左右互搏――老顽童

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左右互搏――老顽童 关于人生、社会、世界的思考。尤其是组合数学、数学教育、思考问题的方法的讨论。 首页 Why The Professor Can (16) 方法 (28) 妙文拾趣 (33) 网站日志 (5) 影视音乐 (8) 数学 (3) 经典欣赏 (11) 小说连载 (34) 评论 (8) 2004 年 5 月 Sun Mon Tue Wen Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 最后更新 沙僧日记――秀逗前年9月30日 此间的少年――乔峰（Ｉ）：一次篮球赛 沙僧日记――秀逗前年9月28日 此间的少年――乔峰（Ｉ）：关系铁 沙僧日记――秀逗前年9月27日 此间的少年――郭靖（Ｉ）：图书馆 沙僧日记――秀逗前年9月14日 此间的少年――郭靖（Ｉ）：朋友 窃喜即偷欢 风雨再访金文明 最新评论 冷峻散势 : 左右互搏就是自己. 金庸 : 多少风情， . 涓涓流水 : 这篇文章不长，但. darkevan : 搞个链接不就行了. ivy : that is a good o. mojaves : 用了我最喜欢的海. isgaryzhu : 你是纪晓岚剧组的. 存档 内人 E-Mail to 老顽童 我的链接 分页: 第一页 [7] [8] [9] [10] [11] [12] [13] [14] [15] Why The Professor Can't Teach――第三章　当前数学研究的特性 - 2003-05-23 10:40 CHAPTER 3: The Nature of Current Mathematical Research. "Even victors are by victory undone." John Dryden We have traced the widespread rise of vigorous mathematical research in this country. We have also observed that the flourishing of research promises not only direct benefits but also indirect beneficial influence on all levels of education. However, the values that might accrue from research depend on the quality of research being done. Let us therefore look into the nature of current mathematical research.* Though we shall discuss mathematical research, many of its features, as earlier noted, apply to other academic disciplines as well. In mathematics, research has a very special meaning. Specifically, it calls for the creation of new results, that is, either new theorems or radically different and improved proofs of older results. Expository articles, critiques of trends in research, historical articles or books, good texts at any level, and pedagogical studies do not count. Thus, the criterion of research in mathematics differs considerably from what is accepted in, for example, a subject such as English. In this area, in addition to the creation of fiction, essays, poetry, or other literature, criticism, biographies that shed fresh light on important or even unimportant literary figures, histories of literature, and texts that may be primarily anthologies are considered original work. Perhaps this distinction between what should be accepted in the respective fields is wise, but let us see what it has led to in mathematics. Because the United States entered the world of mathematical research several hundred years after the leading Western European countries had been devoting themselves to it, our mathematicians, in an endeavor to compete, undertook special directions and types of investigations. One move was to enter the newer fields, such as the branch of geometry now called topology. The advantage of a new field for tyros in research is that very little background is needed and the best concepts and methodologies are only dimly perceived. Hence, because criteria for value are lacking, almost any contribution has potential significance. Publication is almost assured. Of course, the ease with which one can proceed in a new field is somewhat deceptive. New fields generally arise out of deep and serious problems in older fields, and anyone who wants to do useful work must know much about these problems and grapple with them at length in order to secure significant leads. On the other hand, if all one is trying to do is prove theorems, then it is sufficient to start with almost any potentially relevant concept and see what can be proved about it. And if one gets a result that the other fellow didn't get, one may proceed to publish it. The United States was not the only country that took such a course. After World War I, Poland was reconstituted as a nation and the Polish mathematicians undertook a concerted effort to build up mathematics in their country. They decided to concentrate on a narrow field, the branch of topology called point set theory. Why point set theory? Because at that time the subject was still new. One could therefore start from scratch, introduce some concepts, lay down some axioms, and then proceed to prove theorems. This example is offered not to malign Polish mathematicians. There were and are some very good men among them, and good men, even starting from very shallow beginnings, will make progress and produce fine work. What is significant is the deliberate and openly stated decision to start with point set theory because one did not have to know much mathematics to work in it. Generalization is another direction of research that promises easy victories. Whereas the earlier Greek and European mathematicians were inclined to pursue specific problems in depth, in recent years many researchers have turned to generalizing previous results. Thus, while the earlier mathematicians studied individual curves and surfaces, many twentieth-century mathematicians prefer to study classes of curves - and the more general the class, the more prized any theorem about it. Beyond generalizing the study of curves, mathematicians have also carried most geometric studies to n-dimensions in place of two or three. Some generalizations are useful. To learn how to solve the general second degree equation ax2 + bx + c =0, where a, b, and c can be any real numbers, immediately disposes of the problem of solving the millions of cases wherein a, b, and c are specific numbers. But generalization for the sake of generalization can be a waste of time. A lover of generalization will too often lose sight of desirable goals and indulge in endless churning out of more and more useless theorems. However, those for whom publication is the chief concern are wise to generalize. Hermann Weyl, one of the foremost mathematicians of this century, expressed in 1951 his contempt for pointless generalizations, asserting: "Our mathematics of the last few decades has wallowed in generalities and formalizations." Another authority, George Polya, in his Mathematics and Plausible Reasoning, supported this condemnation with the remark that shallow, cheap generalizations are more fashionable nowadays. Mathematicians of recent years have also favored abstraction, which, though related to generalization, is a somewhat different tack. In the latter part of the nineteenth century mathematicians observed that many classes of objects - the positive and negative integers and zero; transformations, such as rotations of axes; hypernumbers, such as quaternions (which are extensions of complex numbers); and matrices - possess the same basic properties. Let us use the integers to understand what these properties are. There is an operation, which in the case of the integers is ordinary addition. Under this operation the sum of two integers is an integer. For any three integers, a + (b+c) = (a+b) + c. There is an integer, 0, such thata+ 0=0+ aa. Finally, for each integer, a, for example, there is another integer, -a, such that a + (-a) = -a + a = 0. These properties are more or less obvious in the case of the positive and negative integers. But if in place of the integers we now speak of a set of objects, which might be transformations, quaternions, or matrices, though the particular set is not specified; and of an operation, whose nature depends on the particular set of objects but is also not specified, we can state in abstract language that the elements of the set and operation possess the same four properties as those of the integers. The abstract formulation defines what is called technically a group. A group, then, is a concept that describes or subsumes the basic properties of many concrete mathematical collections and their respective operations under one abstract formulation. If one can prove, on the basis of the four properties of the abstract group, that additional properties necessarily hold, then these additional properties must hold for each of the concrete interpretations or representations of the group. The concept of a group, very important for both mathematics and physics, is only one of dozens of abstract systems or structures - the latter is the fashionable word - and many mathematicians devote themselves to studying the properties of these structures. In fact, the study of structures is flourishing; the work done on groups alone fills many volumes. Abstraction does have its values. One virtue, as already noted, is precisely that one can prove theorems about the abstract system and know at once that they apply to many concrete interpretations instead of having to prove them separately for each interpretation. Further, to abstract is to come down to essentials. Abstracting frees the mind from incidental features and forces it to concentrate on crucial ones. The selection of these truly fundamental ones is not a simple matter and calls for insight. Nevertheless, there can be shallow and useless abstractions as well as deep and powerful ones. The former are relatively easy to make, and one must distinguish this type of creation from that involved in solving a new and difficult problem - such as proving, as Newton did, that the path of each planet, moving under the gravitational attraction of the sun, is an ellipse or the far more difficult problem, which has still not been solved, of finding the paths of three bodies when each attracts the others under the force of gravitation. Unfortunately, many recent abstractions have been shallow. Beyond the shallowness of some abstractions, there are other negative features of all abstractions. Although unification through abstraction may be advantageous, mathematics pays in loss of resolution for the broadened abstract viewpoint. An abstraction omits concrete details that may be vital in the solution of specific problems. Thus, the manner of executing the processes of adding whole numbers, fractions, and irrational numbers is not contained in the group concept. The more abstract a concept is, the emptier it is. Put another way, the greater the extension, the less the intension. Abstraction introduces other objectionable features. As a theory grows abstract it usually becomes more difficult to grasp because it uses a more specialized terminology, and it requires more abstruse and recondite concepts. Moreover, unrestrained and unbridled abstraction diverts attention from whole areas of application whose very investigation depends upon features that the abstract point of view rules out. Concentration on proofs about the abstraction becomes a full-time occupation, and contact with one or more of its interpretations can be lost. The abstraction can become an end in itself, with no attempt made to apply it to significant concrete situations. Thus, the abstraction becomes a new fragment of mathematics, and those fields that were to receive the benefits of unification and insight are no longer attended to by the unifiers. Weyl spoke out against unrestrained abstraction, maintaining that "in the meantime our advance in this direction [abstraction] has been so uninhibited with so little concern for the growth of problematics in depth that many of us have begun to fear for the mathematical substance." The inordinate attention given to the study of abstract structures caused another mathematician to warn, "Too many mathematicians are making frames and not enough are making pictures." Another popular direction of research may be described roughly as axiomatics. To secure the foundations of their subject the late-nineteenth-century mathematicians turned to supplying axiomatic bases for various mathematical developments, such as the real number system, and to improving those systems of axioms where deficiencies had been discovered, notably in Euclidean geometry. Since there are dozens of branches of mathematics, there are dozens of systems of axioms. Quite a few of these contain ten, fifteen, or twenty axioms. The existence of such systems suggests many new problems. For instance, if a system contains fifteen axioms, is it possible to reduce the number and still deduce the same body of theorems? Given a system of axioms, what would be the effect of changing one or more of them? The classic and notable instance of this last-mentioned type of investigation is, of course, the change in the Euclidean parallel axiom and the resulting creation of hyperbolic non-Euclidean geometry. Changes in several of the axioms led to elliptic non-Euclidean geometry. Clearly, if a system contains as many as fifteen axioms, the changes that can be considered are numerous. The investigation of the consequences of changing the Euclidean parallel axiom was indeed sagacious. By contrast present-day mathematicians, with little reason to do so, pursue all sorts of axiomatic investigations so that in the eyes of many practitioners, mathematics has become the science of axiomatics. The current activity in this area is enormous and overstressed. When axioms were believed to be selfevident truths about the constitution of the physical world, it was laudable to simplify them as much as possible so that their truth could be more apparent. But now that axioms are known to be rather arbitrary assumptions, the emphasis on deducing as much as possible from, say, a minimum number of axioms, which are often flagrantly artificial and chosen merely to reduce the number, is not warranted. The objective seems to be to produce more theorems per axiom, no matter how distorted and unnatural the axioms may be. Consequently, one finds long papers with tedious, boring, and ingenious but sterile material. Nevertheless, the popularity of axiomatics is readily understood. It does not call for the imaginative creation of new ideas. It is essentially a reordering of known results and offers many minor problems. In his 1951 critique of current features of mathematical research, Weyl included axiomatics: "One very conspicuous aspect of twentieth centur mathematics is the enormously increased role whic the axiomatic approach plays. Whereas the axiomatic method was formerly used merely for the purpose of elucidating the foundations on which we build, it has now become a tool for concrete mathematical research.... [However] without inventing new constructive processes no mathematician will get very far. It is perhaps proper to say that the strength of modern mathematics lies in the interaction between axiomatics and construction." Still another questionable activity in modern axiomatics, derogatively termed "postulate piddling," involves the adoption of axioms merely to see what consequences can be derived. A prominent mathematician of our time, Rolf Nevanlinna, has cautioned: "The setting up of entirely arbitrary axiom systems as a starting point for logical research has never led to significant results.... The awareness of this truth seems to have been dulled in the last few decades, particularly among younger mathematicians." Felix Klein, a leading German mathematician who was active from about 1870 to 1925, remarked that if a mathematician has no more ideas, he then pursues axiomatics. Another distinguished professor once remarked that when a mathematical subject is ready for axiomatization it is ready for burial and the axioms are its obituary. The several directions research has taken point up the fact that there are soft and hard problems - or soft and hard research. In the days when the density of good mathematicians was high, soft problems were not often tackled. Moreover, nineteenth-century mathematicians, who were the first to grasp the advantage of abstract structures, faced a higher order of difficulty than present-day mathematicians face in that type of research. In recent times soft problems have been the ones most often tackled, and even if the proofs are complicated, the results may still be merely difficult trifles. There is still another feature of mathematical