Why The Professor Can't Teach Mathematics and the Dilemma of University Education by Morris Kline First published 1977 by St Martin's press. This edition 2000 Copyright © Helen M. Kline & Mark Alder 2000 All rights reserved. I am very grateful for the kind permission of Professor Kline's widow, Mrs Helen Kline for this book to be reproduced.
NOTE: [5],[6],[7], etc. indicate page numbers. Contents Preface 1 1 The Vicious Circle 6 2 The Rise of American Mathematics 17 3 The Nature of Current Mathematical Research 41 4 The Conflict Between Research and Teaching 70 5 The Debasement of Undergraduate Teaching 96 6 The Illiberal Mathematician 111 7 The Undefiled Mathematician 139 8 The Misdirection of High School Education 161 9 Some Light at the Beginning of the Tunnel: Elementary Education 183 10 Follies of the Marketplace: A Tirade on Texts 208 11 Some Mandatory Reforms 235 Bibliography 272 Whether 'tis nobler in the mind to suffer William Shakespeare Melancholia by Albrecht Dürerne Bettmann Archive, Inc. Preface
JeanJacques Rousseau This book indicts liberal arts undergraduate education. In view of rather oftcited failings and shortcomings of education at all levels, why focus on undergraduate education? The liberal arts colleges are the heart of our entire educational system. They address the youth who become our acknowledged leaders or will maintain our cultural level; they offer the prerequisite backgrounds to those who enter the professions of law, medicine, dentistry, and accounting; they provide the basic academic courses for our scientists and engineers; and they educate the primary and secondary school teachers and thereby influence education at these levels. The liberal arts colleges do not explicitly train college teachers, but these men and women tend to perform in the manner in which they have been instructed as undergraduates. The emphasis in this book is on undergraduate mathematics education. [1] However, much of what will be discussed about the teaching of this subject applies to many other subjects. Numerous conversations with colleagues and a few books that have appeared recently assure me that this is the case. Even if this were not so, mathematics education is by no means just a matter of teaching one of the three R's. Mathematics is a major branch of our culture, the backbone of our scientific civilization, and the basis of our technology and financial and insurance structures. Its value to the social sciences, biology, and medicine is also by no means negligible. For all students, whether or not they will ever use the knowledge, sound mathematical pedagogy is vital: Failure in mathematics causes them to lose confidence in themselves, and this loss affects their attitude toward all education.Mathematics education has been a debacle. One need only ask a typical college graduate how much twothirds of threequarters is; and if perchance he should give the correct answer, one need only challenge it to discover his uncertainty. The only generally understood fact about the subject is that it is ununderstandable. Adults? reaction, based on schooling in the subject, is a mixture of awe and contempt; usually it is more of the latter. Even otherwise welleducated people boast about their ignorance of the subject as if it were beneath notice. Why is undergraduate education poor? The prime culprit is the overemphasis on research. The universities justify this emphasis on the grounds that research not only advances knowledge but that proficiency in research is essential to good teaching. I shall examine these contentions and propose to establish that, in today's world, research and undergraduate teaching are in direct conflict. 'Publish or perish' is not a threat to professors only, for its actual [3]interpretation is 'Publish, and perish the students.' To secure research professors, the universities are obliged to offer high salaries, low teaching loads, and the freedom to pursue specialties. Because funds are limited, the universities adopt devious practices to handle other functions, notably undergraduate education. Although the policies of universities are the root of our educational shortcomings, and professors are subject to these policies, professors cannot be completely exonerated. They have power, a measure of independence if tenured, as most are, and moral obligations to the students. They need not give inordinate attention to research. But professors are all too human. They respond to enticements such as better salaries for research; they compete fiercely for the status accorded to research; and they use that status for material gains  such as royalties from miserably written texts. Nor does their neglect of pedagogy seem to trouble many of them. Professors are content to offer courses that reflect their own values at the expense of student needs and interests. Insofar as these courses are for high school and elementary schoolteachers, they affect adversely the quality of education at those levels. There is no panacea for all the ills. But an analysis of what is taking place in our centers of learning points the way to the considerable improvement that could be made with presently available manpower, facilities, and funds. While the objective arguments offered herein have obvious force and are borne out by statistical data to be found in the bibliographic references, many assertions and recommendations are to some extent personal judgments. But the latter are based on experiences of almost five decades in teaching, research, and administration. For most of that period I have taught on both the undergraduate and graduate levels, and I have served for eleven years as chairman of undergraduate mathematics [4] at a major center. As for research, I spent two years as a research assistant at the Institute for Advanced Study, over three years in war research, and twenty years as director of a research division in one of the best graduate mathematics departments in this country. Numerous invited lectures at other colleges and universities and a few visiting professorships have also enabled me to become directly informed about what these institutions are doing. Criticism of university policies and practices is by no means new. Many of the books on higher education in America discuss poor management, lack of educational effectiveness, waste, and the gulf between administration and faculty. Unfortunately, the authors seem to believe that objectivity necessitates wishywashiness; they argue pro and con and present vague opinions about what might be done. One gets the impression that the authors are afraid to draw conclusions or to come to grips with the real issues. Perhaps the reason for the fencestraddling is the timidity ascribed to academics  or perhaps it is an honest failure to recognize priorities. However, we cannot in good conscience continue to vacillate, tread softly, and settle for bland statements. Millions of students pass through the universities, and their lives are seriously affected by that experience. It is true that the universities have many obligations and roles in our society, but the education of liberal arts students is undeniably the fundamental one, and any equivocation about the primacy of this function is an abdication of responsibility. I hope that this book will serve several groups of people. Parents, many of whom sacrifice even some necessities of [5] life to send their children to college, ought to be interested in what treatment their children may receive; the prestige of a university is no assurance of a sound educational process. Legislators who apportion funds for education should be informed on how they are used.Even research professors cannot stand aloof. The greatest threat to the life of mathematics is posed by the mathematicians themselves, and their most potent weapon is their poor pedagogy. Mathematics as a part of liberal arts education may disappear just as Greek and Latin have disappeared and as modern foreign language studies are now gradually disappearing. Finally, citizens who believe that the education of our people is the surest guarantee of the effectiveness of a democratic government should concern themselves with the operation of our key educational institutions. I wish to thank Professor Wilhelm Magnus of the Polytechnic Institute of New York, Professor George Booth of Brooklyn College, City University of New York, and Professor Murray S. Klamkin of The University of Alberta for their critical reading of the manuscript. Their willingness to help in no way implies their endorsement of the contents of the book. I am also deeply indebted to Miss Julie Garriott for her thorough and thoughtful editing and for other contributions to the publication, which extended far beyond her obligations. Lastly grateful acknowledgment to my wife Helen for her critical reading of the manuscript and her punctilious proofreading. Morris Kline
