On Undergraduate Science andEngineering Education

   (I)On science and engineering education, thelarge background we need to clarify is: the talent and thinking ability ofstudents around us can be divided into two basic levels-normal and excellent;for instance, in the mathematics department of Fudan University where I studiedbefore (Fudan is one of the best universities in China), the majority ofstudents belong to the ‘normal’ level and I am in this category, while a smallnumber of them belong to the gifted, bright level, like Weixiao Shen and WenjunWu. The difference of thinking ability between these students is huge: thelearning efficiency of ‘good’ students is tens of times higher than ‘normal’ones. (We all know few students who can well learn everything by only studying3 or 4 hours each day) The big gap of thinking ability embodies in 3 majoraspects: the depth of understanding, the proficiency and the creativeutilization of the same content. From the perspective of actively solvingproblems and mastering specific knowledge (these two things are actually thesame), the ‘normal’ students only grasp few key points, thus, they can barelysolve a small part of after-school exercise, while ‘good’ students can solvealmost all the problems, namely, they have grasped most content inundergraduate courses.

   Over my four years of undergraduate life, Istudied almost day and night due to the enormous enthusiasm for mathematics, inthe daytime, I intensely learned in the 2ndand 3rdTeachingBuilding except class and meal time, and I also worked against time in the 4thand 5thTeaching Building at night; from 8 a.m. to 12 p.m., mylearning plan is always compactly arranged. During this time, I read manyreference books with three aims: deepen the understanding of particularcourses, solve the after-school exercise and prepare for the final exam; forinstance, I carefully read 2 or 3 reference books of higher algebra and many ofmathematical analysis, complex analysis in the library. To work out homeworkassignments, we had to devote a great deal of time and energy in learning, andI still remember the stressful scenarios of studying functional analysis,algebraic topology, partial differential equation (PDE), differential geometryand other courses in the classroom for handing in homework on time. In theundergraduate stage, we earnestly listened to our teachers in the class andrepeatedly read books after class, and learning is the keynote melodythroughout our college days, and I think most of scientific students haveexperienced a similar life condition. However, even with such long-termhardworking, my learning effect was still very bad at graduation; back then,for mathematical analysis, I could just solve about 10%, simplest problems, andI also made little sense of PDE (from today’s higher point of view) since Icouldn’t solve most of relevant problems. To sum up, during undergraduate, Iwas very distressed about the central issue of being unable to actively solveproblems. This basic phenomenon arises from many reasons, and among them, thecore reason is that my thinking ability was too low, which led to a verysuperficial understanding of particular knowledge, and I just had a vague,shallow impression after learning it once. Broadly speaking, the reason of mypoor 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 themajority 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 coursearrangement of Chinese university, we don’t have new classes to learn exceptwriting senior thesis and interning in that year), I did not relax too muchlike some students, instead, I still kept learning, and at that time, since Ihad a lot of free time, I embarked on relearning undergraduate courses, and thecourses I relearned include real analysis, partial differential equation, complexanalysis, etc, and I listened to the teaching again in the classroom then,thus, my knowledge and independent views still grew, and my thinking abilitywas still enhancing every day, but the effect was not very ideal, I stillcouldn’t actively solve problems, and from today’s viewpoint, my understandingthen was still very scattered and vague. Over a long period of time, the senseof anxiety stemming from being unable to actively solve problems was the majorpsychological condition when I faced mathematics.

   Until graduation, I did not master theTaylorexpansion of manycommon functions, and just had a very superficial, disordered and vagueimpression of many basic contents, like separation variable method in partialdifferential equation and permutation group, which is rather ashamed, but it isa basic fact.

   In the graduate stage, the situation changedqualitatively, and I did not concentrate my energy in research like many others,instead, I focused on relearning undergraduate courses, and this broad strategyhas a very good effect. (The reason for this is that I was very distressedabout being unable to solve after-class problems and I think it is an essentialissue I can’t evade) In the first one year and a half, to pass the qualifyingexam, I repeated functional analysis, numerical PDE and abstract algebra forover 10 times, and from the spring semester of the 2ndyear (January,2013), I repeated mathematical analysis (calculus and its theoreticalfoundation), half of higher algebra, 2/3 of abstract algebra, specialrelativity, 2/3 of partial differential equation, C++ and numerical PDE forabout 50 times (I repeated them simultaneously). There are three basic reasonsfor relearning undergraduate courses: firstly, expansion of knowledge;secondly, improvement of ability; thirdly, accumulation of independent ideasabout related courses. These 3 aims are highly unified and intertwined. Withthe constant repetition for over 3 years, in the summer of 2016, my abilityfinally improved beyond the ‘normal’ level, which is a qualitativebreakthrough. The reason for this breakthrough is that my thinking ability has greatlyimproved, and this improvement happens day by day; in the short term, theimprovement is small, but we can feel it; in the long term, I can clearly feelthe giant leap of my thinking ability. For instance, from freshman to junior,my thinking ability was always improving, though it was not ideal then, theprocess of improvement was clear, and from the 1styear of graduateto 5th, my thinking ability was still enhancing, and at the summerof my 5thyear, it finally improved to a good condition.

   For courses I didn’t repeat in the 5 yearsafter undergraduate, my impression is very hazy and vague, like algebraictopology (key points like mapping lifting) and higher algebra (vital contentlike Jordan canonical form), which clearly proves that my thinking ability thenwas too bad and there are serious problems of my internal understanding aboutthese knowledge. I think this kind of intellectual condition is a basicexperience shared by many students: when we learn in undergraduate, our mainfeeling is fully dim and specious.

   In the summer of 2016 (at the end ofAugust), 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 theirquestions and I can actively solve over 2/3 of related problems (in the past, Ifelt that these questions were very hard and I could not find the clues, butnow, I realize that they are all basic problems). This hard but profitableexperience gives me a deep enlightenment: if we spend a great deal of timerepeating one course, then sooner or later we can master them. In mathematicallearning, complex and simple, abstract and concrete are all relative; when wedon’t learn these courses well, we will naturally feel that a number oftheories and questions are hard and abstract, but if our overall understandingdeepens, we will feel them concrete and familiar.

When looking back this longprocess, I can surely say that for these courses, like abstract algebra, Icouldn’t solve most of their problems even I repeated them for 49 times, anduntil the 50thtime (Jan, 2016), I began to be able to solve a smallnumber of the problems; since then, I still had a great improvement after everyrepetition, and at the 55thtime ( August, 2016), I felt that mymastery of these courses was already deep, organized and lucid (I could solveover 2/3 of the problems, and moreover, these problems became natural andfamiliar for me). When I repeated the 55thtime, I not only feltthat I could solve most problems, also felt I had brought the major pointstogether, and my line of thinking became much more clear when facing theseproblems and I also felt that they were much easier, correspondingly, I had arelaxed feeling, and moreover, this is a holistic phenomenon. Therefore, I hearsome students in physics say that they suffer from learning theoretical mechanicssince they can’t solve most problems in it, and this basic fact is easy tounderstand because they just repeated this course for 20 times which is farfrom 50 times, thus, they inevitably could not solve its problems and theysuffered. We know that when we read professional literature, we will feel that asif we learned nothing in the first time when we read it again, and I have asimilar intellectual experience when relearning undergraduate courses, and evenmore serious, because I felt that I had learned nothing in the previous 49times when I read the 50thtime, and until the 51sttime,I finally felt that I really had learned something. In the intermediate stage,when I repeated these courses, I found that there were so many key points I hadcompletely 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 overthree years in learning just 4 undergraduate courses. ( We need to point out thatit is not enough to solve over 2/3 problems, and we have to solve almost all ofthem, especially some hard questions, and only in this way can we achieve thereal goal of relearning undergraduate courses because we can get sufficientdepth of understanding only by solving many hard questions.)

