CHAPTER 11: Some Mandatory Reforms.
Gottfried Wilhelm Leibniz
 The defects in our educational system cannot be eliminated by one measure. There is no one cure for all diseases. Yet, little by little, medicine has conquered some and alleviated the gravity of others. In the educational field the universities' insistence on research as the qualification for appointment and tenure of professors (despite the low quality of much of the research and its irrelevance to teaching), large lecture classes, the use of teaching assistants on a wide scale, and inadequate textbooks are all highly detrimental to the progress of mathematics and to the effectiveness of education. Some helpful steps are apparent, and we have to be willing to take them.
The first remedy lies in recognizing scholarship as well as research. Research in mathematics means the creation of new results or, at least, new methods of proof. Scholarship - which fundamentally implies breadth, knowledge in depth,  and a critical attitude toward that knowledge - is currently deprecated. This distinction is not made in the social sciences, the arts, and the humanities. The person who digs up facts about an older civilization, who writes a detailed and perhaps critical biography of some major or minor historical or literary figure, or who puts together various theories of economics or government is considered creative, though there may be no single new fact in a given work. Of course, there are seminal thinkers in the nonmathematical fields. Some of their work is as novel, as creative, as anything produced in mathematics. But the distinction between old and new cannot be made as readily. In any case, in these fields new work is only a small part of what is accepted and even honored as research. The re-search of what has been done is accorded as much distinction as new work. In fact, a critical biography or evaluation of a man or an era is often lauded far more than the man or men whose work is being assessed. A lucid explanation or interpretation of mathematical research is worth far more than most research papers. Unfortunately, such presentations, even if of high quality, are held in low esteem. But it is scholars - people with a deep and broad knowledge of mathematics and an ability to communicate whether or not they contribute new results - who can correct many evils and perform many vital tasks.
In recent times the overemphasis on research has forced the production of tens of thousands of papers. Beyond the unreasonableness of compelling teachers and scholars to do research, it has also obliged mathematicians to seek specialties so that they can keep abreast of what is being done in the area in which they seek to publish. The consequences have been a proliferation of obtuse papers of dubious value and the fragmentation of mathematics into an incoherent mass of details. More than ever, it has become necessary to decipher the cryptic research papers, to salvage the gems from the sludge, to connect in a coherent account  the mass of disconnected results appearing in the hundreds of journals, and to give prominence to deserving contributions.
More specifically, scholars can elucidate the inscrutable results contained in research papers. Because they can spend more time reading the published papers, they also can detect duplication of results; and the very knowledge that such duplication will be detected would deter those who would consciously publish old results disguised in new terminology and symbolism. Scholars can serve as critics of writing and here, too, force authors and publishers who would fear criticism to aim higher.
There are other valuable functions for the scholar. He can write an expository paper on results obtained in one area and so make the methodology and results accessible to those in other fields. Such a paper can tie together in one intelligible account many research articles that individually mean little or that are of discernible significance only to the specialist. Expository papers would not only broaden the knowledge of currently narrow researchers but might also reveal and synthesize the relationships between works in different areas. They might, in addition, aid scientists and engineers to learn about work that can be useful to them. Survey papers do appear occasionally in the literature, but these are written by specialists for specialists in one particular area and do not aid outsiders.
Today researchers hold innumerable conferences on specialized topics. Physicists and engineers wisely do not attend the usual conference, because it is unintelligible. Instead, they spend time creating what they need, even though their creations may already exist in the mathematical literature. Scholars could lead conferences that explain the research to a broad spectrum of mathematicians, physicists, and engineers.
Lucid, critical, perceptive synthesis, so sorely needed in  the present deluge of publications, would evaluate the import of the findings of the various specialized disciplines and assess them in the light of the overriding questions which prompted the development of the specializations in the first place and which parochial specialists have often lost sight of. Thereby synthesis and synthesizers would keep alive and in the forefront what the discipline as a whole is trying to do or should be doing. When one thinks of the massive manpower, money, and space devoted to mathematical research and the conflict between it and education, then certainly the evaluation of mathematical research and the determination of for what and for whom it is intended is of vast importance.
Scholars, raising questions about the worth or direction of a particular specialization, would keep alive the spirit and activity of dissent. Specialists should be called upon to defend their activity and not be permitted to hide behind such shibboleths as "I am creating art," or "I like what I do." There is a function for the gadfly who poses questions that many specialists would like to overlook. Polemics are healthy. They not only tend to reduce fashions to a scale that their worth warrants, but they also may support the worthy unfashionable and the only seemingly ridiculous. Without scholarship - the organization, explanation, interpretation, and criticism of research - the currently vast number of proliferating disciplines steadily gain in quantity as they lose in quality, vision, and effective use of the little in them that is worthwhile.
Scholars would, by definition, be thoroughly informed in the history of mathematics, which has many lessons that should be brought to the attention of all mathematicians. For example, faddism determined some directions of research. The pointlessness of some of what was done in past centuries might serve to dampen the fire of rampant current fads. History might also remind mathematicians of the major  problems and goals. It is even possible that researchers, informed by scholars of the history of mathematics, might learn humility by becoming acquainted with the truly great works of the past.
Scholarship is not easy or shallow, as researchers would have us believe. It requires a mentality more precious than most original research requires, a judgment of intellectual values that enables one to discriminate between the significant and the incidental, a sophisticated and well-developed common sense, and a sympathetic imagination.
Victor F. Weisskopf, a distinguished professor of physics at the Massachusetts Institute of Technology, has expressed the need for scholars in all of the sciences. Writing in Science (April 14, 1972), he says:
Why aren't scholars performing the functions we have described? Because scholarship, as opposed to research, is  not valued, professors do not cultivate it. There is no place for the scholar in the mathematical world. He either receives no appointment or, if by chance he should acquire appointment and tenure, his work is not recognized. Nor does he find a ready outlet for his work. Current policy bars the use of space in prestigious journals unless one has something new to report.
