problems by using existing tools or developing new tools like new ideas, concepts and approaches.

In a word, this type of workers only have a very small chance to make significant contributions in the future, and they are almost certain to just be able to make some simple, peripheral innovations.

   (XXIV) Considering the breadth of undergraduate basic courses, it is understandable that we do not learn some courses well and shallowly master them, which won’t prevent us from doing the best research, but if we superficially learn all the undergraduate basic courses (this makes up a certain proportion in scientific workers), it is definitely unacceptable.

   Take mathematics as an example, a well known fact is that for some scholars specializing in analysis, their algebra and topology are somewhat bad, while for some scholars specializing in algebra and geometry, their analysis foundation is a little weak, and other scholars studying geometry also have certain internal defects in their knowledge structure; these defects of their knowledge structure will not prevent relevant scholars from making the most outstanding achievements, and mathematicians who make remarkable contributions in all three major fields-analysis, algebra and geometry, like Hermann Weyl, are only few. Combining this basic fact, in this paper, what we emphasize is just: if a scientific worker superficially learns all the basic courses, then his research and work’s quality cannot be good.

   (XXV)Then why do we need to repeat dozens of times in the initial stage, to learn certain course well and solve many hard problems? I think experienced scientific practitioners all know that we need to do a certain amount of hard questions when learning some course; the deeper reason is not hard to understand, and we think there are at least three basic reasons: 1 As we all know that, in the courses we learn, the proof of most theorems and approaches are difficult, and it sufficiently proves that, in most cases, innovations are based on creatively solving hard questions, and due to this basic feature of innovation, solving hard questions is very important. 2 Only by solving hard questions can we understand the spiritual essence of one course, because, to solve hard questions, we need to integrate one part of knowledge and need to creatively use some important concepts, moreover, the computations are usually complex, thus, this can train our deeper and overall understanding of certain course; therefore, if we just do some simple questions, we will miss most key points of one course. It is well known that after we can solve some hard problems, we will feel especially easy to do easy ones. 3 Only by solving relatively hard questions can we feel the pleasure of study and research, and simple ones often just mechanically copy the formulas and their lines of thought are standard, and they do not need flexible techniques and deep ideas, thus, we cannot feel the charming of particular knowledge. To sum up, solving hard questions is an indispensible part of scientific learning; though simplicity is also a basic feature in science and engineering fields, solving hard questions is a necessary step which is hard to evade.

   The shallow information and deep ideas in scientific fields is an interesting and significant basic problem. We all know some science and engineering practitioners just have shallow understandings in every course and they very shallowly master concrete theories in all courses, meanwhile, they can just solve some easy questions and cannot figure out most hard ones; therefore, their research can just deal with some shallow, minor problems, and this superficial research method is wasting everyone’s valuable life. The reason is that one important feature of scientific fields is every course has certain depth, and this requires us to do some hard questions; a lot of knowledge and theories are not that shallow as we mistakenly think, in fact, they all include many deep skills.

   Indeed, for scientific fields, in many cases, it is simple ideas that open a new situation (for example, the Taylor expansion in calculus, congruent standard form in higher algebra, etc), but these simple ideas are actually based on the deep understanding of the whole field, based on the researcher’s solid knowledge and thought foundation. To sum up, we need to have sufficient understanding of the depth of scientific courses. [8]

   In a broader sense, as is widely known, in scientific research, like mathematics, physics and chemistry, ‘deep’ is a very important characteristic, then corresponding to the way of repeating for dozens of times in this paper, if we do not repeat reading some courses and papers for many times, how can our research have depth? I think this point is not difficult to understand. In fact, many good scholars will repeat some courses and papers for quite a few times.[9]

   (XXVI) The large group this paper mainly aims at is scientific workers whose thinking ability is in the ‘normal’ level, and many of them do not systematically well learn undergraduate courses, instead, they just superficially read recent papers, and the final results of this research method are: firstly, their ability is still in a low level; secondly, for the concrete knowledge in undergraduate courses, these practitioners also do not proficiently master, and they just superficially master all the undergraduate courses. The superposition of this kind of shallow knowledge structure and low overall ability leads to that they can only make some peripheral, unimportant innovations, which is surely unacceptable. It is well known that, in most scientific fields, the most creative period of relevant workers is before 45 years old, thus, this process of well learning all the related undergraduate courses needs to start as soon as possible. The reason why these workers do not start this process is not they are not willing to do so, but they don’t realize that their learning of undergraduate basic courses is somewhat bad.