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

   In repeating undergraduate courses, I havethe following 5 guiding principles: 1improvement of thinking ability every day;2fewer but better, I don’t want to repeat 10 courses together, instead, I justrepeated 4 to 5 courses at the beginning, and I will not learn other coursesbefore completely mastering them, and obviously, we have learned them well onlyif our understanding is thorough and clear, and the main criterion of fewer butbetter principle is that we can solve most after-class problems; moreover, dueto the precise characteristic of scientific knowledge, to solve a problem, itis not enough if we just get the overall clue, we must get a precise result ora 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 ourhigh level professional qualities; 4keep repeating and proficiency, repeat over55 times at the beginning; 5the main philosophical idea of Grothendieck:solution of one problem is based on overall foundation and intuition, and ourunderstanding becomes mature only after we feel the problems are natural andtrivial, and the solution then naturally emerges, and the process of solutionshould break up into a series of small and natural steps. (This insight canexplain why we can’t solve problems, the answer is we haven’t repeated enoughtimes, thus our understanding is not deep enough and also not proficientenough, thus, the intuition of solution cannot emerge. The essential cause ofnot being able to solve problems is that our understanding of a block ofknowledge is bad and our overall foundation is defective.)

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

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

   (II) To summarize our analysis, we can realizethat for most Chinese scientific graduates (including those in top Chineseuniversities), their fundamentals are rather weak and they can’t solve mostafter-class problems. Therefore, we should pay sufficient attention to thesebasic courses, and graduates of many majors, including computer science,mechanical engineering, electronic engineering, statistics, aerospaceengineering, civil engineering, petroleum engineering, communicationengineering, chemical engineering, physics, chemistry, etc, should repeat theundergraduate courses if necessary. In my opinion, many electronic engineeringgraduates in Chinese Academy of Sciences have two problems about undergraduatecourses: firstly, their mastery of concrete knowledge is bad; secondly, theycannot solve the majority of problems in them, and this may seem difficult tobelieve for the laymen, but in fact, more than half of the students have thisbasic issue.

   Take professors who have worked for manyyears 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 inmathematical analysis (which is one most basic course); therefore, they mustrepeat undergraduate courses and learn as many basic skills as possible. Intoday’s graduate school, many students begin to do independent research afterpassing the qualifying exam in the 2ndor 3rdyear, whiletheir fundamentals are not solid enough and they can’t actively solve mostproblems, and therefore, we think this research method can only get unimportantresults.

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

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

   (III)By laying down related foundation, wecan better understand subsequent courses; for example, the deep impacts ofmathematical analysis[1]to real analysis include:1 the type of Riemann integrable functions is not resolved in mathematicalanalysis, which is thoroughly resolved in real analysis; 2 function series hasa high demand for uniform convergence, while in real analysis the demand issomewhat lower; 3 exchange of integral is too strict in mathematical analysiswhich is improved by Fubini theorem in real analysis. The impacts ofmathematical analysis to functional analysis include: 1 Baire category theoremcan solve problems like discontinuous points of functions, the convergence ofFourier series, etc; therefore, Baire category theorem not only has a farreaching impact on the main theorems in functional analysis, it also has rootsin mathematical analysis; 2 the part of Fourier series is generalized inHilbert space of functional analysis, and the best square convergent propertyof Fourier series and the Parseval equation are the most important special caseof corresponding theory; 3 some central themes in mathematical analysis isdeepened in functional analysis, including sequence convergence, topologicalproperty, differential operators, integral operators and etc. It is widelyknown that probability theory has a deep influence over statistics, and inprobability theory, we need to compute many probability distributions which areclosely related with multiple integral and series. As to the directenlightenment of concepts and methods of mathematical analysis to point settopology are more familiar to us. Before repeating the undergraduate coursesfor 55 times, I just had a vague impression of mathematical analysis’fundamental influence on these courses, however, with the solid foundation inboth knowledge and problems, I then get a deep and natural feeling. Forinstance, I did not have a deep feeling with the importance ofTaylorexpansion to error analysis of finite difference and the value of residuetheorem to abnormal integral computation until I did a lot of problems aboutTaylorexpansion andintegral in mathematical analysis, and there are innumerable similar examples.The influence of mathematical analysis to relevant courses can be divided intotwo levels: direct and indirect, and we can easily list many concrete examplesof direct impacts, while indirect impact permeates into aspects likemathematical sense and quality, though hard to describe by language and we usethem every day without noticing, they are also very important. As correctlypointed out by Zhiwei Yun, the impact of mathematical analysis and higheralgebra can be extended to graduate courses, and those following courses areeasy to learn if we completely grasp these two courses. The basic courses arethe bedrock of thought system in modern mathematics and if we do not learn themwell, our understanding of mathematics will have many original defects. My ownfeeling is that I feel especially relaxed in learning partial differentialequation (PDE) and numerical PDE after I have a solid foundation in calculus, andthen I feel very natural and strict about many complex deductions in them.

   A concrete point in basic course, like thefundamental theorem of homomorphism in group theory, has 3 levels ofenlightenment 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 themathematical points may have these three levels of value. To sum up, layingdown the foundation mainly has 3 purposes: firstly, improvement of overallability (for instance, to enhance our thinking ability from the ‘normal’level); secondly, form an orderly, original understanding of the holisticthought system in the conceptual level; thirdly, the accumulation of concreteknowledge, which we are familiar with; the whole process is an organic unity ofthese three aspects.

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

 (IV)How to solve problem is clearly a centralproblem 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 4thgraduate year, and I still could not actively solve problems then, particularlyhard problems; I think many chemical engineering and physics PhDs also have asimilar beset. Until the end of the 5thyear after graduation, with9 years of struggle and grope, I gradually overcame this problem; the centralreason of not being able to solve problems is a shallow understanding of thecourses and a low thinking ability (we have pointed out it for many timesabove), and due to a shallow understanding of knowledge, we cannot grasp theessence, central spirit and deep context of the courses, thus, we cannot solveproblems. In my own experience, after 4 years of graduation, more precisely,before October, 2015, I could not solve most problems in multiple variableimplicit functions, the analytical property of parametric variable integral,multiple integral, the first type of surface integral, etc, and I felt thatthese problems were out of reach for me, but when I repeated these courses forthe 51sttime, I finally managed to solve some problems, and I realizedthat they were just easiest and most basic problems, and I think thispsychological experience is quite universal. When I studied the 51sttime, I finally realized that for the theoretical foundation and concreteexamples of the second type of surface integral, I did not master them at allbefore, and in the condition of failing to completely master them at knowledgelevel, I naturally could not actively solve problems. Problem solving is basedon three major aspects: knowledge points (which is often a holisticphenomenon), mindset and depth of understanding, and in many cases, for aconcrete problem, if we do not learn relevant points (for example, if we cannotflexibly use concepts like superior limit and infimum) or the depth of idea isnot enough, or we do not have relevant mathematical mindset, then we can’tsolve it even we think about it for 1 year. When I repeated for 50 times, Icould just solve problems scrappily, and when I repeated the 55thtime (August, 2016), I could extensively solve problems then. My ability ofproblem solving then was based on deepening of foundation and improvement ofthinking ability, namely, my thinking ability had enhanced a lot, and Isuddenly felt that the holistic line of thought and key details were veryclear. In a word, problem solving are based on the solid overall knowledgefoundation and high level of thinking ability, and only with these twoconditions together can we solve problems.

