CHAPTER 10  The Deeper Reasons for the New Mathematics.
Marshall H. Stone
In view of the defects in the modern mathematics program and the failure to remedy the defects of the traditional curñculum, why was the modern mathematics program devised and promoted? Moreover, since the claimed superiority of this program is not sufficiently supported by the contents or other evidence, what accounts for its acceptance? To understand why the modern mathematics curriculum rather than some wiser version was promulgated it is necessary to note first the interests which modern mathematicians pursue. There is no question that up to the late nineteenth century the chief concern of the great mathematicians was to understand the workings of nature. We need not review here the relevant history because the assertion is not disputed. Mathematics was regarded as one of the sciences and indeed during the seventeenth, eighteenth, and most of the nineteenth centuries the distinction between mathematics and theoretical science was rarely noted. In fact, many of the men who have been ranked as the leading mathematicians of the past did far greater work in astronomy, mechanics, hydrodynamics, elasticity, and electricity and magnetism. Mathematics was simultaneously the queen and the handmaiden of the sciences. Nor did these men hesitate to put to practical use the scientific knowledge that they and others had gathered. Newton studied the motion of the moon to help sailors determine their longitude at sea. Euler studied the design of ships and of sails, made maps, and wrote a masterful text on artillery. Descartes designed lenses to improve the telescope and microscope. Gauss not only made a survey of the electorate of Hannover, but worked on the improvement of the electric telegraph and the measurement of magnetism. These few examples could be multiplied a hundredfold. Almost all of these men not only saw the potential in the scientific knowledge they were helping to amass but were keenly concerned that the knowledge be utilized. However, most mathematicians of the past hundred years have broken away from science. They know no science, and what is more, are no longer concerned with the utilization of mathematical knowledge. It is true that some, aware of the noble tradition that motivated mathematical research in the past and that warranted the honor accorded to men such as Newton and Gauss, still claim potential scientific value for their mathematical work. They speak of creating models for science. But in truth they are not concerned with this goal. In fact, since most modern mathematics professors know no science they can't be creating models. They are quite willing to shine by reflection of the light shed by great mathematicians over the past and even justify support of their present research by citing the accomplishments of their predecessors. Mathematics now is turned inward; it feeds on itself; and it is extremely unlikely, if one may judge by what happened in the past, that most of the modern mathematical research will ever contribute to the advancement of science. When confronted with this charge, the mathematicians dare not deny it  but then defend their creations on the ground that they are beautiful. Whether or not there is beauty in the creations need not be argued here. The important point is that this value is used to justify the work. Another feature of the current mathematical activity is the narrow specialization. Mathematics has expanded enormously, as has science, and most mathematicians are almost obliged to concentrate on limited areas in order to keep abreast of other people's creations and produce new results of their own. Needless to say, the training of new mathematicians, which is conducted by professors who are themselves specialists in narrow fields, follows the same course. Doctoral candidates are forced to burrow into obscure corners in order to produce satisfactory theses. They are no longer broadly educated in mathematics, to say nothing of science. Emphasis on mathematics proper and on specialization is especially strong in the United States. The primary reason is that in this country research is a relatively new phenomenon, and American professors anxious to shine and to train students who will shine, specialize in order to produce results quickly. The classical mathematical efforts, which have scientific goals, require extensive background because the subjects involved have been explored for several hundred years. Consequently, only a small percentáge of the mathematicians, those often labeled applied mathematicians, continue to pursue the traditional goals. Most have turned to purely mathematical problems and to the formalization, axiomatization and generalization of what is already known. Such tasks are far easier. The break between mathematics and science was deplored by the famous mathematical physicist John L. Synge as far back as 1944.
Some mathematicians, instead of pretending concern for the utility of their work, brazen forth a new declaration of independence. Professor Marshall H. Stone, then at the University of Chicago, in his article "The Revolution in Mathematics", decided to take the bull by the horns.