   For those individuals whose thinking ability is in the ‘normal’ level, perhaps some of them once scatteredly repeated some undergraduate courses, but they did not embark on the huge process of relearning all the knowledge related to their research in undergraduate, and there are 3 reasons: firstly, they did not realize the hardness of this process, even for mathematical analysis, I was finally able to solve most of its problems after repeating it for 3 years and 7 months, and obviously, this is a long and difficult process. Secondly, they did not realize the breadth of this process, and some students probably repeated some parts of the undergraduate courses, but they did not realize that they needed to repeat all the basic courses related to their own research. Thirdly, they did not realize the high importance of this process, because these workers didn’t have an overall and deep understanding of the situation that their foundation is not solid enough, they maybe thought that it was just a minor problem, but the fact is not like this, the fundamental knowledge in undergraduate courses has a significant meaning to every individual. Among scientific workers, there is a universal misunderstanding: perhaps our learning of undergraduate basic courses is not very good, but this does not affect too much, and moreover, if we want to learn them well, it is not so hard; thus, this is not a very serious problem, however, in our opinion, this is a very wrong view and there is an enormous difference in the mastering of basic courses (even for mathematical analysis in the freshman year) between good students and normal ones. In actual life, these three reasons are often intertwined, and therefore, they lead to the fact that not too many workers embarked on the highly important process of relearning undergraduate courses. 

   We also need to point out that, among scientific workers, a small part of them do possess some fundamental knowledge and they can solve some of the after-class problems (for instance, some students in this level can solve the majority of problems in abstract algebra and algebraic topology, but they cannot figure out basic problems in functional analysis and PDE); for these students, their thinking ability also needs to improve and they need to be able to solve almost all the hard problems, and only by doing this can they achieve the improvement of their depth of thought; otherwise, what we do is just a simple accumulation of knowledge width in the same thinking depth.

   (XXVII)In the meanwhile, we need to point out 8 basic facts: firstly, many scientific workers actually realize the basic fact that the thinking ability of different people has big differences, thus, this is not a new insight; experienced scientific teachers, students and workers often know that the talents between different people has large differences, however, they do not know that these differences in thinking ability can be overcome, and, to some extent, must be overcome by us. Secondly, in the process of relearning undergraduate courses, due to the difference of foundation among people, the repeating times needed perhaps vary widely, but they must meet the standard of solving most after-class problems. Thirdly, the learning and work methods of different people often vary, and this paper values and emphasizes the holistic property of knowledge, but many people’s work style is not like this (as revealed by Dyson in the essay , some mathematicians like general intellectual framework, while other mathematicians pay more attention to isolated and concrete problems), thus, we need to use our own creativity and seek work method which fits our own personalities. Fourthly, about the process of relearning undergraduate courses, different people may work in different ways, here what I adapt is the way of repeating textbooks for dozens of times (with watching videos and discussing with classmates), for different people, we can choose the ways which best suit ourselves, but the final criterion is unified, namely, we need to solve almost all the problems. Fifthly, this paper emphasizes the importance and benchmark of solving problems, but the intension of learning is much richer than simply solving problems; take mathematics as an example, understanding the latent thought essence in knowledge is probably a more important basic step behind solving problems, and meanwhile, the importance of problem solving should also not be overestimated, as the great mathematician Atiyah once said: “I don’t pay very much attention to the importance of proofs. I think it is more importance to understand something.” “A proof is important as a check on your understanding. I may think I understand, but the proof is the check that I have understood, that’s all. It is the last stage in the operation-an ultimate check-but it isn’t the primary thing at all.”[10] This view of Atiyah is very reasonable. Sixthly, the problem solving we deal with in this paper is active problem solving, namely, we can actively solve these problems, not understanding the line of thought after reading the solutions, and such kind of figuring out problems by seeing the solutions is not very meaningful, since it does not include the process of creative thought and positively active thought, and it won’t cost us so much energy; namely, we must actively solve related problems, otherwise, it doesn’t have essential meaning. Seventhly, we may not strictly follow the standard of solving all the after-class problems, but we must be able to solve most hard questions. Eighthly, for the students in ‘normal’ level, they actually do not master almost all the contents in undergraduate courses, namely, many science and engineering students actually know very little about their own major at graduation, which is a somewhat shocking basic fact.