   Many people have the following two feelings:firstly, in many cases, we cannot solve problems, then some days later afterseeing the keys, we still cannot solve them, and a few days later after seeingthe keys again, we think that we will be able to solve these problems, but whendoing them we find that we still cannot solve them. (When reading books, wewill have a similar experience in facing knowledge points) Secondly, for thoseproblems we can solve (or concepts and approaches we have already mastered), wewill also have a new feeling when we read them again: firstly, we are moreproficient; secondly, our mastery is more deep and solid; thirdly, ourparticular 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 andwith a sense of difficulty; finally, we can easily solve the problems. Thesethree stages is a natural process everyone will go through.

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

   An easily observed fact is that scientificproblems have 5 basic characteristics: 1Stubborn, complex, many problems needcomplicated computations. 2Deep, a number of after-school problems requiredepth of thought, and we need to get enough depth of relevant knowledge tosolve them. 3Highly flexible, many relevant problems require flexibletechniques and are full of changes, and they cannot be solved by routine andstandard process. 4Diverse, the after-school problems often test all theimportant points in one chapter, not merely aimed at some particularinformation, thus, they often have a rich, diverse character, and the commonsituation is that nearly every after-school problem has its own feature, and itrequires particular concepts or techniques, namely, every two differentproblems will use different solution methods. 5Delicate, many problems arerelated with delicate information and subtle analytical skill, and in fact, allexperienced science and engineering workers know that nearly every problem inhigher education is very delicate. It is understandable that these 5 basicfeatures of science and engineering problems are often intertwined. (Theinternal connection of these basic characteristics of scientific problems andbasic features of scientific knowledge discussed above is pretty interesting.)

   About this crucial point, we can list someconcrete, vivid illustrations. For example, for Taylor expansion, it is wellknown that it includes many proof problems, and when I repeated the 35thtime (January, 2015), I realized that this part has many inequalities usingsymmetrical ideas, but I felt these problems were disordered, rambling, andlacked inherent law, thus, I could not prove similar new questions; when Irepeated the 55thtime, due to proficiency and independent thinking,I had combed the inner context of this part and my understanding was morecomprehensive and mature, and I realized that it was not made up of one singleproof skill, but a combination of many proof ideas, and I finally grasped allthe proof ideas then. A similar example also happens in the separation variablemethod of partial differential equation, in the past, I was not confident aboutits after-school problems and my computation was very messy; and when I reallyunderstand all the details later, I finally understand the solution method ofthis type of problem, then when facing this kind of problem I already havemature confidence, since I have made sense of all the important points. (Myspecific computation may have some errors, but I have indeed grasped the wholeline of thinking and key sectors about it)

(V)As most of us can feel, inoccasions of thought exchange like seniors’ experience sharing meetings, twocentral problems will be repeatedly raised: 1How to learn professional courseswell? 2 How to plan for the future? What social practices should we take partin? How to balance the relationship between learning and accumulation of lifeexperience? Due to the limit of this paper’s theme, we will mainly answer thefirst question:

   Broadly speaking, we can say that ourprofessional ability will not essentially improve in undergraduate no matterhow diligent we are, because it took me 9 years to achieve an enhancement fromthe ‘normal’ level, which is the large background in our discussion. I rememberin one seniors’ experience sharing meeting in my sophomore year, one studentasked: “abstract algebra is very hard, I cannot make very much sense of it, isthere a good solution?” Our senior answered: “do problems”, but the issue ismuch more complicated, in fact, no matter how many problems we think hard aboutand how diligent we are, due to our low thinking ability, we are impossible tounderstand the key spirit of the problems, thus even though we do manyproblems, we can’t actively solve most of them, namely, we may pay a lot butget very little, thus, we cannot learn these courses well, which is ratherembarrassing, but hard to evade. We all know that we understand very littleabout professional courses at the beginning of undergraduate, thus we suffer alot, but since our junior year, the situation will be improved and we are notso painful, therefore, learning is a process with less and less pain. For thosestudents without strong motivation and enough willingness, we can say thatlearning is a process of bearing, and the first three years are the mostdifficult, we should devote much energy in learning even though we don’t likeit because our thinking ability will unconsciously enhance even we learnwithout interest, and when it comes to the later part, we will feel especiallyrelaxed after our thinking ability enhances. However, for most students, ourthinking ability does improve in undergraduate, but it cannot be a qualitativechange.

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

  When I repeated mathematical analysis andabstract algebra for the 49thtime, I found that I still could notsolve most problems, and later I realize that it is because I did not reallyunderstand most ideas, concepts, approaches and techniques in these coursesthen.

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

  (VII)Broadly speaking, the process ofrepeating undergraduate courses for 55 times is a process of graduallymastering concrete content and details (including numerous ideas, concepts,approaches and techniques), and also a process of gradually deepening specificunderstanding, and these two processes are completely intertwined; roughlyspeaking, the process of repeating these courses is a complex process ofaccumulating ideas, concepts, approaches and techniques and also a process ofintegrating disordered knowledge into a coherent thought system with depth,breadth,  delicacy and organicconnections.

 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 themwill be a mixed and interweaved complex condition (our actual learning processis certainly not by the order of knowledge points and we learn the theorems,problems, concepts, techniques one by one, but a half-orderly process whichgradually permeates), and with the mastery of concrete information, includingmany theorems, problems and skills, we will deepen the overall understanding ofparticular knowledge points. In the beginning, when we are exposed to someknowledge, since there is much information, and lots of ideas, skills, details,concepts mingle together, we will have a confused and jumbled feeling; withmore and more repetitions, we can gradually sort these information out, and ourunderstanding of their inner context will be clear and these knowledge will beordered and organized.

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

  Since every scientific course has twofeatures: firstly, its content is very broad; secondly, it is also complex anddelicate; the overlapping of these two basic features together determines thatwe need to spend a great deal of time in learning related knowledge, and wecannot learn it well if just spending 2 hours each day to study. We all knowthat, for scientific work, we need to learn it over 8 hours each day for over10 years to learn it well, since it is a slow process and we must devote muchtime in learning to well grasp one specific course. (From 2007 in undergraduateto 2018, I probably spent about 8 hours each day in these 10 years in studyingmathematics.)

(IX)Now I want to moredelicately analyze the complex psychological and intellectual experience inrepeating the undergraduate courses for 55 times. When facing these courses, inJanuary, 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 55thtime) do I realize that this conditionstemmed from two basic reasons: firstly, my depth of understanding was notenough, and my understanding of various knowledge and ideas then was shallow,and my understanding of the whole course was also superficial; secondly, myconcrete accumulation was also not enough, and my mastery of the major ideas,concepts, approaches and skills was specious and I did not really grasp theseconcrete content.[2]When I repeated the 55thtime, my overall foundation was alreadydeep 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, inthe past stage, for instance, when I repeated for the 30thtime,since my knowledge structure was fragmentary and shallow, thus, when I didthese problems, I mainly relied on luck and casual guess, trial; while inAugust, 2016, when I repeated the 55thtime, since I already hadmuch relevant knowledge, thought and experience accumulation, thus, when I didthese problems, I felt that my thoughts, intuitions, concrete details, skillsand other aspects were much more clear. To sum up, in problem solving, thepsychological conditions generated by solid foundation and shallow one are twoquite 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 conditionlasted until I embarked on relearning undergraduate courses in 2013. When inJanuary, 2016, I had repeated undergraduate courses for 50 times, and I hadstudied the theory and problems of multiple integral for many times, but myunderstanding was still not clear enough; when it was August, 2016, based onsolid foundation, I finally could solve most problems in multiple integral, andthen I realized that my previous understanding was not deep enough (thus, mypsychological condition of unconfidence before was right, and it naturallyreflected my immature mastery of concrete knowledge): in fact, the use ofcylindrical coordinates and spherical coordinates was more complex than myprevious understanding (it is not very obvious and direct on how to selectwhich coordinates, instead, it requires precise analysis), the variablesubstitution 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 bestsquare convergence theorem. Since January, 2015, I relearned the relevanttheory or rewatched the relevant video every two month (I had repeated thesecontents for many times before), and for many times I thought that I hadcompletely understood this part of knowledge, but in September, 2016, Irealized that I did not really understand these theorems before, since theirconclusions and proofs have rich properties, and my understanding of thesetheorems then was finally mature and reasonable; the reasons behind includethree aspects: firstly, my understanding of the block of Fourier series (itnaturally includes these theorems) was much deeper; secondly, my holisticunderstanding of mathematical analysis was deeper; thirdly, my overallmathematical quality also greatly improved. In a word, due to the overlappingof three basic aspects, my understanding of these theorems was proficient andsatisfactory.