Important Note!The title of this book, which seems to cast aspersions on the competence of professors, is misleading. The author submitted a dozen or so more appropriate titles but the publisher was intransigent. Obviously he wished to trade on the name of his earlier publication WHY JOHNNY CAN'T ADD, which had wide recognition. As the author's wife and sometime secretary, I can testify that Morris Kline was keenly unhappy with the publisher's choice. This book is not an attack on professors but is rather a wideranging critique of undergraduate education. Indeed an appropriate, less jazzy title would have been A CRITIQUE OF UNDERGRADUATE EDUCATION. Helen M. Kline
[6]
In a selfcentered circle, he goes round and round, Sarah Fells Peter Landers found himself caught in a vicious circle. He had just secured a Ph.D. in mathematics from Prestidigious University and, having been well recommended, readily secured a faculty position at Admirable University. Thereupon Peter faced the problem of teaching mathematics to prospective engineers, social scientists, physicists, elementary and secondary school teachers, the general liberal arts students, and those who, like himself, had chosen to become mathematicians. Peter was fully aware of these varied career interests, and he also knew that students came to college with different drives and preparation. But he was confident that his education, typical for Ph.D.'s, had prepared him for the tasks ahead. To put himself in the proper frame of mind he reviewed his own education. The elementary school courses had been acceptable. After all, one did have to know how much to pay for five candy bars if he knew the price of a single bar. True, some operations were baffling. It had not been clear why the division of two fractions had to be performed by inverting the denominator and multiplying  but the teacher seemed to know what was correct. He had constantly referred to rules, principles, and laws. Rules, like rules of behavior, apparently applied to arithmetic, too. For all Peter had known, principles were laid down by the principals of the schools, and certainly they were authorities. As for laws, everyone knew that there were city laws, state laws, federal laws, and even the laws of the Ten Commandments. Certainly laws must be obeyed. Though under some tension as to whether he was violating laws, Peter was young and resilient. In any case, what to do was clear and the answers were right. In his review of his high school education Peter did recall so me doubts he had had about the value of what he was being taught. He hadn't understood why the teacher had to stress that the sum of two whole numbers is a whole number, or why he had to prove that there is one and only one midpoint on every line segment; but evidently the teacher was trying to make sure that no one could be mistaken on these elementary matters. After all, teachers knew best what had to be done. Peter also recalled one teacher's enthusiasm about the quadratic formula. 'You see,' the teacher proclaimed triumphantly after he had derived the formula, 'we can now solve any quadratic equation.' But Peter had been perverse and had asked the teacher why anyone wanted to solve any quadratic equation. The teacher's reply was a disdainful look that caused Peter to shrink back. His question must have been a silly one. He remembered a similar experience in geometry. After a long and apparently strenuous effort, the teacher proved that two triangles are congruent if the sides of one are equal respectively to the sides of the other. Then he turned to the class as if expecting applause. Again Peter dared to speak up: 'But isn?t that obvious' A triangle is a rigid figure. If you put three sticks together to form a triangle, you cannot change its size or shape.? Peter had learned this at the age of five while playing with Erector sets. The teacher's contempt was obvious. 'Who's talking about sticks? We are concerned with triangles.' Despite a few other disagreeable incidents Peter continued to like mathematics. He believed in his teachers. It was easy to comply with their requests, and the certitude of the results gave him, as they had given others before him, immense satisfaction. And so Peter moved on to college with the conviction that he liked mathematics and was going to major in it. His first experiences were disturbing. After his program was approved by an adviser who did not understand what an Advanced Placement Examination Grade of 4.5 meant  the adviser had thought that 10 was a perfect grade so that 4.5 was a poor one  Peter was finally registered. He entered his first college classroom for a course which happened to be English. To his surprise he found about five hundred students already seated. The professor arrived, delivered his lecture, and, obviously very busy, rushed out of the room. Peter never found out what his name was, but apparently names were not important, because the professor never bothered to ask any student his name either. Nor, Peter thought, would the professor have noticed had a different group of five hundred students appeared each time. Term papers were required, and these were graded by graduate students who insisted that 'Who shall I call next?' was correct, though Peter had been taught otherwise in high school. The size of the class and the impersonal character of [9] the instruction disturbed Peter at first, but he soon realized that the requirements of the English course could be met merely by listening. And so he relaxed. Peter's second class, one in social science, surprised him for different reasons. At the professor's desk was a young man not much older than Peter. As the instructor conducted the lesson he was obviously nervous. Somehow the lessons throughout the semester were confined almost entirely to the first part of the text. And the instructor did not welcome questions. The third class  mathematics  was a shock. Peter entered the room and found that it was a large auditorium. At the bottom of the room, at the professor?s desk, was not a man but a box, which proved to be a televisioh set. Shortly after Peter's entrance the box began to speak and the students took notes feverishly. From many seats one could not see clearly, if at all. But by coming early one could get a good seat. And so Peter managed to learn some of his college mathematics by listening and looking at a TV program. Though it was not a requirement, Peter decided to take some physics. He had heard somewhere that mathematics was applied to physics, and he thought he should find out what these applications were. The physics professor constantly talked about infinitesimals and which infinitesimals could be neglected. The mathematics professors, however, had warned that such concepts and procedures were loose and even incorrect. But Peter listened attentively. He was sure that even though the mathematics and the physics professors apparently did not communicate with each other and so did not talk the same language, their methodologies could be reconciled. He did seek counsel from his professors on this matter, but unfortunately they were not available. One was actually living out of the city, in Washington, D.C.; another was always involved in consultations [9] outside the university; and a third had office hours only on Sundays, from 6:00 to 8:00 A.M. In the junior and senior years the classes were smaller, and the courses were usually taught by older faculty. Many blithely ignored the texts they had assigned and spent the period transferring material from their notes to the board. The professors copied assiduously and the students did likewise. When the professors looked up from their notes they looked into the blackboard as though the students were behind it. Nevertheless Peter persevered, received his bachelor's degree, and proceeded to graduate school. His experiences there paralleled those of most other students. Professors were hard to contact. The bulletin descriptions of the courses bore no relation to what the professors taught. Each professor presented his own specialty as though nothing had been done or was being done by anyone else in the world. And so Peter learned about categories, infinite Abelian groups, diffeomorphisms, noncommutative rings, and a variety of other specialties. Prospective Ph.D.'s must write a doctoral thesis. Finding a thesis adviser was like hunting for
water in a desert. After many trials, including writing theses on topics suggested by his professor that, it turned