Mathematicians have always constituted a clannish, elitist, snobbish, highly individualistic community in which status is determined, above all, by the presumed importance of original contributions to mathematics; and in which the greatest rewards are bestowed upon those who, at least in the opinion of their peers, will leave a permanent mark on its evolution. To most mathematicians scholarship is a confession of failure as a researcher, a tacit admission of inability to compete in the arena of pure mathematics. Thus, in the highly status-conscious world of mathematics it takes courage and even sacrifice to violate the canons of respectability; one incurs derision or ostracism.
A scholar performing the functions we have just described may not be a good teacher. The assimilation, synthesis, and evaluation of research are as demanding in time, energy, and special cast of mind as research itself - in fact, more so than most of the research done. True, the scholar would be more broadly informed and, in this respect, better qualified to teach. But knowledge alone does not make a teacher. What is the solution to our pedagogical problem? How are we going to get capable, dedicated men and women who will fashion the courses and write the texts that will inspire and meet the needs of students with diverse interests; spend time counseling students; prepare motivating themes and routines of discovery; know and use in their teaching attractive applications to the physical, social, and biological sciences, psychology, actuarial work, and other fields; take advantage  of new teaching devices such as laboratory materials and films; stimulate thinking; study the rationale and methodology of examinations and grades; and, of course, be able to communicate with young people? Such men and women must also be concerned with what the elementary and high schools are teaching, so that college courses can be built upon what the students actually know when they enter college. The desirable men and women will have to know how much young people can understand about abstractions and what motivations will appeal to students at various ages. (See also Chapter 4.)
Clearly, we need a third class of faculty - namely teachers - to perform functions that are quite different from those of researchers and scholars. We need. teachers of whom students can say, with Chaucer, "Gladly would he learn and gladly teach." Specialists may believe in the value of their subject, but their beliefs usually stop short at this point. Of course, just as scholarship feeds on research, so teachers would look to scholars for sustenance. Otherwise, teaching would become a repetition of stale, outmoded, and downright incorrect prattling. The magnitude of the problem of supplying college teachers is rarely appreciated; there are now about ten million undergraduate students in colleges and universities.
College administrators and professors accept, or pretend to accept, as axiomatic that a Ph.D. well prepared in his specialty, competent as a researcher, intelligent, and sincerely devoted to his subject will certainly be a fine teacher; preparation, training, and special qualifications for teaching are unnecessary. They hold to these beliefs despite the oft-made observation that college teaching is a major learned profession for which there should be a well-defined program to develop the skills that practitioners must possess and to select those who can acquire the skills. Many  professors regard with disdain the training needed for teaching.
To interest men and women in a college teaching career, we shall have to change radically the attitudes and policies of universities. The shibboleth of research, the glorification and the superior emoluments accorded to it, must be countered if we are to combat the low esteem in which teaching is held. Teaching must be rewarded in every way as much as research. Teachers influence thousands of students during their careers and are far more vital to society than researchers except for a Newton. At present, teachers in universities, if retained at all, are tolerated, and their salaries are far below those of researchers.
The argument is often advanced that teachers stagnate. But if the functions described above are required and are performed, there would be little likelihood of stagnation. Indeed, the risk of stagnation is greater in the case of researchers, who, as we have noted (Chapter 4), can and do burn out. Since their function is to strengthen the research capacity of a department, if they do deteriorate they are truly useless for many years of their tenure.
Many people have called for recognition of scholarship and teaching. Dr. Alvin Weinberg, Director of the Oak Ridge National Laboratory, has spoken out forthrightly:
*Reflections on Big Science, M.I.T. Press, 1967.
 In an article in Science (May 6, 1965), Dr. Weinberg stressed the need for teaching:
Should scholars and teachers be expected to publish? Though exceptions should be permitted, some publication is in order. However, the word publication must be understood in its broadest sense. It should embrace far more than research papers containing new results. A good expository paper will benefit far more people than most research papers. A good text is worth a thousand of the usual trifles that appear in research journals. Significant articles on pedagogy are sadly needed and should be welcomed.
Will all scholars and teachers perform as expected? Of course not. The variation of quality among such people will be as great as that among researchers. Some pressure to pursue self-development and up-to-dateness can be exerted by using promotion and salary as incentives. But these measures are all that one can invoke to oblige researchers to continue to produce high-quality research.
Scholars and teachers must be trained. The present staffs of graduate mathematics departments cannot do the job, and changes will have to be made. Research professors  cannot teach the appropriate courses and even regard it as beneath their dignity to do so. Professors who are sympathetic to training scholars and teachers will have to be appointed, and graduate courses that stress breadth of knowledge rather than specialization will have to be instituted. Moreover, there must be collateral courses in the physical and social sciences that will enable teachers to acquire knowledge that can be used to motivate students and to teach applications of mathematics. Such professors and courses would educate college teachers and provide the proper instruction for prospective elementary and high school teachers.
The broadening of graduate education is necessary for other reasons as well. One of the new features of the current educational process is the rise of community (two-year) colleges. Some figures are highly relevant. The University of California has 100,000 students at eight university centers. There is a California state college system with 250,000 students on nineteen campuses. In addition, there are 100 community colleges in California that cater to 700,000 students. Nationwide, about 50 percent of the full-time undergraduate college students are in about 1,200 community colleges - as opposed to about 1,600 four-year colleges and universities - and the percentage in the community colleges is sure to increase.
