Teaching: The New Mathematics
Professor JOHN G. KEMENY has pioneered a completely new program of mathematics instruction. With the encouragement of the Mathematics Association of America, he has radically changed the curriculum at Dartmouth to produce students better suited to the modern demands of industry, the sciences, and the new mathematics.
JOHN G. KEMENY
WHILE mathematics has always been recognized as one of the cornerstones of our educational system, we are entering an era in which an understanding of mathematics will be of even greater importance to all educated men. As our civilization grows in complexity and science plays a more and more vital role, the man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization.
Within a single lifetime a dozen major new areas have been opened in mathematics, and today mathematics is one of the most rapidly growing fields of human knowledge. But new discoveries often take several generations to reach the level where the majority of educated people hear of them.
The first definitive book on mathematical logic was written in 1910 by Bertrand Russell and A. N. Whitehead. In the 1920s we find a handful of pioneers, destined to become world-famous, proving the fundamental theorems of the subject. These same pioneers trained the first group of disciples in the 1930s, making it possible for a small number of universities to offer graduate courses in mathematical logic. These disciples turned out a large number of students after World War II. Today a fourth generation of mathematical logicians is teaching the subject on a broad scale in the colleges.
Modern mathematics is impossible to define in terms of its subject matter. The new mathematics is typified by its approach. Whenever a mathematician observes a pattern in nature, he immediately asks, “What are the essential features of this pattern?” Then he asks what one can deduce from these features. At this stage he forgets all about the example that motivated his research and considers the problem in its pure, abstract form. The advantage of this approach is that his results will have far-reaching applications. Every pattern that shares the given features will also have to share whatever additional features the mathematician has inferred.
During this period in which mathematics is growing and maturing in its basic research, entirely new demands are being made on the world of mathematics. The biological and social sciences are beginning to vie with physical sciences for the services of mathematicians. Many research papers in medicine, psychology, and economics are unreadable for the mathematically uninitiated.
Ideas invented out of intellectual curiosity often turn out to have far-reaching practical applications. The study of geometries of more than three dimensions may at first appear impractical. But relativity theory now requires four dimensions, of a non-Euclidian type. Modern economic theory may employ an abstract geometry of a hundred dimensions to describe the interaction of a hundred different industries. Quantum mechanics has made use of infinite-dimensional geometries. But these applications, while certainly profiting from the new mathematics, have nowhere the abstraction with which the topologist views geometry. We may be assured that when science is ready to make demands for a more basic understanding of geometries, the field of topology will provide answers.
Large industries are now interested in hiring a staff of mathematicians to help in their planning. In some cases this is due to the development of branches of mathematics specially suited for longrange planning, such as linear programming and the theory of games. But even more important is the realization that the man with mathematical training is often best qualified to solve problems of almost any type.
A staff of mathematicians may now solve accurately a scheduling problem that was solved by hit-or-miss methods before. A statistician may find a way to improve the quality of a product, and at the same time lower the cost of supervision. A mathematical logician may be in charge of designing electric circuits. And a telephone company may find it profitable to have a large staff of abstract mathematicians on hand. We even find mathematicians working on military strategy.
All of these demands on the mathematician are modest compared with the requirements of computing machines. With hundreds of these giant brains manufactured annually, there is a danger that mathematicians may have to spend all their time instructing the machines. For example, a single installation of the U.S. Navy advertised for five Ph.D.’s, ten M.A.’s, and twenty-five B.A.’s for a computing center. The request sounds perfectly reasonable until one realizes that — based on the average output of the last decade — the Navy is proposing to hire .5 percent of the B.A.’s, 1 percent of the M.A.’s, and 2 percent of the mathematics Ph.D.’s graduated in a year throughout the entire country!
During a period of revolutionary change, men need the comfort of something unchangeable and hallowed by tradition. Apparently, the undergraduate curriculum has fulfilled this need for mathematicians. I can think of no other reason for the fact that many leaders of mathematical research taught the same dull, unenlightening, and completely outdated topics year after year. Of course, the staffs at the majority of institutions of higher learning had little choice; they taught what they knew. Since their own professors had neglected to bring them into contact with presentday developments, they could hardly be expected to go back to graduate school late in their careers. Nor is it reasonable to expect a man in his fifties to turn his advanced students over to a youngster with a fresh degree. The most human reaction is to decide that the newer mathematics is not worth teaching.
But the pressure of future employers, of the engineering schools, and even of the new men in the socialand biological-science departments is having its effect. More and more mathematics departments have decided that they must change the curriculum, whether they are ready to teach the new subjects or not.
