A Survival Kit for the College Mathematician

A Survival Kit for the College MathematicianAuthor(s): Harley FlandersSource: The American Mathematical Monthly, Vol. 78, No. 3 (Mar., 1971), pp. 291-296Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2317532 .

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1971] MATHEMATICAL EDUCATION 291

Open Conference, conducted immediately after the N.S.F. funded Invitational Conference in March 1970, so that most of the CRICISAM instructor partici- pants could remain and join others in considering new horizons for students who have completed a computer-oriented calculus sequence.

(1) CRICISAM (Center for Research in College Instruction of Science and Mathematics) Calculus: A Computer-Oriented Presentation, Parts 1-5, Florida State University, 1968-69. [The 1970 Edition has been combined into 2 volumes.]

(2) Committee on the Undergraduate Program in Mathematics (CUPM), Newsletter: Calculus with Comnputers, August 1969.

(3) Committee on the Undergraduate Program in Mathematics (CUPM), A General Cur- riculum in Mathematics for Colleges (GCMC) 1965.

(4) E. P. Miles, Jr., A Summary of the ACM-FSU Symposia on the Impact of Computing on Undergraduate Mathematics Instruction CACM, May 1966, pp. 388-389.

(5) E. P. Miles, Jr., et al. The papers of Atchison, Givens, Macon and Murray from (4) plus a detailed summary of the other papers and the discussion at the Symposia. CACM, September 1966, pp. 662-670.

(6) A. Ralston, A critical review of the papers in (5). Computing Reviews, November- December 1966, pp. 462-463.

(7) Proceedings of an Invitational Conference on Calculus and the Computer, CRICISAM and Florida State University, 1970, mimeographed, 36 pages.

A SURVIVAL KIT FOR THE COLLEGE MATHEMATICIAN

HARLEY FLANDERS, Purdue University

1. The problem. I believe that those who teach mathematics have a professional obligation to stay alive as mathematicians. The man teaching in a college, away from the frontier of research, sometimes does not clearly understand this obligation, but more often is under various pressures to do nothing about it.

For contrast, let us first look at the mathematics department in a research oriented university. The professor, under constant publish-or-perish pressure, assigns first priority to doing research. He may not devote sufficient time to class preparation (except for courses and seminars likely to produce research students). He doesn't give a hoot about administration.

Because his effectiveness in research is so dependent on keeping in touch, he spends much of his time in mathematical bull sessions, scanning new journals in the library, getting colloquium speakers, participating in seminars, and at- tending Society meetings.

In this atmosphere, the assistant professor does no administration; the associate professor on the make in research does almost no administration; relatively few full professors-and associate professors who have abandoned the research rat race-handle almost all of the administrative chores, internal and external to the department. (It is often surprising how much those who represent the mathematics department on university-wide committees and legislative bodies are not representative of the department-the research people won't take those assignments.)

In sum, the research man will probably assign these priorities to his working



292 HARLEY FLANDERS [March

time: (1) doing research, (2) reading the literature, bull sessions, seminars, meetings, etc., (3) graduate teaching, (4) undergraduate teaching, (5) administration.

Now let us look at the college mathematics department. The professor there gives, and certainly should give, first priority on his time and energy to (undergraduate) teaching. What I am afraid happens too often is that all his remaining time and energy gets channeled into administration.

I consider this wrong, and believe that staying alive as a mathematician and nurturing a vital interest in mathematics should receive second priority, and be considered almost as important as teaching.

The research mathematician is loyal first to his profession and second to his employer. I expect the college mathematician to be loyal first to his college and second to the mathematics profession. But this second loyalty should be a close second, and he must not allow his college loyalty to overwhelm his responsibil- ity to mathematics, because that will harm his most important work, teaching mathematics.

2. Research publication. I suspect that most college mathematics professors do not produce publishable research. The atmosphere at most colleges, the strong emphasis on teaching as primary purpose, the lack of library facilities, and so on, make it pretty hard to stay near the rapidly expanding research frontier.

A few college mathematicians manage to produce publishable research in spite of these obstacles; this is certainly commendable. Some college mathematicians put considerable energy into unpublishable research. Indeed, the MAA is under some pressure to provide an outlet for what might be called minor research (research not acceptable to the regular research journals because it is too elementary, too small a contribution, or too uninteresting). One of the argu- ments pro is that college men in experimental sciences can publish occasionally, so the college mathematician is at a disadvantage re salary and promotion. Another argument pro is that getting results even in an insignificant area is of real value to the mathematics teacher. I disagree with this second argument; I have reached the conclusion, based on personal examination of hundreds of minor research manuscripts, that much of this work is so far from what mathematics is really about, that the writers are deluding themselves and, indirectly, harming their teaching.

