Volume 10, Number 6 THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA November-December 1990

The Calculus Turmoil Challenges for College Mathematics Paul R. Halmos Lynn Arthur Steen

What's all this calculus turmoil about? Have all mathematicians This issue of FOCUS contains as a special insert the report of except me taken leave of their senses? Surely it isn't just a racket• a joint Task Force of the MAA and the Association of American a way of getting money from the NSF? But if it is not, then why are Colleges (AAC) entitled "Challenges for College Mathematics: An otherwise honest and intelligent people screaming about it? Will Agenda for the Next Decade." This report was prepared as part somebody please explain to me what's going on? of a multidiscipline AAC project to explore effective approaches to undergraduate majors. Issues arising from the project will be the The two main symptoms (or are they the causes?) of this new subject of discussion at the January 1991 AAC meeting of college disease are that lots of students flunk calculus and that most text• and university deans and presidents. books are bad. In August 1990, the MAA Board of Governors endorsed the Task Well, let's see. "More than a third of the 600,000 or so peo• Force report "as a statement about the major on behalf of MAA" ple who study calculus in college every year fail to complete the and recommended it to MAA members "as a framework for cam• course." (Peter D. Lax, "Calculus Reform: A Modest Proposal," pus discussion about the undergraduate major in mathematics." UME Trends, 2 (Number 2, May 1990). For a different view, see The MAA-AAC report follows in the footsteps of the "David II" re• Leonard Gillman's "Two Proposals for Calculus," FOCUS, 7 (Num• port "Renewing US Mathematics: A Plan for the 1990s," and pre• ber 4, 1987).) I am prepared to believe that, and I am neither sur• cedes by several months the publication of the MS 2000 report prised nor horrified. Teachers have been discussing failure rates "Moving Beyond Myths: Revitalizing Undergraduate Mathematics." with one another ever since the concept was invented, and 40% Together these three documents will provide an agenda for action is not unusual among the numbers that they bandy about. The to help stimulate departments of mathematics to improve the effec• failure rate varies a lot, of course-it depends on the quality of the tiveness of undergraduate mathematics for all students regardless university in question as a whole, and, derivatively, on the quality of background, interests, or intended major. of the teachers and the students at the university. A part of the reason is the reason that students take calculus in the first place. Lynn Arthur Steen, professor of mathematics at St. Olaf College The word has a mystique; it's a Mount Everest to climb. It is an in Northfield, Minnesota, former MAA president (1985-1986), and essential part of the tool kit of many who wish to study science, chair of the Association's Committee on the Undergraduate Pro• and science is a glamorous and well-paid(?) and highly respected gram in Mathematics (CUPM) chaired the joint Task Force of the (Turmoil continues on page two.) MAA and AAC that prepared the report described above.

Inside This Issue

Summer Institutes for Minorities and Women...... 4 Calculus, Computers, Concepts, and Cooperative Learning 4 Regional Technology Centers. .... 4 Board Actions in Columbus. 5 Nominations for Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Mathematics, .5 Goodbye. Columbus. 6 and 7 FOCUS on MAA Journals The American Mathematical Monthly. .8 Chaos in the College Mathematics Journal. .8 Mathematics Magazine . . . 9 1990 MAA Journal Awards. Paul R. Halmos . 9 2 FOCUS November-December 1990

(Turmoil continued from front page.) of the classical drill (that I for one went through in the 1930s) can profession. Students who take calculus "because it's there," or for be left to the button pushers-pocket calculators do very well what the glamor, the money, or the respect, are taking it for the wrong for a long time we haven't been wanting to do at all. But I am still in reason, and their failure is neither surprising nor worrisome. favor of teaching some (many?) rigged integration techniques and tedious calculations-they are as rigged and as tedious as finger What is the failure rate for people taking piano lessons? I don't exercises for the pianist, and just as indispensable. know, but I could make up some statistics as well as the next guy, and I wouldn't be surprised if the true statistics pointed to a situation Is computing to be "an integral part" of calculus? (I can't help won• as "lamentable" as the one in mathematics. dering whether that's an intentional pun.) Yes, of course, that's just what I've been saying. By computing, to be sure, I mean As for how the books are bad: they contain "inert knowledge," what calculus teachers have always meant-computing with ratio• including "integration techniques that work only in specially rigged nal, trigonometric, exponential, and logarithmic functions, and get• cases," "fake applications," and "tedious calculations." ting used to their behavior and their relations to each other. I do not Are those comments new? And, new or not, are they correct? Yes, mean computing in the sense that computing machines are good they are correct, and no, they are not new: they have been made for-horrors, no! for many decades (centuries?), and they point to something bad. The greatest "mathematical" invention of the last few centuries is To something that is all bad? I am not sure. not electronic but notational-I refer to decimals and their relation Very few people advocate epsilons and deltas in elementary cal• to arithmetic. An electronic typewriter that can crunch numbers is culus, and, for sure, I am not one of them, I confess that when it useful-but I don't want to teach spelling in courses in literature, comes to arithmetic I am still in favor of the multiplication table being decimal calculations in courses on number theory, and computing drummed into innocent heads (at least through 9 x 9), together with techniques in courses on calculus. I like spelling and decimals and some of the elementary and almost universally needed techniques computers-but they should be kept in their places. of estimating orders of magnitude. (If each of 1,000 students writes 10 themes a week for each of the 30 weeks of the academic year, What then should a calculus course emphasize, if not integral and if, on average, the labor of grading each theme costs 20 cents, rigging and number crunching? Qualitative understanding of then we should be able to decide whether the money we need from functions?-asymptotics and approximation as implied by the dif• the administration is $6,000, $60,000, or $600,000.) Similarly, in ferential equations they satisfy? Fine-sure-why not. Substantial calculus, I am still in favor of teaching the classical basic techniques applications-in engineering, in economics, in psychology?-yes, of the differential and integral calculus. I want a student who sees maybe-but our focus should always remain at the mathematical x 3 or sin x to snap back 3x 2 or cos x, and I want a student who center. We, teachers of calculus, are not usually practicing engi• neers, economists, or psychologists, and just as we might worry sees f 1~:2 and f 1~:2 to know the difference between them about the competence of those people to teach our stuff, they have and to be able to come up with the indefinite integral in both cases, the full right to worry about our encroaching on their expertise. But even if a pocket calculator can do it more quickly. It is true that much I don't really think there is anything serious to worry about as far as the content is concerned. Most calculus courses (certainly the ones I took, the ones I taught, and the ones I hear colleagues tell about) are pretty much freewheeling-they emphasize the mechan• ical techniques to the extent that drill is necessary (or a little too much, or not quite enough, but more or less to the right extent), and they contain the illustrations and applications that the teacher is competent to explain and the students are ready to receive. Ft'JCUS Do bad books contribute to making the courses bad? Yes and no• mainly no. At the calculus level, in my experience, students simply FOCUS is published by The Mathematical Association of America, 1529 Eigh• don't read the book-most of them cannot and none of them has teenth Street Northwest, Washington, DC 20036, six times a year: January• February, March-April, June, September, October, and November-December. to. They cannot because they just don't know how to read-the formal language of the text is repulsive to them, and boring, and Editor: Andrew Sterrett, Interim Associate Director of Programs, MAA Associate Editors: Donald J. Albers, Menlo College and difficult-most of our class time has to be spent in repeating the David W. Ballew, Western Illinois University book in the vernacular. And they don't have to because most of Managing Editor: Harry Waldman our class time is spent in repeating the book in the vernacular. Advertising Manager: Siobhim B. Chamberlin Chairman of the MAA Newsletter Editorial Committee: A famous (infamous?), old (ancient?), bad (dreadful?) calculus Susan L. Forman, Bronx Community College CUNY book is Granville, Smith, and Longley-and its badness is irrelevant The subscription price for FOCUS to individual members of the Association is $3, to using it as a text. It tells some lies, yes, but who reads them?! included as part of the annual dues. Annual dues for regular members (exclusive of subscription prices for MAA journals) are $29. Student, unemployed, emeri• What matters is that it has a table of contents, to which you can tus, and family members receive a 50% discount; new members receive a 30% add and from which you can delete-and it has a large collection discount for the first two years of membership. of numbered problems that you can assign for homework. Being @ by The Mathematical Association of America (Incorporated), 1990. Educa• fearful of a largely imaginary and hugely exaggerated disease al• tional institutions may reproduce articies for their own use, but not for sale, pro• legedly caused by the enforced use of uniform texts is a rather silly vided that the following citation is used: "Reprinted with permission from FOCUS, phobia-any calculus teacher who has taught the subject once can the Newsletter of the Mathematical Association of America (Incorporated), 1990." ISSN: 0731-2040 use any book and do as well with it as with any other.