research that affects seriously the interaction of research and teaching - the chief concern of this book. Whether it involves generalization, abstraction, or axiomatics or pursues some other direction, modern research is commonly acknowledged to be almost entirely pure - as opposed to applied. Pure research may be characterized as mathematics for mathematics? sake. That is, however the theme or problem is obtained, the reasons for undertaking it may be aesthetic interest, intellectual challenge, or sheer curiosity: "Let's see what we can prove." This is the motivation in axiomatics when a researcher rather arbitrarily decides to change an axiom just to find out what changes this entails in the resulting theorems. Applied mathematics, on the other hand, is concerned with problems raised by scientists, or with a theme that a researcher believes is potentially applicable. There is no doubt that the problems of applied mathematics are more difficult. The branches of mathematics customarily associated with applied mathematics are now several hundred years old, and the giants of mathematics have worked in them. Anyone who wants to do something significant today in partial differential equations, for example, must have quite a background. And for the processes of idealization and model building in applied mathematics one must have intimate knowledge of the relevant physical field in order not to miss the essence of the phenomenon under study. (See also Chapter 7.) Pure mathematics is more accessible for another reason. Whereas in applied work the problem is set by scientific needs and cannot be altered, the pure mathematician tackling problem A may, if unable to solve it, convert it to problem B, which could be A with more hypotheses or a related but actually different problem suggested by the work on A. He may end up solving problem B, or while working on it he may find unexpectedly that he can solve problem C. In any case he has a result and can publish it. In other words, the applied mathematician is required to climb a rugged, steep mountain, whereas the pure mathematician may attempt such a climb, but if he finds the going tough he can abandon it and settle for a walk up some nearby gentle hill. Traditionally mathematics had been concerned with problems of science. But these, as we have noted, are far more difficult to solve. Only relatively few men today pursue them. The abandonment of tradition and of the rich source of problems has been justified by a new doctrine: Mathematics is independent of science, and mathematicians are free to investigate any problems that appeal to them. The research done today, so it is claimed, will be useful ten, fifty, and one hundred years from now. To support this contention the purists distort history and point to alleged examples of such happenings. But a correct reading of history belies the contention. Practically all of the major branches of mathematics were developed to solve scientific problems, and the few that today are pursued for aesthetic satisfaction were originally motivated by real problems. For example, the theory of numbers, if one dates its beginning with the Pythagoreans, was undertaken for the study of nature. Nevertheless, the break from science has widened sharply since about 1900, and today most mathematicians no longer know any science or even care whether their work will ever have any bearing on real problems. Marshall Stone, formerly a professor at Yale, Harvard and Chicago, in an article Mathematics and the Future of Science (1957) admits that generality and abstraction - pure mathematics generally - are the chief features of modern mathematics in our country. The best applied mathematics, he concedes, is done by physicists, chemists, and biologists. He might well have added that mathematics developed in a vacuum proves to be vacuous. Quite a different feature of modern research is specialization. The worldwide spread of scientific and technological pursuits has made it impossible for any individual to keep pace with a broad spectrum, and the desire to avoid being beaten to results by an ever-increasing number of competitors, and thus lose the fruit of months of activity, has almost forced mathematicians to seek out corners of their own. Mathematics is now fragmented into over a thousand specialties, and the specialties multiply faster than amoebas. The many disciplines have become autonomous, each featuring its own terminology and methodology. A general meeting of mathematicians resembles the populace of Babel after God had confounded their efforts. Pure mathematicians are unable to communicate with applied mathematicians, specialists with other specialists, mathematicians with teachers, and mathematicians with scientists. It is almost a certainty that if any two mathematicians were chosen at random and shut up in a room they would be so unintelligible to one another as to be reduced to talking about the weather. Consequently, general meetings are now far less numerous than colloquia and conferences on particular topics. Illustrations of the narrowness of modern research are so abundant that almost any article in any journal can serve as an example. Let us note one or two simple ones. One article treats powerful integers. An integer is powerful if whenever it is divisible by a prime p it is divisible by p2. Several papers on this less-than-enthralling theme have already appeared and more are sure to follow. Would that the papers be more powerful than the concept. Still another theme deals with admirable numbers. The Pythagoreans of the sixth century B.C. had introduced the concept of a perfect number. A number is perfect if it equals the sum of its divisors (other than the number itself). Thus 6 = 1 + 2 + 3. If the sum of the divisors exceeds the number, the number is called abundant. Thus, 12 is abundant because the sum 1+2+3+4+6 is 16. One can, however, ask about the algebraic sum of the divisors; that is, one can consider adding and subtracting divisors. Thus 12 = 1+3+4+6-2. Numbers that are the algebraic sum of their divisors are called admirable. One can now seek admirable numbers and establish properties about them, which no doubt are equally admirable. In this same vein are a superabundance of theorems on superabundant numbers. These very trivial examples are, of course, chosen merely because they can be presented quickly to illustrate the narrowness and pointlessness of much modern-day research. Just as everyone who daubs paint on canvas does not necessarily create art, so words and symbols are not necessarily mathematics. Specialization began to be common in the late nineteenth century. Now most mathematicians work only in small corners of mathematics, and quite naturally each rates the importance of his area above all others. His publications are no longer for a large public but for a few colleagues. The articles no longer contain any indication of a connection with the larger problems of mathematics, are hardly accessible to many mathematicians, and are certainly not palatable to a large circle. Mathematical research today is spread over so many specialties that what was once incorrectly said of the theory of relativity does apply to the research: Any one topic is understood by no more than a dozen people in the world. Each mathematician today seeks to isolate himself in a domain that Ime can work for himself and resents others who might infringe on his domain and secure results that might rob him of the fruits of his work. Even Norbert Wiener, one of the great mathematicians of recent times, admitted that he "did not like to watch the literature day by day in order to be sure that neither Banach nor one of his Polish followers had published some important result before me." And the late Jacques Hadamard, the dean of French mathematicians until his death at the age of ninety-eight in 1963, said, "After having undertaken a certain set of problems and seeing that several other authors had begun to follow the same line, I would drop it and investigate something else." There is a way of joining the crowd and yet keeping aloof from the hurly-burly. A favorite device is to introduce some new concept and develop endless theorems whose significance is, to say the least, questionable. The creator of such contrived material may even train doctoral students who, young in the ways and judgment of mathematics, may really believe in the worth of the material and so spread the name of the master. Most of those working in specialties no longer know why the class of problems they are working on was originally proposed and what larger goals their work is supposed to aim at. The modern topologist may not know Riemann's and Poincaré's work. The modern worker in Lie algebras is not likely to know what purpose Lie algebras serve. Of course, these specialists are putting the cart before the horse. The limited problems should contribute to and illuminate the area in which they lie. But the specialists would seem to be taking the position that the major areas exist in order to provide problems on which to exercise their ingenuity. Nor do they recognize that specialization promotes one?s degeneration into a narrow, uncultured person, a craftsman but nothing more. The specialist becomes what José Ortega y Gasset called a "learned ignoramus." As the process of subdivision progresses, specialized research makes less and less provision for synthesis, for pulling strands together, for asking the basic, overriding questions, for stepping back from the easel and looking at the whole picture. Indeed, specialized research does not concern itself with synthesis. Though it may foster localized competence, it may simultaneously rationalize, and even glorify, general ignorance and deliberate unconcern for those questions that transcend the narrow bounds of specialism. Yet these questions are the ones that make sense of the whole enterprise. Rampant specialization turns out to be a misfortune for the specialized pursuits themselves, although it seems to arise through concern for their exclusive needs. One obvious reason is that specialization encourages uninhibited intellectual inbreeding and it is a law, not only of human genetics, that inbreeding increases the incidence of undesirable characteristics. Furthermore, the process of unlimited specialization tends to bar a subject from the interest and participation of anyone outside, even when the outsider could make an essential contribution toward maintaining relevance in the questions asked and the methods used to pursue them. It also dims awareness of the fact that the pursuit of truth is indivisible, that all creative scholars, writers, and artists are ultimately engaged in one great common enterprise - the search for truth. In other words, specialization curtails the basic commitment of the scholar. The evils of specialization have been noted by many wise men. In his history of nineteenth-century mathematics (1925), Felix Klein said that academic mathematicians grow up in company with others like trees in a woods, which must remain narrow and grow straight up in order even to exist and reach some of the light and air. Weyl said in 1951, "Whereas physics in its development since the turn of the century resembles a mighty stream rushing in one direction, mathematics is more like the Nile delta, its waters fanning out in all directions." In the preface to his book, The Classical Groups (2nd ed., 1946), he expressed concern about too much specialization in mathematics: "My experience has seemed to indicate that to meet the danger of a too thorough specialization and technicalization of mathematical research is of particular importance for mathematics in America." David Hilbert, the greatest mathematician of this century, was also concerned about specialization. He wrote: "The question is forced upon us whether mathematics is once to face what other sciences have long ago experienced, namely, to fall apart into subdivisions whose representatives are hardly able to understand each other and whose connections for this reason will become ever looser. I neither believe nor wish this to happen; the science of mathematics as I see it is an indivisible whole, an organism whose ability to survive rests on the connection between its parts." The trend to specialization has already caused mathematics departments to split into four or more departments - pure mathematics, applied mathematics, statistics and probability (with antagonism between the two groups in this area portending a future split), and computer science. Communication among these departments is, of course, almost nonexistent, and competition for money, faculty, and students is keen. Clearly, the evils of specialization lead to inferior work. Specialists define their own area of interest and, as we have already noted, choose areas in which they can avoid competition and the larger, more vital problems. Publication is the goal, and whatever results can be published are published. Ortega y Gasset remarked in his The Revolt of the Masses that specialization provides what the biologist would call ecological niches for mediocre minds. Since specialization is the order of the day, why not journals for specialists? These are now by far the most numerous, and specialists read only the journals in their own areas, thus precluding even their awareness of anything outside their specialty. There are few journals that cover - and none that unify - developments in several fields, to say nothing about all fields of mathematics. Mathematical research has always suffered from another evil: faddism. Like all human beings mathematicians yield to their personal enthusiasms or are ensnared by the fashions of their times. The directions of research are often determined by mathematicians with prestige and power who themselves are subject to whims or the search for novelty. In the nineteenth century, for example, the study of subjects such as elliptic functions, projective geometry, algebraic invariants, and special properties of higher-degree curves was carried to extremes. Most of this work, considered remarkable in its time, would be considered insignificant today and has left almost no trace in the body of mathematics. It is no criticism of mathematicians that an area of research pursued vigorously for a time should prove unimportant in the long run. Mathematicians must use judgment as to what may be worthwhile, and even the wisest can make mistakes. Research is a gamble and one can't be sure that the work will pay off. However, faddism tends to carry a subject beyond any promise of significance. Fads flourish today because usefulness to science is no longer a standard, and the standard of beauty is purely subjective. The most pointed criticism of faddism was made by Oscar Wilde: A fad is the fantastic which for the moment has become universal. Another evil of faddism is that possibly valuable but nonfashionable ideas are disparaged. Hence, brilliant work is often neglected, though it is sometimes belatedly and often posthumously recognized. The classid example is found in the work of Gauss. Gauss, though already acknowledged as great when still a young man, feared to publish his work on non-Eudidean geometry because he would have been condemned by his fellow mathematicians or, as he put it, because he feared the clamor of the Boeotians, a reference to a dull-witted ancient Greek tribe. Fortunately, Gauss's work on non-Euclidean geometry was found among his papers after his death. By that time his reputation was so great that his ideas were accorded the utmost respect. Researchers who place high value on their