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

 When I repeated for the 20thtime,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, namelyit was not based on deep understanding with holistic foundation, thus, I justcould fragmentally solve problems, which proves that we can only locally solveproblems by petty trick, to globally solve all the problems we must have asolid, delicate foundation. When I repeated for the 55thtime, I wasable to globally solve most problems and feel the organic connections betweenthese problems and the concepts, ideas, approaches, techniques in thesecourses, namely, the isolated knowledge information began to converge into anoverall knowledge foundation, and moreover, the problems I mastered began to beintegrated into the overall foundation of one chapter. To sum up, only werepeat enough times and are sufficiently proficient can our understanding ofconcrete knowledge and concrete problems promote from isolation to integrity.

Also at this stage, Ieventually 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 doesinclude some hard contents (like the proof of variable substitution of multipleintegral), but only at this highly proficient stage, I am able to tell whatcontents are really hard. At the initial stage of learning one course, I feelthat almost all the points are hard, but after repeating 55 times, I find thatmuch content is actually quite easy (like the normal plane of curve, thetangent plane of surface, the computation of partial derivative, thecomputation of Euler integral in calculus, they just follow standard solutionmethods), but some parts of knowledge are really hard. About this point, we canlist many other examples, for instance, about “Legendre polynomials areorthogonal polynomials” in calculus, before 2015, I felt that it was hard andcomplicated, 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 Icompletely master it, I find that it is really deep. However, even for hardcontent, at this time, with the deepening of overall foundation, my mastery ofthem is more relaxed, for instance, some computations of parametric variableintegral are rather complex, but I have clearly mastered their main ideas,technical details and spiritual essence. In a word, no matter for simplecontents or hard ones, my understanding has both enhanced a lot.

(X)Another important point weneed to add is that in January, 2016, when I repeated the 50thtime,I found that I still could not solve most problems, and then I was prettydepressed; then in May, 2016, when I repeated the 52ndand 53rdtime, I finally could solve some problems, but my understanding was notthorough; in August, 2016, when I repeated the 55thtime, I finallyhad a lucid feeling, until then also, I truly felt the wonder and pleasure ofthese mathematical knowledge, and finally had a mature and confident mentalityof truly mastering some knowledge. Namely, to scientific knowledge, we canunderstand its real intension only when we learn it well, and only then can wefeel its marvel, and it often requires some time of accumulation.

In summary, in most time ofrelearning undergraduate courses, I was rather depressed; until the last 2 or 3months, since I felt that I had proficiently mastered much precise knowledgeand was close to completely master related courses, meanwhile, my comprehensiveability had enhanced a lot, I began to have a delighted state of mind.Moreover, in the last 2 or 3 months, I not only mastered most ideas, approachesand problems of the courses I repeated but also felt that they becameespecially plain and clear, and I felt easier to understand them, and thus Ihad a light-hearted mentality, which is a strong contrast with my depressedcondition in the first 50 times. To sum up, for the whole process of learningthese 3 to 4 courses, in the initial long period, due to the weak foundation, Ifelt rather difficult to think about many knowledge points, techniques andproblems, but in the last 2 or 3 months, due to the deepening and refining ofholistic foundation and improvement of proficiency, I began to feel quiterelaxed to think about most ideas, concepts and problems in them, which is avery real and pleasant feeling, and I believe practitioners who truly mastersome courses will all have this proficient, simple and stable overall feeling.Namely, in this long journey, after the first cloudy long road, I was finallyexposed to beautiful sunshine at the last stage. In brief, for scientificknowledge, we can feel its real pleasure only when we learn it very well, andthen we can flexibly and fully use it, and half-digested knowledge is not verymeaningful.

The deeper reason for theabove phenomenon is that every section of scientific knowledge includes a lotof complex information, such as ideas, techniques and details, and we can accumulatesome concrete information, like ideas and details, for every more repetition, meanwhileour depth of understanding will also deepen; when I repeated the 30thtime, I only mastered 40% of all the related ideas, skills and details, thus, Ihad a specious feeling and could not solve most problems; when I repeated the51st  time, I had mastered 80%of all the ideas, skills and details, thus, I had learned the main essence ofsome parts of knowledge and could solve some of the problems; when I repeated the55thtime, I had mastered over 95% of relevant ideas, concepts andskills, and my overall understanding was deep and clear enough, thus Inaturally had a lucid holistic feeling. This fully demonstrates that the depthof understanding is based on concrete information, including many ideas,concepts and details, thus, one people is not eligible to say deepunderstanding if he does not truly grasp enough concrete information.

Correspondingly, thisnaturally explains the universal condition when we solve problems, and it iswell known that we can only solve the easiest questions at the beginning, whichis because we haven’t truly mastered most of the approaches, skills and detailsof one part of knowledge then, and with the further accumulation of ideas, concepts,approaches, details and deepening of understanding, we can gradually solve moredifficult and really hard questions.

(XI)If we take the proportionof after-class problems we can actively solve as a clear criterion, inSeptember, 2014, when I repeated the 30thtime, I could justactively figure out 20% of the after-class problems and they are naturally thesimplest 20% problems; in January, 2016, when I repeated the 50thtime, I could actively figure out about 60% of the problems; while inSeptember, 2016, when I repeated the 55thtime, I could alreadyfigure out over 80% of the problems and they included most hard questions. WhenI can actively solve 80% after-class problems, this indicates that myunderstanding of one specific course is stable and mature, while for somestudents, if they can just actively solve 20% of the problems, this naturallyshows that their mastery is shallow, coarse and very unstable. To sum up, theproblems we can actively solve (proportion and level of difficulty, etc) is a goodindicator of our mastery of certain courses.

(XII)Here, it is somewhat meaningfulto articulate the mental and psychological condition when fully mastering somecourses. In September, 2016, when I repeated these courses for the 55thtime, I finally got a sense of mental stability. Before, knowledge of somecourses, like mathematical analysis, abstract algebra and C++ made me veryanxious since I could not address relevant problems and I was not confidentabout them both in concrete knowledge and holistic understanding, and myoverall feeling of these courses was directionless, fragmentary and specious;when in August, 2016, I finally had a relaxed, proficient overall feeling, and finallycould control much information included in them, and then, I had a relaxedfeeling of overlooking from above, since I truly mastered most of the ideas,concepts, theorems, approaches, skills in these courses, and moreover, I didsome hard questions; thus, I know that I had truly understood these courses andno longer had a sense of fog and puzzle. Meanwhile, when we truly master mostcontent in one course, since we have thoroughly mastered such broadinformation, we will have a sense of fullness, delight and achievement. 