  （XXVIII）For repeating over 55 times (it will cost over 3 years’ time) and the difficult task to solve most hard problems, it is somewhat unexpected, but it also fits some basic rules of human society, such as: science and engineering work should always strive for excellence, and any valuable thing is not easy to get and requires arduous effort;[11] hardworking is the foundation of modern society; anything should begin by laying a solid foundation and should start from the root and proceed step-by-step (from this principle, we should repeat undergraduate courses by firstly repeating freshman and sophomore years’ courses), and we cannot make important innovations without a good foundation ( minor innovations are still possible). Broadly speaking, the process of relearning all the courses related to our own research in the undergraduate stage is a hard process with constant struggles, meanwhile (somewhat complementarily), it is also a pleasant journey.

   (IXXX) We think the problems we discuss in this paper are meaningful to the following several kinds of people: 1 some mathematical workers still cannot proficiently write down the second mean value theorem for integrations at 37 or 38, such a condition of weak foundation is prevalent in many majors’ PhDs, faculties and workers, like electronic engineering (some basic courses like digital analog circuit, circuit principle, engineering mathematics), chemistry (courses like organic chemistry), mechanical engineering (courses like material mechanics, mechanical principle), physics, statistics, aerospace engineering, computer science, chemical engineering, etc; we should pay sufficient attention to this phenomenon. 2 To some workers who need mathematical and physical knowledge, like architecture (structural mechanics, elastic mechanics, etc), they should also realize the long process of learning these courses. For the above several kinds of people, the problems discussed in this paper may help them to open a magnificent intellectual horizon.

   Meanwhile, some science and engineering workers may do not need original activities, while laying a solid foundation in basic courses are still of rich value for them. Because mastering basic knowledge in undergraduate and improving their thinking ability can make them better face their work, for instance, learning the pointer, array and other contents in C language, mastering some central contents in higher algebra, like the solution of linear equations, the method of figuring out Jordan standard form, well learning important knowledge in calculus, like definite integral, improper integral, multivariate differential calculus, function series, multiple integral, Fourier series, etc, all of these can enable them to better solve various problems in their work since the affected area of these knowledge is so broad. Many scientific workers do not really master these knowledge, thus, this has a negative impact on their practical work, while some ideas put forward above can change some workers’ situation. For me, before repeating undergraduate courses, I am always not confident about C language, but in August, 2016, I feel that my mastery of C language is already in place, at the same time, my thinking ability has enhanced, and due to the basic changes of these two aspects, my confidence about C language has greatly improved (this is already 9 years after I entered college); considering the broad application of C language, we think the ideas raised in this paper is valuable. Namely, if we just superficially master 10 courses, when facing concrete problems in real work, we often cannot solve them, thus, we don’t have executive ability, and it is less valuable than thoroughly learning 3 to 4 courses. (However, for innovation, only well learning 3 to 4 courses is not sufficient.)

   To sum up, the ideas we articulate in this paper have double meanings: research and learning.

   Obviously, our discussion is very applicable to scientific fields, like mathematics and physics; while for engineering fields, if we relate the actual industrial world, we think this paper also has rich values; firstly, it is of direct meaning to the practitioners in a number of companies in electronic engineering, like Sumsung, Cisco, IBM, etc, and also including many small and medium-sized enterprises in this field. For many companies whose major business is big data and artificial intelligence, our discussion is very applicable to their workers too, since we think there is a deep difference between a worker who has solid foundations in mathematical analysis and higher algebra and another one who just know R software; for instance, in the summer of 2017, I worked in a big data company for a short time, at that time I fully felt the great benefit brought by my deep foundation in calculus and linear algebra (stemming from many repetitions which cost much time), because I felt especially easy when learning some new knowledge and skills and it was easy for me to well master them. In the meantime, it also applies to workers in mechanical engineering, like medical instruments (like Philips, Siemens, Johnson and Baxter) and large instruments. The latent impact of our discussion to other fields, like civil engineering, is also similar. To sum up, the basic issues we deal with in this paper have high potential values to both the science and engineering fields.

