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THE COLLEGE MATHEMATICS JOURNAL VOLUME 16, NUMBER 1 January 1985 WHAT DO I KNOW ? THIS ISSUE ? FORUM?The Algorithmic Way of Life is Best ? What Do I Know? A Study of Mathematical Self-Awareness ? Inverse Functions fj Generalized Pythago? rean Triples ? Picturing Functions of a Complex Variable ? An Official Publication of the Mathematical Association of America This content downloaded from 185.205.48.180 on Fri, 11 Nov 2022 17:32:01 UTC All use subject to https://about.jstor.org/terms
PURPOSE THE COLLEGE MATHEMATICS JOURNAL exists to serve all who are interested in the earlier years of college-level mathematics, the primary focus being on the first two years. The CMJ aims at strengthening the links between teachers of mathematics at all levels. Contributions to this JOURNAL are welcome from teachers in high schools, two-year colleges, and teachers of freshman-sophomore level courses in four-year colleges and universities. EDITORIAL POLICY THE COLLEGE MATHEMATICS JOURNAL seeks lively, well-motivated articles which can enrich instruction and enhance classroom learning. The JOURNAL also invites expository papers which stimulate the thinking and broaden the perspectives of all who teach undergraduate-level mathematics. Articles involving all aspects of mathematics are welcome: history, philosophy, prob? lem-solving, applications, etc. Computer-related mathematics articles and papers which highlight the interface between different branches of mathematics are espe? cially welcome. The JOURNAL also invites other types of contributions, most notably, proofs without words, mathematical poetry, quotes, cartoons, and other mathematical facetiae. SUBMITTING AN ARTICLE It is assumed that manuscripts submitted to THE COLLEGE MATHEMATICS JOURNAL have not already been accepted elsewhere, nor are they under consideration for publication elsewhere. Manuscripts should be typewritten double spaced with wide margins on 8^ x11 paper. Authors should submit the original and three copies along with the author's full address to: Warren Page CMJ?New Manuscript New York City Technical College 300 Jay Street Brooklyn, NY 11201 Authors planning to submit manuscripts may find it helpful to refer to Guidelines for Authors, which appears on p. 61 of the January 1985 CMJ issue. The COLLEGE MATHEMATICS JOURNAL (ISSN 0746-8342) is published by the Mathematical Associa? tion of America at 1529 Eighteenth Street, N.W., Washington, D.C. 20036 and Montpelier, VT, five times a year: January, March, June, September, and November. The annual subscription price for the COLLEGE MATHEMATICS JOURNAL to an individual member of the Association is $14 included as part of the annual dues. (Annual dues for regular members, exclusive of annual subscription prices for MAA journals, are $22. Student, unemployed and emeritus members receive a 50% discount; new members receive a 30% dues discount for the first two years of membership.) The non-member/library subscription price is $33 per year. Bulk subscriptions (5 or more copies) are available to colleges and universities for classroom distribu? tion to undergraduate students at a 50% discount ($7.00 per copy?minimum order $35.00). Subscription correspondence and notice of change of address should be sent to the Membership/ Subscriptions Department, Mathematical Association of America, 1529 Eighteenth Street, N.W., Wash? ington, D.C. 20036. Back issues may be purchased, when in print, from P. and H. Bliss Company, Middletown, CT 06457. Microfilmed issues may be obtained from University Microfilms International, Serials Bid Coordinator, 300 North Zeeb Road, Ann Arbor, Ml 48106. Advertising correspondence should be addressed to Ms. Elaine Pedreira, Advertising Manager, The Mathematical Association of America, 1529 Eighteenth Street, N.W., Washington, D.C. 20036. Copyright ? by the Mathematical Association of America (Incorporated), 1984, including rights to this journal issue as a whole and, except where otherwise noted, rights to each individual contribution. Reprint permission should be requested from A. B. Willcox, Executive Director, Mathematical Associa? tion of America, 1529 Eighteenth Street, N.W., Washington, D.C. 20036. General permission is granted to Institutional Members of the MAA for noncommercial reproduction in limited quantities of individual articles (in whole or in part) provided a complete reference is made to the source. Second class postage paid at Washington, D.C. and additional mailing offices. Postmaster: Send address changes to Membership/Subscription Department, Mathematical Associa? tion of America, 1529 Eighteenth Street, N.W., Washington, D.C. 20036. Printed in the United States of America This content downloaded from 185.205.48.180 on Fri, 11 Nov 2022 17:32:01 UTC All use subject to https://about.jstor.org/terms