The community colleges face a number of special teaching problems. A third of their courses are essentially remedial. About 75 percent of the students seek to learn only technical subjects that will prepare them for jobs, and they terminate their education at the end of two years. Such fields as computing, statistics, probability, and applied mathematics at a low level are the ones that must be emphasized. Clearly, the teachers must have a broad background, possess interdisciplinary skills, and be willing to work at the level in question.
 Many community colleges are wary of hiring Ph.D.'s and even prefer to hire high school teachers. Certainly, Ph.D.'s geared to research are not the proper teachers for community colleges. Those who really want to do research will not take any interest in teaching and will leave as soon as possible for a university position, and those who are willing to devote themselves to the requisite tasks are poorly prepared. Even most Ph.D.'s who teach at four-year colleges are not properly prepared. They too are asked to teach statistics, probability, computer science, and physical applications, but they have little idea of how to cater to such interests.
Beyond training two- and four-year college teachers, graduate programs should serve other interests. The variety of graduate students, high school and college teachers wishing to improve their backgrounds, housewives returning to teaching, practicing engineers and statisticians recognizing the need for more mathematical training, and adults seeking advanced or updated education, must be catered to. History does not ensure prophecy, but the growth in technology seems to indicate an increased demand for graduate education by nonspecialists in mathematical research.
Unfortunately, the many fine intellects that could be devoted to scholarship, teaching, and the training of scholars and teachers are now hindered or even wasted through miseducation. In an article in the American Mathematical Monthly (September 1969), Professor I. N. Herstein of the University of Chicago pointed out that 75 percent of the students trained to do mathematics research never do so after acquiring the Ph.D.; they become teachers at two- and four-year colleges that do not demand research. Other studies confirm this fact. Moreover, surveys made of the research done by Ph.D.'s report that fewer than 20 percent published even one paper a year, and this says nothing about  the quality of those papers.
No doubt many factors disincline or deter Ph.D.'s from pursuing postdoctoral research. Most of those men and women, intelligent enough to complete a doctoral program, could be trained to be scholars and teachers. But currently they are forced to take a research-oriented Ph.D. The practice of training researchers and then asking them to spend a good deal or all of their professional lives at teaching contributes to the debacle of undergraduate teaching. The universities are wasting an enormous amount of professorial and student time, energy, and resources in failing to recognize the misdirected education of the 75 percent.
Several independent commissions have urged the reorganization and broadening of graduate school objectives. One suggestion in particular has received much attention, namely a new degree, to be called Doctor of Arts, that would serve scholars and teachers. As far back as 1946, Howard Mumford Jones, in his Education and World Tragedy, advocated a separate degree for scholars and teachers. The Carnegie Commission on Higher Education, in Less Time, More Options, a report of 1971, not only supports the Doctor of Arts degree but also points out why it is far more desirable than the Ph.D. for prospective teachers. The report states:
The report stresses the need for a degree program that qualifies people to become teachers, to declare by its very existence that teaching is also important and will be equally rewarded. Narrower and narrower specialization should not dominate higher education. In fact, the Doctor of Arts should be the standard liberal arts advanced degree; the Ph.D. could be retained for the relatively few who train for research. Both the Carnegie Corporation of NeW York and the Sloan Foundation have supported the initiation of the D.A. degree in several universities. As of 1975, about twenty-three universities, none of the most prestigious group, offer the D.A. in one or more departments.
Though the specific requirements for the D.A. degree may vary from one institution to another, it should call for a broader knowledge than the Ph.D. degree, perhaps include a course on the nature and problems of a liberal arts program, and require an expository thesis that would also evidence ability to write lucidly. The thesis might synthesize existing knowledge or even tackle many unsolved problems of education, such as motivation.
Many people believe that a Doctor of Arts degree is a 7 simplified doctoral program, a second-rate Ph.D. It is, in fact, the Ph.D. program that is second-rate: Candidates take routine courses, show some ability to learn these in an examination that any D.A. could equally well be required to pass, and then take specialized courses or seminars that require no more than a capacity to learn. There is, of course, the thesis, which in mathematics at least calls for original  research. But the problem is set by a professor, and usually the student gets continual help from the professor. Adolf Hurwitz, a leading mathematician in the first quarter of this century, rightly said, "A Ph.D. dissertation is a paper written by a professor under aggravating circumstances."
The usual thesis is not a sizable contribution to new knowledge or, if it is, is not necessarily the student's contribution. Thus, the Ph.D. does not certify research potential or promise a continuing interest in research. What is finally produced depends very much on the standards of the professors. These are often low, because some professors are too kind and others are anxious to show how many doctoral students they are turning out. The criterion of publication in a respectable journal also ensures nothing, because almost anything can be published these days - and, in fact, publication is no longer a common requirement. In many subjects, the Ph.D. thesis requirement of an original and significant contribution to knowledge is almost meaningless. Graduate students in the physical and biological sciences often work as members of a team, and the students are allowed to choose some detail of the team?s results as the substance of their theses no matter who did the real thinking. As for Ph.D. theses in the social sciences and the humanities, the less said about originality and significance, the better.
On the other hand, an expository, critical, or historical thesis such as might be required for the D.A. degree could be demanded of the student as his work. There would be far less question of who is accomplishing what. The quality of the Doctor of Arts degree, as in the case of the Ph.D., will depend on the quality of the institutions and standards of the professors. The qualities and performance required of the candidates could be so demanding that a professor could  justifiably say to an Arts degree candidate, "If you do not feel up to this program you can settle for the Ph.D."
The usual product of the Ph.D. program is a person of narrow knowledge and broad ignorance. The D.A. may not be any wiser, but he will have some breadth and will be trained for the work that he and most Ph.D.'s subsequently do.