The views expressed by Stone and others have not gone. unopposed. Richard Courant, formerly head of the mathematics department at the preHitler worlds center for mathematics, the University of Göttingen, and then head of the Courant Institute of Mathematica1 Sciences of New York University, replied to Stone (see the reference to Carrier in the bibliography),
The trend toward abstraction, toward mathematics tor mathematics sake, led the worldrenowned mathematician John von Neumann to issue a warning. In his essay "The Mathematician" he stated,
Still another protest was voiced in 1962 by a prominent mathematician, Professor James J. Stoker of New York University:
The danger to mathematics of the break from science has been stressed by many other men. The leading American mathematician George D. Birkhoff, professor of mathematics at Harvard University, said as far back as 1943: "It will probably be the new mathematical discoveries which are suggested through physics that will always be the most important, for, from the beginning Nature has led the way and established the pattern which mathematics, the Language of Nature, must follow." To the argument that mathematics is now potentially more powerful for science because it is free to follow its own course, the "applied mathematicians" counter with the evidence of history. A11 the applications of mathematics to science came from mathematica1 ideas which were inspired by science. No mathematician ever cooked up ideas useful to science by sitting in an ivory tower. It is true that ideas inspired by science later found unexpected application, but the ideas were sound to start with because they derived from genuine physical problems. In the article already mentioned Synge remarks on this point too.
What does the nature of current mathematical research have to do with curriculum reform? The relevance lies in the fact that the leaders in this reform have been college professors. These men were educated in a mathematica1 world that has departed radically from the concept of mathematics that animated the great matheinaticians of the past. About eightyfive per cent of the Ph.D's in mathematics are not only narrow specialists but are concentrated in corners of mathematical logic, algebra and topology, fields which, on the whole, are remote from science. These men do not know even freshman physics nor have they any desire to know it. Because they have no idea of the role that mathematics has played in history they are ignorant as mathematicians and certainly as educated human beings. Most presentday professors pursue abstractions, generalizations, structure, rigor, and axiomatics. Since this is what most mathematicians do, it is not surprising that this is what they think mathematics education should train young people to do. These college professors, competent or incompetent, when called upon to help in the preparation of curricula, can suggest as subject matter only the narrow, specialized abstract topics that they are familiar with or, making some concession to the elementary level of instruction, they suggest somewhat watereddown or abbreviated versions of the more sophisticated treatments of traditional mathematics. This fact accounts in large part for the content of the modern mathematics curricula. Professor Stone, whose characterization of modern mathematical research was described previously, did. not . hesitate to say in the same article that the curriculum must be refashioned to teach that kind of mathematics.
The consequences of having university professors lead curriculum reform are even more harmful. It is generally conceded that college professors are chosen largely for their knowledge of subject matter and research strength and not for their pedagogical skill. Trained to do research, they are illprepared for teaching even on the college level. Mathematicians are not pedagogues. In fact the two classes are almost disjoined sets. It did not occur to these men that the goals of elementary, high school and even undergraduate education and the inerests and capacities of students at these levels have little to do with mathematical research. Having become wise through the acquisition of a Ph.D. and possibly a prestigious position at a major university they believe themselves to be experts in areas in which they are in fact totally ignorant. Despite pedagogical failings when teaching on their own level and despite the fact that most of the professors who participated in curriculum reform had not been inside an elementary or high school since their own student days, the mathematics professors did not hesitate to take on a task that calls for considerable pedagogica1 acumen One can say that they were presumptuous. They acted as though pedagogy was only a detail, whereas if they had really learned anything at all from their studies, they would have known that almost any problem involving human beings is enormously complex. The problems of pedagogy are indeed more difficult than the problems of mathematics, but the professors had supreme confidence in themselves. The trouble with most men of learning, as one wit put it, is that their learning goes to their heads. Dr. AIvin M. Weinberg, Director of the Oak Ridge Nationa1 Laboratory, in his article "But Is the