out, had been done elsewhere and even published, Peter wrote a thesis on almost perfect numbers that completed
his work for the degree. But the world soon began to close in on Peter. As a novice he was assigned to teach freshmen and sophomores. His first course was for liberal arts students, that is, students who do not intend to use mathematics professionally but who take it either to meet a requirement for a degree or just to learn more about the subject. Recognizing that many of these students are weak in algebra, Peter thought he would review negative numbers. To make these numbers meaningful he reminded the students that they are used to represent temperatures below zero; and to emphasize the physical significance of negative temperatures he pointed out that water freezes at 32° F.,so that a negative temperature means a state far below freezing. Though the example was pedagogically wise, Peter could see at once that the students' minds had also frozen, and the rest of his lesson could not penetrate the ice. In a later lesson Peter tried another subject. As an algebraist by preference he thought students would enjoy learning about a novel algebra. There is an arithmetic that reduces all whole numbers by the nearest multiple of twelve. To make his lesson concrete Peter presented clock arithmetic as a practical example: Clocks ignore multiples of twelve, so that four hours after ten o'clock is two o'clock. The mere mention of clocks caused the students to look at their watches, and it was obvious that they were counting the minutes until the end of the period. And so Peter tried another novelty, the Koenigsberg [12] bridge problem. Some two hundred years ago the citizens of the village of Koenigsberg in East Prussia became intrigued with the problem of crossing seven nearby bridges in succession without recrossing any. The problem attracted Leonhard Euler, the eighteenth century's greatest mathematician, and he soon showed by an ingenious trick that such a path was impossible. The villagers, who did not know this, continued for years to amuse themselves by making one trip after another during their walks on sunny afternoons  but when Peter presented the problem in the artificial, gloomy light of the classroom, a chill descended on the class. Peter's next class was a group of preengineering students. These students, he was sure, would appreciate mathematics, and so he introduced the subject of Boolean algebra. This algebra, created by the mathematician and logician George Boole, does have application to the design of electric circuits. The mention of electric circuits appeared to arouse some interest, and so Peter explained Boolean algebra. But then one student asked Peter how one uses the algebra to design circuits. Unfortunately, Peter's training had been in pure mathematics and he did not know how to answer the question. He was compelled to admit this and detected obvious signs of disappointment and hostility in the students. They evidently believed that they had been tricked. In his attempts to explain and clarify other mathematical themes Peter also learned that engineering students cared only about rules they could use for building things. Mathematics proper was of no interest. Nor were the premedical students any more kindly disposed to mathematics. Their attitude was that doctors do not use mathematics but take it only because it is required for the physics course, and even the physics seemed of dubious value. The physical and social scientists had a similar attitude. Mathematics was a tool. They were interested in the [13]real world and in real people, and certainly mathematics was not part of that reality. Peter was soon called upon to teach prospective elementary and high school teachers. He did not expect much of the former. These students were preparing to teach many different subjects and so could not take a strong interest in mathematics. However, high school teachers specialize in one area, and Peter certainly expected them to appreciate what he had to offer. But every time he introduced a new topic, the first question the students asked was, 'Will we have to teach this?' Peter did not know what the high schools were currently teaching or what they were likely to teach in any changes impending in the high school curriculum. Hence, he honestly answered either 'No' or 'I don't know.' Upon hearing either response the prospective teachers withdrew into their shells, and Peter's teachings were reflected from impenetrable surfaces. Peter's one hope for a response to his enthusiasm for teaching was the mathematics majors. Surely they would appreciate what he had to offer. But even these students seemed to want to 'get it over with.' If he presented a theorem and proof, they noted them carefully and could repeat them on tests; however, any discussion of why the theorem was useful or why one method of proof was likely to be more successful or more desirable than another bored them. A couple of years of desperate but fruitless efforts caused Peter to sit back and think. He had projected himself and his own values and he had failed. He was not reaching his students. The liberal arts students saw no value in mathematics. The mathematics majors pursued mathematics because, like Peter, they were pleased to get correct answers to problems. But there was no genuine interest in the subject. Those students who would use mathematics in some [14] profession or career insisted on being shown immediately how the material could be useful to them. A mere assurance that they would need it did not suffice. And so Peter began to wonder whether the subject matter prescribed in the syllabi was really suitable. Perhaps, unintentionally, he was wasting his students' time. Peter decided to investigate the value of the material he had been asked to teach. His first recourse was to check with his colleagues, who had taught from five to twentyfive or more years. But they knew no more than Peter about what physical scientists, social scientists, engineers, and high school and elementary school teachers really ought to learn. Like himself, they merely followed syllabi  and no one knew who had written the syllabi. Peter's next recourse was to examine the textbooks in the field. Surely professors in other institutions had overcome the problems he faced. His first glance through publishers' catalogues cheered him. He saw titles such as Mathematics for Liberal Arts, Mathematics for Biologists, Calculus for Social Scientists, and Applied Mathematics for Engineers. He eagerly secured copies. But the texts proved to be a crushing disappointment. Only the authors' and publishers names seemed to differentiate them. The contents were about the same, whether the authors in their prefaces or the publishers in their advertising literature professed to address liberal arts students, prospective engineers, students of business, or prospective teachers. Motivation and use of the mathematics were entirely ignored. It was evident that these authors had no idea of what anyone did with mathematics. Clearly a variety of new courses had to be fashioned and texts written that would present material appropriate for the respective audiences. The task was, of course, enormous, and it was certain that it could not be accomplished by one man over a few years' time. Nevertheless Peter became [15] enthusiastic about the prospect of interesting investigations and writing that would lure students into the study of mathematics and endear it to them. The spirit of the teacher arose and swelled within him. As these pleasant thoughts swirled through his mind, another, dampening thought, like a dark cloud on the horizon, soon entered. He was a recently appointed professor. Promotion and, more important, tenure were yet to be secured. Without these his efforts to improve teaching would be pointless  he would be unable to put the product of his work to use. But promotion and tenure were obtained through research in some highly advanced and recondite problems almost necessarily chosen in the only field in which he had acquired some competence through his doctoral work. Such research was no minor undertaking. It demanded full time and total effort. Clearly, he must give the research precedence, and then perhaps he could undertake the improvement of teaching. And so for practical reasons Peter decided to devote the next few years to research. But the struggle to publish and to remain in the swim for promotion and salary increases caught Peter in a vortex of neverending spirals of motion; and the closer he came to the center the deeper he was sucked into research. In the meantime Peter continued to teach in accordance with