The Doctor of Arts need not refrain from research. If he finds that he likes research, he can pursue it to the extent that time permits; he might later take a Ph.D. program, engage in research, and publish. Universities could even grant the Ph.D. to such a candidate solely on the basis of publications, which are more likely to be valuable than the hothouse-forced, professor-guided Ph.D. theses. Alternatively, the Doctor of Arts could pursue research training in an organization such as The Institute for Advanced Study in Princeton, which gives no degrees or credits but does offer courses and contact with leaders in research. Training in research could be a postdoctoral undertaking, just as specialization in a branch of medicine is post-M.D. training.
Another group, the Panel on Alternate Approaches to Graduate Education, in its report Scholarship for Society (Educational Testing Service, 1973), has also advocated broader programs in the graduate schools. In the light of modern needs and changing circumstances, it concluded that new elements need to be added to the graduate programs and the horizons of concern expanded. Interdisciplinary studies were also stressed. The Panel scored the graduate schools for their concentration on training only for research and publication. The faculties of the graduate schools, now under the control of research professors, were urged to take a broader view of their professional roles and not to make decisions on tenure, promotion, and salary even  for graduate faculty only on the basis of research and publication:
The Panel concluded that there must be a "re-evaluation of the basic objectives and organization of each graduate school and the disciplines essential to it."
President Derek C. Bok of Harvard University, in his annual report of 1972-73 to the Board of Overseers, stated quite frankly that Harvard's Graduate School of Arts and Science
President Bok agrees with other surveys that more and more of even Harvard Ph.D.'s will be devoting their careers  to teaching and adds, "But surely it is odd to continue placing such exclusive emphasis on research when so many of our students will spend large parts of their careers in predominantly teaching institutions." He also points out that the problem is hardly new and has been raised many times in the past by presidents, deans, and iconoclastic professors.
Yet little has changed, either at Harvard or elsewhere. There is a good chance, however, that the universities will broaden their graduate school programs in the near future. But if they do, in most cases it will not be because they wish to make amends for the narrowness of their present programs or because they have become conscience-stricken about their indifference to the needs of society. It will be to alleviate financial problems by admitting more students and collecting more tuition.
The current academic effort is dichotomized. When research in the United States became a viable activity, the universities, falling for the prestige of research and financial gains made possible thereby, strained their manpower and other resources to win that prestige. But the country as a whole, dedicated to universal education at higher and higher levels, needs broadly informed and competent teachers and scholars - which it has not yet produced. Just as two men pulling on opposite ends of a rope fall to the ground when the rope splits, so our mutually competitive educational efforts have suffered. The research has sunk to fruitless specialization and our classrooms on all levels are staffed with poorly trained or mistrained teachers. The empty space that separates the two ends of the split rope is, in the educational area, the vacuum that remains to be filled by teachers and scholars.
The problem of what to do with research professors who cannot teach even on the graduate level also remains to be solved. Actually, in some cases teaching and research could complement each other; some professors do derive stimulation  from teaching. Moreover, teaching can keep a professor psychologically at ease during an otherwise unproductive period, because the time is being profitably used. Further, it is the desire of any professor who has a worthwhile direction of research to see it pursued, and his students are the most likely candidates for such pursuit. But researchers who cannot or will not teach should be employed in institutes that are devoted entirely to research, conduct no formal teaching, and give no degrees. In such institutions the researchers would be free to pursue their interests full time.
The benefits to be derived from segregating excellent researchers who cannot or will not teach are immense. To burden such people with formal teaching is to deprive them of time and energy and to distract them from their valuable work. It is equally important that such researchers be dislodged from dominance in formulating university policies. Although they may soar to great heights of thought much as a towering mountain peak may rise among hills, they are limited in range. Their role in the educational process may be detrimental. As long as researchers are regarded as the cream of the university personnel, more valuable than scholars, teachers, and administrators, their needs and judgments will usurp those of others. The elitism and snobbery of some are an affront to sincere and capable colleagues. Recognition of the shortcomings of researchers is as important as helping them to function effectively.
Special institutes for researchers abound in Germany and the Soviet Union, where they are government-supported. In Germany, for example, the Max Planck Gesellschaft (the Max Planck Society for the Advancement of Science) directs about fifty research institutes called Max Planck Institutes. They employ about ten thousand people and cover many areas of research. (The Max Planck Gesellschaft is the direct successor of the Kaiser Wilhelm Gesellschaft, which was  founded in 1911. One of the institutes, devoted to physics, was founded for Albert Einstein in 1917.) These institutes are free to choose their own direction of research. Though there are government-supported laboratories in the United States, these have specific missions and are not free to determine their research programs. The Institute for Advanced Study in Princeton* is the only organization that approximates the German and Soviet institutes; however, our own Institute is privately endowed and cannot be the haven for all who should be accommodated. This is not to belittle its contribution. For instance, when Einstein was forced out of Germany by Hitler he became one of the six chosen to inaugurate the mathematics division. Certainly, a number of such institutions for fine research people would keep research active and permit universities to confine themselves to researchers who would promote the numerous functions of the graduate schools.
Still another measure could be adopted by some universities. A few declare openly that their chief function is to do research and train research people. Since the faculties of these universities either cannot teach undergraduates or, even if capable of doing so, cannot take the time and energy to do so properly because research is so demanding, these universities should discontinue undergraduate education. Of course, they would be reluctant to do so. They want the undergraduate courses to help provide jobs for the researchers and for graduate students, and they want the tuition that undergraduates pay, or that the states provide, to support research. But it is just these abuses of undergraduates that must be eliminated, and any justification offered by these universities for retaining undergraduate schools (such as the opportunity they give young students to come into  contact with "great" minds) is factitious. The model for purely research-oriented universities could very well be Rockefeller University in New York City. It is outstanding in research strength and offers only Ph.D. degrees.
*The Institute has no formal connection with Princeton University. It has its own administration, objectives, and funds.
