Introduction to: A Fresh Start, National Mathematics Magazine (1938).
The second semester began today and, with it came the challenge - a class of "repeaters" in Differential Calculus. Not a new challenge for, each year, throughout the collegiate world, boys "flunk" Calculus and then have to repeat it.
"Why talk about it?", you ask? Because I'd like to tell you what we did today and get your reaction to it, based on your experience in similar situations.
At the bell the room was only half full but the back row was solid. Students in this row were ushered to more desirable seats. Late comers took the vacated seats, were invited to move, and saw later comers fill the seats again. Somehow it seemed that they didn't care to get too close to Calculus.
Most of the names and faces were all too familiar. Like the "March of Time", past records flashed across the screen of memory. But memory was unnecessary to predict their attitude, their appearance of complete boredom did that. Not a flicker of interest pierced the wintry gloom.
Probably it was a bad beginning to mention past records but it seemed necessary to announce that they would not count against anyone. This provided the proper opening for the statement that we (the class) were going to discuss Calculus together just as though we knew nothing about it - and we did just that. However, the teacher, at least, added the mental note that we didn't have any very perfect understanding of the prerequisite mathematics either.
Introduction to: We Discover the Meaning of Curvature, National Mathematics Magazine (1940).
It is a matter of considerable satisfaction to any teacher to feel that he has presented a new topic to his class in the best expository style, with the steps of his demonstration properly arranged and supported. He leaves his class room with the conclusion that he has covered a lot of ground in a very neat fashion. Much of his satisfaction evaporates or changes to irritation when, at the next meeting of the class, he finds that his fine presentation has failed to produce any great increase in knowledge on the part of his students. Then, if he is an experienced teacher, he realizes that he has accumulated another bit of experimental evidence of the weakness of the lecture method as a tool for promoting learning.
It is not the purpose of this note to provide any detailed appraisal of the lecture method. It has virtues as well as vices. Usually, however, it seems to have the characteristic indicated above; it gives the teacher the illusion of making rapid progress. Another method, which produces the opposite illusion, is the so-called "heuristic" method. In a class where this method is in use teacher and students discuss together the topic of the day, with the students expressing their thoughts as ideas develop. The teacher directs the discussion and sometimes accelerates it by judicious suggestions, but by no means dominates it. Often very little ground is covered, but, unlike the situation with the lecture method, the advanced position is thoroughly consolidated. The purpose of this note is to outline the manner in which the heuristic method has been used to develop the concept of Curvature with a class in Calculus.
It is futile for any one person to attempt to give a comprehensive answer to so broad a question. First we need to recognize that any answer is a function of several variables, namely: the definition of "learn" as to quality, the specification of "mathematics" as to type, and, particularly, the needs of the person for whom the answer is given. Second, we need to decide whether the answer is to be a superficial one, couched in terms of salesmanship platitudes and implications, or whether we have enough interest in the function to study it with care in the various regions of its definition. And, third, we should try to discover proper applications for our findings.
All of us are quite sure that we know what it means to "learn" mathematics, but I doubt the possibility that we have any precise agreement as to the connotation of the word "learn". Perhaps you will agree that after a person "learns", he "knows". However, that doesn't help much because we have no better agreement on the meaning of "know", or "understanding", or "appreciate". This fact becomes most apparent when the college teacher states that the freshman doesn't know algebra, in spite of the fact that the high-school teacher has given grades to certify the possession of knowledge. Likewise the statement by the grade-school teacher that arithmetic is known is often questioned. Furthermore, there is some suspicion that many teachers, college, high-school, grade, do not know arithmetic, algebra, calculus. Perhaps it is not fitting to dwell longer on this painful subject but the observation might be made that, while ignorance, like poverty, is no disgrace, it is quite unnecessary for a young and normal person in a land of plenteous opportunities. Certainly learning is something more than accumulating a list of facts like beads on a string; it is more than the ability to arrange the facts in intelligent patterns. Perhaps it is better compared to the establishment of interconnections like the intricate wiring of a submarine, such that the throwing of a switch sets in operation a chain of adjustments which combine to produce a desired result.
It might seem easier to define mathematics, but here we are foiled again unless we fall back on definition by illustration and give arithmetic, algebra, geometry as examples. Some people would define mathematics as a tool for measurement, others as a mode of thought by which logical conclusions can be reached. Euler has called it the Queen of the Sciences; some of our students speak of it as the bane of their existence. When you have worked out a good definition for mathematics, test it by seeing whether it separates mathematics and physics, or mathematics and logic. In many institutions of learning, business arithmetic, accounting, mechanics, descriptive geometry, statistics, astronomy are not classified with mathematics. If you explain that these subjects are applied mathematics, you expose yourselves to the question as to what you are applying. Also you bear the burden of proving that subjects traditionally classified as pure mathematics, such as plane geometry, escape the stigma of being called "applied". So mathematics is rather elusive and ethereal when we try to pin it down.
Let us turn to a consideration of our third independent variable, the needs of the person for whom the answer is to be given. Most of you have studied some mathematics and, in the eyes of the world, at least, have "learned" mathematics. If you ask as to why you should have learned mathematics the question is much easier to answer than it would have been when you began to learn to count, or when you started geometry, or when you decided to major in mathematics in your college course. We can count results now which we could not have guaranteed then. In the first place you are making a rather comfortable living by teaching what you have learned; second, you have derived a certain amount of satisfaction from the acquisition of a body of knowledge which you can contemplate as a thing of beauty and can utilize for your entertainment and for the acquisition of other related experiences; third, you have had the advantage of possessing a powerful tool for daily living which has helped you to understand better and cope with the physical world into which you were born, has assisted you in your determined effort to stay in that world as long as possible, and has added to your capacity to contribute your share to the progress of civilization. So, on the whole, we may conclude that it was a good thing for you to learn mathematics. And, since the past seems to provide the safest guide for the future it seems satisfactory to predict that those students of ours who plan to teach mathematics will do well to devote a large share of their attention to learning mathematics. How large this share should be is a question of much difference of opinion, as is apparent when we compare the requirements for state certification with the recommendations made by the Commission on the Training and Utilization of Advanced Students of Mathematics.
Since the answer to our question seems much too easy when we consider only prospective teachers of mathematics, let us focus our attention on a domain where the answer is much harder to obtain. Namely the general case of the student who is in the threshold of exposure to some educative process. At any point, from the time he is born until the time that he leaves this vale of tears, the child and his sponsors have a right to ask the question, "What mathematics should I study, and why?" Now you see that the answer becomes very hard to give because it is so involved with the answers to a number of other questions, yet we must, at least in general terms, be ready to provide a satisfactory answer at every point of time from the cradle to the grave. Of course, by now, you have probably guessed the truth that this paper will provide no answer. It is designed, only, to guide your thoughts along paths where possible answers may lie, leaving to you the choice of satisfactory conclusions, reached on the basis of your own moral, political and educational philosophies.