Teacher Also a Citizen" criticized the narrow professional point of view of mathematicians and scientists. Speaking of both groups he said, Thus, our science tends to become more fragmented and more narrowly puristic because its practitioners, harried as they are by the social pressures of the university community, have little time or inclination to view what they do from a universe other than their own. They impose upon the elementary curricula their narrowly disciplinary point of view, which places greater value on the frontiers of a field than on its tradition, and they try to put across what seems important to them, not what is important when viewed in a larger perspective. The practitioners have no taste for application or even for interdisciplinarity since this takes them away from their own universe; and they naturally and honestly try to impose their style and their standards of value. . . At another place in the article Weinberg noted the trend in both the new science and mathematics curricula. But insofar as the new curricula have been captured by university scientists and mathematicians of narrowly puristic outlook, insofar as the curricula reflect deplorable fragmentation and abstraction, especially of mathematics, insofar as the curricula deny science as codification in favor of science as research, I consider them to be dangerous. . . . . . The professional purists, representing the spirit of the fragmented, researchoriented university, got hold of the curriculum reform and, by their diligence and aggressiveness, created puristic monsters. But education at the elementary level of a field is too important to be left entirely to the professionals in that field, especially if the professionals are themselves too narrowly specialized in outlook. It is perhaps unnecessary to add that the professiona1 mathematicians are so intent on making
their careers through mathematica1 research that they take little or no time to acquire any knowledge of the history
of their subject or of its human and cultural significance. Some even boast of their ignorance of science. A few
may be informed in the broader values of mathematics but do not think it is necessary to teach it. Hence mathematicians
are not really prepared to put their subject in an interesting light and thereby attract to the subject students
who might very well take to it if the classroom material were appealing. Even if anxious to attract students, the
limited professors are unable to do so and will not Professor Feynman, whose article. "New Textbooks for the New Mathematics"has already been cited, also scathingly criticized the new mathematics because the texts were written by pure mathematicians who are not interested in the connections of mathematics with the real world nor in the mathematics used in science and engineeing because it is on the whole not new but old. None of the above criticisms is intended to challenge the good intentions of the college professors but good intentions often succeed in doing no more than paving certain roads. Nevertheless their mistakes were so gross that one cannot but ask, How could they have gone so far wrong? In part, we have already accounted for their errors, but there is more to the story. As professionals with extensive training in mathematics they had acquired some understanding of the subject. Forgetting that they themselves had required years to achieve this understanding they believed that they could impart it at once to young minds. Moreover, their interest was to develop future mathematicians, but because they overlooked the pedagogy they failed even in that task. They concentrated on the superficia1 aspect of mathematics, namely, the deductive pattern of weIlestablished structures, instead of emphasizing how to think mathematically, how to create and how to formulate and solve problçms. Moreover, professional mathematicians are aheady motivated to pursue mathematics. Hence they failed to take into account that other people do not see the point of studying mathematics. The professional mathematicians are the most serious threat to the life of mathematics, at least so far as the teaching of the subject is concerned. They resent students who do not take to the subject at once and are impatient with students who want to be convinced that the subject is worthy of interest. Yet mathematics proper can be a deadly subject especially because the courses are arranged in the order determined by the proper logica1 sequence and this means much drudgery to progress in the subject. If mathematics professors were required to spend eight years in elementary school, three or four years in high school and then another year in college on ceramics they would object strenuously. But they do not see that mathematics for mathematics sake has even less appeal to young people than ceramics has for themselves. Mathematicians of this century are very much concerned with rigor. There are historica1 reasons for this preoccupation. However, we have already pointed out how damaging rigorous proof is to the student. Why does the mathematician insist on it? The answer is that he is favoring his professional interest, as, for example, in building on a miniinal set of axionis. He is not willing to consider the pedagogy.