Second-class postage paid at Washington, DC and additional mailing offices. Yes, there are good books and they have been with us perhaps Postmaster: Address changes to Membership/Subscriptions Department, Math• even longer than the bad ones that water them down-Landau, ematical Association of America, 1529 Eighteenth Street Northwest, Washington, Courant, and Apostol may have lived in vain, but they did live. DC 20036-1385. Printed in the United States of America. (Turmoil continues on next page.) November-December 1990 FOCUS 3

(Turmoil continued from previous page.) baseball pitchers and airplane pilots. Such respect, such a social No, good books don't catch on-and a large part of the reason is background and atmosphere, have been eroding in the US for most that our calculus students are not ready for them. Most students of the twentieth century-since, as a rough approximation, the time are more than intelligent enough to cope, but they've been held of John Dewey, my favorite villain, one of whose effects appears to back-pushed back!-since their early years in grade school and have been to cause people to sneer at abstractions and admire the high school, and they arrive at the university not just untrained or "practical." That, I think is why the US is at or near the bottom of badly trained but almost negatively trained. Some of them in their many school achievement lists in which the countries of the world turn become poorly trained grade school teachers and high school are compared-that is why we have a "calculus crisis" but Japan teachers and go on blissfully to give their students training at least does not. as bad as they had received. Yes, there is a disease, but calculus is neither its cause not its The results of the bad training are visible everywhere. It shocked main symptom. We mathematicians can do our small bit to cure it, me to find, as I was teaching a course on analytic functions, that but not by rewriting calculus books. All that we can do, all that we my students (all of whom had credit in a course called "advanced are professionally able to do, is to insist on raising the quality of calculus") not only did not know the sum of a geometric series but primary and secondary education by establishing and maintaining maintained that they had never even heard of it-that's bad col• a high quality in college courses, by insisting on and strictly enforc• lege calculus. But, in the same course, I was even more shocked ing severe prerequisites, and by encouraging and properly training to learn that they knew neither the factor theorem (the relation be• prospective grade school and high school teachers. That we can tween the roots of a polynomial equation and the factors of the do, and I hope we will. polynomial) nor its name-once again they had never even heard of it-and that's bad high school algebra. And, incidentally, and strongly and relevantly, the same sort of ignorance is visible in other subjects, notably in the use of language. When I require that the solutions of problems be communicated to me not just by scribbled FOCUS 1990 Statement of Ownership, equations but by sentences that begin with a capital letter, end with Management, and Circulation a period, and have a reasonable sprinkling of commas between, my students groan and tremble-they don't believe me, they think I am unreasonable, and they simple don't (can't) do it.

Did all this start "in the fifties"-is it all a new disease caused largely by "the vast majority of research mathematics?" Balderdash! The U,"'~., .. problem is not new. Practicing mathematicians have known about I", :t'~'I.(::::::,,~ YOCVS ... it long before the 1930s, when I began to dip my toes into the I' 1;_' MI,I...\lDRfJlR'Qf>"'- :::~;b~rp'~' mathematical ocean, and they will keep talking about it, I predict, • .od 00 e, "'-.-,Six Indiv. Member $5 '':~~~~' .... ItO long after we are all gone. Old people always say that the world 152' $1...... tb St. ,.oW. W••bL..gt.o,n.".2l)ll" is going to hell in a handbasket, and I'm ready to subscribe to that ailil\gAcldt.uof· H..,___,. 011;.**. tenet-things might always have been bad, but maybe, maybe, 1529 Eigbtuntb st.• H.W., Wuhingt()n, e. "uIl.N...... -.d~~M.alng Mdt... of PUblithw; iditOl'; '"' Min they are getting worse. Y\lDOi""er fl'lilfltlf_ .. _ .~A6liI/IIf--J Th. Matheutical Auociatiotl of America, 1529· Eighteenth St. ~N.W•• Washington. D.C. 20036 Students know less and less about language (because they are MibraJ Dt. Andrew Sterrett. HM, 1529 Eighteenth St•• N.W., Washington. D.C. 20036 taught less and less), and they know less and less about calculus _w~._ ... c_ ....._, (because they are given less and less preparation, and, in fact, Harry Wald1ll4n. MAA. 1529 li$heeench St., H.W., W..bington~ D.C. 20036 7. OWMl'fl/a-.Jb1G"'~,In ___#lMNu_"' __ iuloi."~"""'''''''''':'''"_~'of'~~'''~ progressively worse preparation, in high school). Is calculus the l~fI'.~.f/IfIIM __tif""'lJl\IIf __ by.~.Jhi:~_~i{".~~~H,~l/o-d/1r."fI"'f"Iff" (W"""~/l""'_""_~''''·''''''''_ii,,,iMiritAiil''''''Ii_(f·~uPflblUllltl.,·-.mfJt.~,Jn only mathematical sufferer? No-absolutely not-and I suppose ...... ~'·".,H~I·I1uM_"tt/lflllktf4.J that the reason calculus is usually singled out is that it's a huge ...... C~ ...... industry-it is the bread and butter of many of us. But precalcu• lus (multiply and divide polynomials and sines and cosines) and postcalculus (add vectors and multiply matrices) are in equally bad • 0, shape. CO'nipitJ...M4lIInt ......