At this time, when I readbooks of these courses, for every theorem, every problem and every illustrationentering my horizon, I have a sense of proficiency and am familiar withnumerous details included in them, and I have a mature confidence when facingthese courses, and now I realize that there may be some concrete information Ihave not grasped, but I am familiar with the major part. At this time, I feelproficient in both concrete details and thought essence of every theorem,concept and example, and I profoundly realize that only by being familiar withall details of one knowledge part can we learn it well, and if we are notsufficiently proficient about some problems and theorems, then it indicatesthat our understanding is not mature; in a word, for one particular knowledgepart, we must be proficient with every concept, detail, idea and skill and thenwe can study it well.

Before, when facing manytheorems and hard questions in some courses, I often had a sense of fear andsome sense of mystery, I felt that they are hard, and at the same time, theyare 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 theyare actually easy, and the ideas and skills included are very clear, I thenhave 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. (Thisagain demonstrates that a part of knowledge is often en bloc, and it isunlikely to solve some problems isolatedly, even though we want to solve fewlimited problems, we must be familiar with the relevant whole part) Also atthis time, I realize for the first time what psychological condition is when wewell learn one particular course in higher education, and the overall mysteriousfeeling which covers one part is replaced by a sense of simplicity, proficiencyand sufficient confidence. At this point, when facing the content of thesecourses, I have a similar feeling as facing knowledge in high school, likefunction 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 highlyfamiliar with every concept, skill and detail of one part, it naturally createsa sense of simplicity; meanwhile, only by learning one part of knowledge to asimple degree can it demonstrate that our understanding is mature enough.

(XIII)In undergraduate, thebasic issue we face is that our study time is limited; in higher education, thecontent of knowledge is huge, and only one course, like calculus, has broaderinformation than the sum of high school mathematics, and moreover, it is muchharder, let alone we have dozen of courses to learn in college; thus, thecommon condition of most students is to passively follow the curriculum, tiredof learning newly instructed classes, and we do not have time to go over thecourses; 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 abstractthinking ability then is not very strong, and we also do not have many independentviews then, thus we do not actively solve problems then, while in college, ourproblem solving is based on intuition and deep understanding of relevantcontent, and the thoughtfulness also enhances a lot. Accordingly, in terms ofproblems need to be addressed, since the knowledge points in elementaryeducation are somewhat few, thus, for every concrete point, there are oftenmany questions to repeatedly test it, while in higher education, eachafter-class problems often has its own unique feature. Namely, for all majorsof science and engineering, from high school to higher education, both thecontent we need to learn and the problems we need to creatively address unconsciouslygo through an enormous change.

An obvious basic fact is thatcompared with high school, some major basic features of college knowledge, likethe complexity, degree of abstract and difficulty, also have greatly risen.(For example, the quadratic form in higher algebra is more complex, abstractand delicate than high school mathematics like sequence) When facing complexknowledge in undergraduate, we often do not well prepare for it mentally andpsychologically, and meanwhile, in undergraduate, we begin to seriously thinkabout 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 aqualitative change, which is a basic fact we need to recognize.

In a word, due to the timepressure of study and increase of difficulty in knowledge, we often passivelyfollow the course progress and do not have sufficient time to digest and absorbinformation 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 tolearn, we still cannot master these concrete knowledge, and I finally masteredsome courses in freshman year, like mathematical analysis, at the 5thyear in graduate.)

(XIV)For those who havegraduated from college or graduate school and entered the society, likegraduates of electronic engineering and mechanical engineering, we can surelysay that the foundation of the majority of them are rather bad; thus, no matterhow disordered and turbulent life is in actual work, they all probably need torepeat undergraduate courses for over 55 times to improve their thinkingability, professional quality and the ability to solve actual problems, andonly by doing this can they better qualify for their own work. Since thecurriculum in undergraduate is very crowded, and we are constantly preparingfor the new exams, thus, we do not have enough time to repeat basic courses for55 times; therefore, this crucial learning process needs to be taken up aftergraduation. In the meantime, though our thinking ability is low inundergraduate, and we will feel hard and painful when facing some concreteknowledge, the great deal of time devoted in this stage is also indispensible,since the accumulation in this low thinking ability stage is the necessarypremise for the latter part. Therefore, in undergraduate, we must devote muchenergy in learning, and without this stage’s sustained accumulation, thesubsequent learning cannot achieve the improvement of thinking ability.

(XV)In the complex process ofscientific learning, we should pay attention to look at problems from the independentthinking perspective, which is all important for science and engineeringworkers, and if we master these concrete information thoughtfully andartistically in our own way, we can doubly enrich our understanding; in thisprocess, our independent thinking is tremendously vital, and we need tounderstand particular knowledge in our own mindset, since everyone has differentmode of thinking, thus, we need to brand the knowledge with independentnotions, 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 ofparticular knowledge and can we innovate in the future, namely, one course hastwo different levels: knowledge and thought, and the latter basic aspect isabsolutely indispensable.

In the process of relearningundergraduate courses, only by forming many independent views can we preparefor our independent research in the future, and independent views are the mainbedrock for the future innovation; if we just master concrete knowledge withoutindependent ideas, then our independent research’s quality has no guarantee.Only by mastering knowledge in our own way, these thoughts and information canbe full of life in our mind, and it can lay a solid foundation for our futurecreative 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 makesignificant contributions in the future, since the amount of information is notrich and deep enough.

In particular, independentthinking mainly includes two aspects: speculative level and artistic level. Thespeculative thinking can enable us to understand particular knowledge fromspeculative perspective, and artistic thinking can enable us to master concreteknowledge creatively and full of novel vitality, since art requires creativityand creativity also requires art. For instance,Men of Mathematicsby ET Bell is well known to us, Bell’s writingis definitely elegant, but his writing lacks depth of speculation, thus, he didnot make the most outstanding contributions (As the central fact we have pointedout elsewhere[3]:in the mathematical and physical world, only the masters’ expressions have bothtrue deep thought and high artistry, like Laplace, Dirac and Heisenberg, andmost of the Nobel Prize winners’ writings also do not achieve this level, afterall, 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 are other intellectual capabilities.

(XVI)As we have repeatedlyemphasized, in the process of repeating undergraduate courses, the mastery of knowledgeis the first aspect, and in this process, our accumulated independent ideas arethe second aspect, meanwhile, in this process, the improvement of our thinkingability is the very important third aspect. The huge improvement of thinkingability is also a beautiful gift which repeating undergraduate courses bringsto us, for instance, if we repeat basic courses, like mathematical analysis,after we repeat dozens of times, our overall thinking ability will greatlyimprove, thus, when facing knowledge in computer science (like algorithm anddata structure), our absorbing efficiency and depth of understanding willgreatly enhance, and then we will feel these particular knowledge is especiallyeasy. Namely, repeating courses like mathematical analysis will lead to anessential improvement of our thinking ability, and will also lead to a greatimprovement in efficiency when we learn other fields. While if we justsuperficially read recent papers, our scientific thinking ability will alsoimprove, but not too much, moreover, this kind of superficially learning willnot 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 deepenough.