However, there are measures that can enable a university to educate undergraduates and still retain the research and graduate training. To eliminate robbing undergraduates in order to favor graduate teaching and research, the two missions of mathematics departments (and other departments) - graduate and undergraduate - should be administered separately and their budgets maintained separately. Undergraduate tuition should be used solely for undergraduates. Moreover, appointments to undergraduate teaching should not be subject to approval by the head of the graduate department. The segregation of funds and authority would permit the undergraduate divisions to hire full-time competent teachers in place of the low-paid, immature graduate students who now do the bulk of undergraduate teaching. Any surplus could be used to increase scholarships and facilities.
The suggestion that the undergraduate and graduate functions be separated may appear radical. Actually, it would be a reversion to an older order. When graduate education and research were first undertaken in this country, during the last quarter of the nineteenth century, the graduate faculties were separate. It was President Charles W. Eliot of Harvard who, in 1890, first established a single faculty for both divisions. This move was wise at that time, because the teaching faculties were weak and the researchers were more able teachers of the advanced courses. Also, through close contact with the teachers, the researchers exerted a largely beneficial influence.
The separation recommended for today's universities does not mean that one division should not influence the  other, or that the faculties must be entirely distinct from one another. A wise undergraduate chairman will utilize the services of a graduate professor if he believes it will improve the undergraduate program, and the graduate chairman wiii do the corresponding thing. A wise dean would suggest or encourage such cooperation.
Researchers oppose the total separation of graduate and undergraduate schools. The arguments are, first, that "A good graduate school provides the best way of assuring high academic and intellectual standards." Just how these standards work their influence on the undergraduate schools is not made clear. The research professors are not at all interested in the undergraduate activity. What is clear is the disregard of the undergraduates in favor of the graduate faculty (Chapter 5). Second, "A good graduate school provides the best way of recruiting a distinguished faculty. The strength of a faculty depends on its creativity - on the opportunity to do research, write books and work with advanced students." Again, what does this have to do with undergraduate teaching?
And then comes the giveaway. "Third, the graduate school is responsible in great part for the reputation of a university, for the impression outsiders have of it" (italics added).* But what value does the reputation of a university for graduate work or the impression this reputation creates have for undergraduate teaching? And if one does attract good undergraduates but does not do justice to them, how can one justify attracting the students and then fouling up their lives? The prestige of the graduate school ensnares the best undergraduates, springs the trap, and then lets them perish while struggling to free themselves. One is quite safe in assuming that the more prestigious the university, the  fewer its educational concerns and the poorer its educational effectiveness. Certainly, as long as research is the criterion of faculty value, few benefits will accrue to teaching.
*These arguments were presented, as quoted above, on the Op-ed page of the New York Times of December 17, 1975.
Still another argument for the status quo is that graduate schools "attract the best teachers." If the purport is that the presence of a graduate school attracts good undergraduate teachers, then the facts deny this contention. A good graduate school attracts young people who seek to promote their own research more readily, and these people aspire to become prestigious researchers rather than great teachers. Moreover, at most universities they have no choice; their futures depend only on their achievements in research.
The claim that undergraduate professors need the backing, stimulus, and knowledge of the researcher is false. There are some excellent four-year colleges in this country where the best teaching takes place (Chapter 5). The professors in such schools are entirely isolated from research professors. A testimonial to the superior teaching performance of the four-year colleges was recently given by a professor at one of the most prestigious universities. Speaking of his own institution he said, "It wants to be Amherst to its undergraduates and Yale to its graduate students." Surely, then, independent undergraduate divisions in universities would not suffer.
The objection that research professors advance to the separation of graduate and undergraduate colleges of a university is entirely self-serving. Political colonialism of the nineteenth century has been ref ashioned into academic colonialism by which the graduate division preys on the undergraduate just as surely as wolves prey on sheep.
If the graduate schools are forbidden to use undergraduate funds, how will they maintain themselves? The answers are to cut down the number of research professors, require more teaching, cut out the competition for high-salaried  researchers, and eliminate those men and women who, though called professors, contribute no more than their names. Money would also be saved on supporting personnel and space. The indirect gains would also be appreciable. Fewer specious or trivial papers would be published, and the truly worthy ones would stand forth. Further, because the number of journals in all fields is now so large that libraries cannot find the space and funds for them, reduction in the size and number of journals would reduce library costs. These costs are now so great that a current study financed by major foundations (New York Times, June 23, 1975) is seeking new ways of preserving and disseminating research that would eliminate journals.
Research and graduate teaching were dire needs in 1876. In 1976 both must be re-examined. While a greater variety of graduate programs is needed, the total number of graduate degrees can be reduced now and for the forseeable future; all predictions of undergraduate enrollment agree that it will decline. The universities are even now competing fiercely for students, and the competition will be stiffer as college enrollment drops. Quality of education is in the long run the best attraction, whereas the roseate bubble of prestige, which has enveloped many institutions and enabled them to look attractive, may be pierced and collapse. In any case, the sacrifice of the undergraduates to favor the graduate activities cannot be tolerated.
The separation of undergraduate and graduate functions would have another positive value. At present the running of the undergraduate activities -decisions on courses to be offered, selection of teachers (whether graduate students or mature people), choice of texts, and teaching schedules - is assigned to an administrative assistant by the head of the department. The low quality of the resulting undergraduate education does not worry the head. He knows that his  performance will be judged by the research activity. On the other hand, the undergraduate chairman, if properly chosen and a power in his own right, would seek out competent teachers. Thus, teachers would finally find a place in the universities, and the money to pay them their due would also be available. The graduate departments, in turn recognizing that their existence would depend in part on getting jobs for their graduates, might be obliged to introduce a doctoral program that would actually train teachers.