Is the message coming through? I am saying that there is a lot of unjustified turmoil, but I am not saying that there is nothing wrong. 9,.ForCOn'>P"tionbyNOnPfOfitOfotnlll~AlltMtllitdToNliiat .v,I:,.", I am saying that the calculus turmoil is rather silly-it is both mis• T"'Pllt9o... fUllC1ioft,ltldnonl)foflUl:.NlIofthitlJlllMl~.tlon_mil fOr't9i'.. ~I"IPUfl»OPtfOltd:_' ,~ [!]H"NetCh"'8edOvr~ H.. Cttenll'd During (If ..IwItfttI·~,-NIMl#~of placed and exaggerated-but there is something wrong, and what's .....,:edintl'" 2 Monm. o ~1ZMontht dI4tv~..m.II\U_,1 ' ,c. '~I_Durint wrong is much broader and deeper and more serious than calculus En.trI .nd Nllute of Cifcu"tiOft A_..... Copi.. Ac1ull No. CopiH 01SinO" '"ue fSl,t/ufvcflolv(lft""fHaa, ~12~ .....~ Nnrnt to'ilint 0.'. students who flunk and textbooks that are dull. A. Toul No. CopiU ~ l'rm /fwoJ :JliS3:3 29 000 e. P-'dand/Oo'AeQ\MI,adClrcUlittlQn , ..5,,111 throueh dnItIfI,.1IId c.riIra...... , vtndorlIl'ldCGUI\W Ill" 0 0 Are most human beings too stupid to learn calculus and grammar? a. MaiI&ubJe,ill1iOn 11'tIUI~~.MI 29.31$ 27 497 No, the trouble is not genetic-it is, for want of a better word, so• c. Tourl>1id lIftfJIorRaQ_tedCirculllloll I$Mlitlll'l/JIItiilldlOJJ1l ninS 27 497 ciological. In Japan, we are told, students learn calculus, and they D,'i'.. Ci'~,bvMIlI,C.".,. 01'OU\etMtl... are the same kind of human beings as the ones in Manhattan, New 5_111.., C~irIIM1wY, ...a0thl;, F_COf)\Itj .356 1 080 t.1'ItQIDlI1fiblltlOflawtoofC-'OJ York or Manhattan, Kansas. But in Japan there seems to be an al• 30,671 28 577 ,. COll!.' Not DilftrilWt.d most universal respect for learning, a social background from which l,OI~ ....,I"jto"",Iffi-eI;:_ted,~tftt#p11ntint 861 423 ,.hill"' from N,_ AOIfltI children absorb the feeling that intellectual effort can be pleasant 0 0 G.TOT"'lls-.ofl.FlIltll4Z~~""".tlWl~loi.tJ 11l$33 29,000 and rewarded and applauded-an atmosphere in which scholarship .1$i""~t/lCl.fittaOf.l!~~~ ..."e:\l~t.lIfI""O'O_ H'I eerttfy th8t thestatementt IYUldI by .. is encouraged as much as (more than?) athletics and commerce, l1MlttbcM .. oorrect 8nd ~ i ..~. t;."···"?MembershiP Manager a social structure in which professors are as good as (better than?) ,. form 362,8. Fell. 1919 4 FOCUS November-December 1990

1991 Summer Mathematics Institutes Regional Technology Centers

• University of California at Berkeley Ohio State University invites applications for a National Science Intensive Mathematics Program Foundation (NSF) project to train school/university teams to estab• for Underrepresented Minority Undergraduates lish regional technology centers. Regional team training will be provided through summer inservice sessions and academic year • Mills College follow-up conferences at Ohio State University. These regional Intensive Mathematics Program centers, in turn, will train teachers as school technology special• for Women Undergraduates ists at their regional sites. Local living expenses with some stipend and travel support are available. In addition, Ohio State University In an effort to increase the number of underrepresented minority will offer partial first-year support for a few regional sites. Dead• and women students seeking careers requiring a PhD degree in line for completed applications: 15 February 1991. For further the mathematical sciences, the University of California at Berke• information, contact: Frank D. Demana and Bert K. Waits, Ohio ley (UCB) and Mills College will each offer a residential summer State University, Department of Mathematics, 231 West Eighteenth program for undergraduate students, 17 June-26 July 1991 (thirty Avenue, Columbus, Ohio 43210. students at UCB; twenty-four at Mills). The programs are being organized by Uri Treisman of the Department of Mathematics at Swarthmore College and the Dana Center for Mathematics and Sci• ence Eduction at UCB, by Leon A. Henkin of the Departments of Calculus, Computers, Mathematics at UCB and Mills, and by Lenore Blum of the Depart• ment of Mathematics at Mills College and the International Com• Concepts, and Cooperative Learning puter Science Institute (ICSI). Funding for these programs is being Purdue University will host a summer workshop partially funded sought from the National Science Foundation. by NSF and West Educational Publishing Company in late May We are asking faculty members to seek out ethnic minority and and early June 1991. The two-week, intensive, total-immersion women students who could benefit from our programs and to en• program will focus on the use of computers, research into learning courage them to apply. The UCB Institute is designed for African• theory, and a cooperative learning environment to help students American, Mexican-American/Chicano, Latino, and American In• learn calculus concepts. A three-day preworkshop tutorial on the dian undergraduates. The Mills Institute is designed for women of basic use of the necessary computer systems will be offered as an all ethnicities. All applicants must have completed two years of option. Program participants are expected to return the following collegiate mathematics with distinction by June of 1991. summer for two days to discuss their teaching experiences based on the workshop. Each participating student will explore two areas of mathematics. Part of this exploration will take place in seminars of twelve stu• The program will provide hands-on experience with computer soft• dents each, organized by senior mathematicians. Students will be ware (both MS DOS and Macintosh), class materials, and semi• encouraged to tackle many challenging problems individually, in nars on learning theory, including viewing and discussing videos small groups, and in consultation with graduate student mentors. of Purdue classroom and laboratory experiences in cooperative In addition, there will be weekly colloquia designed to provide par• learning. The program will feature the use of the mathematical ticipants with a broad view of current work in mathematics as well programming language ISETL and its graphics package, in addi• as of career opportunities for mathematicians. Each student se• tion to the symbolic computer systems Maple and Derive. Partic• lected will receive room and board, a $2,000 stipend, and travel ipants will be encouraged to consider piloting the calculus course allowance to and from the summer institute. presented in the workshop during' academic year 1991-92. Partial support for attending the workshop will be available. For informa• Previous Berkeley seminar leaders have included Ani Adhikari, tion about the program and application material, contact: Ed Dubin• Efraim P. Armendariz, Oscar Moreno, James W. Pitman, Carl sky or Keith Schwingendorf, Department of Mathematics, Purdue Pomerance, Carl L. Prather, Louise A. Raphael, Don P. Rawlings, University, West Lafayette, Indiana 47907; [email protected] or and Edwin H. Spanier. Previous Berkeley colloquium speakers [email protected]. Completed application deadline: 15 Febru• have included Hajnal Andreka, Richard A. Baker, Lenore Blum, ary 1991. Successful applicants will be notified by 15 March 1991. Andrew Casson, Alice Chang, Amy Cohen, Martin D. Davis, Persi Diaconis, F. Alberto Grunbaum, Leon A. Henkin, Morris W. Hirsch, Wu-Yi Hsiang, Irving Kaplansky, Serge Lang, Thomas M. Liggett, Istvan Memeti, Kenneth C. Millett, Brad G. Osgood, Robin A. Pe• mantle, Diane Resek, James A. Sethian, Stephen Smale, Martha In Memoriam K. Smith, Michael Starbird, and P. Emory Thomas, Lamberto Cesari, Professor Emeritus, University of Michigan Faculty for the 1991 Summer Mathematics Institutes will be drawn at Ann Arbor, died 12 March 1990 at the age of 79. He was from universities and research laboratories nationwide. Mathemati• an MAA member for 36 years. cians interested in participating in either institute as seminar leaders or colloquium speakers are invited to contact Professor Uri Treis• Lawrence D. Gould, Instructor, North Carolina School of Sci• man (University of California at Berkeley): Department of Math• ence and Mathematics, died 9 May 1990 at the age of 58. He ematics Swarthmore College, Swarthmore, Pennsylvania 19081, was an MAA member for 25 years. (215) 328-8224; or Professor Lenore Blum (Mills College) ICSI, William R. Orton, Jr., Professor Emeritus, University of 1947 Center Street, Berkeley, California 94704, (415) 643-9153. Arkansas, died 9 May 1990 at the age of 68. He was an Projected deadline for applications is 22 February 1991. Further in• MAA member for 37 years. formation and application forms for both institutes can be obtained S. Thomas Parker, Professor Emeritus, Kansas State Uni• by telephoning: (415) 642-5881; or writing to: 1991 Summer versity, died 14 March 1990 at the age of 76. He was an Mathematics Institutes, c/o Dana Center for Mathematics and Sci• MAA member for 45 years. ence Education, 230 B Stephens Hall, University of California at Berkeley, Berkeley, California 94720. November-December 1990 FOCUS 5