Here, we can give one concreteexample. Before the summer of 2014, though I had repeated bubble sorting in Clanguage for about 10 times, I felt that it was hard and abstract, while, inthe spring of 2016, with the constant improvement of thinking ability, I then feltthat bubble sorting was simple, and I easily mastered it with just onerepetition. This example actually belongs to a broader illustration, namely,for the entire C language course, in my sophomore year (the autumn of 2008), Ionce 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 Ifailed to grasp with hundreds of hours’ study in undergraduate, includingcharacter array, two-dimensional array, pointer, structure, linked list, macro(parametric macro definition, nonparametric macro definition), stringmanipulation function, array initialization, etc, and then, my understanding ofthese much concrete knowledge was more clear. In summary, the improvement ofthinking ability will improve our absorbing and understanding speed of localinformation, and it will also enhance our mastering efficiency of thelarge-scale knowledge, and these two aspects are both very valuable for ourwork and life. Here we also explain the basic phenomenon why talented studentsaround us can learn all sorts of knowledge well by only working 3 to 4 hourseach day, and the reason behind is that their thinking ability is strong.

Similarly, as to thestatements of “the function series of convergence in measure has almosteverywhere convergent subsequence” in real analysis and “n order symmetricmatrix 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 easilygrasped them by only one repetition in a short time.

In summary, the enhancement ofthinking ability, the improvement of knowledge basis and the accumulation of independentideas are the three major thought treasures brought by systematically repeatingundergraduate courses. As most scientific workers can feel, differentscientific workers have huge differences in talent, and one of the basicconclusions of this paper is: this gap in thinking ability can be overcome (theway is to repeat undergraduate courses in suitable situations), but it requiresgreat effort.

As mentioned above, the basicstarting point of this paper is to well learn undergraduate courses, and somepeople may argue that some prominent scientific workers’ foundation is somewhatweak, like Grothendieck and Smale, but they also make first-classcontributions; here, we should not overlook one basic fact: indeed, thesebrilliant mathematicians may be bad at fundamentals, but their thinking abilityis very strong and in the ‘good’ level, thus they can quickly learn much newknowledge and integrate them, while the majority of people whose fundamentalsare weak also do not have strong thinking ability, thus, they need to repeatundergraduate courses to enhance their ability, otherwise if they justsuperficially read recent papers, their thinking ability will not have a basicchange even in their 40s, moreover, even in 40s, they do not well learn anyundergraduate course.

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

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

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

In the meanwhile, our abstractgeneralization ability will also gradually improve. For many similar questions,similar approaches and similar concepts, we will gradually discover generalrules hidden behind, and our capability of using various ideas, knowledge andtechniques together will clearly improve, and we will have more and moreholistic generalization to the basic features of some courses. At this time, wewill gradually digest a part of knowledge and will naturally find more and moreknowledge linkages, and moreover, we will gradually form an organicunderstanding of one whole course. In a word, we will get enormous pleasure inmany kinds of theoretical generalization.

With this complex process, ourcapability of capturing key information and concretely thinking aboutparticular contents will also enhance. When I spent 3 years and 7 months inrepeating and thoroughly mastered 4 to 5 undergraduate courses, I realized: inundergraduate, ‘good’ students absorb much concrete information, while ‘normal’students only absorb some superficial, simple knowledge; here, by relearningundergraduate courses we can solve this basic question. To begin with, throughthe immense accumulation of concrete knowledge, we will gradually realize whichtechniques, ideas and concepts play a central role in one proof or question.Meanwhile, our particular understanding to one course will become moreconcrete, and every course is actually made up of many concrete details, butwhen our thinking ability is low and our understanding of one course isshallow, we cannot notice much important concrete information at all, only afterour thinking ability improves and knowledge foundation deepens can theynaturally come into our horizon. To sum up, with repeating undergraduatecourses, our mathematical thinking ability will become both more abstract andmore concrete, and they will mutually reinforce and promote-the improvement ofconcrete thinking ability will give more further detailed and further elaboratematerials to our abstract generalization, while the improvement of abstractthinking ability will enable us to more easily find huge, diverse concreteinformation.

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

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

With the increase ofrepetitions, our depth of thought will hugely promote. It is easy to understandthat in undergraduate, ‘good’ students’ understanding of one specific course ismuch deeper than ‘normal’ ones, and when our knowledge foundation is weak andhollow and our ability is bad, our understanding of one particular course isdoomed to be superficial, while after our knowledge foundation broadens anddeepens, we will begin to be able to understand some profound ideas in thebooks, meanwhile, we can gradually understand some deep experience of previouspeople and good students. At this time, if we read relevant books, we will havea deep understanding, not a superficial impression like the past; since we cantell the difference between superficial and deep ideas in one course, we willform a deep mindset and can independently solve many hard problems, and in themeantime, we can lay a good thought and knowledge foundation for the futuremeaningful innovations.

In this process, ourcapability of analyzing and comprehensively solving problems will also greatlypromote. Due to the enhancement of foundation and improvement of capability, wecan more deeply analyze lots of theorems and knowledge, and when facing newproblems, we can creatively find some approaches and paths to solve them by anumber of methods, like analysis and synthesization. At this time, the depthand delicacy of our analysis will obviously enhance, and we can more flexiblyuse important ideas in other distant parts. In summary, our abilities ofconcretely analyzing particular knowledge and problems and comprehensiveutilization of many methods and skills will both apparently promote.

To sum up, the concept ‘thinkingability’ has two levels of meaning: firstly, as an overall concept, it ismeaningful, and an appropriate example is, in undergraduate, hundreds of myclassmates sat in the same classroom and listened to the same courses likecomplex analysis, real analysis, functional analysis, algebraic topology at thesame time, but the difference of understanding between ‘normal’ students and ‘good’ones was tremendous (I realize this point at 5 years after graduation, afterproficiently 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 richconnections of these knowledge, and moreover, their mastery was much moreproficient, thus, their creativity of solving problems was much stronger (forthose knowledge the ‘good’ students mastered in one hour, we would perhapsspend 50 hours to get the same level of understanding, however, this comparisonis not appropriate, because even we spent that time, we can truly understandthem only after 8 or 9 years since we enter the college). In summary, thisconcept can be used to well describe our velocity and efficiency in absorbingknowledge. Secondly, ‘thinking ability’ is a general term for the 8 to 9capabilities discussed above, and these capabilities represent most basicabilities 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, theygradually have these good capabilities since high school), thus, their learningefficiency is dozens of times to ‘normal’ ones. Only by combining these twolevels together can we get a reasonable understanding of ‘thinking ability’.

Finally, we must say that theaims of relearning undergraduate courses are not merely for the promotion ofour thinking ability, in this process, the concrete knowledge we master is alsohugely important, and the popular notion that education is mainly for thecultivation of ability is very wrong, because: firstly, knowledge and abilityis unified, and we can hardly achieve the overall improvement of thinkingability without much concrete knowledge accumulation; secondly, concreteknowledge is very important for students of every major, for instance, calculusis very important and many fields of business need it, and finite element andfinite difference methods are also important for many engineering majors; if wejust have the empty so-called ability without these concrete knowledgeaccumulation, how can we creatively treat the concrete problems in our actualwork? In a word, accumulation of knowledge is a basic part of education, andits importance is not under the promotion of students’ thinking ability andcomprehensive ability.[4]

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

For example, in March, 2014, Ithought I had completely grasped the analytical properties of function series,while in July, 2016, with two more years’ accumulation, I realized that I didnot really master these content in 2014, and my previous understanding had twobasic defects: firstly, it was one-sided, namely, I didn’t organicallyintegrate 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 aboutmaximal ideal and ring in abstract algebra, I could not solve it then, and Iwas distressed for a period of time, but four years later, I realized that evenI could solve that problem in 2012, my understanding of ring and the wholeabstract algebra still had big problems, and therefore, my fundamental problemwas that there were serious problems about my overall understanding of thispart of knowledge, not merely an isolated particular problem. The second thinghappened in the autumn of 2015, I felt reluctant in solving one problem whichinvolves energy, kinetic energy, velocity and momentum in special relativity,and that problem was a little complicated for me; later, after repeatingspecial relativity for several more times (August, 2016), I realized that myoverall understanding of special relativity before was not proficient enough,and my understanding of much concrete knowledge had a big problem, and when myoverall mastery of special relativity further enhanced, I felt especially easyin solving that problem I was unconfident about in 2015. (This experiencesufficiently proves that if we want to master one part of knowledge, we must beable to solve almost all the problems in it) In summary, these cases can helpus to clarify one misunderstanding: it is just accidental if we cannot solvesome 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 withdozens of repetitions between a specious understanding and a really thoroughone.