Much minor research consists either of axiom systems for unnatural struc- tures, or minute generalizations of known' theorems and their proofs. I believe there are far more profitable activities for the college mathematician, and shall list some later. Incidentally, Prof. Klee started the Research Problems section of the Monthly to provide a source of meaningful (dare I say relevant?) open problems. I expect that few college mathematicians will ever crack one of these problems, but I know it is far healthier mathematically to work on a concrete problem than on an abstract generalization of a generalization....



1971] MATHEMATICAL EDUCATION 293

3. Administration. Over the years I have visited many colleges. I am always surprised at how much administration is done by mathematicians, at how much time they spend in staff and committee meetings, and at how seriously they take their administrative problems, often to the point where there seems to be little else they talk about. Obviously this is counter-productive to mathematics and must be resisted.

Unfortunately there are both faculty members and professional administrators who see administration per se as the most important activity of the college. There is a strong underlying belief that you can actually legislate quality, that by sufficient work on permuting course numbers, changing prerequisites, curriculum, entrance requirements, by administrative restructuring, and by the rest of the lot, the same teachers will make infinitely more out of the same students. Nonsense!

The good university with 2000 faculty members has perhaps 200 faculty administrative activists, but the college with 150 faculty members can have such overwhelming problems that it may require 149 faculty administrative activists. I have visited small colleges where the faculty as a whole meets almost weekly, and the meetings, often at night, are 3 or 4 hours long in order to get in all of the committee reports.

We have all seen how thoughtlessly a committee is appointed. Here is a typical example: The chemistry department decides its freshmen need computer coding, so it wants the history breadth requirement dropped. The committee on committees, realizing this is a matter of basic educational philosophy, appoints a 15-man committee, representing every department, to meet 4 hours per week for 10 weeks, and then to report to a special faculty meeting. Thus 15 X4X 10 = 600 man-hours are spent on the committee alone. Of course the faculty meeting (150 menX4 hours=600 man-hours) will reject the report, the committee will reconvene, etc.

If your college allocates too many of your hours to administration, then you must educate your administrators (whether autocratic central or autono- mous faculty) to the plain fact that you have other work to do besides meeting classes, preps, and committees, that the regular time parcels scheduled for your own mathematical enrichment are just as inviolable as your classroom hours, and that you are hired to be a mathematics professor, not a college administra- tor.

This is where your chairman comes in. I believe one of the most important functions of a mathematics department chairman is protecting his staff from administrative chores, both within and without the department. Particularly, if a staff member volunteers for excessive administrative work, the chairman must protect the man from himself. If your chairman does not see this as an important part of his job, perhaps it is time for a change.

Actually it is often wise to rotate the chairmanship every 5 years or oftener. The chairman who considers his post as permanent, sooner or later must identify his interests with the administration's interests. Instead of representing the




mathematics department and fighting for its interests, he is likely to consider himself the administration spokesman in the department.

Two final suggestions: Make a fixed rule: no college business after 5 P.M. or on weekends. If there really is an overwhelming problem facing your college (department), find another college (mathematics department) you consider better than yours, which has faced the same problem, and use its solution. Corollary: adopt Harvard's curriculum and save time.

4. Time allocation. Suppose you teach 12 contact hours per week. Your preparations, office hours, and grading will take another 12 hours, so 24 hours are committed to your primary job, teaching. Assuming you want to work a 40 hour week-no more, no less-and that you are not the chairman, what happens to the other 16 hours?

I suggest an absolute maximum of 4 hours per week per man for administration, both in and out of the department. Thus, for example, if the chairman of an 8 man department (excluding himself) needs about 8 hours per week help in the department, then he should allow a maximum of 24 hours per week from mathematics for college-wide business. He should closely scrutinize the assign- ment of this time, and scream if it is frittered away.

This leaves a solid 12 hour per week period for mathematical survival. The point of my article so far has been to buy this time. Now I want to spend it.

5. What to do. I take it as axiomatic that if you cease to learn mathematics and cease to work at mathematical problems, then you will lose your enthusiasm for mathematics and become first a dull and then a lousy teacher. However, if mathematics is vital to you, if a beautiful theorem and proof excites you, if solving a problem thrills you, if seeing a problem challenges you, if mathematics keeps popping into your head, then you have passed survival and you might be a great teacher. The goal then is clear. I shall now list ways to spend the 12 hours per week (I so carefully extracted) working towards the goal.

(1) Work on MONTHLY problems; try to solve some and try to propose some. This is the closest you may come to original research, and when you actually solve a problem, you get much of the same satisfaction. Other problem sources are the Mathematics Magazine and the Siam Review.