IF ADOPTING new courses is difficult, designing a new curriculum from scratch is incomparably harder. A college needs experts in a variety of mathematical branches (no small group could hope to speak for all of mathematics), all willing to participate in a time-consuming experimental program. Even if such a group existed, it is unlikely that its efforts would be unhampered by the rest of the department or the administration. Vested interests are as strong inside ivy-covered walls as outside, and they seem to last longer in the protected atmosphere.
Happily, a strange series of circumstances produced just the right combination of factors at Dartmouth College. After thirty years of continuous service, almost the entire mathematics department was ready to retire, and the college decided to bring in a young Ph.D. to take charge of planning for the future. And so I came to Dartmouth in 1953, at age twenty-seven, to take part in the rebuilding program. Today we have seventeen Ph.D.’s, three of whom have devoted over thirty years of their lives to Dartmouth, and the rest of whom have come since my appointment. We have been extremely fortunate in the caliber of the young men attracted to Dartmouth. In part they were drawn by the reputation of the college, in part by the youth of their colleagues (the median age dropped thirty years), but mostly by the opportunity of building a department and a curriculum to fit their own dreams. We have also had the encouragement and support of the senior members and of an enlightened administration.
Since it is easier to describe one program than to try to speak of national trends, I will concentrate on the Dartmouth mathematics program. I do not say that it is the best in the nation, but we have worked on it very hard. We do not claim to have invented all the ideas — indeed, we are indebted for them to a long list of mathematicians — but we have tried to combine the best of their ideas with some innovations of our own. At least our program points in the right direction, and many colleges and universities have paid us the great compliment of copying parts of it.
Our basic assumption is that no one or two programs can fill the needs of all students of mathematics. We have, therefore, designed four programs. We offer an honors major for the student aiming for graduate work in mathematics, and a major for the students who will make mathematics their career, but will do little or no graduate work. Included in this group are prospective high school teachers, computer programmers, and industrial mathematicians. We also provide a strong program for physicists and engineers, and we try to meet the ever-increasing needs of biological and social scientists.
Let me start with the last program, since this is entirely new in college curricula. In the past, the social-science student brave enough to elect mathematics in college soon found out that most of the mathematics he needed was attainable only among the courses reserved for mathematics majors, and that he was forced to wade through a morass of courses designed exclusively for engineers. We had to face up to the fact that almighty calculus is not nearly as important in the social sciences as in the physical sciences. Indeed, it may not even be as important in the physical sciences as the curriculum would lead us to believe. While all students should know some calculus, if they have but one year to imbibe, they should also taste mathematics of a different vintage. That is why we developed a course called Finite Mathematics, to supplement an introduction to the calculus.
The history of this course is most interesting. When we asked social scientists what mathematics other than calculus they use, they mentioned topics that were never taught below the junior year and often were reserved for graduate courses. We were certainly worried about teaching high-level mathematics to liberal-arts freshmen, but decided that it was better to fail to get across the material they needed than to succeed in teaching them something useless. Therefore, we chose some mathematical logic, probability theory, and linear algebra as our basic material, and for further study such ultramodern topics as linear programming, game theory, and Markov chains. But how were we to teach freshmen what some of our Ph.D.’s did not know?
The very nature of our subject matter came to our rescue. Since abstract mathematics develops from simple concrete examples, we found it possible to illustrate all the basic concepts in terms of familiar examples. We relied on the student s intuition, which is usually sound in the finite case but often misleads him when he is forced to contemplate infinity.
For example, in probability theory all questions concerning the outcome of an election are finite questions, in the sense that all possible variations are finite in number. But in picking a point on a line by some random device, we must allow infinitely many possibilities. The predictability of elections can be discussed in our freshman course; the picking of a point on a line cannot. I do not consider this a major loss for the students.
The course in finite mathematics was an immediate success at Dartmouth and has been a major factor in the decision of 95 percent of all freshmen to study mathematics in college. It, and other courses similar to it, has been widely adopted throughout the country. But I did not fully realize how well we had succeeded until I overheard one of our freshmen telling his parents about the “fascinating new mathematics course, which has no mathematics in it at all.” Indeed, the course had nothing in it that fitted his high school conception of the nature of mathematics.
STUDENTS interested in mathematics or in the physical sciences need a firm foundation in classical mathematics, though not necessarily an oldfashioned calculus course or three years of nothing but calculus. We not only have shortened the calculus sequence, but also have made it more interesting, we hope. We examined all the traditional topics to see whether there was a reason for including them, other than that they have always been taught. Second, we decided to spend more effort on teaching fundamental ideas and less on specialized skills. This decision has created the most controversy.