the syllabi and texts handed down to him by his chairman. His few, necessarily limited efforts to stir up some activity among his older colleagues, who were in a better position to break from the existing patterns, were futile because these professors had accepted the existing state of affairs and chose to shine in research. Success there was more prestigious and more lucrative. Ultimately, Peter, like other human beings, succumbed to the lures that prominence in research held forth. As for the students—well, students came and went, and they soon [16] became vague faces and unremembered names. Education might hope for an epiphany, but Peter was not ordained to be the god of educational reformation. By the time he had acquired tenure he had joined the club. Like others before him he concentrated on research and the training of future researchers who would also be compelled to resort to perfunctory and ineffective teaching. Peter had taken his place in the vicious circle. The history of Peter Landers' aborted teaching efforts, real enough, seems exaggerated. One might conceive of its taking place in nineteenthcentury Germany or France. But the United States is devoted to education. We were the first nation to espouse universal education and to foster the realization of the potential of every youth. Our Founding Fathers, notably Benjamin Franklin and Thomas Jefferson, stressed the necessity of this policy, and it was adopted. Even today no country matches the educational opportunities and facilities that the United States provides for its youth. But the practices within educational institutions seem to be in marked variance with the principles and policies of our country. How has it come to pass that Peter and the many thousands of his colleagues find themselves enslaved
by research, while education, the major goal of our vast educational system, is being sacrificed? Does the pressure
to do research stem from the professors because they prefer the prestige and monetary rewards? Or does it come
from the university administrations? In either case, does not research make for better teaching? Or is there a
conflict between the two, and if there is, how can we resolve it? Since the crux of the problem lies with the universities
which train the teachers of all educational disciplines and at all levels we must examine the policies and practices
of our higher educational institutions. [17] CHAPTER 2: The Rise of American Mathematics
JeanJacques Rousseau Clearly, the relationship between teaching and research  their compatibility or incompatibility, their possible mutual reinforcement or opposition  must be investigated. To carry out these tasks we shall first examine the rise and present status of American education. In particular we must look into the role that research has come to play. This history will at least tell us why we are where we are. One of this country's marks of greatness, as Ralph Waldo Emerson pointed out in Education, is that schooling at many levels was undertaken almost immediately by the first settlers. The greatness is underscored by the fact that the United States was founded in the main by poor, uneducated people of vastly different backgrounds and languages, people who came here to improve their lot, whether by gaining political freedom, spiritual freedom, or sheer material necessities. It is true that many sought only the[18] freedom to practice their own brand of religious intolerance. However, no matter what values were sought, education was undertaken at the very outset. Elementary schools, set up at once, were soon followed by secondary schools. The earliest of the latter were private Latin grammar schools, the first of which, the Boston Latin Grammar school, was founded in 1635. Even colleges were established early. Harvard College opened its doors in 1636 and William and Mary was next in 1693. Yale started in 1701 as the Collegiate School of Connecticut in Old Saybrook, and then moved to New Haven in 1716 at which time it adopted the name of its benefactor, Elihu Yale. The numerous colleges founded in the seventeenth and eighteenth centuries were sectarian and in fact at first were devoted to supplying clergymen for the several faiths. Harvard, for example, trained Puritan ministers. The only nondenominational school in the colonies up to 1765 was the University of Pennsylvania. The policy of separation of church and state, which was adopted by the Republic in 1787, made public support of sectarian schools illegal. Even where indirect support might have been possible, religious differences and animosities caused legislators to refuse public support to sectarian colleges of faiths adverse to their own. Hence, many nonsectarian colleges were founded. The Morn!! Act, passed by Congress in 1862, marked a major turning point by supplying public funds to 'godless' universities for 'the Benefit of Agriculture and the Mechanic Arts.' This Act enabled states to found the landgrant institutions, among which the Midwestern universities became leaders. Many more public and private colleges and universities were established in the succeeding decades, with church affiliation still the more common. As late as 1868, Cornell opened amid much criticism because it had no sectarian ties. [19]To be sure, education was not widespread at the outset. In the seventeenth century only one colony, Massachusetts, insisted on a few years of compulsory education for all its children. As the country became somewhat settled this practice spread, and by 1910 all but six states had adopted it. Concurrently the age to which students had to remain in school was raised to fourteen, sixteen, and ultimately eighteen, though it differed from state to state. It is remarkable that in the early part of the nineteenth century the United States was the only country with some compulsory education. Moreover, by the late nineteenth century the right to free primary and secondary education was recognized, though this right was not widely utilized. In 1890 only 7 percent of the children of high school age actually attended schools, and only about 3 percent of the appropriate age group attended a college or university. As the population grew not only the number of students, but also the percentages increased rapidly. The right of high school graduates to lowcost college education was granted first by the big Midwestern universities. The privilege was specious, however, because these universities admitted all high school graduates and then flunked 50 to 70 percent in the first year. Nevertheless, the right to a college education gained ground all over the United States. What was the quality of the education? In colonial times and in the early decades of the Republic, because food, shelter, and clothing had to be given precedence, education amounted to little more than the transmission of ignorance. Certainly the pursuit of mathematics and the physical sciences, which might contribute in the long run even to the increase of material goods, had to be sacrificed to immediate needs. The country concentrated on reading, writing, arithmetic, and religion. Pedagogy, which had [20]already been receiving some attention in Europe, received no attention here for a couple of centuries: primers showing the common man how to do simple arithmetic can hardly be dignified with that term, especially since copying numerals and rote counting were the most that was taught. The need for arithmetic in commerce, exploration, surveying, and navigation motivated the introduction of fullfledged courses in that subject—but surprisingly, not in the elementary schools. In fact it was not until the eighteenth century that it was taught and, if at all, in the colleges and universities. Before 1729, the arithmetic texts were reprints of texts from England. After that date such texts were written and published in America. (The first one was written by Isaac Greenwood, a professor at Harvard College from 1728 to 1738, who had the advantage of studying in England and the disadvantage of being fired for intemperance.) The rest of the eighteenthcentury curriculum was a hodgepodge of cultural materials and moral imperatives. Keeping young people within the fold was a major concern; preparation for careers, the ministry, law, and medicine continued to be important. To this was added education in agriculture and forestry. A century later the landgrant colleges were founded to teach the latter two subjects specifically, though they soon expanded to include academic work. Utility and personal success, Alexis de Tocqueville observed in his Democracy in America (1835), were the chief concerns. Gradually the colleges and universities adopted liberal