Of course, there are disadvantages and even dangers in separating graduate and undergraduate administrations and budgets. Friction, competition for funds and facilities, and even opposition may develop. It is trite to say that we must choose the lesser of two evils, but when the lesser evil is to do justice to the millions of undergraduates and, at that, indirectly provide some benefits to graduate teaching and research, the choice is clear.
It is not unlikely that even government intervention may force improvements. The most prestigious universities practiced racial and religious discrimination for decades. Present federal regulations have compelled elimination of this and other unfair practices. The likelihood of further governmental action is implied by the statement made by President Derek C. Bok of Harvard in his 1972-73 annual report to the Board of Overseers:
State departments of education have been rather lax in imposing standards of quality, but the call for a crackdown has already appeared in newspaper editorials.
Americans might well consider another measure. Elementary school teachers used to receive training in what were called normal schools, and secondary school teachers were taught in colleges devoted to teacher training. Both types of  schools were indeed poor, but only because all levels of education and knowledge in this country were poor. By 1960, the normal schools and teachers colleges had converted to liberal arts colleges, often with graduate and research programs. The latter have sought to emulate the large universities and have enlisted faculty who are sub jectoriented and who seek to obtain recognition in their respective professional circles. Teacher education has been sacrificed. Prospective teachers now attending these altered colleges or the liberal arts colleges of the universities are taught content by the typical Ph.D. or by a graduate student, neither of whom is informed in the specialized needs of the students. We might now reverse history and re-establish four-year colleges devoted to teacher training. Such colleges today could demand and obtain knowledgeable faculty that would devote itself to the education of teachers. Perhaps the wisdom of such a move might be more evident by considering the situation in engineering schools. These schools must be separated from liberal arts colleges because the engineering students have to learn many technical subjects. Nevertheless, in most universities all mathematics courses are taught by liberal arts professors who care little and know less about what kind of mathematics is best for engineers. Hence, as noted earlier (Chapter 7), many engineering colleges of the large universities offer their own mathematics courses rather than have their students take those offered by the academic mathematics department. Though the proper education of prospective school teachers is closer to liberal arts than is that of future engineers, nevertheless there are features that demand specially oriented professors.
The need for such specialists is underscored by events of the 1960s. College professors should know what students learn in high school, and professors who train prospective  high school teachers should certainly know what these teachers will face. During the 1960s the high schools most commonly taught the New Math. When the subject of the New Math was broached in conversations among professors, even men who were specifically undergraduate teachers would ask, What is the new math? Whether it was good or bad, the high school teachers were obliged to teach it.Certainly, then, college professors involved in training high school teachers should have been fully informed about that curriculum. Indeed, many apologists for the New Math blame its failure not on the content but on the fact that the teachers were not prepared to teach it. There is some substance to this contention.
Since the universities do not on their own initiative undertake to meet the needs of most of the students, one might look to other agencies to exert pressure on the universities. The most prestigious agency is undoubtedly the National Academy of Sciences. The Academy should certainly be concerned with the scientific health of the nation. But apparently, the most important function of this organization is to elect members - or, as one member put it, to bore those who belong and to embarrass those who do not. The election process itself may give some indication of what can be expected of the Academy. Election must be initiated through nomination by a member. Whom would a member know? Most likely his own colleagues. And he would prefer such men because honored colleagues imply honor to oneself. On the other hand, a rival in research is clearly not worthy of honor. The actual election of nominees is a political bargaining session. Because the members, coming from numerous, diverse, and highly specialized fields, really cannot judge most of the men they vote for or against, many biases determine the decisions.
The election process is incredthly complex and devious.
When the academicians Norbert Wiener and Richard Feynman, both distinguished in their respective fields of mathematics and physics, witnessed these elections, they resigned in disgust. The topologist Stephen Smale, after attending his first Academy meeting, remarked irreverently, "It was really fantastic as these days passed to see how this group of America's most celebrated scientists meeting together could be so dominated by the question of just how to increase their membership and ways to remember their dead." The outcome of elections is that the in-group perpetuates itself. This is also evident from the fact that no engineers, medical people, or social scientists were among the Academicians until about ten years ago when, after protests, the Academy decided to elect members from these fields. It is significant that in the years between 1950 and 1973 twelve American scientists who won Nobel prizes - surely, some deserved it - were not members of the Academy at the time. On the whole, the membership consists of researchers who do not possess breadth of vision; one would not, therefore, expect the Academy to help in pedagogical matters. But one can expect the members to be concerned about the enormous waste in research and the damage that the flood of publication is causing. However, the Academy does nothing about these problems.
Once in a while the members are stirred into action and appoint a committee to perform a particular function. In the late 1960s, when the United States government was still pouring money into mathematical research, the members of the Academy were not satisfied that the amount was enough and appointed the Committee on the Support of Research in the Mathematical Sciences, known as COSRIMS. The Committee decided that its best argument for increased support would be that the nation needs more Ph.D.'s for college and university teaching. Even by 1965, independent  studies of future college populations were estimating that the college population would stabilize by 1970 and, somewhat later, decrease. These estimates were reliable, because the number of children who would be of college age in the next two decades was known, and the proportion of high school students entering college had not changed appreciably in several decades. Moreover, the mathematics departments were already turning out Ph.D.'s in greatly increased numbers. In fact, Dr. Allan Cartter, at that time head of the American Council on Education, warned the academic community during the boom years of the mid1960s that there would be an oversupply of Ph.D.?s and this was evident, some said, even in 1961.