Before I relearnedundergraduate courses (in 2012), sometimes I felt that my mathematical analysiswas not very bad, and it seems that I had mastered its main parts; later, afterI spent 3 years and 7 months in repeating it, I began to realize that mymastery then was very bad, and I nearly missed all its essence. This case fullyproves that it is indeed very easy to have specious understandings inscientific learning.

(IXX)As described above, weemphasize the enormous importance of creatively independent thinking in theprocess of laying a solid foundation in undergraduate courses all the time,namely, in the process of laying a good foundation, we need to develop ourinitiative spirit and absorb broad undergraduate knowledge through our ownimagination, and we should brand these knowledge with our own mindset.Meanwhile, we should also view the rich values of laying a good undergraduatefoundation imaginatively: firstly, after laying a solid foundation, we can moreefficiently control our work and life and can better develop our creativespirit in technology and make real contributions to the development of society,however, in a broader intellectual horizon, we should not be too utilitarianabout laying a good knowledge foundation, and its ultimate goal is not forbetter wages or higher social status, but with broader value of life, namely,laying a good foundation can enhance our intense interest to a particular fieldand can enable us to enjoy the rich tastes of intellectual food in professionalknowledge; meanwhile, solid foundation can broaden our horizon of life, thus wecan better develop our imagination and originality, which can make our lifemore full, rich and profound, to sum up, promoting our interest, imagination andcreativity in our own major and moreover, promoting the imagination,exploration enthusiasm and creative attitude towards our whole life is theessential goal of laying a solid undergraduate foundation. To sum up, no matterin the process of laying a solid foundation or after that, we need to seebeyond the narrow goals of realistic level, and we should move into broaderknowledge and intellectual fields, and should always view our professions andlives with deep curiosity, while laying a solid knowledge foundation can helpus better follow our own enthusiasm and interest and can enable us to viewlife, society and universe with broader and better imagination.[5]In a word, realistic andutilitarian goals will greatly narrow the rich values of laying a goodundergraduate foundation. A famous western proverb “All work and no play madeJack a dull boy” is partially consistent with the basic spirit here.

(XX)In the meanwhile, weshould be clear about the broad goal of well learning undergraduate courses,namely, relearning undergraduate courses should aim at relearning all the basiccourses relevant to our own research; since all the basic courses of everyscientific major form an organic whole, and these courses have strong internalconnections; therefore, we need to lay a solid foundation in all the basiccourses related to our research, and only by doing this can we have a broadprofessional foundation for our future independent work. The interconnection ofvarious mathematical courses is a well known basic fact, for instance: infunctional analysis, we can use Banach contractive mapping theorem to solve theexistence and uniqueness of the solution of ordinary differential equations,which is a central problem in ODE, and if we are not familiar with ODE, ourunderstanding of the internal value of Banach contractive mapping theorem willrelatively narrow, namely, functional analysis and differential equations areclosely related; similarly, in abstract algebra, when we solve theimpossibility of the trisection of an angle by using field extension, we alsouse knowledge in complex analysis; while in higher algebra, the proof of thefundamental theorem of algebra can be based on ideas from multivariablecalculus and can also be solved by using ideas of algebraic topology andcomplex function, while the fundamental theorem of algebra is naturallyfundamentally important in algebra, namely, algebra have deep connections withmany courses, like mathematical analysis and topology. To sum up, the integrityof mathematics is a basic characteristic of modern mathematics, and theinterplay of its various courses is deep and extensive and it also touches thecentral part of these courses, and I think many other scientific majors alsohave such a characteristic.[6]Thus, if we want to standout in any scientific major, we have to lay a solid foundation in all the basiccourses related to our own research in our major; here, another major aim inrelearning undergraduate courses is already clear: due to the central featureof integrity of every scientific major, thus, we need to learn all theundergraduate basic courses related to research well, and the overall processof relearning undergraduate courses mainly is: when we repeat courses infreshman and sophomore years (study should follow a step-by-step approach,thus, we should first relearn courses in freshman and sophomore years), due toour low thinking ability then, our learning rate will be slow and it will costa long time, but after we finish repeating basic courses in freshman year, dueto the great improvement of our thinking ability, the time we spend in relearningcourses of junior and senior years will be relatively less.

In the meantime, though thispaper insists on the basic view that we should learn most undergraduate courseswell, considering the complexity of reality, some people may think that we donot need to pay special attention to those courses irrelevant to our ownresearch, which is naturally understandable.[7]However, we should befamiliar with basic courses closely related to our research; for instance, forthe students studying PDE, they must be highly proficient with mathematicalanalysis, functional analysis, differential geometry and ODE, and they need tosolve almost all the after-class problems.

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

In the autumn of 2016, when Ithoroughly mastered some courses, like mathematical analysis, I found that theywere not so hard, but I had gone through such a long and complicatedintellectual journey, this gave me a complex feeling, and I was doubting : isit necessary for me to spend so much energy in them? After all, these knowledgeis very natural, but I clearly know that until 2015, these knowledge was stillobscure and deep for me, thus, this intellectual journey is definitely a realexperience, and moreover, I think this long learning experience is quiteuniversal in all science and engineering fields. In the autumn of 2016, whenthe internal context of these courses naturally emerged, and when many partswhich I thought were highly tricky before showed an unadorned appearance, I hada plain and natural feeling about this tortuous, interesting and continuouslydeepening learning experience.

Later, I realize that thereason for which I spent 3 years and 7 months in learning 4 to 5 undergraduatecourses well is simple: firstly ,the range of these courses is very broad, andthey all include tens of thousands of ideas, thousands of techniques and lotsof concepts and approaches, moreover, these concepts and techniques are ofteninterrelated, thus, mastering such a huge amount of information naturallyrequires a long time; secondly, these courses all have certain depth, and lotsof problems and content have considerable difficulty, therefore, our understandingof 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, Ispent about 3 years and 7 months in total to learn these courses well.

(XXII) Here, we need todetailedly analyze the basic reasons which lead to superficially read recentpapers (this shallow research approach happens among some professional workersat 37 or 38 years old, and also among lots of PhDs in many majors). Manyworkers who study PDE and numerical PDE master very superficially in basiccontent of related courses, like PDE and finite element; about PDE, they don’ttruly master much content, including the deduction process of D’Alembertformula, separation of variable method, Duhamel’s principle (Duhamel’sprinciple emerges in many different occasions, like one dimensionalnonhomogeneous equation, n dimensional nonhomogeneous equation, its combinationwith separation of variable method, etc) and energy method (like the deductionof energy inequality) in wave equation, the Dirichlet’s principle in harmonicequation, and they cannot solve relevant questions and also do not reallymaster the complex details of these contents. (Indeed, after we truly masterthese contents, we will find that they are not really hard, but the superficialresearch way of these workers leads to their bad learning effect which is farfrom true understanding). About numerical PDE, they also have a shallow anddisordered understanding of much basic content, like the discretization ofequation, the deduction process of stiffness matrix and mass matrix ofmultidimensional equation, the deduction of error analysis of multidimensionalfinite element, the essence of stability of finite difference equation, theerror analysis of finite difference method, etc.