   (XXX)In the following part, we want to further analyze the possible actual impacts of this paper, and the complex reactions possibly triggered by it:

   For many students in mathematics and physics, if they have normal talent, then it is very difficult for them to make first-class contributions by systematically repeating undergraduate courses and laying a good foundation, because the golden age in scientific fields is before 45 years old. The above example, Carleson, is a successful case, and Hardy is another example, he said that (also, it is an objective fact) his professional career truly began at 34 years old when he met Littlewood, but, even with some past cases, for these students, it is still considerably difficult to make outstanding contributions. Therefore, our paper is just partially applicable to this type of students. However, our paper is of certain value to another type of workers in mathematics and physics, namely, for a few excellent professors in some top universities, like Harvard and Yale, their knowledge structure has certain internal defect-their analysis side is probably very fantastic, but they are afraid to do algebra and topology problems and are afraid to touch them, or they do very well in algebra, but they are afraid to touch any analytical questions (Mr. Shing-Tung Yau once mentioned this phenomenon); this way of doing research may be not bad, but there is certainly a limitation in it, and for this type of workers, the analytical framework and deep experience we provide in this paper perhaps have some value.

   For the workers in engineering fields, like electronic engineering, aerospace engineering and statistics, their research and work may not require exceptional genius, and their courses are not too deep and broad, since part of their job is routine, and they can better qualify for their work by laying down a solid foundation. Meanwhile, the number of workers in these engineering fields is much larger than mathematics and physics, therefore, many of the practitioners in these fields have a weak foundation, and for these people, they need to have a broad plan and should not only consider short-term (2 or 3 months) goals. Considering the enormous number of engineering workers, the impact to engineering majors is probably the major impact of this paper.

   The reaction to this paper may be: (1) About those students whose foundation is weak, their learning about all the basic courses is superficial, and they did not realize this point before (also probably realized it), but they quickly realize this basic fact with our extensive analysis, namely, they have an objective assessment about their weak fundamental; considering that most scientific workers have a clear understanding of their own fundamental, thus, they have a sober judgment about whether they can solve problems; since they realize this point, they will probably systematically relearn some courses and build certain foundations for their own work, meanwhile, they may also choose the way of superficially learning and continue to publish some ordinary papers, which is also understandable. (However, a small group of students will perhaps have a disordered and false judgment about their knowledge foundation and real ability, meanwhile, when one person is 35 years old, the crowd will form a somewhat objective judgment about the ability of a particular worker, since the judgment of many experts together will be objective.) (2) About those good students whose foundation is solid, they do not have any experience of the phenomenon described in this paper, and they just need to learn graduate’s courses and directly embark on independent research and work. However, another basic issue analyzed in this paper, namely, the thought in speculative level and artistic level is extremely important for their long-term and deep development (we have analyzed it in another paper). We should note that there are so many science and engineering majors, and the students’ foundation and professional pursuits are also very complex; thus, our discussion in this paper will create different reactions in different individuals.[12]  

   What we really want to stress is that scientific workers should have long-term views about their own work. For instance, for practitioners in electronic engineering, if they keep superficially learning (hastily read recent books, papers and related literature without thinking, roughly learn new techniques and softwares), they will still lack a mastery of some necessary basic courses, and this kind of foundation is problematic in actual work. If one worker spends 3 years in doing so-called frontier research or work, after 3 years, his foundation won’t change too much and he also cannot achieve essential overall progress; not as good as spending 3 years’ time in learning several basic courses and truly improve one’s ability; after all, three years’ time is not so short and also not so long, and spending 3 years in well learning several courses will give us enough confidence in our own work.