work should be obliged to read a somewhat detailed history of mathematics, a subject most mathematicians do not know. They would be amazed to find how much that was regarded as vital and central in the past has been dropped so completely that even the names of those activities or branches are no longer known, Though the lesson of history is rarely learned, fads do not, fortunately, dominate the directions of research for long. What individuals create is destined to live only insofar as it is related to the evolutionary development of mathematics and proves fruitful in its consequences. One additional source of research papers of dubious value should be mentioned - Ph.D. theses and their offspring. The students, beginners in research, cannot tackle a major problem; what they do tackle is not only suggested by a professor but is performed with his help. The results are generally minor and, in fact, usually the professor can see in advance how to solve the problem. If he could not, he might worry about whether he is assigning too difficult a problem to a beginner. The new Ph.D. is, in today's world, forced to produce low quality research. If he enters or seeks to enter the university field, where researchers are now more sought after, he is under pressure to publish. Under these conditions what will he publish? He is at a stage in life where he is really not prepared to publish a paper of quality. Typically, his one experience in research was his doctoral thesis, in which he was guided by a professor and gained only enough knowledge to produce an acceptable thesis. Hence, all he actually is prepared to do is add tidbits to his thesis. But he cannot afford to be deterred by the knowledge that his publications may be insignificant. Some publication is better than none. Were he to try to solve a deep problem requiring extensive background and several years to complete, with the danger of failure all the greater, he would have nothing to show for quite some time, if ever. Hence, he must tackle and publish what can be done readily, even if the solution is labored and the result pointless. The results of pressure on faculty and young scholars to publish, the natural expansion of research in our scientifically oriented culture, the entry of the Soviet Union, China, and Japan during this century into the group of countries leading in research, and the expansion of Ph.D. training to meet the needs of universities and colleges (which in recent years has meant 750 to 1,000 mathematics Ph.D.'s per year in the United States alone) are reflected in the volume of publication. There are now over a thousand journals devoted wholly or partially to mathematical research. About five hundred are devoted solely to mathematics, and new ones are appearing almost weekly. Summaries of the articles are published in Mathematical Reviews, which does not cover all articles and in fact neglects pedagogy and much applied mathematics. In 1970 there were 16,570 reviews; in 1973, about 20,000. If all applied mathematics had been covered there would have been about 40,000 reviews in 1973. The expansion of publications has been going on at the rate of 5 percent annually. In the period 1955 to 1970 the volume of publication equaled the volume in all of the rest of recorded history. The published papers are about one-fourth of those submitted to journals. Hence, one can see how much effort is put forth by faculty to climb the ladder of research. To help mathematicians keep track of what has been published, secondary and even tertiary aids, such as indices and lists of titles, have been developed. There is an Author Index of Mathematical Reviews, which lists by author and subject the summaries published in Mathematical Reviews. For the years 1940 to 1959 The Index has 2,207 pages. For 1965 to 1972 it has 3,032 pages and 127,000 items, whereas the Indexes of the previous twenty-five years, 1940-1964, covered 156,000 items in all. There is also a journal, Contents of Contemporary Mathematical Journals (biweekly), that offers an index classified by subject of all current papers and books in mathematics. About 1,200 journals are covered and these do not include some in applied mathematics. We may await momentarily an index of the Contents and an index of all indices. The volume of publication has evoked critical comments from prominent mathematicians. One of them, Peter Hilton, has written, ". . . we are all agreed that far too many papers are being written and published. We are turning into a community of writers who do not read simply because we have no time to do so. It is a terrifying thought that if we were to spend eighteen hours a day reading new mathematics we would have substantially more to read at the end than at the beginning." In addition to zero population growth this country should aim for zero publication growth. It was generally agreed in the 1930s, when the pace of research was much slower, that nine out of ten papers had little to say and had no impact on mathematics. Some significant quantitative information was supplied by Kenneth 0. May, a professor at the University of Toronto, who studied the nearly two thousand publications from the seventeenth century to 1920 on the limited topic of determinants. He presents the following data: New ideas and results 234 14% Duplication (beyond independent 350 21% simultaneous publication) Texts and education 266 15% Applications of results 208 12% Systematization and history 199 12% Trivia 737 43% Totals including overlap 1994 117% The explanation of the 117 percent is that some papers fell into two or more categories; actually there were 1,707 separate papers. Professor May estimated that the significant information about determinants, including the main historical accounts, is contained in less than 10 percent of the papers. He also mentions that in 1851 there were ten duplications of a paper published in a leading journal. Today (1977), with far more papers published and far less concern for the significance of the research, one might estimate that no more than 5 percent of the publications offer new material. The duplication is endless. Some of it is noted in Mathematical Reviews. The American Mathematical Monthly occasionally reports duplications and errors and even cites instances of purportedly new research material that has already appeared in texts.

• See for example, the issue of December 1976, pp. 798-801 This is not to say that all of the other 95 percent are wasted. A few have educational value. Nevertheless, the journals are filled with papers of flea-sized significance, and these pollute the intellectual world as noxiously as the automobile pollutes the air we breathe. Authors deliberately publish minor variants of older research or repeat older results in new terminology. Unfortunately, the introduction of new terminology is a never-ending game, and a translation of old material can pass undetected, just as a French paper must be accepted as new by one who can?t determine whether it has appeared in German. One famous nineteenth-century German mathematician did simply translate English papers into German and publish them as his own. Some researchers take one reasonably coherent paper and break it up into three or four smaller ones. This stratagem permits much repetition, thus resulting in more published pages and giving the impression of a teeming mind. The profusion of articles and the ever-increasing number of journals make it impossible for even the specialist to read what is published in his own area. Hence, though he may pretend to know what has been done, he actually ignores the literature except for the few papers that he happens to know bear directly on his immediate goal - publication of his own paper. Months or years later some observer may note a duplication and call attention to it. Apart from the expense involved, the flood of papers seriously hampers research. A conscientious researcher will try to keep abreast of what is being done in his area, partly to utilize the results already obtained and partly to avoid duplication. He must then wade through a vast number of papers at the expense of considerable time and effort, and at that he will not cover all the relevant literature. The problem of keeping abreast of the literature had already begun to bother Christian Huygens in the late seventeenth century. In 1670 he complained, ". . . it is necessary to bear in mind that mathematicians will never have enough time to read all the discoveries in geometry (a quantity that is increasing from day to day and seems likely in this scientific age to develop to enormous proportions) if they continue to be presented in a rigorous form according to the manner of the ancients." Leibniz at the end of the seventeenth century deplored "this horrible mass of writing which continually increases" and which can only "drive away from the science those who might be tempted to indulge in it." In mathematics, where the newness of a result should be readily recognized and the difficulties overcome in proof readily apparent, it would seem that papers would be easily and accurately evaluated. Most journals do send manuscripts to referees before accepting them. But the good mathematicians who might serve as referees are so busy doing their own research, and the volume of publication they must follow is so enormous, that most do nothing about judging work in their own specialty, to say nothing of other areas of mathematics. Moreover, most papers are so sparse in explanation that their correctness is hard to judge. The narrowness of mathematicians also renders them unfit to discriminate between what is fundamental and what is trivial, between basic insight and mere technical byplay. For interdisciplinary papers it is almost impossible to find competent referees. Personal factors also intervene. Individuals favor friends and discriminate against rivals. The state of refereeing is revealed by the reactions to a recent decision of the American Mathematical Society. Up to 1975 all papers submitted for publication in the several journals supported by the Society were sent to referees with the names and affiliations of the authors recorded on the papers. The Society decided to try, for one of its journals, blind refereeing, that is, submitting the paper to the referee without the name and affiliation of the author. The protests of referees and even of two of the associate editors of that journal were vehement. They pointed to the thanklessness of the work, the difficulty in finding competent referees, and the problem of judging the correctness and worth of a paper. In the ensuing debate, partly through published letters, the opponents of blind refereeing admitted that the name and affiliation of the author helped immensely in the refereeing process. What these opponents were really saying is that they were not judging papers on their merits but were relying on the reputation of the author and his institutional affiliation to aid in determining the correctness and value of his work. If one may judge by the protests, many referees used no more than this information to make their decisions. This debate brought into the open all the weaknesses of the refereeing process. Moreover, today many papers are published without judgment by referees. There are countless symposia each year, and the papers read there are published automatically in the proceedings. Some universities produce their own journals, in which faculty members can publish at will. Publication in the Proceedings of the National Academy of Sciences is automatic not only for members but also for nonmembers whose papers are submitted through a member. The extent of the Academy's publications may be judged by the fact that in 1970 the editors decided to restrict each member to no more than ten papers per year. The present situation contrasts sharply with what prevailed in the seventeenth, eighteenth, and nineteenth centuries. Of course, there were fewer publications. But papers were sent to referees who were not only distinguished mathematicians but also broad scholars. Even then there were slips in both acceptance and rejection. R.J. Strutt, the son of one of the greatest mathematical physicists, Lord Rayleigh, relates in his life of his father that a paper by Lord Rayleigh that did not have his name on it was submitted to the British Association for the Advancement of Science and was rejected as the work of one of those curious persons called paradoxers. However, when the authorship was discovered, the paper was judged to have merit. Nevertheless, on the whole the refereeing of earlier times was competent and critical. Moreover, the editors took pride in the quality of the work published in their journals and were anxious to maintain excellent reputations. They therefore took pains to secure competent criticism of articles submitted. It is also relevant that usefulness to science served as the major standard by which most papers were judged. Actually, what is major or minor in research can be very difficult to determine. François Vieta, who first taught us to use letters to stand for a class of numbers, as in ax2 + bx + c an idea that now seems trivial but was not advanced until after two thousand years of first-class mathematics had been created, gave mathematics the basis for all proof in algebra and analysis. Surely this idea was as valuable as any major result of Newton. The assertion that quality of research is difficult to judge may seem to contradict our earlier assertion that most papers have little, if any, value. The worth of a few papers - for example, those that solve a long-standing problem that had baffled great minds - is certainly great. In other cases the authors state why the results they have obtained are important, so that the work can be more readily judged. When Vieta introduced letters for classes of numbers he stated that he could now make the distinction between numerical algebra and a science of algebra (to use modern terminology). Perhaps many a seemingly worthless paper has merit, but if that merit is not apparent to knowledgeable mathematicians, only an adverse judgment is in order. Some sociologists of science are trying to measure the quality of research papers by the number of times a given paper is cited by later papers. Toward this end they invented and use the Science Citation Index. But this measure is almost childish. Very good papers are often soon superseded by ones that advance the subject still further. Even when the advances are minor, the later papers will surely be the ones cited. A fad will be cited many times over a period of years. Many young researchers cite their professors, even at the expense of the true creator, in order to curry favor. Accepting citations, then, would seem to require first measuring the honesty of scientists. Modern mathematical research seems impressive. There is a vast and growing structure. Recent work has delineated more sharply the nature of the older subjects and has pointed the way to almost endless paths of new developments. Abstractions and generalizations have linked apparently unrelated subjects, giving mathematics some measure of unity, and have put some difficult classical theorems in a new setting where they become more natural and meaningful, at least to a trained mathematician. Mathematics now has a more qualitative character, in contrast to the manipulative and quantitative character of much of classical mathematics. Many new subjects have been created; and areas of older subjects that no longer seem significant have been discarded. We no longer learn all 467 theorems in Euclid's