Board Actions in Columbus For secretaries and program committee chairs in the Sections, there was good news emanating from the Board meeting in Columbus. Gerald L. Alexanderson, Secretary, MAA A new Polya Lectureship was approved. I quote from the report of the Committee assigned the task of formulating the standards and policies for this Lectureship: "George Polya embodied the high The Mathematical Association of America has had an active year, quality of exposition which the Mathematical Association of America with a new administrative team in place in the Washington of• seeks to encourage. To further this goal, the Board of Governors fice (June 1990 FOCUS), the dedication of the newly renovated of the Association hereby creates the George Polya Lectureship. Polya Building in June (September 1990 FOCUS), and the anniver• This lectureship will be held by an individual representing the high sary celebration this past summer in Columbus (January-February standards set by George Polya. Each Section will be entitled to a 1990, March-April 1990, June 1990, and this issue of FOCUS). All Polya Lecture approximately once every five years." This program of these events have been or will be reported on elsewhere, in sep• will provide to Sections speakers of national prominence. When a arate articles in FOCUS and in the annual report of the Executive Section is eligible for a Polya lecturer, it may also invite a speaker Director. In these remarks I shall comment briefly on recent actions from the list of national officers, editors, etc., as in the past. The of the Board of Governors and other related activities. Polya lecturer will not have to substitute for a speaker from the list normally available to Sections. As I remarked in my report at the Business Meeting in Columbus, we need not fear that the MAA is an organization of the increas• Summer meetings remain a concern. Attendance at the Colum• ingly aged. Our Student Chapters program has been an enormous bus meetings ran over 1,300, a large number by recent standards, success. In the most recent list of new members to go before the but that was probably due to the special nature of the anniversary Board of Governors, 81% were students. This dramatic growth meetings. If attendance were to fall again to the 60D-800 level ex• in the number of student members in the Association (mainly un• perienced at some recent summer meetings, there will probably be dergraduate students, it would seem) is surely in large part due further negotiations with the American Mathematical Society (AMS) to our student chapters. The Membership Committee is currently to adjust the format of summer meetings or to go on some sort of discussing ways of encouraging graduate students to join the As• schedule like that proposed earlier, with summer meetings only in sociation. Along with this growth, we are seeing more activities for those years without international congresses (ICM or ICME). A joint students at our national and sectional meetings. AMS-MAA committee met in Washington, DC in June 1990 to dis• cuss summer meetings. Some of the ideas that were developed at At its Columbus meeting in August, the Board of Governors elected the meeting will be evident in the plans for the summer meetings a new editor for the American Mathematical Monthly, to replace in Orono, Maine, in August 1991. Herbert S. Wilf of the University of Pennsylvania. The new editor will be John H. Ewing of Indiana University. Professor Ewing has At the Business Meeting and Prize Session in Columbus, awards been associate editor of the Monthly and editor of The Mathemati• were made to winners of the Carl B. Allendoerfer, Lester R. Ford, callntelligencer, which, in the words of the nominating committee, and George Polya Awards. (Names and citations appear in this he "transformed ... into a lively, widely praised mathematical mag• issue of FOCUS, page nine.) John W. Kenelly gave a report on the azine." It is perhaps premature to thank Herb for his fine service drive for the Building Fund, for which the Association is actively so• as editor of the Monthly, since he has over a year remaining in his liciting contributions to cover the renovation of the buildings housing term, but it is comforting at this point to know that the Monthly will the Washington, DC offices of the Association. Earlier that week remain in capable hands following 1991. The present appointment a President's Breakfast honored those who had contributed at the is an important one, because the new editor will preside over the "hexahedra" level. centennial of the Monthly in 1993. The next meeting of the Board will take place in San Francisco, The Board also elected a nominating committee to develop slates California, Tuesday, 15 January 1991. of candidates for the offices of president-elect, first vice-president, and second vice-president. The committee members are Doris W. Schattschneider (chair), Wade Ellis, Jr., Martha J. Siegel, Lynn Arthur Steen, and Alan C. Tucker. Suggestions of names of pos• sible candidates are solicited for the Committee in the September Nominations Sought for 1990 issue of FOCUS, page four. Vueh-Gin Gung and The Board approved a report, "Challenges for College Mathematics: Dr. Charles V. Hu Award An Agenda for the Next Decade," of a joint Task Force of the MAA and the Association of American Colleges (AAS). The center insert The committee to select a recipient for the Yueh-Gin Gung and in this issue of FOCUS reproduces this report in full. Dr. Charles Y. Hu Award for Distinguished Service to Mathemat• ics seeks nominations for that award. The principal clause in the The Board also considered at some length a draft of a report regulations governing the award states: from the Committee on the Mathematical Education of Teachers (COMET), "A Call for Change: Recommendations for the Mathe• "The award is to be made for outstanding service to mathematics. matical Preparation of Teachers of Mathematics." Suggestions were The period of service may be long or short, and the award may solicited; a final version of the report will be presented for approval be made on the basis of one or several activities. The contribution at the January 1991 meeting of the Board. should be such as to influence significantly the field of mathematics or mathematical education on a national scale. The recipient should A proposal for restructuring the many committees of the Association be a member of the Association who is a resident of the United into six groupings referred to as "councils" or "areas" has been States or Canada." discussed at previous Board meetings and meetings of committee chairs. In Columbus the Board approved the restructuring plan, with Nominations must include supporting information and should be approval of a final version deferred to the January 1991 meeting. sent by 26 November 1990 to Professor Deborah Tepper Haimo, Final details involving the assignment of some committees to these Department of Mathematics and Computer Science, University of "councils" or "areas" remained to be worked out. Missouri at St. Louis, St. Louis, Missouri 63121. Goodbye, Columbus

Meeting participants gather in the lobby of Mershon Leonard Gillman, 42nd President of the M.4A Auditorium on the campus of Ohio State University.