To prepare curricula at any level one anust know the objectives of education at that level. For example, it is far more important in the lower grades to interest students in learning than it is to develop proficiency in one subject. To know these objectives one must devote a great deal of thought to the whole problem of education. And this mathematicians do not and will not do. Most mathematicians are not at all interested in the psychology of learning. Th,is is a very difficult subject, more difficult than mathematics. How much can young people learn? Should they be induced by an offer of candy to learn something by rote? Do abstactions come easier to young people? Would negative numbers seem less artificial if taught early? A pedagogue will do his best to find out what psychologists have established and a!so learn from his own experience. mathematicians will not take the trouble to find out what psychology can offer, nor will they take the trouble to develop their own skill in the art of teaching. It is a1so easy .to see why the texts are so poorly written. Professional mathematica1 writing has a style of its own. It is succinct, monotonous, symbolic and sparse. The chief concern is to be correct. On the other hand, good texts must have a lively style, arouse interest, tell students where they are going and why. Writing is an art and mathematicians do not cultivate it. One of the basic reasons that mathematicians fail as pedagogues stems from the nature of the mathematical mind. The common belief is that the mathematician is the epitome of intelligence and hence should always be able to act wisely and to prescribe solutions to all problems. Most people believe this because they are scared by the mere appearance of symbols and conclude that if a man can master these symbols he inust be intelligent. One might just as well conclude that every Frenchman must be intelligent because he masters French. But I shall venture to draw a distinction between a mathematician's intellectual capacities and his wisdom. The mathematician does have the ability to make sbarp distinctions in the meanings of words, the capacity to learn and apply the laws of logic, and the capacity to retain and compare a number of facts. He has what I sball call a rational mind. He may a1so be creative in mathematics. Wisdom may indeed include these rational qualities but it also includes much more: judgment, the capacity to learn from experience, the perception of values, tho understanding of human beings; and the capacity to use knowledge for the solution of human problems. These latter qualities are not possessed by mathematicians any more than by any other group of people selected at random. It is rational to present mathematics logically, but it is not wise. Consider the teaching of calculus. One knows that the calculus is built on the theory of limits and so may conclude that the way to teach calculus is to start with the theory of limits. The wise man would also consider whether young people can learn the theory of limits from scratch and whether they will want to learn it without motivation and prior insight. On a sheer probability basis, wisdom would be distributed among mathematicians as among lawyers, doctors, engineers and businessmen. But a little analysis seems to raise the question of whether mathematicians are likely to have their share. What attracts people to mathematics? Mathematics is a simple subject compared to economics, psychology or physics. It is a narrow subject. One does not have to have an extensive background to do pure mathematics. Moreover, mathematics per se does not deal with human beings and the complex problems that dealing with human beings pose. As Bertrand Russell said, "Remote from human passions, remote even from the pitiful facts of nature, the generations have created an ordered cosmos, where pure thought can dwell as in its natural home and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world." Hence mathematics is likely to attract those who do not feel competent to dea1 with people, those who shy away from the problems of the world and even consciously recognize their inability to dea1 with such problems. Mathematics can be a refuge. Mathematicians as a class are overrated in another essential respect. One tends to assume that professors are superior in character and that they will therefore espouse only those causes and movemen that are helpful to society. Unfortunately, mathematics has its share of opportunists, bandwagonjumpers, reactionaries, prestigeseekers, powerseekers, and moneygrabbers. It is sad to read in the history of mathematics that even many great mathematicians stooped to presenting as their own results they took from others. This appraisal of mathematicians is terribly negative, but it seems necessary to discredit the belief that mathematics professors are infallible and a truly superior group. In view of their overriding concern with personal. advancement through research, their unwillingness to devote time to the problems of pedagogy, and possibly limited development as wise human beings, it is not likely that college professors can lead curriculum work on the high school and elementary school levels. How did college professors become involved in e1ementary and high school curriculum reform? There is no question that some help from this source was needed. Elementary and high school teachers do not have enough time to follow the advances in mathematics, to keep abreast of any new materia1 that shou1d be integrated into the curriculuin, to profit from any investigations in the learning process, and to incorporate such materiaI in any largesca1e curriculum revision. They need to be apprised of these matters by professors, whose function it is to be informed in these areas. Hence the mathematics professors were called upon to participate in reform. There is another group that might have served usefully in curriculum reform. This is the group of education professors. However, most of these men were not informed in advanced mathematics and concentrated on how to teach the traditional mathematics. The plight of these professors was expressed openly and honestly by Professor Max Beberman, who was a mathematics education professor. In the Proceedings of a University of Illinois Committee on School Mathematics Conference held at the University of Illinois in 1964 and devoted to The Role of Applications in a Secondary School Mathematics Curricu1um", he said,