Elements or all 487 theorems in Apollonius' Conic Sections. But a critical look produces dismay. The proliferation of new themes, generalization, abstraction, axiomatics, and specialization may yield easy successes, but they divert attention from more concrete and difficult problems concerned with ideas of substance. Abstractions and specialties abandon reality to enter clouds of thin and diffused themes. An overweeningly arrogant antipathy to papers that do not follow the modern fashions also encourages less valuable activity. Mathematicians today care less and less about why mathematics should be created and pursued. They pay far less attention to what is worth knowing or what benefits society; nor do they question why society should support them. One of the most disturbing facts about current research is that graduate students, young Ph.D.'s, and even many established mathematicians no longer ask, Why should I undertake this particular investigation? Any inquiry that promises to produce answers and publication is regarded as worthwhile. No commendable purpose need be served except, perhaps, to advance the career of the researcher. A problem is a problem is a problem, and that suffices. Though criticism is rarely voiced, one past president of the American Mathematical Society and the Mathematical Association of America did have the courage to deprecate much modern research. No doubt much worthless research is done in all academic fields. But remoteness and pointlessness are far more prevalent in mathematics. The reason stems from the nature of the subject, especially as it is currently pursued. Mathematics deals not with reality but with limited abstractions. In past centuries these did come largely from real situations, and the prime motivation for the mathematics was to learn more about physical reality. It was recognized that the pursuit of well-chosen problems in mathematics proper must directly or indirectly pay dividends in scientific work, and mathematicians were obliged to keep at least one eye on the real world. But today mathematicians know better what to do than why to do it. The pointlessness of much current research is evident in the very introductions to papers. Students and professors seeking themes for investigation scan the publications and tag onto them. Many a paper begins with the statement, "Mr. X has given the following result. . . . We shall generalize it," or, "Mr. X has considered the following question. . . . A related question is ..." There may be no point to either the generalization or the related question. Another common introduction states, "It is natural to ask ..." ; a most unnatural and far-fetched question follows. The consequence is a wide variety of worthless papers. Mathematical research is also becoming highly professionalized in the worst sense of that term. Research performed voluntarily and sincerely by devoted souls, research as a relish of knowledge, is to be welcomed even if the results are minor. But hothouse-grown research, which crowds the journals and promotes only promotion, is a drag on science. Intellectual curiosity and the challenge of problems may still provide some motivation, but publication, status, prizes, and awards such as election to the National Academy of Sciences are the goals, no matter how attained. Deep problems that call for the acquisition of considerable background, years of effort, and the risk of failure are shunted aside in favor of artificial ones that can be readily tackled and almost as readily solved. This indictment of current research may surprise many people. Surely mathematicians are men of intellect and would not write poor or worthless papers. But the quality of the intellects engaged in research runs the gamut from poor to excellent. Francis Bacon in his Novum organum (1620) sought to mechanize research and was rightly and severely criticized. His own contemporary Galileo demonstrated through his work the extent to which originality and serendipity must enter. Bacon may indeed have oversimplified the task of research, but his expectation that anyone can do it, even "men of little wit," is not far from what happens today in mathematical research. Professor Clifford E. Truesdell, an authority in several applied fields and a man of vast knowledge, has had the courage to speak caustically. In his Six Lectures on Natural Philosophy he says: "Just as the university has changed from a center of learning to a social experience for the masses, so research, which began as a vocation and became a profession, has sunk to a trade if not a racket. We cannot tight the social university and mass-produced research. Both are useful - useful by definition, since they are paid, if badly.... The politician, the lawyer, the physician, the general, the university official are all modest men, more modest than most mathematicians. Research has been overdone. By social command turning every science teacher into a science-making machine, we forget the reason why research is done in the first place. Research is not, in itself, a state of beatitude; research aims to discover something worth knowing. With admirable liberalism, the social university has declared that every question any employee might ask is by definition a fit object of academic research; valorously defending its members against attacks from the unsympathetic outside, it frees them from any obligation to intellectual discipline...." Though each mathematician must be free to pursue the research he prefers, he does have the responsibility to produce potentially applicable papers or papers that offer high aesthetic quality, novelty of method, freshness of outlook, or at least the suggestion of a fruitful direction of research. But far too many mathematicians take advantage of the facts that potential use is difficult to judge and aesthetic quality is a matter of taste. Hence, the good is swamped by the bad. Of course, as in the past, history will decide what is of lasting value. It is the deserved fate of inferiors to fall into oblivion. But the temporary profusion of ideas they introduce constitute today a hampering and almost insuperable obstacle to real progress. Whether or not current research will lirove more hindrance than help to the advancement of mathematics is, however, not our primary concern. We undertook to survey the nature of this research because its maturation seemed to promise improvements at all levels of our educational system. Let us, then, look into the relationship of research and teaching. C&O 发表于 10:40 | 阅读全文 | 评论(0) | 引用(trackback0) 成语和学习――它山之石 - 2003-05-22 10:08 11．它山之石 成语“它山之石”来自诗句“它山之石，可以攻玉”，意思是借助别的山上的石头，可以来琢磨玉器。意味别国的贤才可作为本国的辅佐。后用以比喻能帮助自己改正缺点错误的外力，一般多指朋友。 我们在学习和生活中，经常使用它山之石的手段。小孩子打架时经常会搬出父母这块石头。在同学们的争论中，如果有谁说，老师就是这样讲的，其他人大概就会被老师的话这块石头压爬下了。许多文章和谈话为了阐明自己的观点，经常会用古今中外名人的话当石头，会引用他人的观点和材料当石头。这里也不例外。 学习某学科知识时，“它山之石”可以是其它学科的思维方式、处理方法、使用的工具和正确结果等。在科学研究中，更少不了它山之石。就拿数学来说数学工具被广泛地应用到各个领域。另一方面，数学研究的许多问题都是其它学科提出的题。现在科学技术发展的一个趋势是各学科交叉渗透，由此产生许多边缘学科。产生新的学科可是一个非常了不起的问题。 大家知道，诗是一幅画，画是一首诗。画家看到一首好诗，如果想把诗的意境用画体现出来，就给自己提出了一个问题。反过来，诗人面对如画的景色，就有了如何以诗咏景的问题。在这些情况下，那诗那画就是“它山之石”。类似地，把中国的古典名著，如《红楼梦》，《西游记》，《三国演义》和《水浒传》改编成电影或电视剧也就是把一种艺术形式表现的东西用其它艺术形式再现的问题 《孙子兵法》是我国古代的一部奇书，讲的是用兵打仗的策略。但是，这本书在美国的书店里与经济类图书放在一起。尽管有些人说商场如战场，但是经商和打仗还是不一样的。人们运用军事策略于经济活动也是利用它山之石。无独有偶，据说日本有一家大企业要求其职工读《三国演义》。 上面谈到的“它山之石”，都是从彼处汲取有用的东西用到此处，有点像“进口”。从反向思维的角度来看，彼和此的位置是对称的。就像国和国之间的贸易一样，有进口也要有出口。所以，当我们学习了一些新的东西或者在某方面有所体会时，可以试着找找在其它方面相应的东西。最简单的做法是，把这方面的术语换成另一方面的术语。这样，不但可以加深对其它方面内容的理解，有时还可能提出新的问题。 例如，艺术家丹纳说过：“一个人之所以成为艺术家，是因为他惯于辨别事物的根本性质和特色；别人只见到部分，他却见到全体，还抓住它的精神。”丹纳的话是关于艺术家的，如果我们把这些话从艺术换到科学，我们就可以得到一些做科学家的道理. 这个例子非常简单，只是把“艺术”换成“科学”就可以了。但是，有的时候并非如此简单。譬如，把中文的成语翻译成英文，只说字面的意思不行，说出它的隐喻也是不够的，还要把成语的韵味体现出来。 把某方面的认识搬到其它类的方面并不意味着事情就结束了。至少还要思考一下类似的内容是否正确或者合适。把丹纳的话中的艺术家换成科学家后，我们再来想想“别人只见到部分，他却见到全体，还抓住它的精神。”这段文字。作为科学家，是否看到事物的全体并不重要，关键在于抓住了本质。因此，我们得到这样的话“一个人之所以成为科学家，是因为他惯于辨别事物的根本性质和特色。” C&O 发表于 10:08 | 阅读全文 | 评论(0) | 引用(trackback0) 成语和学习――求同存异 - 2003-05-22 09:56 第四篇 意味深长 对于某些问题，采用人们都会的方法往往是最简单、最直接、最有效的方法，像哥仑布把鸡蛋立在桌子上的方法没有人不会做。事情就是那么简单，关键是你能不能想到可以那样做。 简单的方法经常被忽视。使用简单的方法解决简单的问题是容易的，不容易的是：使用简单的方法解决不简单的问题。提出与众不同的观点自然显示出与众不同，而对与众相同的观点有自己独到的理解更可以显示出与众不同。 10．求同存异 成语“求同存异”中的“求”字是寻求的意思，“存”字是保存、保留的意思。该成语的意思是寻求共同之处，保留不同意见。讲的是不因个别分歧而影响主要方面的求得一致。有时“求同存异”也说成“求大同，存小异”。 现在将“求同存异”的意思理解成这样：寻求共同之处，保存有差别的地方，不再强调大同小异。则“求同存异”是一种学习和思维方法。 学生在学校的学习是循序渐近的，新的东西总是建立在学过的旧东西的基础上的。新旧内容总要有一些共同的东西，新，体现在与旧的差异上。其中数学内容最为明显。理解了旧内容，又知道了新旧内容的差异，也就学会了新内容。 例如： ・分数的加、减、乘、除运算与整数的加、减、乘、除运算的差异就在分数的概念上。 ・有理数的加、减、乘、除运算与整数的加、减、乘、除运算的差异就在有理数的概念上。 ・实数的加、减、乘、除运算与有理数的加、减、乘、除运算的差异就在负数和无穷不循环小数的概念上。 ・整式的加、减、乘、除运算与整数的加、减、乘、除运算的差异就在整式的概念上。 因而学习新知识的过程应该是求同存异的过程。 求同存异就是找到相同和不同的东西，相同东西已经知道了，再记住不同的东西就可以了。很多所谓新的东西不过就是有那么一点新的地方。 记一条路时，只要知道起止点、路上的一些特殊的地方，如转弯处、特殊标志等，就可以了。不必记住路上所有的地方。类似地，学习新知识时，自己不知道、不易理解或觉得困难的地方就是“异”，而知道或易于理解的地方就是“同”，知道了原来不知道的内容或克服了难点就是存了“异”。读懂了一本书后回头总结理出眉目时，那些属于纲目的东西就是异，由纲目展开的内容就是同。做事情的时候，别无选择的地方就是“同”，可以选择的地方就是“异”。 求同存异是发现、提出和解决问题的一种思维方法。 当人们把某些事物联系在一起时，对它们进行求同存异，就会加深对这些事物的理解，就有可能发现新的问题。 求同就是寻求某些事物的共同特征。共同特征可以是表面上，但是最重要的是内在的共同特征。求同的一个基本方法就是归纳方法。归纳方法又称实验归纳法，它是通过对个别的一些经验事实和感性材料进行概括和总结，从而获得普遍的结论等的一种思维方法。数学中的许多猜想就是通过归纳的方法提出的，譬如，到目前为止还未得到证明的哥德巴赫猜想。在中小学的数学教学内容和习题中，也有一些寻找规律的东西。求同的另一个方法是类比。类比就是把某个事物较熟悉的性质转移到和它相似的事物上去，从而做出相应的判断和推理，导致发现新的东西。 发现某事物的独特的东西就是存异，在某些具有一定相同性质的事物中发现它们之间的差别也是存异。发现某事物与其它事物有根本不同的性质的重要意义不但不小于发现共同特征，而且往往意义更大。在文学艺术创作中，文学家和艺术家的存异往往是在人们司空见惯的地方挖掘出闪光的东西，发现美好的东西。 求同存异由求同和存异现方面构成。求同的同时不能忽略存异，存异的同时不能忘记求同。无论是从求同角度出发寻找某些事物的规律，还是从存异的角度出发寻找某些事物的差别，最终得到的可能是大同小异，也可能是小同大异。 对于世界上的事物，从某个角度或某个层次观察时可能会是大同，而换一个角度或层次去看时却是大异，反之亦然。例如，对于今天和明天的太阳，人们不会觉得有什么不同，然而有人却说，每天的太阳都是新的。对于河流也有类似的说法，昨天的河流已不复存在了。再比如人吧，作为人的共同特征不必多说，而世上的人的指纹却没有相同的。所以，认知世界时既要求同也要存异。 在进行创造性的工作时，人们追求的是标新立异，而标新立异的前提是求同存异。此时的求同存异是总结别人的出发点、得到的结果、处理的方法和工具等方面的相同和不同之处。在此基础上提出自己的与众不同的东西。 C&O 发表于 09:56 | 阅读全文 | 评论(0) | 引用(trackback0) Why The Professor Can't Teach――第二章 美国数学的兴起 - 2003-05-22 09:44 CHAPTER 2: The Rise of American Mathematics It must be observed, I do not esteem, as public institutions, those ridiculous establishments that go by the name of Universities. Jean-Jacques Rousseau Clearly, the relationship between teaching and research - their compatibility or incompatibility, their possible mutual reinforcement or opposition - must be investigated. To carry out these tasks we shall first examine the rise and present status of American education. In particular we must look into the role that research has come to play. This history will at least tell us why we are where we are. One of this country's marks of greatness, as Ralph Waldo Emerson pointed out in Education, is that schooling at many levels was undertaken almost immediately by the first settlers. The greatness is underscored by the fact that the United States was founded in the main by poor, uneducated people of vastly different backgrounds and languages, people who came here to improve their lot, whether by gaining political freedom, spiritual freedom, or sheer material necessities. It is true that many sought only the freedom to practice their own brand of religious intolerance. However, no matter what values were sought, education was undertaken at the very outset. Elementary schools, set up at once, were soon followed by secondary schools. The earliest of the latter were private Latin grammar schools, the first of which, the Boston Latin Grammar school, was founded in 1635. Even colleges were established early. Harvard College opened its doors in 1636 and William and Mary was next in 1693. Yale started in 1701 as the Collegiate School of Connecticut in Old Saybrook, and then moved to New Haven in 1716 at which time it adopted the name of its benefactor, Elihu Yale. The numerous colleges founded in the seventeenth and eighteenth centuries were sectarian and in fact at first were devoted to supplying clergymen for the several faiths. Harvard, for example, trained Puritan ministers. The only nondenominational school in the colonies up to 1765 was the University of Pennsylvania. The policy of separation of church and state, which was adopted by the Republic in 1787, made public support of sectarian schools illegal. Even where indirect support might have been possible, religious differences and animosities caused legislators to refuse public support to sectarian colleges of faiths adverse to their own. Hence, many nonsectarian colleges were founded. The Morn!! Act, passed by Congress in 1862, marked a major turning point by supplying public funds to 'godless' universities for 'the Benefit of Agriculture and the Mechanic Arts.' This Act enabled states to found the land-grant institutions, among which the Midwestern universities became leaders. Many more public and private colleges and universities were established in the succeeding decades, with church affiliation still the more common. As late as 1868, Cornell opened amid much criticism because it had no sectarian ties. To be sure, education was not widespread at the outset. In the seventeenth century only one colony, Massachusetts, insisted on a few years of compulsory education for all its children. As the country became somewhat settled this practice spread, and by 1910 all but six states had adopted it. Concurrently the age to which students had to remain in school was raised to fourteen, sixteen, and ultimately eighteen, though it differed from state to state. It is remarkable that in the early part of the nineteenth century the United States was the only country with some compulsory education. Moreover, by the late nineteenth century the right to free primary and secondary education was recognized, though this right was not widely utilized. In 1890 only 7 percent of the children of high school age actually attended schools, and only about 3 percent of the appropriate age group attended a college or university. As the population grew not only the number of students, but also the percentages increased rapidly. The right of high school graduates to low-cost college education was granted first by the big Midwestern universities. The privilege was specious, however, because these universities admitted all high school graduates and then flunked 50 to 70 percent in the first year. Nevertheless, the right to a college education gained ground all over the United States. What was the quality of the education? In colonial times and in the early decades of the Republic, because food, shelter, and clothing had to be given precedence, education amounted to little more than the transmission of ignorance. Certainly the pursuit of mathematics and the physical sciences, which might contribute in the long run even to the increase of material goods, had to be sacrificed to immediate needs. The country concentrated on reading, writing, arithmetic, and religion. Pedagogy, which had already been receiving some attention in Europe, received no attention here for a couple of centuries: primers showing the common man how to do simple arithmetic can hardly be dignified with that term, especially since copying numerals and rote counting were the most that was taught. The need for arithmetic in commerce, exploration, surveying, and navigation motivated the introduction of full-fledged courses in that subject―but surprisingly, not in the elementary schools. In fact it was not until the eighteenth century that it was taught and, if at all, in the colleges and universities. Before 1729, the arithmetic texts were reprints of texts from England. After that date such texts were written and published in America. (The first one was written by Isaac Greenwood, a professor at Harvard College from 1728 to 1738, who had the advantage of studying in England and the disadvantage of being fired for intemperance.) The rest of the eighteenth-century curriculum was a hodgepodge of cultural