arts subjects with an emphasis, following the English model, on the classics. Molding character and teaching an aristocratic style of life to the wellborn were also objectives, at least until 1900. During the eighteenth century arithmetic was gradually moved down into the Latin grammar schools and high schools, where it and the other subjects were taught mainly [21]as preparation for college. In turn the colleges Harvard, William and Mary, Yale, Princeton, Pennsylvania, and others  began to require arithmetic for admission. Yale did so in 1745 and Princeton in 1761; however, Harvard did not do so until 1803. In the nineteenth century arithmetic finally became an elementary school subject, mainly because the growing industrialization of the country required more knowledge from workers. But mental discipline was also a reason for teaching the subject. Elementary schools in Massachusetts and New Hampshire were the first to teach arithmetic. This was in 1789. Only by the end of the nineteenth century was arithmetic firmly a part of the elementary school curriculum. The increasing need for mathematics in manufacturing, railroading, engineering, cartography, and the study of science—especially mechanics and astronomy  motivated the introduction of algebra, geometry, and trigonometry. These subjects entered the curriculum on the college level. They were taught as junior and senior courses at Harvard, Yale, and Dartmouth from 1788 on and were retained by most colleges until the end of the nineteenth century. Thus, the mathematics curriculum at the University of Pennsylvania in 1829 reviewed arithmetic and taught the beginnings of algebra and geometry in the freshman year. The sophomore year covered more of algebra and geometry, some plane and spherical trigonometry, surveying and mensuration. Only in the third year were what are now beginning college subjects, such as analytic geometry and differential calculus, introduced, along with perspective and mathematical geography. The senior year completed elementary calculus and initiated work in the essentially mathematical subjects of dynamics and astronomy. Early in the nineteenth century a few colleges began to [22] require for admission some of what we now call high school mathematics. Thus in 1820 Harvard upgraded its admission requirements to include elementary algebra. Yale did so in 1847 and Princeton in 1848. As for Euclidean geometry, Yale in 1865 was the first to require it for admission. Princeton, Michigan, and Cornell followed suit in 1868 and Harvard in 1870. During the last part of the nineteenth century algebra and geometry finally became high school subjects. Preparation for college, as colleges gradually began to require algebra and geometry for admission, and mental discipline were stressed as the reasons for teaching these subjects on the high school level. From 1820 to 1900 the universities gradually raised their level of mathematics instruction. By 1900 trigonometry, analytic geometry, and the calculus became standard college subjects. Although most colleges went no further in the early part of the nineteenth century, a few moved on to differential equations and more advanced subjects. Nevertheless,the general level of mathematical knowledge was low. The status of American mathematics in 1900 can be judged by an incident that is ludicrously revealing. A country physician, Edward J. Goodwin, who possessed no sound knowledge of mathematics, submitted a bill to the Indiana Legislature in 1897 that called for declaring the value of to be 4. A comparison of the common formulae for the area of a circle and the area of a circumscribed square shows immediately that cannot be 4. Nevertheless, the Legislature considered the bill. Fortunately the Senate, thoroughly modern in one respect, postponed action and the bill was never passed. One may not be too surprised by the actions of elected officials, but the American Mathematical Monthly, a journal founded by leaders of that time, in its first volume, [23]July 1894, published Goodwin's proposal and in 1895 dutifully printed other absurd Goodwinisms. The pressure in the United States to raise the levels of mathematics education and to educate more and more of American youth increased sharply as the country became a greater world power. World War I certainly showed the need for more mathematics education, and everincreasing technological uses of mathematics added to the pressure. The universities responded by raising the requirements for admission and by adding advanced courses. This brief sketch of the rise of mathematics education has not addressed the question of how teachers were procured. The upgrading of the formal level of education and the increase in the number of students did not in itself provide a supply of teachers. In colonial times almost anyone could set himself up as a teacher, and since the population of this country was made up mainly of uneducated people, one can readily appreciate the level of teaching. As late as 1830, Warren Colburn, one of the earlynineteenthcentury American educators, said in an address:
Although the colonial colleges were not founded to train teachers, many did so. But the colleges
admitted only men [24]whereas, because of the low
salaries, women were the most likely candidates.* Also, most colleges were still sectarian and so could not be
the main source of public school teachers or be favored by public money for the education of teachers. Further,
the better colleges insisted on more qualifications for admission than prospective school teachers could afford
to obtain. The problem of supplying teachers for a public school system that kept extending universal compulsory education was never really solved. It was aggravated by the fact that more and more immigrants, uneducated and poor, entered the country. Nor was the situation any better in the colleges. When Charles William Eliot became president of Harvard College in 1869, he emphasized that one of the outstanding problems was to induce ambitious young men to adopt the calling of professor. 'Very few Americans of eminent ability,' he said, 'are attracted to this profession.' The supply of good teachers continued to fall short of the need. Certainly many who entered teaching in the nineteenth and early twentieth centuries were not the best choices. Teaching remained very poorly paid, had little or no prestige, and required few qualifications. Of course, some capable people, unable to afford the education that other professions required, turned to teaching. The colleges and universities, which were not themselves[25]well staffed, gradually took over the function of educating teachers. Professors of education became members of college faculties beginning in 1832 at New York University and at other colleges soon after. Up to 1920, however, the training of elementary school teachers was still done in the normal schools, which by that time required a two to four year course of study. The colleges trained the nation's high school teachers. Then the colleges and universities began to establish departments or schools of education to train both elementary and high school teachers. Beginning around 1950, the normal schools were converted to fouryear liberal arts colleges with emphasis on teacher training. High school mathematics teachers now take a fouryear liberal arts course with a major in mathematics, but elementary school teachers take little academic mathematics and a great deal of instruction in pedagogy. The rise of institutions and faculties devoted to teacher education did not improve the knowledge of mathematics proper that the professors should have possessed. In fact, the level of mathematics instruction was very low until well into the twentieth century. American texts were poor; good ones had to be imported. A few people educated themselves. One of the outstanding examples was Nathaniel Bowditch (17731838), who translated and added explanatory notes to Laplace's Mécanique céleste, one of the great works of European mathematics. This translation was the first substantial mathematics book in the United States and, in fact, in the entire Western hemisphere. The first truly great American scientist was the selftaught physical chemist Josiah Willard Gibbs (18391903), who is also known for his contribution to vector analysis. Many professors went to Europe for their education; these men were certainly the best educated and [26] were actually the only ones qualified to teach mathematics and train mathematicians. Gradually the quality of education and available books improved sufficiently for the United States to train its own mathematicians, though hardly in numbers sufficient to serve the