The appeal to professors of mathematics for information on what subjects sbould be pursued is not in itself a mistake and in fact, as we have pointed out, is necessary. Unfortunately, the ones who participated ìn curriculum work were not in the main the choice ones. Those few who still worked in the traditional areas of mathematics, the areas concerned with mathematics and its relationship to the sciences, were in the 1960's heavilyengaged in research on the problems of the rapidly expanding sciences. Hence it was the professors from the more remote, abstract and pure side of mathematics who felt free to devote themselves to curriculum work. Moreover, apart from their own onesidedness, these men, like college professors generally, had had no experience or contact with elementary and high school curricula. In fact, they had disdained such interests in the past and so had no idea of what should be taught at these levels or how young people think. Many professors of questionable competence, noting that they would have to deal only with elementary mathematics and seeking some activity that might add to their prestige, gladly joined in. One might think that the pedagogical weaknesses of the college professors would be offset by the high school teachers and the education professors. Surely the latter two groups shou1d know what can be taught to elementary and high school students and what might motivate these young people. But the high school teachers and the professors of education were overawed by the mathematicians. One who can write a research paper is regarded in our society as a person of extraordinary ability. How could lowly members of the fraternity question such men of distinction and unquestionable knowledge? It seems fair to say on the basis of what transpired that in every curriculum group the college professors dominated the educators and the schoolteachers. The educators and teachers bowed down to the idols, not knowing that most had feet of clay. This account of what transpired does not argue against the involvement of intellectuals in the problems of the schools. The need for the involvement has been acknowledged. But it may show that school systems have to be strong enough to recognize when they are being helped and when they are being led astray. The value of the aid the schools will get will depend entirely on the competence of those being helped. They must judge whether the advice they get is sound. Put otherwise, college professors can be used as consultants but certainiy should not lead and dominate the fashioning of curricula for the elementary and high schools. We have accounted for the direction which the new mathematics curriculum took, but this does not explain its relatively widespread acceptance. In view of the manifold defects of this curriculum one would think that it would be rejected outright by the country at large. Several factors account for its adoption. Probably the largest single factor is that the curriculum groups were organized and well financed. Hence these groups undertook active campaigns to put the new curricula across. Not only the leaders but members of the many groups began to speak for the new curricula at various meetings of teachers, principals and administrators. Since the traditional curriculum was not successfu1, these people were at least receptive to a new curriculum. When assured that mathematicians, education professors and high school teachers had collaborated and were agreed on the merits of the new curriculum, those addressed were impressed. Beyond speeches, the curriculum groups issued literature. The document already mentioned, The Revolution in School Mathematics, and subtitled, A Challenge for Administrators and Teachers, was issued by the National Couiicil of Teachers of Matheatics in 1961. Ostensibly the document was a report on conferences held around the United States to inform school administrators and mathematics supervisors of the nature of the curricula. But it described the curricula as ones that would enable them to provide leadership in establishing new and improved mathematics programs. It implied that administrators who failed to adopt the reforms were guilty of indifference or inactivity. But in 1961 this country had had very little experience with the new curricula. A booklet championing and advocating them at that time can be fairly accused of propaganda. Unfortunately, the propaganda was effective. Most school administrators do not have the broad scientific background to evaluate the proposed innovations. They do not know whether these innovations are models of educational knowhow combined with superlative subject matter or are the enthusiasms of subjectmatter specialists without much relevance to the needs of students. The pressure does put the administrators on the spot. They can, seem to show interest and progress by adopting one of the modern programs, or they can be thoroughly honest and admit that they are not competent to judge the merits of any one. What actually happened is that many principals and superintendents urged the modern curricula on their teachers just to show parents and school boards that they were alert and active. The very adoption of the term modern mathematics is pure propaganda. Traditional connotes antiquity, inadequacy, sterility, and is a term of