materials and moral imperatives. Keeping young people within the fold was a major concern; preparation for careers, the ministry, law, and medicine continued to be important. To this was added education in agriculture and forestry. A century later the land-grant colleges were founded to teach the latter two subjects specifically, though they soon expanded to include academic work. Utility and personal success, Alexis de Tocqueville observed in his Democracy in America (1835), were the chief concerns. Gradually the colleges and universities adopted liberal arts subjects with an emphasis, following the English model, on the classics. Molding character and teaching an aristocratic style of life to the well-born were also objectives, at least until 1900. During the eighteenth century arithmetic was gradually moved down into the Latin grammar schools and high schools, where it and the other subjects were taught mainly as preparation for college. In turn the colleges -Harvard, William and Mary, Yale, Princeton, Pennsylvania, and others - began to require arithmetic for admission. Yale did so in 1745 and Princeton in 1761; however, Harvard did not do so until 1803. In the nineteenth century arithmetic finally became an elementary school subject, mainly because the growing industrialization of the country required more knowledge from workers. But mental discipline was also a reason for teaching the subject. Elementary schools in Massachusetts and New Hampshire were the first to teach arithmetic. This was in 1789. Only by the end of the nineteenth century was arithmetic firmly a part of the elementary school curriculum. The increasing need for mathematics in manufacturing, railroading, engineering, cartography, and the study of science―especially mechanics and astronomy - motivated the introduction of algebra, geometry, and trigonometry. These subjects entered the curriculum on the college level. They were taught as junior and senior courses at Harvard, Yale, and Dartmouth from 1788 on and were retained by most colleges until the end of the nineteenth century. Thus, the mathematics curriculum at the University of Pennsylvania in 1829 reviewed arithmetic and taught the beginnings of algebra and geometry in the freshman year. The sophomore year covered more of algebra and geometry, some plane and spherical trigonometry, surveying and mensuration. Only in the third year were what are now beginning college subjects, such as analytic geometry and differential calculus, introduced, along with perspective and mathematical geography. The senior year completed elementary calculus and initiated work in the essentially mathematical subjects of dynamics and astronomy. Early in the nineteenth century a few colleges began to require for admission some of what we now call high school mathematics. Thus in 1820 Harvard upgraded its admission requirements to include elementary algebra. Yale did so in 1847 and Princeton in 1848. As for Euclidean geometry, Yale in 1865 was the first to require it for admission. Princeton, Michigan, and Cornell followed suit in 1868 and Harvard in 1870. During the last part of the nineteenth century algebra and geometry finally became high school subjects. Preparation for college, as colleges gradually began to require algebra and geometry for admission, and mental discipline were stressed as the reasons for teaching these subjects on the high school level. From 1820 to 1900 the universities gradually raised their level of mathematics instruction. By 1900 trigonometry, analytic geometry, and the calculus became standard college subjects. Although most colleges went no further in the early part of the nineteenth century, a few moved on to differential equations and more advanced subjects. Nevertheless,the general level of mathematical knowledge was low. The status of American mathematics in 1900 can be judged by an incident that is ludicrously revealing. A country physician, Edward J. Goodwin, who possessed no sound knowledge of mathematics, submitted a bill to the Indiana Legislature in 1897 that called for declaring the value of to be 4. A comparison of the common formulae for the area of a circle and the area of a circumscribed square shows immediately that cannot be 4. Nevertheless, the Legislature considered the bill. Fortunately the Senate, thoroughly modern in one respect, postponed action and the bill was never passed. One may not be too surprised by the actions of elected officials, but the American Mathematical Monthly, a journal founded by leaders of that time, in its first volume, July 1894, published Goodwin's proposal and in 1895 dutifully printed other absurd Goodwinisms. The pressure in the United States to raise the levels of mathematics education and to educate more and more of American youth increased sharply as the country became a greater world power. World War I certainly showed the need for more mathematics education, and ever-increasing technological uses of mathematics added to the pressure. The universities responded by raising the requirements for admission and by adding advanced courses. This brief sketch of the rise of mathematics education has not addressed the question of how teachers were procured. The upgrading of the formal level of education and the increase in the number of students did not in itself provide a supply of teachers. In colonial times almost anyone could set himself up as a teacher, and since the population of this country was made up mainly of uneducated people, one can readily appreciate the level of teaching. As late as 1830, Warren Colburn, one of the early-nineteenth-century American educators, said in an address: The business of teaching, except in great seminaries, has not been considered as one of the most honorable occupations, but rather degraded; so that few persons of talent would engage in it. Even in our own countr and age, it has been too much the case that persons wit a little learning, and unwilling to work and unfit for anything else, have turned schoolmasters, and have been encouraged in it. They have been encouraged in it because the pay of school teachers, in most instances, has been just sufficient to obtain that class of persons.... Although the colonial colleges were not founded to train teachers, many did so. But the colleges admitted only men whereas, because of the low salaries, women were the most likely candidates.* Also, most colleges were still sectarian and so could not be the main source of public school teachers or be favored by public money for the education of teachers. Further, the better colleges insisted on more qualifications for admission than prospective school teachers could afford to obtain. Recognition of these facts prompted Horace Mann, in the 1830s, to advocate special means for training teachers. State normal schools were founded that required only an elementary school education, gave teachers a modicum of training, and then sent them out to teach. These schools were organized under Mann's leadership in the late 1830s and they spread throughout the country. The problem of supplying teachers for a public school system that kept extending universal compulsory education was never really solved. It was aggravated by the fact that more and more immigrants, uneducated and poor, entered the country. Nor was the situation any better in the colleges. When Charles William Eliot became president of Harvard College in 1869, he emphasized that one of the outstanding problems was to induce ambitious young men to adopt the calling of professor. 'Very few Americans of eminent ability,' he said, 'are attracted to this profession.' The supply of good teachers continued to fall short of the need. Certainly many who entered teaching in the nineteenth and early twentieth centuries were not the best choices. Teaching remained very poorly paid, had little or no prestige, and required few qualifications. Of course, some capable people, unable to afford the education that other professions required, turned to teaching. The colleges and universities, which were not themselves well staffed, gradually took over the function of educating teachers. Professors of education became members of college faculties beginning in 1832 at New York University and at other colleges soon after. Up to 1920, however, the training of elementary school teachers was still done in the normal schools, which by that time required a two- to four- year course of study. The colleges trained the nation's high school teachers. Then the colleges and universities began to establish departments or schools of education to train both elementary and high school teachers. Beginning around 1950, the normal schools were converted to four-year liberal arts colleges with emphasis on teacher training. High school mathematics teachers now take a four-year liberal arts course with a major in mathematics, but elementary school teachers take little academic mathematics and a great deal of instruction in pedagogy. The rise of institutions and faculties devoted to teacher education did not improve the knowledge of mathematics proper that the professors should have possessed. In fact, the level of mathematics instruction was very low until well into the twentieth century. American texts were poor; good ones had to be imported. A few people educated themselves. One of the outstanding examples was Nathaniel Bowditch (1773-1838), who translated and added explanatory notes to Laplace's Mécanique céleste, one of the great works of European mathematics. This translation was the first substantial mathematics book in the United States and, in fact, in the entire Western hemisphere. The first truly great American scientist was the self-taught physical chemist Josiah Willard Gibbs (1839-1903), who is also known for his contribution to vector analysis. Many professors went to Europe for their education; these men were certainly the best educated and were actually the only ones qualified to teach mathematics and train mathematicians. Gradually the quality of education and available books improved sufficiently for the United States to train its own mathematicians, though hardly in numbers sufficient to serve the needs of the country. The first of the outstanding American-trained mathematicians was Benjamin Peirce (1809-1880), a Harvard graduate and professor at Harvard from 1831 to his death. Other Harvard mathematics professors of his time, such as William Fogg Osgood and Maxime Bocher, got their Ph.D.'s in Europe before joining the university faculty. Despite the increase during the early decades of this century in well-trained mathematicians, one could not be complacent about mathematics education. There were too few teachers to staff the increasing number of elementary schools, high schools, colleges, and universities. The states, cities, and towns lagged in their appreciation of the value of education and were miserly in funding. Consequently, salaries remained low and potentially fine teachers continued to choose other jobs and professions. Some indication of the quality of the teachers comes from data compiled by the Educational Testing Service. As late as 1954 the elementary school teachers who were interviewed feared and hated mathematics. Naturally this influenced their teaching. Half of 370 teachers tested could not tell when one fraction was larger than another. Thus, the teachers knew less than they were required to teach. High school teachers in most states could qualify for a license to teach mathematics with only ten hours of college mathematics. In many states a license to teach in high school is still sufficient to qualify for teaching any subject. Since until recently ignorance was the outstanding characteristic of American educators at all levels, not only was the factual content of mathematics courses low, but also the educational goals were either imperfectly or mistakenly perceived. We need not pursue here the conflicts, disagreements, and theories advanced over the several centuries of American education. That social utility should be an objective, particularly when the colonies and even the young Republic were struggling for survival, could hardly be challenged; but arithmetic was also advocated as a mental discipline that would extend to other areas. Though portions of secondary school mathematics could be defended on the ground of their usefulness in applications such as surveying and navigation, it was impossible to defend most of high school mathematics on this ground. Instead, mental discipline, knowledge of a branch of our culture, the inherent beauty of the subject, and preparation for college were advanced as justifications. As to the pedagogy, intuitive approaches using concrete materials and sensory experiences, espoused, for example, by Johann Heinrich Pestalozzi (1746-1827), and abstract rigorous approaches were both advocated to impart the values and even the meaning of mathematics. In practice, however, the teachers, barely understanding what they were taught, handed down the processes and the proofs mechanically. Drill was the order of the day. Any questions were answered with the dogmatic reply: This is the way to do it. To divide one fraction by another, invert and multiply. The product of two negative numbers is a positive number. If x + 2 = 7, transpose the 2 and change the sign so that x = 7 - 2, or 5. The rationale of geometric proofs was never given. Students memorized them and handed them back on examinations. Thu departments and colleges of education attached to universities were not helpful. The reasons are patent. The mathematics educators were themselves taught subject matter in the same way as others were taught, and so their understanding of mathematics was no better than that of good students. It was in fact worse, because the educators did not deem it necessary to go as far in their subject as those who planned to be professional mathematicians and scientists. Furthermore, very little was known then (and even now) about the psychology of teaching and learning. Hence, the educators had little to contribute. They taught prospective teachers how to carry out the drill and memorization, the very processes by which they had been taught. Another unproductive effort was to call upon psychologists, who commonly offered a standard course in the psychology of education. But the psychologists also had little to offer. Professor Edward Lee Thorndike (1874-1949) of Columbia University?s Teachers College advocated massive repetition or drill. Students should be trained to respond automatically: 3 + 2 should immediately elicit 5. Understanding would come eventually. But this advice was no more than an endorsement of the rote teaching that had been practiced for generations. (See also Chapter 9.) This method of teaching is still advocated by educators such as Professor B.F. Skinner of Harvard - now it is called programmed learning. To improve education, innumerable commissions and committees were appointed by mathematical organizations and state and local governments. The committees were to study goals, content, and pedagogy. The history is extensive but irrelevant. One factor that hampered the reforms suggested by the many committees, conferences, symposia, and commissions is that no sound evaluation was employed to test the recommendations. Another was that changes were introduced too quickly, making it difficult for teachers to implement them properly. The efforts and sincerity of the mathematics educators are not in question. However, the nature and goals of mathematics education, including appropriate subject matter, the best methods of teaching it, the role of applications, and