Thomas W. Tucker, current First Vice-President, Alan C. Tucker, First Vice-President, 1988-1989, Andrew M. Gleason, and Alice C. Beckenbach

Shirley Willcox, David P. Roselle, MAA Secretary, 1975-1983, and Alfred B. Willcox, MAA Executive Director, 1969-1989

Ronald L Graham, Danalee Buhler, Peter Frankl, Brad W. Jackson, and Joe P. Buhler demonstrate their juggling talents at the Mathematical Circus. Henry L. Alder, 37th President ofthe MAA, introduces G. Baley Price, the Association's 27th President.

Wade Ellis, Jr., Speaker, MAA Day

Donald J. Albers, Second Vice-President, 1983-1984, and Saunders Mac Lane, 24th President ofthe MAA

Ronald L. Graham, MAA First Vice-President, 1982-1983

Eileen L. Poiani and Ivan Niven, 40th President ofthe MAA, and Stan Wagon Henry O. Pollak, 36th President of the MAA FOCUS on MAA Journals

The American Chaos in the College Mathematical Monthly Mathematics Journal

Herbert S. Wilt, Editor Ann E. Watkins and William E. Watkins, Editors

The October 1990 edition of the The January 1991 College Math• American Mathematical Monthly ematics Journal will be a spe• realizes a landmark achievement cial issue on how the study of in the nearly century-long span of dynamical systems can enliven the journal's publication. This is• traditional courses. Authors in• sue, a special, thematic collection clude Robert L. Devaney, Gilbert devoted entirely to geometry, fea• Strang, and James T. Sandefur. tures color on both its cover and its pages (the first time the Monthly Devaney, this issue's coordinator, writes, "The mathematics of has ever done so), to emphasize the subject's visual beauty. The chaos and fractals is at once accessible, alluring, and exciting. cover illustration (above) reproduces a minimal surface David A. Fractal geometry offers a wonderful arena for combining computer Hoffman of the University of Massachusetts at Amherst and Her• experimentation and geometric insight." In the new section on re• search projects for students, Paul C. Matthews and Steven H. Stro• mann Karcher of the University of Bonn discovered. James Riordan gatz suggest problems to get students started exploring the connec• and Jim Hoffman of the Geometry, Analysis, Numerics, and Graph• tion between chaotic mappings and probability distributions. The ics Center (GANG) at the University of Massachusetts at Amherst popular "Fallacies, Flaws, and Flimflam" presents a gentle series of created the image. It is singly-periodic and behaves at infinity much theorems and proofs that culminates with the proof of a theorem, like the union of helicoids. which may be new to you, that the set of real numbers is countable. Furthermore, this addition of color on the cover and inside plates SOME ADVICE FOR THE NEXT EDITOR Although we are only about a complements the journal's superb miscellaney of essays from sev• year and a half into our term as coeditors of the CMJ, we have some eral distinguished authors. Together, their words and pictures in• advice to give to our successor(s). Some of this advice concerns teract to shape a spectacular and historic issue. authors, so if you are a potential author, please stop reading now. • S. S. Chern of the Mathematical Sciences Research Institute at 1. The quickest way to predict whether a manuscript will the University of California at Berkeley asks, "What is Geom• eventually be accepted: Count the number of authors. Of etry?" His essay, the issue's keynote selection, together with the first roughly three hundred manuscripts submitted as main his poem in the original Chinese and in English translation, ex• articles to the CMJ during our term as editors, about 6% of presses the unity of geometry and physics. manuscripts with a single author but 35% of manuscripts with multiple authors were accepted for publication. • Marcel Berger of the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, presents a delightful discourse on convexity 2. The second quickest way to predict whether a manuscript with numerous examples-some of which he locates in art and will be accepted: Count the number of references. Referees anatomy. like authors who have done their homework and papers that have historical perspective and that give proper attribution to • Ronald L. Graham of AT&T Bell Laboratories and Frances Yao those who have already published on the topic. of Xerox Palo Alto Research Center (PARC) lead a "Whirlwind Tour of Computational Geometry" and discuss the major prob• 3. What to do when the trisector comes: Don't try to point lems, both solved and unsolved, in this young, dynamic field. out the mistake. Otherwise, by return mail you will receive a substantially longer revision in which the "carelessness" has • David A. Hoffman and William H. Meeks, III, both of the Univer• been rectified. sity of Massachusetts at Amherst, deliver an especially "color• ful" contribution featuring dazzling, computer-generated, color 4. What to do if you (or the printer) makes a mistake: Don't prints of minimal surfaces. Their article will excite both the eye worry. No one will notice. Has anyone looked at the cover of and the mind. the May 1989 CMJ?

• Robert Osserman of Stanford University explores recent devel• 5. How to deal with irate authors who think that their manu• opments in another branch of geometry: curvature. Indeed, scripts should have been accepted and that the referees everyone interested in curvature should read his examination are out-to-lunch: There is no way to deal with irate authors. of this field's progress and agenda during the eighties. 6. How to find good referees: Thorough and helpful referees • Finally, William P. Thurston of Princeton University considers are gold, and editors hoard them and cherish them. Occa• the possibility of tiling a portion of a plane lattice that is cut out sionally, after reading an article in FOCUS, a potential referee by a simple closed curve. He also discusses some engaging writes to the editor volunteering his or her services. examples he worked out with John Conway. (Journals continues on page nine.) Challenges for College Mathematics:

An Agenda for the Next Decade

Report of a Joint Task Force

of the

MATHEMATICAL ASSOCIATION OF AMERICA

and the

ASSOCIATION OF AMERICAN COLLEGES

LYNN ARTHUR STEEN, CHAIR, St. Olaf College JEROME A. GOLDSTEIN, Tulane University ELEANOR GREEN JONES, Norfolk State University DAVID LUTZER, College of William and Mary PHILIP URI TREISMAN, University of California, Berkeley ALAN C. TUCKER, SUNY at Stony Brook This report was completed in cooperation with a national review of arts and sciences majors initiated by the Association of American Colleges as part of its continuing com• mitment to advance and strengthen undergraduate liberal learning. The Mathematical Association of America was one of twelve learned societies contributing to this review. Each participating learned society convened a Task Force charged to address a com• mon set of questions about purposes and practices in liberal arts majors; individual task forces further explored issues important in their particular fields. Reprints of this report on mathematics are available from the Mathematical Association of America, 1529 Eighteenth Street, NW, Washington, DC 20036.

