not sufficient merely to teach them what the computer can do for them. They must also be taught what the computer cannot do for them. JOHN G. KEMENY, Some years ago the Committee on the Undergraduate Program in Mathematics (CUPM), recommended that the first introduction to computing be part of freshman or sophomore mathematics. At that time implementation did not seem practical, but the coming of time-sharing computing systems has made it possible to implement the recommendation at Darthmouth College. We now train 650 freshmen (80 percent of the class), each year, in the rudiments of computer programing. The program is incorporated into second semester freshman mathematics, which is calculus for the physical science students and finite mathematics for other students. In either case, students attend two 1-hour lectures and read a short manual. After this, they debug four programs, entirely on their own, with a hands-on work on a teletype input to the time-sharing system. The resulting programs are tested by the computer. We now have a careful evaluation of this program. During the 10-week academic term, a typical student spends three-quarters of an hour per week on the teletype and an equal amount of time in planning his programs. Therefore, the entire computer training occupies only 15 hours of a student's time. In spite of the fact that such modest times are used, we are running better than 90 percent success throughout the freshman classes, and most of the students feel that they have learned enough to make use of the computer in other courses. I am of the opinion that no other academic program yields as high a dividend, per time invested, as the freshman computer program. Even if the student never again touches the computer, he will leave the college with a sensible attitude toward the use of high-speed computers. We also know that a significant minority of the students avail themselves of time-sharing in connection with more advanced courses. The ability to assign computer problems as a matter of routine, in any course that has a year of mathematics as a prerequisite, is beginning to show a significant effect on the campus. Our engineering and business schools have made most imaginative use of time-sharing in a wide variety of courses. We also find scattered, but interesting use throughout the science and social science departments. And everyone seems to agree that the use will expand as the faculty gains more experience. It is my personal opinion that the computer will go a long way toward breaching the gap between the abstractness of modern mathematics and practical applications. I hope that, in the future, the basic goal of mathematics education-teaching fundamental principles-will be extended to the point where a student can translate powerful mathematical logarithms into computer programs. In those cases where this has happened on the Dart mouth campus, we find that undergraduate students can solve significant research problems in a variety of applications of mathematics. H. W. JOHNSTONE, JR., Professor of Philosophy and Assistant to the Vice President for Research, Pennsylvania State University Let me begin by agreeing with you (J. R. Pierce) that most undergraduates should be exposed to computers and computing as tools. But my own argument for such exposure is not that most students ought to learn to use the computer in order to solve practical problems, any more than I would argue that they should study scientific method in order to solve practical problems. Scientific method, in the form in which is sometimes a required course (or part of one) for all or most undergraduates, is a liberal study. Its purpose is to acquaint the student with the nature of scientific thinking, so that he will see science not as a kind of familiar magic that he takes for granted, but rather as a human achievement. In my view, a similar liberal course ought to be given on computers. The emphasis would be upon the concept of a computer and upon the general methods of using computers. The student who had been exposed to such a course would see the computer as a human achievement rather than as a black box to be taken for granted. He would see how the possibility of using computers to solve problems has revolutionized the ways in which we think about the problems. A person for whom the computer is merely something to be used gains from his contact with it no appreciation of the nature of the contemporary world. Such appreciation presupposes a certain awareness of the nature and method of the computer as such-an awareness that is quite different from the knack of programing in FORTRAN. No one could say that the course I have in mind is concerned with gadgetry. I would say that the nature and method of the computer are philosophical ideas, like those of the nature and method of science in general. Most of the ideas presented in truly liberal courses are philosophical. The philosopher can be concerned with these ideas in a deeper way than is either the student or teacher of the liberal course. The philosopher is not content merely to expound the ideas; he is interested in stating as clearly as possible what they mean. Both the idea of the computer itself and society's assumptions about the use of the computer need analysis and clarification. The philosopher sees culture as manmade, and indeed as an expression of man's view of his own nature. Medieval culture was the expression of the view that man is at home in the world. When science first arose in the 17th century, it was both the cause and the result of an increasing sense of alienation from the world. Man regarded himself as a creature of subjectivity, whose senses screened him from the invariant mathematical relationships that governed the universe in its infinity. Nowadays we are less sure that these relationships are invariant, less sure that the universe is infinite, and more confident in our own point of view. Some of this confidence has been won through the use of the computer. Its role in our culture is thus an expression of our view of our own nature. At your symposium (the computer symposium presented at Bell Laboratories in June 1966), I was fascinated by the particular applications that you are making of computers. What impressed me most deeply was that in the number of these applications we have reached a turning point. The use of the computer has all at once spread to all aspects of our culture. It is this that struck me as being of primary philosophical relevance. What is relevant is the way the computer has changed the quality of contemporary life—not so much in satisfying our material needs as in causing us to think about ourselves in a new way. H. O. POLLAK, Director, Mathematics and Statistics Research, Bell Telephone Laboratories The two most important current facts about mathematics in the colleges and universities of the United States are the increases in the number of mathematics majors and in the total number of students enrolled in mathematics courses. The number of mathematics majors has multiplied by a factor of five in the last 10 years from 4,000 in 1954 to 20,000 in 1964. This is about three times the rate of growth of the overall student population. Similarly, from 1960 to 1965 the total enrollment in undergraduate mathematics courses has risen by about 50 percent, from 570,000 in the fall of 1960 to a guess of over 800,000 for the fall of 1965. This is one and a half times the rate of growth of the overall student population. Now where do the mathematics majors go after they receive their bachelor's degree? First of all, about a quarter of them seem to go into computing, as either a full-time or part-time occupation. We very much owe these students a feeling for the numerical and algorithmic side of mathematics as part of their undergraduate training. Furthermore, pretty close to half the mathematics majors go on to graduate work (mostly in mathematics itself). Many of these students become secondary teachers of mathematics. Now the signs are very clear that some aspects of computer education are moving down from the colleges and becoming a part of the secondary curriculum. Many experiments are already underway-for example, the States of New York and Pennsylvania are teaching computing to some of the present teachers, and SMSG has written a high school computing course. The current plans for future curriculum reform in secondary mathematics all include the computer. Mathematicians who are most concerned with teacher preparation are now recommending that all prospective secondary teachers should learn something about computers (as well as about applications) as part of their college preparation. Finally, the use of the computer in mathematical research is also a growing phenomenon. There are many examples where new ideas for research, or a need to find a new understanding, have appeared through judicious numerical work which brought into the open quite unexpected and unexplained mathematical phenomena. It is not fashionable among research mathematicians to admit this use of the computer, but I can document many examples of it. Despite the current growth of mathematics as an undergraduate major, most of the undergraduates taking mathematics (roughly 80 percent of the total mathematics enrollments) are students from other disciplines; i.e., prospective users of mathematics. Many of the engineers and physical, biological, or social scientists taking mathematics will apply their knowledge through the computer, and it is very necessary in teaching mathematics to these students to keep before them the relation between mathematics and computing. Thus, the teacher of undergraduate mathematics must also know something about the computer. Future computing specialists, future secondary teachers of mathematics, future research mathematicians, future users of mathematics in other disciplines, and future teachers of undergraduate mathematics make up most of our body of mathematics students. It is clear that most undergraduate mathematics courses should contain questions, problems, and pieces of theory which are motivated by the numerical aspect of the subject. Furthermore, all mathematics majors should have an exposure to the computer. 79 |