While mathematicians are willing to admit that many of the techniques taught to all students in the traditional curriculum have ceased to be interesting, it seems to us that few educators face up to the fact that the remaining techniques are likely to suffer the same fate. For example, we think that our engineering students are being given just the right training for the previous generation of engineers. Often, after taking jobs, they have to be retrained completely.
It is a common complaint that our students do not remember their freshmen skills by the time they need them in their junior year. Since many of these techniques are so complicated that the instructor himself must refresh his memory annually, the student can hardly be blamed. However, the instructor knows his basic principles so well that he can pick up the special tricks in a short time. If entirely new techniques are developed, he does not have to start from scratch; he simply reads an account of the new developments.
The educational implications are clear. Let us teach our students only basic ideas and fundamental techniques. Let us teach them to use reference materials and to read the current literature. They may still need on-the-job-training or periodic retraining, but they can do most of this for themselves. They will not feel that the bulk of their college education was wasted.
One must also take into account the impact of computing machines. Often we train our students to become second-rate computing machines. This may have had some use thirty years ago, but today we cannot possibly compete with electronic computers. Let us leave to the machines all that is dull and purely mechanical and turn our efforts to more enjoyable tasks.
Physical scientists need the modern tools too. We teach a more advanced version of the finite mathematics course to all of them in their sophomore year and have them ready for advanced courses by their junior year. We offer a selection of courses designed to introduce them to any field in mathematics, pure or applied.
The best testimonial for this program is the success of some of our weakest students in their future careers. They may have felt while at Dartmouth that they were lagging behind their classmates, but later they found that their training put them ahead of graduates of classical curricula.
I HOPE that I have conveyed our deep interest in students of all types. But I should be less than honest if I did not admit that the primary educational aim of any mathematician is the training of mathematicians. The honors program has always been our special concern. In no field is the difference between the best students and very good students as great as in mathematics. There is that special gift for mathematics which, though it can be cultivated, can only be provided by nature. The greatest crime of the traditional curriculum, from kindergarten through college, is the neglect of the exceptional student of mathematics. I am quite certain that many more men and women have left mathematics because of boredom than because of the inability to comprehend.
I have followed many fine experiments in ability grouping in grade and high schools and have not been surprised by the amazing success of some of these programs. On the college level, the cure was introduced by Princeton University, and we have tried to follow that outstanding example. The mathematically promising student must be located as soon as he arrives on the campus, must be allowed to study within small groups, among his peers, taught by research mathematicians.
While I have laid great stress on good curricula for other students, for honor students the subject matter is almost irrelevant. These students may be recognized by their attitude toward the material that is not in the syllabus. When an average class is told that they are about to treat a topic that will not appear in the final exam, they take this as a signal to catch up on their sleep. Honors sections come to life when they are allowed to wander freely into topics that are not required.
Since these students can cover the required material with much less drill, there is time for mathematical excursions of the mind. They are encouraged to work out as much mathematics as they can for themselves and to ask the instructors anything at all, as long as it concerns mathematics. We know that in spite of this “wasted time,” they will acquire a much deeper understanding of basic mathematics, and we cultivate their taste for more esoteric material by responding to their own requests. I particularly enjoy teaching a freshman honors section. In a single year I may be forced to tell them about the fourth dimension, infinity, some theorem in topology that has caught their imagination, or just what a mathematician does when he creates.
Since the first two years of the honors program are open to any able student who loves mathematics, it is not uncommon to find a future doctor, lawyer, businessman, or musician studying side by side with the budding mathematician. But we try to identify the latter by his junior year, and then he is allowed to progress at his own speed. He has the opportunity of covering the equivalent of a strong master’s program as an undergraduate. He is exposed to courses in modern versions of the calculus, algebra, and geometry, as well as topology, probability theory, logic, number theory, and applied mathematical fields. As far as possible, honors majors study in seminars, and they are encouraged to explore mathematics on their own.
We are particularly proud of having pioneered in the wide use of undergraduates as research assistants. Perhaps they cannot do as much as advanced graduate students, but mathematics is a field for the young in mind. Our computing center is staffed exclusively with undergraduate assistants; others are assigned to individual faculty members to help them on research projects. Though this may mean simply the checking of computations, or reading of manuscripts, or searching in literature, many assistants have also contributed original ideas to faculty research, and so far three of our assistants have published original papers of their own.
As these students, and students from other institutions, reach our graduate schools, we hope to provide the staff necessary to spread modern mathematical ideas to all colleges. The potential of mathematics may have been dormant, but our colleges are awakening. The next generation will have a chance to become mathematically literate.