   Professional ability and technical foundation is obviously a central problem in scientific practitioner’s work, and considering the complex analysis of this paper, we can get one exhilarating basic conclusion: the professional foundation of scientific workers can be changed, and we can also change our professional capability, but we must start from basic courses and make long-term plans. Admission to top universities and entering good companies cannot change our professional strength, since strength can only stem from long-term (over 3 years) systematic accumulation and it will not change with the change of external environment; the professional knowledge of science and engineering majors is too broad and delicate, and it requires long-term accumulation, how can it be suddenly mastered by us just because we enter top companies? We think that for various scientific majors, systematically learning basic courses is probably the only way to change our professional strength and superficially doing research and work will make us stamp on the same ground, and it is virtually impossible to reinforce our technical foundation by piecemeal learning, and seeking short-term success and playing petty tricks can hardly bring us a solid, fundamental progress; while time (for instance, 2 years, 3 years or 5 years) will quickly pass in the busy life and work. Meanwhile, considering the heavy work of job assignments, the bustle of life and it will cost over 3 years to repeat undergraduate courses, this is indeed a knotty problem and requires some wisdom to deal with it.

   In a word, we think this paper will have a certain degree of impact for scientific workers, but is also limited. Meanwhile, with the passage of time, the impact of this paper will gradually deepen.

   (XXXI) We need to point out that the two categories of scientific students’ talents in this paper-‘normal’ and ‘good’ are just a coarse approximate analysis; the actual situation is much more complicated and there are many different levels of talent. The analysis of talent differences among students is just a basic ingredient of this paper and it is naturally not the major purpose of it, our main purpose is to display the long and complex process of learning some scientific courses.

   (XXXII) Finally, a basic awareness we need to establish is: laying a solid foundation in undergraduate courses is certainly not the eventual goal, and our ultimate aim is to make essential innovations in our own area; thus, for the knowledge we have thoroughly mastered, we should not waste much time in learning them, and we need to think the possible directions of creative developments. In academic fields, we will find some students with good foundation cannot get original results, which is because that they lack a systematic understanding about research and they also lack independent thinking of particular knowledge, thus, they don’t know how to find new research areas and problems, and we need to avoid this kind of behavior. Thus, in the process of relearning undergraduate courses, we need to think about many important issues in other aspects, including how to find novel research direction, the interconnection of many courses, understanding thought systems, fostering deep intuitions, creating unknown tools, developing existing techniques, refining latent concepts, capturing flashed inspirations, etc. We need to creatively understand undergraduate knowledge, and also need to keep independent research views and cutting-edge consciousness; without the rich understanding and broad reserve of academic original awareness, it is hard for us to make important contributions in our research. The nature of research is to find new knowledge, not merely learn existing knowledge, and therefore, we need to look ahead and think about new directions and not stay too long in known knowledge. To sum up, the return to foundation is for better facing the future, if we cannot face the future, return to foundation will lose most of its meaning.

   Here, one part of argument given by Beveridge is very helpful to our discussion, he wrote: “Charles Nicolle distinguished (a) the inventive genius who cannot be a storehouse for knowledge and who is not necessarily intelligent in the usual sense, and (b) the scientists with a fine intelligence who classifies, reasons and deduces but is, according to Nicolle, incapable of creative originality or making original discoveries. The former uses intuition and only calls on logic and reason to confirm the finding. The latter advances knowledge by gradual steps like a mason putting brick on brick until finally a structure is formed. Nicolle says that intuitions were so strong with Pasteur and Metchnikoff that sometimes they almost published before the experiment results were obtained. Their experiments were done mainly to reply to their critics.”[13] This passage is very thought-provoking, some researchers with good foundation cannot make original discoveries, and this fact is widely known to us (one of the reasons is perhaps the lack of accumulation in thought level), thus, we need to pay high attention to original spirit; to keep a blooming condition in all the human scientific fields, innovative spirit and original ability have an overriding importance. Of course, innovation needs solid knowledge and thought foundation, Beveridge’s exposition just reminds us: originality, making new discoveries, finding new phenomenon and creating new technology are the most important and most decisive criteria in all science and engineering fields. One sentence of Whitehead probably gives the best generalization of the fundamental value of innovation: “Passively understanding the past will lose the entire value contained in the past. A living civilization needs learning, but not only learning.” [14]

   (XXXIII)Correspondingly, the basic ideas raised here do not aim at the few students whose talent is ‘good’, because they have well mastered the undergraduate basic knowledge in the four undergraduate years, thus, they do not need to repeat them, and for these students, they don’t need to relearn undergraduate courses, they can just master graduate courses and begin independent research. Even for the scientific students whose foundation is not good, in the actual process of work and research, they are unlikely to spend a whole period of 3 years and 7 months like me in just repeating freshman and sophomore years’ courses; thus, we need to mainly focus on our work and frontier research, however, we also should pay attention to basic courses.