In 1991, the Association of American Colleges published a single volume edition of all twelve learned society reports with a companion volume containing a separate report on "Liberal Learning and Arts and Science Majors." Inquiries about these two publications may be sent to Reports on the Arts and Sciences Major, Box R, Association of American Colleges, 1818 R Street, NW, Washington, DC 20009.

Generous funding for the project and dissemination of the reports was provided by the Fund for the Improvement of Postsecondary Education (FIPSE) and the Ford Foundation. Contents

Preface iv Mathematics in Liberal Education l HISTORICAL PERSPECTIVE 1 COMMON GOALS 2 DIVERSE OBJECTIVES 3 MULTIPLE TRACKS 5 EMPHASIZING BREADTH 6 EFFECTIVE PROGRAMS 7

Challenges for the 1990s 8 LEARNING 8 TEACHING 9 TECHNOLOGY 11 FOUNDATIONS 13 CONNECTIONS 17 VARIETy 14 SELF-EsTEEM 15 ACCESS 16 COMMUNICATION 16 TRANSITIONS 17 RESEARCH 17 CONTEXT 19 SOCIAL SUPPORT 20

Mechanisms for Renewal 20 DIALOGUE 20 ASSESSMENT 21 FACULTY DEVELOPMENT 22 DEPARTMENT REVIEW 23 GRADUATE EDUCATION 24

Summary 25

References 26 Preface

In response to a request from the Association of and provosts at a variety of institutions; and American Colleges (AAC), the Mathematical As• many leaders of mathematical professional soci• sociation of America (MAA) convened a special eties. This draft was also reviewed at meetings Task Force to address a range of issues concerning of AAC and MAA in January 1990. At the latter the undergraduate major as a sequel to the 1985 meeting, the document was discussed extensively AAC report Integrity in the College Curriculum. at an invitational three-hour roundtable meeting Whereas the 1985 Integrity study examined the that involved about forty experienced college and effectiveness of general education, the new AAC university mathematicians, including many who study addresses the contribution that "study in have been working actively to improve opportuni• depth" makes to liberal education. Each of the ties for women and minorities in the mathematical several AAC disciplinary task forces examined how sciences. study in depth relates to goals for the major, as• Sectional Governors of the MAA were asked to surance of intellectual development, and relations nominate exemplary departments whose programs with other fields. illustrate issues discussed in the report. Descrip• The MAA-AAC Task Force operated in the con• tions of some of these programs have been adapted text of two other simultaneous MAA reviews of the as illustrations of promising practices in the final mathematics major. One was conducted by a sub• draft. During the spring of 1989 graduating se• committee of CUPM, the Committee on the Un• niors on several campuses were surveyed as part dergraduate Program in Mathematics; the other by of a multi-disciplinary AAC effort to gather stu• COMET, the Committee on the Mathematical Ed• dent opinion. Subsequently, leaders of several stu• ucation of Teachers. Since these other committees dent chapters of the Mathematical Association of are charged by MAA to provide specific advice to America were asked to review a draft of the report. the mathematical community about requirements Their careful and thoughtful responses underscore for the mathematics major, the MAA-AAC Task our belief in the value to students of the recom• Force dealt only with broader questions that are mendations contained in this report. central to the AAC study. Hence this report is not The report has benefited enormously from these intended as a detailed statement about curricular many external reviews. We believe that it now content, but as a statement of issues and priori• represents a consensus of the informed mathemat• ties that determine the context of undergraduate ical community concerning urgent issues of impor• mathematics majors. tance to the undergraduate mathematics major. In preparing the present draft, the Task Force In August, 1990 the report was unanimously ap• held several open hearings on issues concerning proved by the MAA Board of Governors as an offi• the undergraduate major at national meetings of cial MAA statement concerning the undergraduate the Mathematical Association of America and the major. We hope that widespread discussion of this American Mathematical Society. A draft of this report will help focus the efforts at reform that are report was mailed for review to several hundred already underway on many campuses. persons including every Governor, Chair, and Sec• retary of the 29 sections of the Mathematical As• LYNN ARTHUR STEEN, sociation of America; department heads, deans, Task Force Chair Challenges for College Mathematics: An Agenda for the Next Decade

Mathematics in Liberal Education 800,000; each year about 4,000 students received a bachelor's degree in mathematics, and about 200 The 1985 AAC report Integrity in the College received Ph.D. degrees. Curriculum [6] sets forth a vision of undergrad• Following an unsuccessful initial effort to intro• uate education steeped in the tradition of liberal duce a "universal" first-year course in college math• education. It describes study in depth in terms ematics, CUPM concentrated in its second decade of the capacity to master complexity, to undertake on proposals to strengthen undergraduate prepara• independent work, and to achieve critical sophis• tion for Ph.D. study in mathematics [15]. Spurred tication. To achieve the kind of depth envisioned on by Sputnik and assisted by significant support by the authors of this report, students must grap• from the National Science Foundation, interest in ple with connections, progress through sequential mathematics rose to levels never seen before (or learning experiences, and enhance their capacity since) in the United States. By 1970 total under• to discern patterns, coherence, and significance in graduate mathematics enrollments had expanded their learning. Study in depth should enhance stu• to over three million students; U.S. mathematics dents' abilities to apply the approaches of their ma• departments produced 24,000 bachelors and 1,200 jors to a broad spectrum of problems and issues, doctoral degrees a year. and at the same time develop a critical perspective But then the bubble burst. As student interest on inherent limitations of these approaches. shifted from personal goals to financial security, We are especially concerned in this study with and as computer science began to attract increas• how the experience in the major contributes to the ing numbers of students who in earlier years might education of the great majority of students who do have studied mathematics, the numbers of mathe• not pursue advanced study in the field of their un• matics bachelor's degrees dropped by over 50% in dergraduate major. Hence we focus more on the ten years, as did the number of U.S. students who quality of students' engagement with their colle• went on to a Ph.D. in mathematics. However, total giate major than on curricular content which may undergraduate mathematics enrollments continued be required for subsequent study or careers. This to climb as students shifted from studying mathe• emphasis on general benefits of the major rather matics as a major to enrolling in selected courses than on specific things learned gives the AAC un• that provided tools necessary for other majors. In dertaking a distinctive perspective that is not often 1981, at the nadir of B.A. productivity, CUPM emphasized in discussions of the mathematics ma• published its second major comprehensive report jor by mathematicians. on the undergraduate curriculum [16] which advo• cated a broad innovative program in mathematical Historical Perspective sciences. For over 35 years the Committee on the Under• During the 1980s the number of undergraduate graduate Program in Mathematics (CUPM) has majors rebounded, although both the number and helped provide coherence to the mathematics ma• the percentage of U.S. mathematics students who jor by monitoring practice, advocating goals, and persist to the Ph.D. degree continued to decline suggesting model curricula. Until the 1950s, most [1, 21]. As a consequence, one of the most urgent mathematics departments functioned primarily as problems now facing American mathematics is to service departments for science and engineering. restore an adequate flow in the pipeline from col• CUPM was established in 1953 to "modernize lege study to Ph.D. attainment. Three recent re• and upgrade" the mathematics curriculum and "to ports from the National Academy of Sciences [39, halt the pessimistic retreat to remedial mathemat• 9, 44] document this crisis of renewal now con• ics." At that time total enrollment in mathemat• fronting U.S. mathematics. ics courses in the United States was approximately The precipitous decline in the 1970s and early 2 AAC-MAA TASK FORCE REPORT