the means to attract and involve students were not determined by these studies. Undoubtedly, changing conditions in this country rendered pointless many recommendations that may have been right in their time. The proportion of students attending high school in the United States grew from 12 percent in 1900 to over 90 percent in 1967. Obviously the interests and backgrounds of high school students became far more varied as that population increased. The Bachelor of Arts (or Science) curriculum, designed originally to give American youth some knowledge of classictil (Greek) and European culture, was broadened somewhat after the Civil War to include the social, physical, and biological sciences and the humanities. In 1900, when about 4 percent of the college-age group went to college, this program may have been proper. In 1970 about 48 percent, or seven million students, went to college, for totally different reasons than just the acquisition of knowledge. The colleges had to adapt to many more levels of academic ability, far more diverse backgrounds, and many new goals―notably career training in a wide variety of fields. Beyond these changes was the initiation in 1901 of community or two-year colleges, which were the successors of the private junior colleges first established in the late nineteenth century and which were preparatory for the senior colleges. The community colleges now have many terminal students who seek essentially a vocation. Despite gradual improvement in the quality of the teaching staff on the lower educational levels, a stabilized educational system, reasonable facilities, and the far greater availability of texts and books, mathematics education was on the defensive from 1920 to 1945. Students did poorly. Enrollment in academic mathematics courses decreased, notably in algebra and geometry. The transfer of training was, perhaps rightly, deprecated. During this period the colleges reduced requirements in mathematics for admission, and students took fewer mathematics courses once enrolled. Many institutions even dropped the mathematics requirement, and many high schools also dropped mathematics as a requirement for a diploma. The value of any mathematics education was widely challenged. The status of mathematics in the 1930s was described by mathematics professor Eric T. Bell in the American Mathematical Monthly of 1935: Are not mathematicians and teachers of mathematics in liberal America today facing the bitterest struggle for their continued existence in the history of our Republic? American mathematics is exactly where, by common social justice, it should be - in harnessed retreat, fighting a desperate rear-guard action to ward off annihilation. Until something more substantial than what has yet been exhibited, both practical and spiritual, is shown the non-mathematical public as a justification of its continued support of mathematics and mathematicians, both the subject and its cultivators will have only themselves to thank if our immediate successors exterminate both. The subsequent history and status of mathematics education was affected by the entry of a new factor - research, in the sense of original contributions. In the area of mathematics this necessarily came rather late. Knowledge in this subject is cumulative. To contribute to mathematics in 1800, for example, one had to know the work of the Greeks, Descartes, Fermat, Newton, Leibniz, Euler, Lagrange, Laplace, and many others. In science the situation was roughly similar. Mechanics was well developed by the end of the eighteenth century, and contributions by neophytes could hardly be expected. The newer fields of science offered more opportunities. In electricity, for example, which was just beginning to be cultivated in 1800, it was possible for the American Joseph Henry to share honors with the Britisher Michael Faraday in the discovery of electromagnetic induction. In the nineteenth century, however, American science as a whole was, like American mathematics, below the European level. Researchers in general had no place in the United States before about 1850. They could become teachers at low-level colleges and universities and spend their spare time, of which there was little, in research. But such endeavors received no encouragement. In fact, a research person might even arouse suspicion that he was not paying enough attention to his teaching and might even have radical ideas about curriculum and the method of instruction. In 1857, a committee of the Columbia College Board of Trustees attributed the poor quality of the college's educational efforts to the fact that the professors 'wrote books.' A professor's obligation was not to advance knowledge but to transmit it. Of course the United States at that time was not really prepared to do research, much less train Ph.D.'s for research. Though during the nineteenth century, as we have already noted, many American professors went to Europe to study, their number was relatively small. Moreover, while a few years in a great cultural center helps immensely, it does not produce researchers, particularly if they return to an intellectual atmosphere in which research is neither cultivated nor widely appreciated. Researchers need the stimulus of colleagues with common interests and of students who carry on the work of the masters and press them for problems and In effect, during the nineteenth century good understanding and appreciation of research did not exist. For example, Josiah Willard Gibbs, the physical chemist mentioned earlier, worked at Yale University, where for many years he received no pay. In Europe his research in thermodynamics was well understood and highly valued. He wrote a first-class book on statistical mechanics in 1901 that was appreciated by such masters as Felix Klein and Henri Poincaré but ignored in the United States. When the distinguished German mathematician, physicist, and physician Hermann von Helmholtz visited Yale in 1.893 and was greeted by the top dignitaries of the university, he asked where Gibbs was. The university officials, nonplussed, looked at each other and said, 'Who?' The state of research in the United States even in the early twentieth century may be illustrated by an incident in the life of Walter Burton Ford, an American mathematician who died at the age of ninety-seven in 1971. He submitted his doctoral thesis to Harvard. The paper was judged by Bocher, Osgood, and Byerly, who were leaders in American mathematics in the first few decades of this century, and deemed unacceptable. Ford sent the paper to the reputable French Journal de mathématiques, where it received praise from the editors and was published. The Harvard faculty then reversed its decision and awarded the Ph.D. to Ford. Another of the early great American mathematicians was Norbert Wiener (1894-1964). His fields of endeavor were not understood here; however, when the European mathematicians began to praise Wiener's work, the American mathematicians took notice and fortunately, if belatedly, he received honor here. When research in mathematics and the sciences was first undertaken, about 1850, the university leaders thought that it belonged not in the universities but in special institutions referred to as academies. Advanced degrees were offered only in law, medicine, and theology. Yale, however, did initiate graduate training in 1847 and conferred the first Ph.D. degree in mathematics in 1861. Harvard instituted a graduate program in 1872 and conferred the first degree in mathematics in 1873. Though the universities did begin to undertake research and training for research, the inadequacy of the American research capacity was recognized by some men who had become aware of the high standards of the German universities. It was these men who persuaded wealthy individuals to found universities that would stress research and strengthen the already existing advanced training. David Coit Gilman induced Johns Hopkins, a merchant and banker, to found the institution named after him, which opened in 1876. Leland Stanford, who made his fortune in railroading, established Stanford University, which opened in 1891. And William Rainey Harper convinced John D. Rockefeller to finance the University of Chicago, which opened in 1892. Johns Hopkins and Chicago started as graduate schools but soon added undergraduate colleges. The already existing graduate schools began to take research more seriously and other colleges added graduate schools. Through these moves the German universities made their impact on education here and high-level research was at least launched. Although some advocates and entrepreneurs of research - Andrew Dickson White of Cornell, Charles W. Eliot of Harvard, and Gilman - believed that universities should undertake research, they also believed it should be subordinate to teaching. Imparting truth, White held, is more important than discovering it, and Eliot declared in 1869 that 'the prime business of American professors in this generation must be regular and assiduous class teaching.' During his early years as president he was suspicious of professors who devoted much time to research because he feared that this activity interfered with their teaching. But a generation later the university leaders reversed the emphasis, and research took precedence over teaching. President William Rainey Harper of Chicago led this change, declaring: 'The first obligation resting upon the individual members who comprise it [the university] is that of research and investigation.' David Starr Jordan of Stanford declared, 'The crowning function of a university is original research.' To defend its position the Harper-Jordan group asserted, in Jordan's words, that 'investigation is the basis of all good instruction. No second-hand man was ever a great teacher and I very much doubt if any really great investigator was ever a poor teacher.' Hundreds of professors and scores of university administrators took up the Jordan refrain that no one could be a good teacher unless he also did research. Harper not only made research the first obligation of professors; he also instituted the practice that promotion of the University of Chicago faculty 'will depend more largely upon the results of their work as investigators than upon the efficiency of their teaching.' Harper was also responsible for directing the graduate schools to concentrate on training all graduate students to be researchers. Breadth of knowledge was derogated and training for teaching was entirely ignored. The discovery of new facts or the recovery of forgotten facts became the supreme, prized goal. Whether subordinated to teaching or esteemed more highly, research did take hold. One could say that it was effectively initiated when Johns Hopkins offered a professorship to the already famous British mathematician James Joseph Sylvester (who was refused a job in British universities because he was Jewish). Sylvester served from 1877 to 1883. He and William E. Story founded the first research journal in the United States, the American Journal of Mathematics, in 1878. This journal immediately became the outlet for the earliest significant research papers written by Americans; European contributions added to its prestige. The first volume contained several highly original papers by the American-educated mathematical astronomer George William Hill and in the volume of 1881 Benjamin Peirce published a paper he had written and circulated privately a decade earlier. When the University of Chicago was organized it immediately hired outstanding research professors. In mathematics it appointed Eliakim Hastings Moore, an American who had studied in Germany, and Osker Bolza and Heinrich Maschke, both imported from Germany. This university became the first outstanding American research center, and it trained the first truly great American mathematicians, among them George David Birkhoff, Leonard Eugene Dickson, and Oswald Veblen. The precedent set by Johns Hopkins, Stanford, and Chicago was soon followed by other universities, which now began to demand Ph.D.'s who could carry on research at their institutions. Some colleges also sought Ph.D.'s for the sake of the prestige gained by having such researchers on their faculties. However, the overall quality of the Ph.D. training was generally poor. The degree was aptly described by Jean-Paul Sartre as a reward for having a wealthy father and no opinions. The low quality of the Ph.D.'s, who were expected to become teachers as well as researchers, began to alarm prescient educators. In 1901 President Abbott Lawrence Lowell of Harvard said: 'We are in danger of making the graduate school the easiest path for the good but docile scholar with little energy, independence or ambition. There is the danger of attracting an industrious mediocrity which will become later the teaching force in colleges and secondary schools.' A few years later David Starr Jordan, despite his strong advocacy of research, deplored the lack of professors qualified to teach, which he attributed to the narrowness and triviality of the doctoral dissertation. A more devastating and concerted attack was made by the distinguished Harvard philosopher, William James, in his essay of 1903, "The Ph.D. Octopus." * James was concerned that the rush for the Ph.D. crushed the true spirit of learning in the colleges. He objected to colleges and universities seeking Ph.D.'s as evidence to the world that they had stars on their faculties: Will anyone pretend for a moment that the doctor's degree is a guarantee that its possessor will be a success as a teacher? Notoriously his moral, social, and personal characteristics may utterly disqualify him from success in the classroom; and of these characteristics his doctor's examination is unable to take any account whatever.. . . In reality it is but a sham, a bauble, a dodge, whereby to decorate the catalogues of schools and colleges. And in 1908, the distinguished American educator Abraham Flexner foresaw the greater evil to come, namely, that even though universities might improve the quality of the Ph.D. training, they would be sacrificing college teaching on the altar of research. For better or worse, the emphasis on research grew stronger. To further it mathematicians decided to hold meetings and to support more journals. They founded the New York Mathematical Society in 1888, which became the American Mathematical Society in 1894. Initially the Society devoted itself to research and teaching. In the first decade or two of this century members of the Society - Eliakim H. Moore, Jacob W.A. Young, David Eugene Smith, Earle R. Hedrick, George Bruce Haisted, and Florian Cajori, among others - did take an active interest in education, including secondary school teaching. They made sensible recommendations and seriously attempted improvements. But after a couple of decades the Society concentrated on research, whereupon another group of men founded the Mathematical Association of America in 1915 to cater specifically to undergraduate education. In 1921 still another group, concerned with secondary and primary school education, founded the National Council of Teachers of Mathematics. More and more, research became the major interest of the universities, until in the large universities it gained favor over all other functions. Just how research and teaching might have fared had the United States continued its main reliance upon its own resources cannot be known. But unexpected developments altered the American scene. During the Hitler period many of the leading mathematicians of Germany, Italy, Hungary, and other