   In summary, the 4 central thought themes of this paper are: systematically repeating undergraduate courses, problem solving (as pointed out above, the problem solving we emphasize in this paper must be active problem solving, not solving problems after reading the keys), the holistic improvement of thinking ability and independent thinking; though we mainly discuss the higher education of mathematics, I think the same learning rules also apply to a number of science and engineering disciplines, including physics, chemistry, mechanical engineering, electronic engineering, computer science, statistics, aerospace engineering, petroleum engineering, civil engineering, chemical engineering, communication engineering, etc.

   Our overall belief is: nearly all the science and engineering workers who make real contributions have a solid foundation (for instance, when we explore the field of differential geometry, if our foundation is not strong enough, then even we can vaguely guess the existence of Gauss-Bonnet formula, we will necessarily be unable to complete the complex proof, or when we explore the field of quantum mechanics, even we can guess the general direction of Dirac equation, we cannot detailedly get the overall properties of this equation, or perhaps a more intuitive argument is that for those scientific workers who get good results in independent research stage, most of them are also good students in undergraduate; to sum up, solid foundation is fundamentally important for scientific practitioners, which is a somewhat obvious fact), all the science and engineering workers who make really significant contributions have extremely strong independent thinking ability (in the process of learning, independent thinking can doubly enrich the concrete information, thus, the process of scientific learning is a positive and active procedure). All the conclusions in this paper are based on these two obvious basic facts.

   In conclusion, one secret lurking in the undergraduate science and engineering education for over 100 years is uncovered. (This secret is not discovered for a long time because it looks easy, but in fact it is not, and it is the mergence of five fundamental insights: 1 The difference of thinking ability, most people just realize the basic fact that there are differences of thinking ability among scientific students, but they don’t realize that we need to make enormous effort to improve form the ‘normal’ level to the ‘good’ level. 2 Relearning undergraduate courses, some people may realize this point, but they did not repeat for sufficient times and their repetition was less than 55 times, and they did not insist on the basic principle that most after-class problems need to be solved, thus, they falsely believed that they had mastered these courses after repeating for some times, in fact, there was still a long distance ahead; meanwhile, some students did relearn some undergraduate courses, but they didn’t realize that we need to repeat all the undergraduate courses related to our own research. 3 One purpose of relearning undergraduate courses is to improve our ability, and the process of our learning is one process in which our thinking ability keeps improving, and we can evidently feel this by comparing our intellectual condition between senior year and freshman year, but, almost nobody realized: to thoroughly solve the issue of improvement of our overall ability, we must return to undergraduate courses. After repeating the undergraduate courses for over 55 times, our professional thinking ability will greatly enhance, and we will feel especially easy when facing concrete knowledge and problems, namely, in the process of relearning undergraduate courses, the mastery of knowledge and the improvement of ability are deeply intertwined, which is certainly a pleasant feeling. 4 In learning, the accumulation of knowledge is one aspect, and the understanding in thoughtful and artistic levels of particular knowledge is another indispensible aspect, namely, we need to actively master relevant knowledge in our own way, independent thinking can doubly enrich our scientific information; in a word, when facing basic knowledge, we not only need long-term accumulation, we also need creative accumulation. 5 As to the time dimension, relearning undergraduate courses will require a long time, and even we don’t do research and read any recent literature, and just take on the assignment of relearning undergraduate courses, relearning undergraduate courses will cost us at least 4 to 5 years after graduation. The above 5 points perhaps can be discovered isolatedly by some people, but only by merging them together can we form an overall insight of undergraduate science and engineering education.)  

                                                                                                                                                Sept 29, 2018