Nevertheless, the figures issued by COSRIMS indicated that there would be an enormous shortage of Ph.D.'s in the 1970s. In fact, the Committee recommended an increase of two hundred more in mathematics each year over the preceding year for the next five years after 1968. Because students were thus led to believe that jobs would be available, the number of Ph.D.?s awarded in mathematics increased from 970 in 1968 to 1,281 in 1972. It has since begun to drop, because the shortage of jobs has discouraged students from entering the field. As one article put it, the good news in the summer of 1974 was that unemployment among Ph.D.'s was not much worse than in 1973.
COSRIMS not only did not do its homework but allowed its desire to obtain more funds for research to be defended on simple-minded extrapolation from what was happening in the 1960s. In effect, as one prominent professor observed, the report was a piece of propaganda rather than one based on hard-headed research. The work of COSRIMS was but one activity of the National Academy, and one might be tempted to dismiss this one "mistake" as minor. But those young people who were encouraged to take a Ph.D. in the expectation of obtaining employment as college teachers and are now walking the streets cannot be quite so charitable.*
Another organization that could have significant effect on unbridled, low-quality research and on the programs of the graduate schools is the American Mathematical Society. At its founding in 1894, the Society planned to devote its efforts to teaching, writing, research, and any labor that any member might consider appropriate (Chapter 2). Writing, on all levels, was explicitly recommended. In particular, the Society described itself as an organization of teachers:
Of course at that time research in the United States was in its infancy. As research became more active, the Society turned its attention more and more in that direction, and it began to extol research as the only significant activity of college professors.
One would expect, however, that the interest in good research and in the diffusion of relevant knowledge into the classroom would cause the Society to concern itself with teaching and expository writing and to combat overspecialization, the flood of poor research articles, the imbalance between pure and applied mathematics, and the ever increasing isolation of mathematics. But none of these evils has been tackled.
The measures that the American Mathematical Society could adopt to check excessive and wasteful publication, to aid teaching, and to reduce the inhumane pressure on young Ph.D.'s to publish would require careful consideration. A  radical measure, but one that seems to have at least more merits than defects, would be to prohibit men and women who are only two or three years past the Ph.D. from publishing in its journals. This not only would permit the young professors to adjust themselves to teaching but also would eliminate many of the worthless articles that are now flooding the journals. And it would oblige administrators to judge people rather than to count publications.
*The shortcomings of the Academy are described in Boffee, Phillip M.:The Brain Bank of America, McGraw-Hill, 1975.
Certainly, the Society should advocate a wiser appointment policy. In place of the widespread three-year appointments, new Ph.D.'s should be appointed with no time limitation other than the understanding, as recommended by the American Association of University Professors, that they must serve seven years to acquire tenure, with notification at the end of the sixth year if tenure is not to be granted. Naturally, those who are clearly unsuitable could be dismissed sooner because the appointments are legally on a year-to-year basis. But if a young teacher feels that he has at least six years in which to adapt himself to a faculty position and can therefore allow some time to recognize his natural interests and aptitudes, he is more likely to devote himself to teaching, to building a better background for research, and to determining what his major contribution to the university can be. A new Ph.D. also needs time to recoup psychologically from his doctoral work. The university in turn will have more time to judge what his actual strengths are. If it should be necessary to terminate the appointment of a teacher who has served for five or six years, such a teacher would normally be about thirty years old, still young enough to find another position. In fact, having had the opportunity to examine himself, he could seek an appointment suited to the career most congenial to him. 
The Society could expose other undesirable practices. It could, for example, ask for more integrity in university administrators. Many a department today appoints dozens of new Ph.D.'s for a three-year period knowing that there would be no possibility of a permanent appointment for more than one or two. These appointments are not carefully screened. The reason given for the multiple appointments is that the tyros will be observed during the three-year period and the best retained. But the real reason for the policy is that the university can boast of having a large number of research-oriented Ph.D.'s while paying them about one-third of a full professor's salary.
The Society also should be concerned with the quality of its publications. Since refereeing of papers is now an almost impossible task (Chapter 3), the Society could insist that papers submitted to its journals be readable; thus, at least correctness could be determined. Most authors write as though they were more concerned to be admired than understood. The full worth of a paper may be harder to determine, but even this judgment could be facilitated. Each author should be required to state what, at least in his opinion, is the contribution of his paper - application to science, aesthetic value, or intellectual interest. The most likely objection to such requirements is that more journal space would be needed, and journal space is expensive. But the proposed requirements would result in eliminating a great many of the presently published papers because they are either incorrect, worthless, or duplications. The net result would be a saving of space and easier determination by readers of whether a paper is relevant to their interests. With the present uncontrolled flood, the good is swamped by the bad.
Still another organization, the Mathematical Association of America,  explicitly states its devotion to furthering the interests of undergraduate education. This organization should certainly be fighting the overemphasis on research, the low status accorded to teaching, the lack of training of graduate students for future teaching, and the use of teaching assistants. In fact, the organization was founded by a group of the American Mathematical Society, which in 1915 recognized that the Society had begun to ignore teaching in favor of research. But this organization is no more effective in its sphere than the Society is in research. No more can be expected of it than to maintain the status quo.
Both organizations appoint committees that report on current problems and urge the appointment of more committees and the gathering of information. In the last few years the need to diversify the training of mathematicians has been stressed by panels at meetings and occasional articles in the journals of the two societies. A recently appointed committee even pokes a little fun at one organization by using the title "How to Cope with M.A.A.: Mathematical Avoidance and Anxiety." But a scrutiny of these talks and articles soon yields the underlying reason for urging any reform - to help Ph.D.'s get jobs. For example, a rapidly expanding possible market for Ph.D.'s is the two-year junior college. Hence, the panels and articles address themselves to the needs of these colleges. Industry also needs mathematicians, and applied mathematics is frequently recommended as preparation for such jobs.