European nations fled their countries; a large number of them came to the United States. When the Institute for Advanced Study was founded at Princeton in 1933 for postdoctoral research, three of the six mathematics professors chosen for its faculty were Albert Einstein, John von Neumann, and Hermann Weyl. The numerous refugees soon found places in American universities and added enormously to our mathematical strength. They also trained doctoral students for research, and the number and quality of Ph.D.'s increased significantly. Though the acquisition of competent researchers was a boon to that activity, it contributed little to the teaching at the college level. The refugee mathematicians who came here during the Hitler terror were trained in the German universities, which were - until they lost their best professors - the strongest in the world. However, there was and is no undergraduate education in Germany. Students go directly from a gymnasium (high school) to a university, where they specialize in a subject, usually to obtain the Ph.D. degree. Training for research is the goal of the education (though students who do not complete the degree may pass an examination that qualifies them to be gymnasium teachers). Moreover, the professors, who are research specialists, do not feel obliged to be concerned with pedagogy, since student motivation, drive, and ability are presumed. Though the refugee mathematicians were intelligent and well intentioned they were not, by reason of background, able to apportion their efforts and knowledge among the diverse needs of the American universities, such as undergraduate education. The graduate schools were therefore turned still more in the direction of training researchers and became even less concerned with pedagogy. While the graduate schools were absorbing the great academicians who had come to the United States, World War II broke out. The war was a battle of scientists. Faster ships and airplanes, radar for detection and to advance antiaircraft gun control, improvements in range and accuracy of artillery, better navigational techniques for ships, planes, and submarines, and the development of the atomic bomb proved to be crucial and convinced the country that if it was to remain a major power, more mathematical and scientific research was needed. Hence, during and after World War II our government began to support research on a large scale. Research became a more and more valued and prestigious activity, and professors concentrated on this work. With the rising importance of research came the rising esteem for the researcher. The American professor, once a lowly figure among the elite, had now achieved high social rank. Government money also made an essential difference. Billions of dollars to finance research were given to professors, who thereupon became sources of income to the universities. On paper all research grants and contracts cover only the actual cost of the research performed on behalf of some governmental agency or program. But in effect these grants cover far more. They support graduate students employed to aid in the research. As a consequence more graduate students can attend the university, and more tuition income is received. Professors working on contracts give up most of their teaching duties. In their place poorer paid, often young, instructors conduct most of the classes, though the name of a well-known professor still serves as a drawing card in attracting students. Laboratories and equipment needed to perform contract research are paid for by the government but are used for other research and for instructional purposes. Secretaries to professors, paid by contract money, and a generous overhead also make contract research very attractive. It does enable a university to enlarge its research and graduate programs, though - contrary to claims often made―it does not contribute to undergraduate education. The effect was obvious. The universities, already inclined to favor research, began to compete intensively for research professors. Most of these professors and most graduate schools turned their attention to producing more researchers. The Ph.D. programs were geared solely toward this end. The attention that academic professors had given to undergraduate, secondary, and elementary school education during the first few decades of this century was all but abandoned. Nevertheless, there would no longer seem to be cause for the concern expressed by Lowell in 1901. The United States now has many strong graduate schools staffed by competent research professors, and research, judged at least by the volume of output, is flourishing. Mathematics research reinforces scientific strength, and scientifically and technologically the country has achieved pre-eminence. The researchers can certainly offer sound graduate training, and the knowledge imparted in this training should filter down to all levels of education. Wise professors, concerned not only with the extension of knowledge but also with its transmission, can devise ways of presenting their subjects that their students - the professors and teachers of the future - could use to good advantage. The recently organized large schools or departments of education in the universities and colleges can provide the special instruction and advising that the elementary and secondary school teachers require. Insofar as mathematics in particular is concerned, although the American Mathematical Society has abandoned its earlier interest in teaching, the Mathematical Association of America and the National Council of Teachers of Mathematics have taken over that vital concern for the undergraduate and lower levels. On the face of things, the United States would seem to have reached, if not an ideal, then certainly a very reasonable and even rosy position at all levels of education. Why, then, did Peter Landers face so many problems as a teacher and find himself unable to resolve them? C&O 发表于 09:44 | 阅读全文 | 评论(0) | 引用(trackback0) 《铁齿铜牙纪晓岚》歌词赏析 - 2003-05-21 04:21 开篇设问：谁说书生百无一用？与世俗的观念相悖，颇为悲壮。并试图回答这个问题。 中篇欲进先退、欲扬先抑。大肚能容与难得糊涂，这都是已经饱含为人处世的哲理了，颇有出世之味。然而两对非理性之词：莫笑和偏有，也知道和忍不住，似乎无法这样超脱，先退。原来要为天下苍生，登高一呼，后扬。境界全出。神来之笔。 尾篇用铁齿、铜牙、烟袋和秃笔点出主人公特征，十分工整贴切。 最后再次提出开篇的命题。经过前面的回答，此时已不同彼时，底气十足。以反问收笔。 C&O 发表于 04:21 | 阅读全文 | 评论(1) | 引用(trackback0) What is Combinatorics――什么是组合数学？ - 2003-05-21 04:08 [Combinatorics] has emerged as a new subject standing at the crossroads between pure and applied mathematics, the center of bustling activity, a simmering pot of new problems and exciting speculations. ――Gian-Carlo Rota, [Studies in　Combinatorics, p.vii] The formal study of combinatorics dates at least to Gottfried Wilhelm Leibniz's Dissertatio de Arte Combinatoria in the seventeenth century. The last half-century, however, has seen a huge growth in the subject, fueled by problems and applications from many fields of study.　Applications of combinatorics arise, for example, in chemistry, in studying arrangements of atoms in molecules and crystals; biology, in questions about the structure of genes and proteins; physics, in problems in statistical mechanics; communications, in the design of codes for　encryption, compression, and correction of errors; and especially computer science, for instance in problems of scheduling and allocating resources, and in analyzing the efficiency of algorithms. Combinatorics is, in essence, the study of arrangements: pairings and groupings, rankings and orderings, selections and allocations. There are three principal branches in the subject. Enumerative Combinatorics is the science of counting. Problems in this subject deal with determining the number of possible arrangements of a set of objects under some particular constrains. Existential Combinatorics studies problems concerning the existence of arrangements that possess some specified property. Constructive Combinatorics is the design and study of　algorithms for creating arrangements with special properties. Combinatorics is closely related to the theory of graphs. Many problems in graph theory concern arrangements of objects and so may be considered as combinatorial problems.Also, combinatorial techniques are often employed to address problems in graph theory. C&O 发表于 04:08 | 阅读全文 | 评论(0) | 引用(trackback0) Why The Professor Can't Teach――第一章　怪圈 - 2003-05-21 03:18 CHAPTER 1: The Vicious Circle. In a self-centered circle, he goes round and round, That he is a wonder is true; For who but an egotist ever could be Circumference and center, too. Sarah Fells Peter Landers found himself caught in a vicious circle. He had just secured a Ph.D. in mathematics from Prestidigious University and, having been well recommended, readily secured a faculty position at Admirable University. Thereupon Peter faced the problem of teaching mathematics to prospective engineers, social scientists, physicists, elementary and secondary school teachers, the general liberal arts students, and those who, like himself, had chosen to become mathematicians. Peter was fully aware of these varied career interests, and he also knew that students came to college with different drives and preparation. But he was confident that his education, typical for Ph.D.'s, had prepared him for the tasks ahead. To put himself in the proper frame of mind he reviewed his own education. The elementary school courses had been acceptable. After all, one did have to know how much to pay for five candy bars if he knew the price of a single bar. True, some operations were baffling. It had not been clear why the division of two fractions had to be performed by inverting the denominator and multiplying - but the teacher seemed to know what was correct. He had constantly referred to rules, principles, and laws. Rules, like rules of behavior, apparently applied to arithmetic, too. For all Peter had known, principles were laid down by the principals of the schools, and certainly they were authorities. As for laws, everyone knew that there were city laws, state laws, federal laws, and even the laws of the Ten Commandments. Certainly laws must be obeyed. Though under some tension as to whether he was violating laws, Peter was young and resilient. In any case, what to do was clear and the answers were right. In his review of his high school education Peter did recall so me doubts he had had about the value of what he was being taught. He hadn't understood why the teacher had to stress that the sum of two whole numbers is a whole number, or why he had to prove that there is one and only one midpoint on every line segment; but evidently the teacher was trying to make sure that no one could be mistaken on these elementary matters. After all, teachers knew best what had to be done. Peter also recalled one teacher's enthusiasm about the quadratic formula. 'You see,' the teacher proclaimed triumphantly after he had derived the formula, 'we can now solve any quadratic equation.' But Peter had been perverse and had asked the teacher why anyone wanted to solve any quadratic equation. The teacher's reply was a disdainful look that caused Peter to shrink back. His question must have been a silly one. He remembered a similar experience in geometry. After a long and apparently strenuous effort, the teacher proved that two triangles are congruent if the sides of one are equal respectively to the sides of the other. Then he turned to the class as if expecting applause. Again Peter dared to speak up: 'But isn?t that obvious' A triangle is a rigid figure. If you put three sticks together to form a triangle, you cannot change its size or shape.? Peter had learned this at the age of five while playing with Erector sets. The teacher's contempt was obvious. 'Who's talking about sticks? We are concerned with triangles.' Despite a few other disagreeable incidents Peter continued to like mathematics. He believed in his teachers. It was easy to comply with their requests, and the certitude of the results gave him, as they had given others before him, immense satisfaction. And so Peter moved on to college with the conviction that he liked mathematics and was going to major in it. His first experiences were disturbing. After his program was approved by an adviser who did not understand what an Advanced Placement Examination Grade of 4.5 meant - the adviser had thought that 10 was a perfect grade so that 4.5 was a poor one - Peter was finally registered. He entered his first college classroom for a course which happened to be English. To his surprise he found about five hundred students already seated. The professor arrived, delivered his lecture, and, obviously very busy, rushed out of the room. Peter never found out what his name was, but apparently names were not important, because the professor never bothered to ask any student his name either. Nor, Peter thought, would the professor have noticed had a different group of five hundred students appeared each time. Term papers were required, and these were graded by graduate students who insisted that 'Who shall I call next?' was correct, though Peter had been taught otherwise in high school. The size of the class and the impersonal character of the instruction disturbed Peter at first, but he soon realized that the requirements of the English course could be met merely by listening. And so he relaxed. Peter's second class, one in social science, surprised him for different reasons. At the professor's desk was a young man not much older than Peter. As the instructor conducted the lesson he was obviously nervous. Somehow the lessons throughout the semester were confined almost entirely to the first part of the text. And the instructor did not welcome questions. The third class - mathematics - was a shock. Peter entered the room and found that it was a large auditorium. At the bottom of the room, at the professor?s desk, was not a man but a box, which proved to be a televisioh set. Shortly after Peter's entrance the box began to speak and the students took notes feverishly. From many seats one could not see clearly, if at all. But by coming early one could get a good seat. And so Peter managed to learn some of his college mathematics by listening and looking at a TV program. Though it was not a requirement, Peter decided to take some physics. He had heard somewhere that mathematics was applied to physics, and he thought he should find out what these applications were. The physics professor constantly talked about infinitesimals and which infinitesimals could be neglected. The mathematics professors, however, had warned that such concepts and procedures were loose and even incorrect. But Peter listened attentively. He was sure that even though the mathematics and the physics professors apparently did not communicate with each other and so did not talk the same language, their methodologies could be reconciled. He did seek counsel from his professors on this matter, but unfortunately they were not available. One was actually living out of the city, in Washington, D.C.; another was always involved in consultations outside the university; and a third had office hours only on Sundays, from 6:00 to 8:00 A.M. In the junior and senior years the classes were smaller, and the courses were usually taught by older faculty. Many blithely ignored the texts they had assigned and spent the period transferring material from their notes to the board. The professors copied assiduously and the students did likewise. When the professors looked up from their notes they looked into the blackboard as though the students were behind it. Nevertheless Peter persevered, received his bachelor's degree, and proceeded to graduate school. His experiences there