1980s in the number of undergraduate mathemat• many students are not mathematics majors-they ics majors paralleled both a significant decrease in are enrolled to learn mathematical techniques used mathematical accomplishment of secondary school in other fields. As a consequence, the major has graduates and an erosion of financial support for suffered from neglect brought about in part by mathematical research. Strangely, all this occurred the overwhelming pressure of elementary service at precisely the same time that the full scope of courses. mathematical power was unfolding to an unprece• In spite of the 1965 and 1981 reports from dented degree in scientific research and business CUPM and dozens of collateral studies published policy. These conflicting forces produced a spate in the last decade, there is no national consen• of self-studies within the mathematical community, sus about the undergraduate mathematics curricu• some devoted to school education [13, 19, 40, 41], lum. However, examination of practices at many some to college education [2, 50, 59], a few to grad• campuses reveals common threads that are highly uate education [8, 12], and others to research issues compatible with goals of the AAC Integrity study. [9, 24, 29]. The crisis of confidence among mathe• These include a multiple track system that ad• maticians reflected broad public concern about the dresses diverse student objectives, emphasis on quality and effectiveness of mathematics education breadth of study in the major, and requirements [24,45,32]. Even research mathematicians are now for depth that help students achieve critical sophis• taking seriously the crisis in school mathematics tication. In this report we build on these common [27, 62]. A recent report [43] by the National Re• threads to suggest new goals for study in depth search Council recommends sweeping action based that will both enhance the mathematics major and on an emerging consensus with broad support that better suit students who will live and work in the extends well beyond the mathematical community. twenty-first century. Market-driven demand for undergraduate math• Common Goals ematics majors is strong. There is a shortage Mathematics shares with many disciplines a of mathematical scientists with doctor's degrees; fundamental dichotomy of instructional purpose: there is consistent demand (with high salaries) for mathematics as an object of study, and mathe• those who hold master's degrees in the mathemati• matics as a tool for application. These different cal sciences; there is steady demand for undergrad• perspectives yield two quite different paradigms uate mathematics majors in entry-level positions in for a mathematics major, both of which are re• business, finance, and technology-intensive indus• flected in today's college and university curricula. try; and there is increasing need for well-qualified The first, reminiscent of the CUPM recommenda• high school teachers of mathematics, especially as tions of the 1960s, focuses on a core curriculum a consequence of the national effort to raise stan• of basic theory that prepares students for gradu• dards for school mathematics. Moreover, the in• ate study in mathematics. The second, reflecting creasingly analytical nature of other professions the broader objectives of CUPM's 1981 mathemat• (business, law, medicine) as well as the continued ical sciences report, focuses on a variety of math• mathematization of science and engineering pro• ematical tools needed for a "life-long series of dif• vide strong indicators of the long-range value of ferent jobs." Typically faculty interests favor the an undergraduate mathematics major for students former, whereas student interests are usually in• who are entering other professions. clined towards the latter. Most campuses support Today, mathematics is the second largest dis• a mixed model representing a locally devised com• cipline in higher education. Indeed, over 10% of promise between the two standards, although ac• college and university faculty and student enroll• curate descriptions of present practice are virtually ments are in departments of mathematics. How• non-existent. (Partly in response to the paucity ever, more than half of this enrollment is in high of information about degree programs in mathe• school-level courses, and most of the rest is devoted matics, the National Research Council through its to elementary service courses. Less than 10% of the project Mathematical Sciences in the Year 2000 is total post-secondary mathematics enrollment is in seeking to stimulate a comprehensive examination post-calculus courses that are part of the math• of degree patterns in mathematics.) ematics major. Even in these advanced courses The 1985 AAC Integrity study, echoing themes CHALLENGES FOR COLLEGE MATHEMATICS 3 reflected in similar studies [10, 38, 42], provides undertake intellectually demanding mathemati• a broad context in which to examine practices cal reasoning. of individual disciplines. The AAC report high• •A mathematical sciences curriculum should be lights "study in depth" as an essential component designed for all students with an interest in of liberal education, but not for the reasons com• mathematics, with both appropriate opportuni• monly advanced by students and their parents• ties for average mathematics majors and appro• preparation for vocation, for professional study, or priate challenges for more advanced students. for graduate school. Integrity views study in depth • Every student who majors in the mathematical as a means to master complexity, to grasp coher• sciences should complete a year-long course se• ence, and to explore subtlety. "Depth cannot be quence at the upper-division level that builds on reached merely by cumulative exposure to more two years of lower-division mathematics. and more ... subject matter." The AAC goals for • Instructional strategies should encourage stu• study in depth are framed by twin concerns for in• dents to develop new ideas and discover new tellectual coherence intrinsic to the discipline and mathematics for themselves, rather than merely for development of students' capacity to make con• master the results of concise, polished theories. nections, both within their major and with other • Every topic in every course should be well moti• fields. vated, most often through an interplay of appli• Both previous models advanced by CUPM for cations, problem-solving, and theory. Applica• the undergraduate major reflect these same po• tions and interconnections should motivate the• tentially conflicting concerns. The earlier major, ory so that theory is seen by students as useful which emphasized a traditional core mathematics and enlightening. curriculum as preparation for graduate study, was • Students majoring in mathematics should un• motivated principally by the internal coherence of dertake some real-world mathematical modelling mathematics; the more recent proposal stressed project. an interplay of problem solving and theory across • Mathematics majors should complete a minor in both the broad spectrum of the mathematical sci• a discipline that makes significant use of mathe• ences and the boundaries that mathematics shares matics. with other disciplines. Although many believe Emphasis on coherence, connections, and the in• that the latter model, emphasizing broadly appli• tellectual development of all students are evident cable mathematical methods, is better geared to in these principles from the 1981 CUPM report. graduates' future employment, others-including At the level of broad goals, the prevailing profes• many liberal educators-believe that broader abil• sional wisdom concerning undergraduate mathe• ities such as the art of reasoning and the disposition matics matches well the intent of AAC's Integrity. for questioning are of greater long-term practical value. However, both types of majors-and the Diverse Objectives many mixtures that exist on today's campuses• help students discern patterns, formulate and solve Once one moves beyond generalities and into problems, and cope with complexity. In this sense, specifics of program development, however, main• stream mathematical practice often diverges from present practice of mathematical science majors many of the explicit AAC goals. Most students in U.S. colleges and universities matches well the overall objectives of Integrity in the College Cur• study mathematics in depth not to achieve broad goals of liberal education but for some professional riculum. purpose-for example, to support their study of Certain principles articulated by CUPM in science or to become a systems analyst, teacher, 1981 (reprinted in a recent compendium [17] of statistician, or computer scientist. Others study CUPM curriculum recommendations from the past mathematics as a liberal art, an enjoyable and chal• decade) make explicit areas where Integrity's ob• lenging major that can serve many ends. It is as jectives and those of the mathematical community true in mathematics as in any other field that the align: great majority of undergraduate mathematics ma• • The primary goal of a mathematical sciences ma• jors do not pursue advanced study that builds on jor should be to develop a student's capacity to their major. 4 AAC-MAA TASK FORCE REPORT