Of course, the advice from present-day professors in universities on the needs and problems of the junior colleges and industry is almost worthless. Very few of the men have had experience in either field and so do not know what these organizations require of employees. The advice expresses  pious sentiments motivated solely by the job shortage. The overemphasis on research, the narrowness of the Ph.D. training, and the miserable teaching of undergraduates in most universities are not attacked.
These articles and speeches attempt to cure a serious illness with aspirin. No concrete, effective measures are undertaken. Actually, many members of both societies are influential in their own university departments; some are chairmen. But they do nothing at their own institutions to broaden the nature or goals of graduate education or to secure and retain scholars and teachers. Research is the coin of the realm. Ironically, some of these chairmen sought administrative positions because they did not do research. Though a few chairmen now favor a broader thesis that might be an expository dissertation rather than original research, in the present atmosphere the chances for a person who writes such a thesis to get a job are practically nil. One chairman magnanimously said he would hire such a person for two years, but of course would not keep him beyond that period.
One other organization, the National Council of Teachers of Mathematics, warrants mention. This organization is concerned with high school and elementary school education. It is more effective in its sphere than either of the other two. High school and elementary school teachers do not have the time and opportunity to develop and test new ideas, new techniques of teaching, and new communication media. Through its journals, meetings, and workshops the National Council does provide instruction in content and methodology. But some of its leaders have been gullible. In 1961, before the New Math had received any significant testing, the Council published a pamphlet, The Revolution in School Mathematics, that not only advocated the New  Math but charged that those schools that failed to adopt it were remiss in their obligations.
Moreover, the Council should represent the views of all the members. It has not always done so. For example, during the heyday of the New Math, the Council's journal, The Mathematics Teacher, could not avoid publishing some articles condemning the New Math. Thus, when a critical speaker, invited to give a keynote speech at some meeting, submitted his speech for publication, the editors dared not reject it. But they made sure it was followed in the same issue by a rejoinder. Apparently the editors wished to prevent criticism of the New Math from having any influence on its members. Most editors of the American Mathematical Monthly, the official organ of the Mathematical Association of America, also reject articles critical of the existing conduct of education.
The cause of the shortcomings in the several organizations is readily explained. Leadership in them offers prominence, prestige, and - indirectly -personal advancement in one's university, college, or other institution. Hence, such positions are sought by members of the profession. How does one attain high position in these organizations? The procedure is not unlike what happens in politics. The president appoints the important committees, including the committees that nominate future officers and editors of journals. The committees are sure to be composed of insiders, and they make the choices for lesser committees. An aspirant to office seeks and makes the acquaintanceship of some prominent members of the society and at an opportune time offers to serve on some minor committee. These prominent members, usually already on important committees, will choose this man because his name stands out as opposed to the thousands of names of unknown  members. Going along with the majority and continuing cultivation of prominent members leads to more and more important roles, visibility in society affairs, and almost inevitably to higher and higher office. Finally, the organization sends out an election ballot to its members with Mr. BLank as the only choice for president. Of course, Mr. Blank is elected. The leadership perpetuates its own kind.
That mediocrities and power - or prestige-seekers should rise to the presidency of a national organization may seem incredible. But one has only to think of some of the men who have become presidents of the United States.
Occasionally, a fine researcher is honored for his contributions by election to the presidency of one of these organizations. But it is rare that such a person is deeply concerned about the problems of the mathematical community or, if he is, that he has the administrative capacity to be effective. In fact, most presidents are content to bask in the sunshine of the office and perform perfunctorily during the year or two that they serve. In their behalf one should add that they are not relieved of their normal duties at their universities and so cannot devote much time and energy to organizational work.
However, thoughtful, sincere, and unusually energetic men can be found in the organizations we have just discussed; and if perchance they should come to the fore, they could get their respective agencies to grapple with the problems of research and teaching and certainly aid immeasurably in solving them. This does happen - but all too rarely.
The above recommendations will not solve all the problems of mathematics education. We must know more about how children learn and what motivates young people  at various ages, and we must devise and employ more pedagogical aids; however, these measures call for more study and, unlike the ones recommended, cannot be employed at once.
Reforms are needed not only to improve mathematics education. Both the survival of mathematics in the curriculum and of research itself are at stake. The concentration on pure, esoteric studies will ultimately mean less support from society and, as Richard Courant once predicted, all significant mathematics will be created by physicists, engineers, social scientists, and schools of business administration. Poor education drives away students, and fewer students means fewer jobs for mathematicians. The demand for mathematics education was in sharp decline during the 1930s, and deservedly so. Only World War II revived it. One would hardly hope for a repetition of such a saving measure.
Though reforms of various sorts have yet to be adopted, we need not be too despondent about our educational system. The defects are somewhat excusable in the light of the early handicaps. As the United States grew in technology, power, and world influence, we sought to match other powers in scientific strength. We were the nouveaux riches who wanted scientific status. And so we imitated the western European countries, which had produced the best research. In our zeal to equal them we lost sight of one of our principles that should truly have given us pride and gratification: Unlike the European countries, we pledged ourselves early in our history to universal education and to free or low-cost education at higher and higher levels. The tasks of building and improving the educational system and at the same time striving for equality or even leadership in research were beyond the country's resources. We have attained that leadership, at the expense of quality education for all students.
 It is now time for mathematicians to shed their narcissism, broaden their vision and interests, limit research to worthwhile papers, and thereby release time and energy to the many-sided needs and tasks of education. They must cater to the interests of all students, undergraduate and graduate alike. Only recognition of the interdependence of research, scholarship, and teaching can advance mathematics itself, improve teaching, and further the multitudinous valuable uses of mathematics in our society.
I am very grateful for the kind permission of Professor Kline's widow, Mrs Helen Kline for this book to be reproduced.
Copyright © Helen M. Kline & Mark Alder 2000
Version: 21st December 2018