paralleled those of most other students. Professors were hard to contact. The bulletin descriptions of the courses bore no relation to what the professors taught. Each professor presented his own specialty as though nothing had been done or was being done by anyone else in the world. And so Peter learned about categories, infinite Abelian groups, diffeomorphisms, noncommutative rings, and a variety of other specialties. Prospective Ph.D.'s must write a doctoral thesis. Finding a thesis adviser was like hunting for water in a desert. After many trials, including writing theses on topics suggested by his professor that, it turned out, had been done elsewhere and even published, Peter wrote a thesis on almost perfect numbers that completed his work for the degree. With the Ph.D. behind him, Peter presumed he was prepared for college teaching. Upon taking up his position at Admirable University he received from his department chairman the syllabi for the several courses he was to teach and was told what texts he was to use for these courses. Cheerful, personable Peter went about his assigned tasks with enthusiasm, He had always liked mathematics and had no doubt that he could convey his enthusiasm and understanding of the subject to his students. He had been informed by the chairman that to secure promotion and tenure he would be expected to do research. This requirement in no way dimmed Peter's spirit, because he had been told repeatedly that mathematicians do research and was confident that the training he had received had prepared him for it. But the world soon began to close in on Peter. As a novice he was assigned to teach freshmen and sophomores. His first course was for liberal arts students, that is, students who do not intend to use mathematics professionally but who take it either to meet a requirement for a degree or just to learn more about the subject. Recognizing that many of these students are weak in algebra, Peter thought he would review negative numbers. To make these numbers meaningful he reminded the students that they are used to represent temperatures below zero; and to emphasize the physical significance of negative temperatures he pointed out that water freezes at 32° F.,so that a negative temperature means a state far below freezing. Though the example was pedagogically wise, Peter could see at once that the students' minds had also frozen, and the rest of his lesson could not penetrate the ice. In a later lesson Peter tried another subject. As an algebraist by preference he thought students would enjoy learning about a novel algebra. There is an arithmetic that reduces all whole numbers by the nearest multiple of twelve. To make his lesson concrete Peter presented clock arithmetic as a practical example: Clocks ignore multiples of twelve, so that four hours after ten o'clock is two o'clock. The mere mention of clocks caused the students to look at their watches, and it was obvious that they were counting the minutes until the end of the period. And so Peter tried another novelty, the Koenigsberg bridge problem. Some two hundred years ago the citizens of the village of Koenigsberg in East Prussia became intrigued with the problem of crossing seven nearby bridges in succession without recrossing any. The problem attracted Leonhard Euler, the eighteenth century's greatest mathematician, and he soon showed by an ingenious trick that such a path was impossible. The villagers, who did not know this, continued for years to amuse themselves by making one trip after another during their walks on sunny afternoons - but when Peter presented the problem in the artificial, gloomy light of the classroom, a chill descended on the class. Peter's next class was a group of pre-engineering students. These students, he was sure, would appreciate mathematics, and so he introduced the subject of Boolean algebra. This algebra, created by the mathematician and logician George Boole, does have application to the design of electric circuits. The mention of electric circuits appeared to arouse some interest, and so Peter explained Boolean algebra. But then one student asked Peter how one uses the algebra to design circuits. Unfortunately, Peter's training had been in pure mathematics and he did not know how to answer the question. He was compelled to admit this and detected obvious signs of disappointment and hostility in the students. They evidently believed that they had been tricked. In his attempts to explain and clarify other mathematical themes Peter also learned that engineering students cared only about rules they could use for building things. Mathematics proper was of no interest. Nor were the premedical students any more kindly disposed to mathematics. Their attitude was that doctors do not use mathematics but take it only because it is required for the physics course, and even the physics seemed of dubious value. The physical and social scientists had a similar attitude. Mathematics was a tool. They were interested in the real world and in real people, and certainly mathematics was not part of that reality. Peter was soon called upon to teach prospective elementary and high school teachers. He did not expect much of the former. These students were preparing to teach many different subjects and so could not take a strong interest in mathematics. However, high school teachers specialize in one area, and Peter certainly expected them to appreciate what he had to offer. But every time he introduced a new topic, the first question the students asked was, 'Will we have to teach this?' Peter did not know what the high schools were currently teaching or what they were likely to teach in any changes impending in the high school curriculum. Hence, he honestly answered either 'No' or 'I don't know.' Upon hearing either response the prospective teachers withdrew into their shells, and Peter's teachings were reflected from impenetrable surfaces. Peter's one hope for a response to his enthusiasm for teaching was the mathematics majors. Surely they would appreciate what he had to offer. But even these students seemed to want to 'get it over with.' If he presented a theorem and proof, they noted them carefully and could repeat them on tests; however, any discussion of why the theorem was useful or why one method of proof was likely to be more successful or more desirable than another bored them. A couple of years of desperate but fruitless efforts caused Peter to sit back and think. He had projected himself and his own values and he had failed. He was not reaching his students. The liberal arts students saw no value in mathematics. The mathematics majors pursued mathematics because, like Peter, they were pleased to get correct answers to problems. But there was no genuine interest in the subject. Those students who would use mathematics in some profession or career insisted on being shown immediately how the material could be useful to them. A mere assurance that they would need it did not suffice. And so Peter began to wonder whether the subject matter prescribed in the syllabi was really suitable. Perhaps, unintentionally, he was wasting his students' time. Peter decided to investigate the value of the material he had been asked to teach. His first recourse was to check with his colleagues, who had taught from five to twenty-five or more years. But they knew no more than Peter about what physical scientists, social scientists, engineers, and high school and elementary school teachers really ought to learn. Like himself, they merely followed syllabi - and no one knew who had written the syllabi. Peter's next recourse was to examine the textbooks in the field. Surely professors in other institutions had overcome the problems he faced. His first glance through publishers' catalogues cheered him. He saw titles such as Mathematics for Liberal Arts, Mathematics for Biologists, Calculus for Social Scientists, and Applied Mathematics for Engineers. He eagerly secured copies. But the texts proved to be a crushing disappointment. Only the authors' and publishers names seemed to differentiate them. The contents were about the same, whether the authors in their prefaces or the publishers in their advertising literature professed to address liberal arts students, prospective engineers, students of business, or prospective teachers. Motivation and use of the mathematics were entirely ignored. It was evident that these authors had no idea of what anyone did with mathematics. Clearly a variety of new courses had to be fashioned and texts written that would present material appropriate for the respective audiences. The task was, of course, enormous, and it was certain that it could not be accomplished by one man over a few years' time. Nevertheless Peter became enthusiastic about the prospect of interesting investigations and writing that would lure students into the study of mathematics and endear it to them. The spirit of the teacher arose and swelled within him. As these pleasant thoughts swirled through his mind, another, dampening thought, like a dark cloud on the horizon, soon entered. He was a recently appointed professor. Promotion and, more important, tenure were yet to be secured. Without these his efforts to improve teaching would be pointless - he would be unable to put the product of his work to use. But promotion and tenure were obtained through research in some highly advanced and recondite problems almost necessarily chosen in the only field in which he had acquired some competence through his doctoral work. Such research was no minor undertaking. It demanded full time and total effort. Clearly, he must give the research precedence, and then perhaps he could undertake the improvement of teaching. And so for practical reasons Peter decided to devote the next few years to research. But the struggle to publish and to remain in the swim for promotion and salary increases caught Peter in a vortex of never-ending spirals of motion; and the closer he came to the center the deeper he was sucked into research. In the meantime Peter continued to teach in accordance with the syllabi and texts handed down to him by his chairman. His few, necessarily limited efforts to stir up some activity among his older colleagues, who were in a better position to break from the existing patterns, were futile because these professors had accepted the existing state of affairs and chose to shine in research. Success there was more prestigious and more lucrative. Ultimately, Peter, like other human beings, succumbed to the lures that prominence in research held forth. As for the students―well, students came and went, and they soon became vague faces and unremembered names. Education might hope for an epiphany, but Peter was not ordained to be the god of educational reformation. By the time he had acquired tenure he had joined the club. Like others before him he concentrated on research and the training of future researchers who would also be compelled to resort to perfunctory and ineffective teaching. Peter had taken his place in the vicious circle. The history of Peter Landers' aborted teaching efforts, real enough, seems exaggerated. One might conceive of its taking place in nineteenth-century Germany or France. But the United States is devoted to education. We were the first nation to espouse universal education and to foster the realization of the potential of every youth. Our Founding Fathers, notably Benjamin Franklin and Thomas Jefferson, stressed the necessity of this policy, and it was adopted. Even today no country matches the educational opportunities and facilities that the United States provides for its youth. But the practices within educational institutions seem to be in marked variance with the principles and policies of our country. How has it come to pass that Peter and the many thousands of his colleagues find themselves enslaved by research, while education, the major goal of our vast educational system, is being sacrificed? Does the pressure to do research stem from the professors because they prefer the prestige and monetary rewards? Or does it come from the university administrations? In either case, does not research make for better teaching? Or is there a conflict between the two, and if there is, how can we resolve it? Since the crux of the problem lies with the universities -which train the teachers of all educational disciplines and at all levels -we must examine the policies and practices of our higher educational institutions. C&O 发表于 03:18 | 阅读全文 | 评论(0) | 引用(trackback0) 成语和学习――郑人买履 - 2003-05-21 03:10 ９．郑人买履 郑人买履是一则成语故事，源于《韩非子・外储说左上》，“郑人有欲买履者，先自度其足，而置之其坐，至之市而忘操之，已得履，乃曰：‘吾忘持度’。反归取之，乃反，市罢，遂不得履。人曰：‘何不试之以足?’曰：宁信度，无自信也。’”后用“郑人买履”讽刺不顾实际情况，只相信教条的人。 这个“宁信度，无自信也”的人是有点傻，傻到只相信他量自己脚的那个“度”，而不相信自己的足。但是，必须注意到“度”也是足的正确反映，是他对自己脚的认识，如果买到了鞋，此成语故事大概就不会流传至今了。必须承认，人类对于世界其中包括人类自己的认识是逐步的、不断加深的。谁又能保证我们的后人不笑话我们当今的做法是“郑人买履”呢。 一个人在世一生也是认知世界的一生，自觉地或者不自觉地。传授知识者需要了解接受知识者的“度”才能有的放矢，取得效果。科学家要使一般人明白科学知识，就必须以人们能理解的语言和方式介绍科学知识。作为接受知识者的学生只有以自己现有的“度”理解了新的知识才能不断使其“度”有所提高。也许别人的理解方法比自己的更有效、更简单，但我“宁信度”。 一个人的“度”是与他的知识结构、思维方式和处理事情的方法紧密相关的。当他理解新东西时，总是以自己的已有知识为基础通过自己的思维方式去思考、判断。有的时候，仅仅因为叙述新东西的术语不明白，就使接受者难以理解。偶尔看到股市行情的报道时，对于某某股涨多少、跌多少我还可以理解。可是对什么“牛市”、“熊市”之类的话就不知所云了。 我现在使用英文时，如果不把英文翻译成中文就不知道英文说些什么，原因很简单，我的思维是用中文进行的。 背得滚瓜熟不等于掌握了知识。没有理解而死记硬背住的东西就是喝到肚子里的墨水，只有理解了东西才能变成头脑中的知识。肚子里的墨水太多会消化不良的。读书会使人知道得很多，但也使一些食而不化的人疯疯颠颠。 那个可爱的郑人，不知道也不相信用自己的脚就可以为自己买到合适的鞋。但是他知道用量他的脚的草棍可以买到鞋。圆周率π是个无理数，圆的面积等于圆的半径的平方乘以π。当我们要算出一个圆的面积时，只有取π为3.14或更多位的近似值，否则我们就得不到面积的具体值。这个时候，通过取近似值把我们不会算的东西化成了我们会算的东西。 在科学技术研究中，一件很重要的工作就是把人们不会做的事情通过某些方法转换成人们会做的事情，或者把人们不容易做到的事情借助于某些工具转换成人们容易做到的事情。谁也没有孙悟空的火眼金睛，可是借助于先进的医疗设备人们可以把人的五腹六脏看得清清楚楚。当今时代的信息传递工具完全可以使“秀才不出门”而“全知天下事”。 匈牙利数学家罗莎・彼谈及数学家思维方式时曾经分析过下面的事例。 有人提出了这样一个问题：“假设在你面前有煤气灶、水龙头、水壶和火柴，你想烧开水，应当怎么去做?”对此，某人回答说：“在壶中灌上水，点燃煤气，再把壶放到煤灶上。”提问者肯定了这一回答。但是，他又追问道：“如果其它的条件都没有变化，只是水壶中已经有了足够多的水，那么你又应当怎么去做?”这时被提问题者往往会很有信心地回答道：“点燃煤气，再把壶放到煤气灶上。”但是，这一回答却未能使提问者满意，因为，在后者看来，更为恰当的回答是：“只有物理学家才会这样做；而数学家则会倒去壶中的水，并声称他已经把后一个问题化归成先前的已经得到解决的问题了。 提问者描述的数学家是不是很象“郑人买履”的那个郑人?但是罗莎指出，数学家就是这样，“他们往往不是对问题实行正面的攻击，而是不断地将它变形，直至把它转化成能够得到解决的问题。”（罗莎・彼得，《无穷的玩艺》，南京大学出版社，1985。）当然，罗莎分析上面的事例的目的是为了论述解决数学问题的一种思维方式。并不是数学家在实际生活中，非得把壶中的水倒掉，至少我不会。 C&O 发表于 03:10 | 阅读全文 | 评论(0) | 引用(trackback0) 成语和学习――不求甚解 - 2003-05-21 03:06 8．“不求甚解” 成语“不求甚解”源自陶潜(陶渊明)的《五柳先生传》，“好读书，不求甚解。” 意思是指读书只领会精神，不在一字一句的解释上多花费工夫，本是褒义词。但是该成语现在常用来指学习不认真，不求深入理解。类似的说法还有，“只知其一，不知其二”，“知其然，不知其所以然”。学习语言时，譬如学习中文，都会谈到阅读的方法。阅读的方法有三种: 1．略读：快速地游览，以了解文章的大意和主要情节为目的。 2．查读：迅速地查找到需要了解的信息，也就是为了解决某些特定的问题而阅读。 3．细读：为了解文章的细节而阅读。 陶潜意义下的“不求甚解”读书应该属于第一种读书方法。 我不清楚你们学生掌握陶潜意义下的“不求甚解”读书方法程度如何，只想谈谈不求深入理解意义下的“不求甚解”学习方法。在北京大学数学科学学院，经常有国内外学者做学术报告，其中不乏大师一级的人物。学院要求研究生每学期必须听若干次学术报告，鼓励高年级的学生志愿去听。这些学术报告通常不是科普报告而是非常专业的，就是不做相同方向研究的教授们也常常不懂具体内容。有的学生听过一次就来问题了，“老师，我听不懂报告的内容，怎么办?”其实，老师们明明知道学生们听不懂几乎所有的报告。可是为什么还要求他们去听呢?让学生听报告目的就是让他们知道一点报告内容的皮毛，因为，这些报告的特点是介绍报告人做了些什么工作,通常会涉及到这些工作的起因、意义、用途、特点、使用工具、目前的研究状况、尚未解决的问题、将来的发展等一些内容。通过这些报告可以开拓学生们的眼界、引发研究的兴趣、了解发展动态。 听这种报告的学习方法就是“不求甚解”。不但不需深入了解，搞懂每个细节，甚至只知道几个关键词就够了。如果对某个报告内容感兴趣，可以向报告人索取有关的文献以求进一步的了解、学习和研究。 学生们习惯于用“求甚解”的方法学习和掌握知识，因为我们学校里要求的学习内容大都是些最基本的东西。因而“求甚解”是必要的。那些“不求甚解”的学生是用错了地方。一个人的精力、能力是有限的，在校读书的时间也是有限的，不可能事事都求甚解。要“读万卷书”就不能卷卷细读，要“行万里路”就不可能步步扎实。“不求甚解”是不用教的，你们生来就具有。你们看课外书时不就常常不求甚解吗?只是对那些你们觉得是高深的知识不知道也可以不求甚解地去学。所以，在踏踏实实学好基本或重点内容的同时，应该尽可能“不求甚解”地多读些书、走马花地多行些路。 C&O 发表于 03:06 | 阅读全文 | 评论(0) | 引用(trackback0) 真情真美 - 2003-05-20 11:39 真情真美　孙楠　许茹云 ――电视剧《射雕英雄传》片尾曲 真情真美　　　真如一池春水 风吹点点涟漪　感受细致入微 痴心无罪　　　付出没有不对 就算一生一世　从此相依相随 不必在乎是谁　翻转是是非非 把前尘做白纸　写上无怨无悔 我们都愿意给　只要爱的纯粹 就算有苦有累　我们一起去背 爱是多么可贵　贵在有所作为 只要同去同归　成败也无所谓 爱是多么可贵　贵在有所作为 只要同去同归　成败也无所谓 来日风雨中有没有伤悲 无论怎么样一起来面对 百转千回　　　纵横南北 敞开我们的心扉 爱是多么可贵　贵在有所作为 只要同去同归　成败也无所谓 爱是多么可贵　贵在有所作为 只要同去同归　成败也无所谓 成败也无所谓 C&O 发表于 11:39 | 阅读全文 | 评论(0) | 引用(trackback0) 14.127223014832