THE COLLEGE MATHEMATICS JOURNAL VOLUME 16, NUMBER 1 January 1985 Editor Warren Page, New York City Technical College, Brooklyn, NY 11201 Board of Editors Jeanne L. Agnew, Oklahoma State University, Stillwater, OK 74074 Patrick J. Boyle, Santa Rosa Junior College, Santa Rosa, CA 95401 Donald W. Bushaw, Washington State University, Pullman, WA 99164 Michael W. Chamberlain, U. S. Naval Academy, Annapolis, MD 21402 Leo Chosid, New York City Technical College, Brooklyn, NY 11201 R. S. Cunningham, California State College at Stanislaus, Turlock, CA 95380 Josephine Edwards, Canberra College of Advanced Education, Belconnen A.C.T. Australia Gino Fala, Community College of Philadelphia, Philadelphia, PA 19130 Stanley Friedlander, Bronx Community College, Bronx, NY 10453 Robert M. Geist, Clemson University, Clemson, SC 29631 Sheldon P. Gordon, Suffolk Community College, Selden, NY 11784 Samuel A. Greenspan, Bronx Community College, Bronx, NY 10453 Erwin Just, Bronx Community College, Bronx, NY 10453 James J. Kaput, Southeastern Massachusetts University, North Dartmouth, MA 02747 Alan Mackay, Birkbeck College, London, England Vedula N. Murty, Pennsylvania State University, Middletown, PA 17057 Virgil Nelson, North Plains Community College, North Platte, NE 69101 Peter L. Renz, Bard College, Annandale on Hudson, NY 12504 Ann E. Watkins, Los Angeles Pierce College, Woodland Hills, CA 91371 David H. Wheeler, Concordia University, Montreal (H4B 1R6) Quebec, Canada Milton E. Winger, University of North Dakota, Grand Forks, ND 58202 Technical Consultant Tibor Mingovits, New York City Technical College, Brooklyn. NY 11201 This content downloaded from 185.205.48.180 on Fri, 11 Nov 2022 17:32:01 UTC All use subject to https://about.jstor.org/terms
The Algorithmic Way of Life is Best Stephen B. Maurer Stephen B. Maurer is Associate Professor of Mathematics at Swarthmore College, where he has taught since 1979. He grew up in Silver Spring, Maryland. He received a B.A. from Swarthmore in 1967 and a Ph.D. at Princeton in 1972. He has taught previously at Princeton, the University of Water? loo, Hampshire College, and the Phillips Exeter Academy. His major scholarly interest has been research and curricu- lar development in discrete mathematics, but he has also made forays?sometimes continuous?into mathematical bi? ology, economics, and anthropology. He is an MAA Visting Lecturer, Chairman of the MAA Committee on High School Contests, and a member of CUPM. Professor Maurer is a consultant for the Alfred P. Sloan Foundation, working on projects to increase the role of discrete mathematics in the mathematics curriculum and the role of mathematical and technological "modes of thought" in the general curricu? lum. "Whenever you can settle a question by explicit construction, be not satisfied with purely existential arguments." Hermann Weyl, 1946 Weyl was referring to the well-known penchant of mathematicians to rigidly separate showing that solutions exist from actually finding them. The former endeavor is viewed as a pure, high-minded occupation, abounding in elegance if abstract methods are used. The latter is seen as an applied, rather disreputable activity, to be done on the side if done at all. Weyl was saying that this schism, and the esthetic behind it, are a mistake. WeyPs advice has not been heeded much. Yet today, an even stronger admoni? tion deserves our support: seek the algorithmic way of life. Algorithms go farther than constructions: they involve systematic, mostly itera? tive procedures that are effective for a wide range of inputs, as opposed to one-time, ad hoc manipulations. Moreover, the algorithmic way of life means much more than merely finding solutions and finding them by algorithms; it means adopting an algorithmic frame of mind towards all aspects of one's mathematical work. It means, as Weyl suggested, trying to fuse computation with proof?developing the theory from the algorithms instead of separately or vice versa. It even means letting one's interest in algorithms suggest the questions one will try to answer. Thinking with and about algorithms can unify ail one's mathematical endeavors. It also extends the range of one's mathematics, and provides as satisfying an esthetic as the old existential esthetic?if one will only let it. This content downloaded from 185.205.48.180 on Fri, 11 Nov 2022 17:32:02 UTC All use subject to https://about.jstor.org/terms