(2) Read the MONTHLY. I know (naturally) of no other source of so much useful material for the college mathematician. I suggest you read all of the Notes in each issue. Each contains a nice idea, or a little gem, and each can be worked through in two hours, often less. You will broaden considerably by reading short Notes in other than your major field.

Read one major expository article in each issue (individually, or as a department project).

(3) Have a department seminar. (Nearby colleges can cooperate.) This should meet twice a week for two hours, with a break. The first hour or hour- and-a-half should be a lecture, the rest discussion. This should be regularly scheduled, say every Tuesday and Thursday, 3-5, and should never be cancelled



19711 MATHEMATICAL EDUCATION 295

for college business. The lecturing should rotate among your whole staff, and everyone should prepare in advance by reading the material to be covered. (Of course, invite your better students.)

One possible topic is current MONTHLY main articles. I think working through (part of) a book is a better idea. You must go slowly and very carefully, so that everyone keeps up. Two or three pages a meeting is good progress. Pick a book with exercises and do them. Pick concrete mathematics rather than abstract mathematics. Pick a book that is recent, but in a field in which none of your staff is an expert.

These seminars should be lively. The speaker should be interrupted whenever possible, everyone should be ready to argue. No sissy stuff!

Here is a brief list of possible books: G. Birkhoff and G. -C. Rota, Ordinary Differential Equations, Ginn & Co., 1962. P. M. Cohn, Universal Algebra, Harper & Row, 1965. S. Halgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962. L. Hormander, Introduction to Complex Analysis in Several Variables, Van

Nostrand, 1966. N. Jacobson, Lie Algebras, Interscience, 1962. I. Kaplansky, Commutative Rings, Allyn & Bacon, 1970. W. Magnus, A. Karras, D. Solitar, Combinatorial Group Theory, Jnterscience,

1966. H. Pollard, Mathematical Introduction to Celestial Mechanics, Prentice-Hall,

1966. E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966. J. Todd, Introduction to the Constructive Theory of Functions, Academic Press,

1963. (4) Attend local MAA and AMS meetings. Take notes on the hour addresses

and discuss them in your seminar. (5) Have a colloquium speaker once or twice a semester. Get someone from

the nearest research department and pay his expenses. (You should try to offer a $50 honorariumn, although it isn't essential-it is enough for a small college. The going rate at universities is $75-$100.)

Keep the visitor on campus for a day and talk mathematics with him. This is your chance to get help with difficult textbook problems, to get things you are having trouble with explained, to hear some new ideas on classroom pre- sentations, and so on. The chances are that your visitor would rather talk mathematics than anything else, and that he wants to help you.

(6) If there is a university less than an hour's drive from you, it might be worthwhile going as a group to a weekly seminar. If it is more than two hours away, your total time (5 or more hours) is not worth a single hour. Again, if the university is close by, you should attend the colloquium lectures when you know the invited speaker is a good expositor. Warning: The young colloquium speakers on interviews tend to snow everybody with their deep, deep research results.

(7) Keep abreast of professional activities by reading the Mathematical




Education section and national meeting reports in the MONTHLY, the CUPM reports, the news items and letters in the AMS Notices, and the CBM's News- letter. This is material about mathematics, not mathematics, and is not a sub- stitute for primary sources.

(8) Train a team for the Putnam competition. This means knowing how to work the problems yourself, and teaching your best students how to solve problems, a rewarding activity.

6. Sabbaticals. It is very important to get away from your college once in a while and devote full time to mathematics. Ideal is a sabbatical leave with full pay every seventh year. Full pay is important because college salaries are not so high that one can live for a year on half or two-thirds. And it is important to do no teaching or other paid work during that year off. Perhaps you should settle for every tenth year, but insist on full pay.

The best thing to do with your sabbatical leave is go to a major mathematical center and be a graduate student again. Audit 3 or 4 courses and seminars, and work harder than the registered students. Pick up as much mathematics as you can and hope it will keep you going for the next 10 years.

7. Final remarks. Suppose over a period of a year or two, you and your colleagues work through a hard book and work the problems. This is a fine ac- complishment, and you should be proud of yourselves. No agency will hand you a monetary reward or even a brass medal. Still you should report to your college administration exactly what you are doing and why. Perhaps you will set a good example for other departments.

The goal, briefly is to have a vital attitude toward mathematics. Do not get distracted into thinking that you are learning new things that are directly ap- plicable to your undergraduate teaching. Because you are studying operators in Hilbert space does not mean you teach your junior first course in linear algebra as a special case of Hilbert space, nor because you are studying the Lebesgue integral do you inflict measure and content on your freshman calculus course. Restraint is the hardest thing to learn in teaching mathematics, and we are all sometimes guilty of going too far too soon. The frustration of teaching students who are years away from the things we are dying to tell them is something we have to live with, like it or not.



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A Survival Kit for the College Mathematician