censure. Modern connotes the uptodate, relevant, and vital. The terms modern and new were used for all they were worth. Speakers capta1ized on the fact that the traditional curriculum offered little that was not known before 1700. Of course, as we have seen, the terms modern and new were hardly justified since in the main the new curricula offer a new approach to traditional mathematics. A few speakers actually degraded themselves by resorting to thinly veiled threats. Some of them were on the Commission on Mathematics of the College Entrance Examination Board. This Board formulates the aptitude tests which high school students take for admission to college. The speakers hinted that these tests would contain questions on the modern mathematics topics. Since the teachers were anxious to have their students do well on these tests, they felt compelled to learn what the new curricula contain and compelled to teach these topics. Another device, deliberately employed to put modern mathematics across, was described by Professor Paul Rosenbloom, a very competent mathematician, who participated in the writing of curricula. In the article, "Applied Mathematics: What is Needed in Research and Education" (see the reference to Carrier in the bibliography), Professor Rosenbloom described this device:
Such promotional schemes led one school consultant to say that if the pedagogical insights of the developers of some of the modern programs were equal to their promotiona1 acumen, the millennium oÍ mathematics education would be here. Even though many teachers on the writing teams were disappointed and even chagrined by the compromises they were obliged to make, they nevertheless returned to their home districts proud to have been participants in fashioning curricula and naturally inclined to favor what they had helped to create. They soon became the ardent champions of modern mathematics and took the .lead in promoting it. Many college professors seeking an activity in which they believed they would be competent  surely, they argued, high school materia1 is child's play for us knowledgeable college professors  took up the advocacy of the modern curricula and even initiated courses to train teachers in modern mathematics. Others climbed on the bandwagon because it gave them visibility and even prestige to participate in what had become a prominent activity. High school and elementary school teachers, too, anxious to show leadership in education, have taken up the promotion of what was thrust upon them. Unfortunately, because they are committed to fulltime, exhausting work in their duties as teachers, they have not had the opportunity to learn more about what is significant in mathematics and so have not been able to examine critically the new versions as the road to education in mathematics. Many teachers jumped in because they saw an opportunity to advance their writing efforts by catering to the new mathematics. Often they have done no more than dress up old texts with some sprinkling of the new mathematics and label these books modern mathematics texts. Naturally such teachers will at least outwardly profess advocacy of modern mathematics. One could rationalize such texts as compromises. The argument wou1d be that students are not ready for a radical shift to modern mathematics, especially if they have already had a few years of traditional mathematics. But these compromise texts are really not transitional. They have not worked out a reasonable unification of traditional and modern topics. They are clearly commercial jobs that give the impression of being modern mathematics texts but actually are patches of traditional and modern topics that are entirely unintegrated. The publishers, seeking to gain the edge on the market, put out series of new mathematics texts and, to ensure their adoption, not only joined in the propaganda for  the new mathematics through cleverly worded advertisements but sent speakers to teachers meetings to speak for the new mathematics. The combination of curriculum leaders, teachers who became partisans, and publishers now form a tightly knit web of vested interests preying on the mathematical unsophistication of press, public and even foundations who support this movement. Beyond the factors we have already described there are other reasons that modern mathematics is favored. There is no doubt that some teachers actually believe that the axiomatic deductive approach is the essence of mathematics. Whether they acquired this limited view through the instruction they themselves received or have been induced to adopt it because many texts favor it, they are at least sincere if not effective pedagogues. One also has the sneaking suspicion that a few teachers enjoy presenting the familiar number system in the recondite axiomatic form because they understand the simple mathematics involved and yet can appear to be teaching profound subject matter. Many young teachers believe that, now that we have the correct polished version of mathematics, it is sufficient to give the axiomatic or rigorous approach and that students will absorb it. These very same teachers would have been swamped by such a presentation, but having learned the correct version they can no longer recall and appreciate the difficulties they encountered in learning the rigorous versions. Some teachers, knowing the rigorous proofs, feel uneasy about presenting merely a convincing argument which they, at least, know is incomplete. But it is not the teacher who is to be satisfied; it is the student. Good pedagogy demands such compromises. The natural desire of the teacher is to proffer the completed polished deductive mathematics. This is certainly the more elegant version. But the value to the student is inversely proportional to the elegance and smoothness of the organization, because the final version is a much reworked and unnatural account. Other teachers