needs of the country. The first of the outstanding Americantrained mathematicians was Benjamin Peirce (18091880), a Harvard graduate and professor at Harvard from 1831 to his death. Other Harvard mathematics professors of his time, such as William Fogg Osgood and Maxime Bocher, got their Ph.D.'s in Europe before joining the university faculty. Despite the increase during the early decades of this century in welltrained mathematicians, one could not be complacent about mathematics education. There were too few teachers to staff the increasing number of elementary schools, high schools, colleges, and universities. The states, cities, and towns lagged in their appreciation of the value of education and were miserly in funding. Consequently, salaries remained low and potentially fine teachers continued to choose other jobs and professions. Some indication of the quality of the teachers comes from data compiled by the Educational Testing Service. As late as 1954 the elementary school teachers who were interviewed feared and hated mathematics. Naturally this influenced their teaching. Half of 370 teachers tested could not tell when one fraction was larger than another. Thus, the teachers knew less than they were required to teach. High school teachers in most states could qualify for a license to teach mathematics with only ten hours of college mathematics. In many states a license to teach in high school is still sufficient to qualify for teaching any subject. Since until recently ignorance was the outstanding characteristic of American educators at all levels, not only [27] was the factual content of mathematics courses low, but also the educational goals were either imperfectly or mistakenly perceived. We need not pursue here the conflicts, disagreements, and theories advanced over the several centuries of American education. That social utility should be an objective, particularly when the colonies and even the young Republic were struggling for survival, could hardly be challenged; but arithmetic was also advocated as a mental discipline that would extend to other areas. Though portions of secondary school mathematics could be defended on the ground of their usefulness in applications such as surveying and navigation, it was impossible to defend most of high school mathematics on this ground. Instead, mental discipline, knowledge of a branch of our culture, the inherent beauty of the subject, and preparation for college were advanced as justifications. As to the pedagogy, intuitive approaches using concrete materials and sensory experiences, espoused, for example, by Johann Heinrich Pestalozzi (17461827), and abstract rigorous approaches were both advocated to impart the values and even the meaning of mathematics. In practice, however, the teachers, barely understanding what they were taught, handed down the processes and the proofs mechanically. Drill was the order of the day. Any questions were answered with the dogmatic reply: This is the way to do it. To divide one fraction by another, invert and multiply. The product of two negative numbers is a positive number. If x + 2 = 7, transpose the 2 and change the sign so that x = 7  2, or 5. The rationale of geometric proofs was never given. Students memorized them and handed them back on examinations. Thu departments and colleges of education attached to universities were not helpful. The reasons are patent. The mathematics educators were themselves taught subject [28] matter in the same way as others were taught, and so their understanding of mathematics was no better than that of good students. It was in fact worse, because the educators did not deem it necessary to go as far in their subject as those who planned to be professional mathematicians and scientists. Furthermore, very little was known then (and even now) about the psychology of teaching and learning. Hence, the educators had little to contribute. They taught prospective teachers how to carry out the drill and memorization, the very processes by which they had been taught. Another unproductive effort was to call upon psychologists, who commonly offered a standard course in the psychology of education. But the psychologists also had little to offer. Professor Edward Lee Thorndike (18741949) of Columbia University?s Teachers College advocated massive repetition or drill. Students should be trained to respond automatically: 3 + 2 should immediately elicit 5. Understanding would come eventually. But this advice was no more than an endorsement of the rote teaching that had been practiced for generations. (See also Chapter 9.) This method of teaching is still advocated by educators such as Professor B.F. Skinner of Harvard  now it is called programmed learning. To improve education, innumerable commissions and committees were appointed by mathematical organizations and state and local governments. The committees were to study goals, content, and pedagogy. The history is extensive but irrelevant. One factor that hampered the reforms suggested by the many committees, conferences, symposia, and commissions is that no sound evaluation was employed to test the recommendations. Another was that changes were introduced too quickly, making it difficult for teachers to implement them properly. The efforts and sincerity of the[29] mathematics educators are not in question. However, the nature and goals of mathematics education, including appropriate subject matter, the best methods of teaching it, the role of applications, and the means to attract and involve students were not determined by these studies. Undoubtedly, changing conditions in this country rendered pointless many recommendations that may have been right in their time. The proportion of students attending high school in the United States grew from 12 percent in 1900 to over 90 percent in 1967. Obviously the interests and backgrounds of high school students became far more varied as that population increased. The Bachelor of Arts (or Science) curriculum, designed originally to give American youth some knowledge of classictil (Greek) and European culture, was broadened somewhat after the Civil War to include the social, physical, and biological sciences and the humanities. In 1900, when about 4 percent of the collegeage group went to college, this program may have been proper. In 1970 about 48 percent, or seven million students, went to college, for totally different reasons than just the acquisition of knowledge. The colleges had to adapt to many more levels of academic ability, far more diverse backgrounds, and many new goals—notably career training in a wide variety of fields. Beyond these changes was the initiation in 1901 of community or twoyear colleges, which were the successors of the private junior colleges first established in the late nineteenth century and which were preparatory for the senior colleges. The community colleges now have many terminal students who seek essentially a vocation. Despite gradual improvement in the quality of the teaching staff on the lower educational levels, a stabilized educational system, reasonable facilities, and the far greater availability of texts and books, mathematics [30] education was on the defensive from 1920 to 1945. Students did poorly. Enrollment in academic mathematics courses decreased, notably in algebra and geometry. The transfer of training was, perhaps rightly, deprecated. During this period the colleges reduced requirements in mathematics for admission, and students took fewer mathematics courses once enrolled. Many institutions even dropped the mathematics requirement, and many high schools also dropped mathematics as a requirement for a diploma. The value of any mathematics education was widely challenged. The status of mathematics in the 1930s was described by mathematics professor Eric T. Bell in the American Mathematical Monthly of 1935:
The subsequent history and status of mathematics education was affected by the entry of a new factor  research, in the sense of original contributions. In the area of mathematics this necessarily came rather late. Knowledge in this subject is cumulative. To contribute to mathematics in 1800, for example, one had to know the work of the Greeks, Descartes, Fermat, Newton, Leibniz, Euler, Lagrange, Laplace, and many others. In science the situation was roughly similar. Mechanics was well developed by the end [31] of the eighteenth century, and contributions by neophytes could hardly be expected. The newer fields of science offered more opportunities. In electricity, for example, which was just beginning to be cultivated in 1800, it was possible for the American Joseph Henry to share honors with the Britisher Michael Faraday in the discovery of electromagnetic induction. In the nineteenth century, however, American science as a whole was, like American mathematics, below the European level. Researchers in general had no place in the United States before about 1850. They could become teachers at lowlevel colleges and universities and spend their spare time, of which there was little, in research. But such endeavors received no encouragement. In fact, a research person might even arouse suspicion that he was not paying enough attention to his teaching and might even have radical ideas about curriculum and the method of instruction. In 1857, a committee of the Columbia College Board of Trustees attributed the poor quality of the college's educational efforts to the fact that the professors 'wrote books.' A professor's obligation was not to advance knowledge but to transmit it. Of course the United States at that time was not really prepared to do research, much less train Ph.D.'s for research. Though during the nineteenth century, as we have already noted, many American professors went to Europe to study, their number was relatively small. Moreover, while a few years in a great cultural center helps immensely, it does not produce researchers, particularly if they return to an intellectual atmosphere in which research is neither cultivated nor widely appreciated. Researchers need the stimulus of colleagues with common interests and of students who carry on the work of the masters and press them for problems and [32] In effect, during the nineteenth century good understanding and appreciation of research did not exist. For example, Josiah Willard Gibbs, the physical chemist mentioned earlier, worked at Yale University, where for many years he received no pay. In Europe his research in thermodynamics was well understood and highly valued. He wrote a firstclass book on statistical mechanics in 1901 that was appreciated by such masters as Felix Klein and Henri Poincaré but ignored in the United States. When the distinguished German mathematician, physicist, and physician Hermann von Helmholtz visited Yale in 1.893 and was greeted by the top dignitaries of the university, he asked where Gibbs was. The university officials, nonplussed, looked at each other and said, 'Who?' The state of research in the United States even in the early twentieth century may be illustrated by an incident in the life of Walter Burton Ford, an American mathematician who died at the age of ninetyseven in 1971. He submitted his doctoral thesis to Harvard. The paper was judged by Bocher, Osgood, and Byerly, who were leaders in American mathematics in the first few decades of this century, and deemed unacceptable. Ford sent the paper to the reputable French Journal de mathématiques, where it received praise from the editors and was published. The Harvard faculty then reversed its decision and awarded the Ph.D. to Ford. Another of the early great American mathematicians was Norbert Wiener (18941964). His fields of endeavor were not understood here; however, when the European mathematicians began to praise Wiener's work, the American mathematicians took notice and fortunately, if belatedly, he received honor here. When research in mathematics and the sciences was first undertaken, about 1850, the university leaders thought that it belonged not in the universities but in special institutions [33] referred to as academies. Advanced degrees were offered only in law, medicine, and theology. Yale, however, did initiate graduate training in 1847 and conferred the first Ph.D. degree in mathematics in 1861. Harvard instituted a graduate program in 1872 and conferred the first degree in mathematics in 1873. Though the universities did begin to undertake research and training for research, the inadequacy of the American research capacity was recognized by some men who had become aware of the high standards of the German universities. It was these men who persuaded wealthy individuals to found universities that would stress research and strengthen the already existing advanced training. David Coit Gilman induced Johns Hopkins, a merchant and banker, to found the institution named after him, which opened in 1876. Leland Stanford, who made his fortune in railroading, established Stanford University, which opened in 1891. And William Rainey Harper convinced John D. Rockefeller to finance the University of Chicago, which opened in 1892. Johns Hopkins and Chicago started as graduate schools but soon added undergraduate colleges. The already existing graduate schools began to take research more seriously and other colleges added graduate schools. Through these moves the German universities made their impact on education here and highlevel research was at least launched. Although some advocates and entrepreneurs of research  Andrew Dickson White of Cornell, Charles W. Eliot of Harvard, and Gilman  believed that universities should undertake research, they also believed it should be subordinate to teaching. Imparting truth, White held, is more important than discovering it, and Eliot declared in 1869 that 'the prime business of American professors in this generation must be regular and assiduous class teaching.'[34] During his early years as president he was suspicious of professors who devoted much time to research because he feared that this activity interfered with their teaching. But a generation later the university leaders reversed the emphasis, and research took precedence over teaching. President William Rainey Harper of Chicago led this change, declaring: 'The first obligation resting upon the individual members who comprise it [the university] is that of research and investigation.' David Starr Jordan of Stanford declared, 'The crowning function of a university is original research.' To defend its position the HarperJordan group asserted, in Jordan's words, that 'investigation is the basis of all good instruction. No secondhand man was ever a great teacher and I very much doubt if any really great investigator was ever a poor teacher.' Hundreds of professors and scores of university administrators took up the Jordan refrain that no one could be a good teacher unless he also did research. Harper not only made research the first obligation of professors; he also instituted the practice that promotion of the University of Chicago faculty 'will depend more largely upon the results of their work as investigators than upon the efficiency of their teaching.' Harper was also responsible for directing the graduate schools to concentrate on training all graduate students to be researchers. Breadth of knowledge was derogated and training for teaching was entirely ignored. The discovery of new facts or the recovery of forgotten facts became the supreme, prized goal. Whether subordinated to teaching or esteemed more highly, research did take hold. One could say that it was effectively initiated when Johns Hopkins offered a professorship to the already famous British mathematician James Joseph Sylvester (who was refused a job in British universities because he was Jewish). Sylvester served from 1877 to 1883. He and William E. Story founded the first [35] research journal in the United States, the American Journal of Mathematics, in 1878. This journal immediately became the outlet for the earliest significant research papers written by Americans; European contributions added to its prestige. The first volume contained several highly original papers by the Americaneducated mathematical astronomer George William Hill and in the volume of 1881 Benjamin Peirce published a paper he had written and circulated privately a decade earlier. When the University of Chicago was organized it immediately hired outstanding research professors. In mathematics it appointed Eliakim Hastings Moore, an American who had studied in Germany, and Osker Bolza and Heinrich Maschke, both imported from Germany. This university became the first outstanding American research center, and it trained the first truly great American mathematicians, among them George David Birkhoff, Leonard Eugene Dickson, and Oswald Veblen. The precedent set by Johns Hopkins, Stanford, and Chicago was soon followed by other universities, which now began to demand Ph.D.'s who could carry on research at their institutions. Some colleges also sought Ph.D.'s for the sake of the prestige gained by having such researchers on their faculties. However, the overall quality of the Ph.D. training was generally poor. The degree was aptly described by JeanPaul Sartre as a reward for having a wealthy father and no opinions. The low quality of the Ph.D.'s, who were expected to become teachers as well as researchers, began to alarm prescient educators. In 1901 President Abbott Lawrence Lowell of Harvard said: 'We are in danger of making the graduate school the easiest path for the good but docile scholar with little energy, independence or ambition. There is the danger of attracting an industrious mediocrity which will become later the teaching force in colleges and [36] secondary schools.' A few years later David Starr Jordan, despite his strong advocacy of research, deplored the lack of professors qualified to teach, which he attributed to the narrowness and triviality of the doctoral dissertation. A more devastating and concerted attack was made by the distinguished Harvard philosopher, William James, in his essay of 1903, "The Ph.D. Octopus." * James was concerned that the rush for the Ph.D. crushed the true spirit of learning in the colleges. He objected to colleges and universities seeking Ph.D.'s as evidence to the world that they had stars on their faculties:
And in 1908, the distinguished American educator Abraham Flexner foresaw the greater evil to come, namely, that even though universities might improve the quality of the Ph.D. training, they would be sacrificing college teaching on the altar of research. For better or worse, the emphasis on research grew stronger. To further it mathematicians decided to hold meetings and to support more journals. They founded the New York Mathematical Society in 1888, which became the American Mathematical Society in 1894. Initially the Society devoted itself to research and teaching. In the first decade or two of this century members of the Society  Eliakim H. Moore, Jacob W.A. Young, David Eugene Smith, Earle R. [37] Hedrick, George Bruce Haisted, and Florian Cajori, among others  did take an active interest in education, including secondary school teaching. They made sensible recommendations and seriously attempted improvements. But after a couple of decades the Society concentrated on research, whereupon another group of men founded the Mathematical Association of America in 1915 to cater specifically to undergraduate education. In 1921 still another group, concerned with secondary and primary school education, founded the National Council of Teachers of Mathematics. More and more, research became the major interest of the universities, until in the large universities it gained favor over all other functions. Just how research and teaching might have fared had the United States continued its main reliance upon its own resources cannot be known. But unexpected developments altered the American scene. During the Hitler period many of the leading mathematicians of Germany, Italy, Hungary, and other European nations fled their countries; a large number of them came to the United States. When the Institute for Advanced Study was founded at Princeton in 1933 for postdoctoral research, three of the six mathematics professors chosen for its faculty were Albert Einstein, John von Neumann, and Hermann Weyl. The numerous refugees soon found places in American universities and added enormously to our mathematical strength. They also trained doctoral students for research, and the number and quality of Ph.D.'s increased significantly. Though the acquisition of competent researchers was a boon to that activity, it contributed little to the teaching at the college level. The refugee mathematicians who came here during the Hitler terror were trained in the German universities, which were  until they lost their best professors  the strongest in the world. However, there was and is no undergraduate education in Germany. Students go [38]directly from a gymnasium (high school) to a university, where they specialize in a subject, usually to obtain the Ph.D. degree. Training for research is the goal of the education (though students who do not complete the degree may pass an examination that qualifies them to be gymnasium teachers). Moreover, the professors, who are research specialists, do not feel obliged to be concerned with pedagogy, since student motivation, drive, and ability are presumed. Though the refugee mathematicians were intelligent and well intentioned they were not, by reason of background, able to apportion their efforts and knowledge among the diverse needs of the American universities, such as undergraduate education. The graduate schools were therefore turned still more in the direction of training researchers and became even less concerned with pedagogy. While the graduate schools were absorbing the great academicians who had come to the United States, World War II broke out. The war was a battle of scientists. Faster ships and airplanes, radar for detection and to advance antiaircraft gun control, improvements in range and accuracy of artillery, better navigational techniques for ships, planes, and submarines, and the development of the atomic bomb proved to be crucial and convinced the country that if it was to remain a major power, more mathematical and scientific research was needed. Hence, during and after World War II our government began to support research on a large scale. Research became a more and more valued and prestigious activity, and professors concentrated on this work. With the rising importance of research came the rising esteem for the researcher. The American professor, once a lowly figure among the elite, had now achieved high social rank. Government money also made an essential difference. Billions of dollars to finance research were given to [39] professors, who thereupon became sources of income to the universities. On paper all research grants and contracts cover only the actual cost of the research performed on behalf of some governmental agency or program. But in effect these grants cover far more. They support graduate students employed to aid in the research. As a consequence more graduate students can attend the university, and more tuition income is received. Professors working on contracts give up most of their teaching duties. In their place poorer paid, often young, instructors conduct most of the classes, though the name of a wellknown professor still serves as a drawing card in attracting students. Laboratories and equipment needed to perform contract research are paid for by the government but are used for other research and for instructional purposes. Secretaries to professors, paid by contract money, and a generous overhead also make contract research very attractive. It does enable a university to enlarge its research and graduate programs, though  contrary to claims often made—it does not contribute to undergraduate education. The effect was obvious. The universities, already inclined to favor research, began to compete intensively for research professors. Most of these professors and most graduate schools turned their attention to producing more researchers. The Ph.D. programs were geared solely toward this end. The attention that academic professors had given to undergraduate, secondary, and elementary school education during the first few decades of this century was all but abandoned. Nevertheless, there would no longer seem to be cause for the concern expressed by Lowell in 1901. The United States now has many strong graduate schools staffed by competent research professors, and research, judged at least by the volume of output, is flourishing. Mathematics research reinforces scientific strength, and scientifically and [40] technologically the country has achieved preeminence. The researchers can certainly offer sound graduate training, and the knowledge imparted in this training should filter down to all levels of education. Wise professors, concerned not only with the extension of knowledge but also with its transmission, can devise ways of presenting their subjects that their students  the professors and teachers of the future  could use to good advantage. The recently organized large schools or departments of education in the universities and colleges can provide the special instruction and advising that the elementary and secondary school teachers require. Insofar as mathematics in particular is concerned, although the American Mathematical Society has abandoned its earlier interest in teaching, the Mathematical Association of America and the National Council of Teachers of Mathematics have taken over that vital concern for the undergraduate and lower levels. On the face of things, the United States would seem to have reached, if not an ideal, then certainly a very reasonable and even rosy position at all levels of education. Why, then, did Peter Landers face so many problems as a teacher and find himself unable to resolve
them?
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 22nd March 2001