The reason for the basicphenomenon that scientific courses are easy to be superficially learned is thatknowledge points of scientific courses have some deeper basic features; takePDE as an example, many knowledge points, like the separation of variablemethod, the deduction of membrane vibration equation, spherical mean of waveequation and the Green function of harmonic function all have three important basicfeatures: 1complex, 2delicate, 3deep. If we roughly read these knowledge pointswe often think that we have completely mastered them, but, if we deeply studythem, we will find: firstly, they actually include numerous details, and thesedetails all require careful deductions; secondly, they use many complex ideasand have intricate connections with other points in basic courses, this course andfollowing courses; thirdly, they all have certain depth; obviously, nearly allthe knowledge points in scientific courses have the above three importantcharacteristics. To sum up, these three basic features of knowledge points inscientific courses lead to a huge difference between a real understanding and aplausible one, which also makes it easy for many workers to learn roughly andsuperficially.

From the above analysis, wecan see that, in science and engineering fields, if we just superficially learnall the courses, we can just make some 2ndor 3rdclasscontributions and can never make really significant contributions, which isundoubted. (Because it is hard to really master one course, and only by reallymastering some courses can we make a good contribution)

   You may ask one question: why am I sofamiliar with the intellectual condition of these students who superficiallyread recent papers? The answer is simple, because before I relearnundergraduate courses, my understanding of PDE and numerical PDE is similar astheirs; about these courses, my learning then also lacked enough depth andconcrete, rich understanding. In a word, even for individuals who specialize inone particular direction (such as the researchers who specialize indifferential geometry), their learning about basic courses in their own field isnot deep and delicate enough, then it will be very hard for them to get somevaluable results in their research with such a weak foundation; thus, our viewthat we should reinforce our own foundations is somewhat meaningful.

   (XXIII) Below, we want to analyze somenegative impacts brought by the research method of superficially reading recentpapers. In today’s graduate school, over the five years of PhD study, most PhDstudents often embark on independent research immediately after passing thequalifying exams, namely, they begin the process of reading recent papers andbooks and doing independent research at this moment. This research method is souniversal and popular that almost everyone is used to it, but the negativeeffects of such a research method also requires our deep thinking, at thistime, many students whose thinking ability is ‘normal’ still have seriousproblems about their elementary courses, which usually leads to a somewhatserious internal defect of their understanding with recent papers, andtherefore, the research quality of them almost inevitably has serious problems.Broadly speaking, this research method of superficially reading recent papersstems from a shallow understanding of the basic features of scientificknowledge, learning and innovation.

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

   Broadly speaking, superficially learningelementary courses will create two basic outcomes: 1 An insufficient mastery ofinformation breadth of all courses, take real analysis as an example, itincludes lots of information and details, while a mathematical worker whosuperficially learns probably just master 20% of the ideas, techniques,information and details included in real analysis, and moreover, they are alsothe easiest 20%, and it is easy to understand that a worker who superficiallylearns elementary courses has just mastered 20% easiest content of all courses.2 Due to the mastered information is not rich and delicate, these workers justhave a very shallow understanding of relevant courses, and they can onlyunderstand some shallow ideas and can just solve some easy problems (forexample, 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 intellectualessence in these courses, and a natural phenomenon is that these workers’understanding of all courses is very shallow. The deficiency of informationbreadth and shallowness of understanding will lead to two basic phenomena inresearch: 1 as to theory and problems, due to a weak foundation, a worker whosuperficially reads recent books often cannot realize which theory and problemsare important, central and meaningful in their own research; 2 as to tools likeideas, 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 newdirections, but they are not able to solve these problems by using existingtools or developing new tools like new ideas, concepts and approaches.

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

[1]InChinese universities, the course “mathematical analysis” more or less equalsthe two courses ‘calculus’ and ‘principles of mathematical analysis’ in foreignuniversities, namely, it includes two parts-calculus and its deeper principles.The content of calculus is very broad, and meanwhile, its theoretical foundationis also somewhat complex.

[2]Theexperience described in this paper is very likely a universal phenomenon, likeCarleson, a master of harmonic analysis, once described his personal experienceof learning and research in mathematics, he wrote: “At 19 I got my BS. It allseemed very easy and I still had no idea what mathematics was all about.” “Igot my degree in 1950 and a permanent position as a professor in 1954. Lookingback, I can now say that I still did not know what serious mathematics orproblem solving really meant. It would take me another four years, till 1958,at the age of 30, when I for the first time wrote a paper that I still considerof some interest.” See the essay “It would be Wonderful to Prove Something” inOne Hundred Reasons to be a Scientist,p. 61, ICTP, 2004. The phenomenon described by Carleson was very similar to thebasic problem analyzed in this paper, and it is naturally an enlighteningliving example.

[3]Seepart (IV) of my paper “On the Thought Foundation of Science and EngineeringPractitioners”.

[4]Aboutthis important point, well-known thinker Whitehead once wrote: “But what is thepoint of teaching a child to solve a quadratic equation? There is a traditionalanswer to this question. It runs thus: The mind is an instrument, you firstsharpen it, and then use it; the acquisition of the power of solving aquadratic equation is part of the process of sharpening the mind.” “(thisnotion is) one of the most fatal, erroneous, and dangerous conceptions everintroduced into the theory of education.” “You cannot postpone its (mind) lifeuntil you have sharpened it. Whatever interest attached to your subject-mattermust be evoked here and now; whatever powers you are strengthening in thepupil, must be exercised here and now; whatever possibilities of mental lifeyour teaching should impart, must be exhibited here and now. That is the goldenrule of education, and a very difficult rule to follow.” See the famous paper “TheAims of Education”, Presidential address to the Mathematical Association ofEngland, 1916.

[5]Wecan refer to John Dewey’s argument: “There are two forms of habits, one isroutine form, namely, the activities of organism have a comprehensive,sustained balance with the environment; the other form is to actively adjustour activities to handle new situations. The former habit provides thebackground of growing, and the latter one forms continuous growth. Active habitsinclude thinking, innovation and originality of applying our ability to newgoals. This active habit is opposed to the routine which inhibits growth. Sincegrowth is the feature of life and education is growth, and it has no other goalexcept itself.” The objective of this part is similar to Dewey’s expositionhere. SeeDemocratism and education,the third section “education is growth”, included inCollection of Dewey’s Education Works, p. 158, East China NormalUniversity Press, 1981.

[6]Asonce pointed by Hormander, a leading mathematician of PDE: the field should notbe divided too small and too early, and young people learning PDE should alsohave a solid foundation in other aspects like algebra and topology, or elsethey won’t develop too well in the future. About this point, we can refer tothe introduction of Hormander inContemporary Mathematical Masters, Beijing University of Aeronautics and AstronauticsPress, 2005.

[7]Aboutthis point, the outstanding mathematician, Serre, once said in an interview of1986: “If you are interested in one special problem, you will find that onlyvery little known work is relevant to you. If something is indeed relevant, youwill learn it very quickly, since you have an application aim in your mind. “”Forone given problem, normally you don’t need to know very much.” See “An Interviewwith Jean-Pierre Serre”,Mathematical Intelligencer, 8(4), 1986, 8-13. However, we should note that thissuggestion of Serre is perhaps only applicable to ‘good’ students whosethinking ability is strong and who learn undergraduate basic courses well,since their knowledge foundation is solid and they also have enough depth ofthought, while for the ‘normal’ students, they should take a different work andresearch approach (for this type of students which are the majority of allstudents, about their research method, this paper may provide a partial answer).