want to give students the whole truth at once so that they should not have to unlearn what they once learned. But one cannot teach even English or history by starting at the top. The A that a high school student might earn for an English composition would most likely be rated C at the college level. Further, teaching 2 + 2 = 5 and then having to correct it is one thing but teaching subtraction as "taking away" and then introducing the notion that 2 is the additive inverse to 2 is another. For a youngster the latter is verbiage. Another major reason for the popularity of the axioaatic deductive approach is that it is easier to present. The entire body of material is laid out in a clear, cleancut sequence and all the teacher has to do is repeat it. He has but to offer a canned body of material. I have heard teachers complain that many students, particularly engineers, wish to be told only how to perform the processes they are asked to learn and then want to hand back the processes. But the teachers who present the logical formulation because it avoids such difficulties as teaching discovery, leading students to participate in a constructive process, explaining the reasons for proceeding one way rather than another, and finding convincing arguments, are more reprehensible than the students who wish to avoid thinking and prefer just to repeat mechanically learned processes. Postulating properties has the advantage, as Bertrand Russell put it, of theft over honest toil. Pedagogically it is worse because the theft produces no gain in understanding. We have already noted that many teachers, especially at the college level, prefer to present rigorous axiomatic approaches because they favor their own professional interest at the expense of the student. Even if such systems could be made understandable to young people the time required to teach them should be spent on more significant material. In this matter as well as in presenting sophisticated rigorous proofs the teachers are serving themselves not on1y in the form in which they present mathematics but also in the premature teaching of subjects such as abstract algebraic concepts, linear vector spaces, finite geometries, set theory, symbolic logic and matrices, because these subjects are advanced and satisfy the teacher's ego. Is it any wonder that students become alienated and question the relevance of what they are being taught? For whatever reason teachers insist on presenting to young people modern rigorous proof, they are deceiving themselves. As we have already noted (Chapter 5), there is no ultimate rigorous proof. This fact derives from the very way in which mathematics develops. The superb research mathematician and pedagogue Felix Kiein has described it. "In fact, mathematics has grown like a tree, which does not start at its tiniest rootlets and grow merely upward, but rather sends its roots deeper and deeper at the same time and rate that its branches and leaves are spreading upward. . . . We see, then, that as regards the fundamental inuestigations in mathematics, there is no final ending, and therejore on the other hand, no first beginning, which could offer an absolute basis for instruction. "Poincaré expressed a similar view. There are no solved problems; there are on1y problems that are more or less solved. Mathematics is as correct as human beings are and humans are fallible. At no time in the history of mathematics have we been less certain of what rigor is. Hence no
proof is really complete and the teacher must comproinise in any case. It would be interesting to know how many
teachers are aware that set theory, which they now regard as the indispensable beginning to any rigorous approach
to mathematics, has been the source of our deepest and thus far insuperable logical difficulties. Those who are
not aware of the foundationa1 problems might at least note the words of one of the foremost mathematicians of our
time, Hermann Weyl. The question of the ultimate foundations and the ultimate meaning of mathematics remains open;
we do not know in what direction it will find its final solution nor even whether a final objective answer can
be expected at all. Mathematizing may well be a creative activity of man, like language or music, of primary originality,
whose historicai decisions defy com Though the rigorous axiomatic deductive approach is favored, there are indications that some professors who present rigorous material are really uncertain as to the wisdom of doing so. A number of calcu1us books begin with rigorous definitions and theorems, for example, those conceining Iimits and continuity, and then never refer to this material. Thereafter they use the cookbook presentation. The most charitable view of such books is that the authors wish to ease their own consciences or to give the students some idea of what rigor means. Perhaps a more accurate view is that these bóoks offer only a pretense of rigor in order to appeal to both markets, the one that demands rigor and the one that is satisfied to teach mechanica1 procedures. Other texts adopt another compromise. In the body of the text the presentation is mechanical with perhaps an occasional condescension to an intuitive explanation. The real explanation is given in rigorous proofs, but these are put in appendices and presented so compactly that they are certain to be totally ununderstandable to the student. However, the authors have salved their consciences. Such books are no different from the old mechanical presentations. They do contribute to understanding in one respect, namely, they show that competent mathematicians are inept in pedagogy. For the various reasons we have been citing, modern mathematics has now become quite fashionable. But fashion, as Oscar Wilde put it, is that by which the fantastic becomes for the moment universal. Copyright © Helen M. Kline & Mark Alder 2000 Version 22nd March 2001.