Because so few U.S. students pursue graduate for non-mathematical professions such as law, study in the mathematical sciences, many math• medicine, theology, or public service, or for other ematicians believe that the mathematics major employment which does not directly use mathe• should be strengthened in ways that will prepare matical skills. students better for graduate study in mathematics. There is no lack of sources for advice about ob• Whereas formerly this view may have been based jectives. Recent documents outline priorities for on the myopic academic view of undergraduate ed• teacher training [14], applied mathematics [25], ucation as the first step in the reproduction of uni• and service courses [33]. The diversity of U.S. edu• versity professors, today in mathematics this per• cation ensures that different departments will have spective is reinforced by vigorous argument based different priorities that recognize differences in en• on an impending shortage of adequately trained tering students, in goals of graduates, and in insti• mathematics faculty. Some argue for greater depth tutional missions. to ensure that all mathematics majors are capable For many students the link between their under• of pursuing further study; others argue for greater graduate major and their post-graduate plans is breadth to attract more students to the mathe• very elastic. Such students study mathematics for matical sciences. National need has now reinforced the same reason that hikers climb mountains-for self-interest in emphasizing preparation of the next challenge, for fun, and for a sense of accomplish• generation of Ph.D. mathematicians as an impor• ment. Often mathematics is paired with another tant priority for many college and university math• major to provide complementary strengths. At one ematics departments [20, 39]. institution this has helped strengthen the mathe• This said, there remains considerable room for matics major while enhancing connections to other debate about strategies for achieving the several fields of study: different (but overlapping) objectives that are com• An analysis of the senior class shows that nearly mon among the majors offered by the 2500 math• half of the graduates had double majors. In many ematics departments in U.S. colleges and universi• cases the mathematics major was taken to pro• ties: vide theoretical and computational grounding nec• essary for a modern approach to primary majors • ADVANCED STUDY. Preparation for graduate in economics, physics, chemistry, or biology. In study in various mathematical sciences (includ• other cases availability of a double major gave stu• ing statistics and operations research) or in other dents who enjoyed mathematics an opportunity to continue their studies in mathematics while also mathematically based sciences (e.g., physics, gaining a desired major in an apparently unrelated computer science, economics). field-for example, English, philosophy, or music. Many of our majors chose mathematics as a sound • PROFESSIONAL PREPARATION. Skills required liberal arts approach to a general education. The to pursue a career that requires considerable success of the major program in mathematics is background in mathematics: due in part to the belief of both faculty and stu• dents that the study of mathematics is not just • Natural and Social Science. Background for for those intending to pursue a career in the area. careers in science or engineering, in biology (in• Some argue that the goals of liberal education cluding agriculture, medicine, and biotechnol• are best served by a mathematics major designed ogy), in many computing fields, and in such to prepare students for graduate school. Even emerging interdisciplinary fields as cognitive though most mathematics majors never undertake science or artificial intelligence. further study of mathematics, advocates of pre• • Business and Industry. Preparation for ca• graduate preparation argue that the special com• reers in management, finance, and other busi• bination of robust problem solving with rigorous ness areas that use quantitative, logical, or logical thinking achieved by a solid pre-graduate computer skills normally developed through major also serves well the more general objectives undergraduate study of mathematics. of sequential study, intellectual development, and • School Teaching. Preparation for teaching sec• connected learning. This view is substantiated in ondary school mathematics, or for careers as part by a strong history of mathematics majors mathematics-science specialists for elementary being eagerly sought by employers and graduate school. programs in other areas (e.g., law, economics) for • LIBERAL EDUCATION. General background a wide variety of non-mathematical professions. CHALLENGES FOR COLLEGE MATHEMATICS 5

Others argue that since most of today's college to student aspirations and to institutional goals. students do not foresee graduate study in math• Wherever faculty and students share common ob• ematics as a desirable goal, it is only through a jectives, mathematics can thrive. more general major stressing links to other fields that enough students can be recruited to major Multiple Tracks in mathematics to serve as a proper basis for the Most mathematics departments resolve the di• nation's needs in teaching, in science, and in math• lemma of diverse goals for the major with some ematics itself. In this view, a broad major stress• sort of track system. In some institutions there ing mathematics as part of liberal education is an are separate departments such as Applied Math• effective strategy for strengthening the pool of po• ematics, Operations Research, Statistics, or Com• tential mathematical doctoral students as well as puter Science, whereas in others these options are students in other mathematically oriented profes• accommodated by means of explicit or implicit sions. It is rare to find a department of mathe• choices within the offerings of a single Department matics that would naturally place top priority on of Mathematics or Department of Mathematical a major that specifically serves liberal education, Sciences. Tracks within the major are a sensible as is common, for example, in many departments strategy to respond to competing interests of stu• of English or philosophy. The needs of society and dents, of faculty, and of institutions. the constraints of client disciplines in science and Although tracks do accommodate student inter• engineering do not permit mathematicians this lux• ests-and thereby help sustain enrollments-they ury. But in the present climate, departments are can produce a fragmented curriculum. Whereas rediscovering the strategic value of a broad major, in the late 1960s most mathematics majors took even for those who do continue professionally in six or seven standard courses before branching into mathematics. Here's how one institution expresses electives, by 1981 CUPM found that there was no its objectives for the mathematics major: longer any national agreement on such an exten• sive core of the undergraduate mathematics ma• The mathematics major is designed to include stu• jor. At that time, only elementary calculus and lin• dents with a wide variety of goals, tastes, and backgrounds. Mathematics is an excellent prepa• ear algebra were universally recognized as required ration for fields from technical to legal, from sci• courses within the mathematics major. Branching entific to managerial, and from computational to occurred after the third or fourth course, rather philosophic. It is also a source of satisfaction for people in every line of endeavor. Recognizing this, than after the sixth or seventh. we have constructed a program to welcome inter• Today however, despite institutional diversity, ested students of all sorts. Our comparatively un• there is striking uniformity in the elementary and structured program reflects not only the diversity of interests of our students, but their increasingly intermediate courses pursued by mathematics ma• diverse backgrounds, and the increasingly diverse jors: all begin with calculus for two, three, or nature of what are now being called the "mathe• four semesters; most introduce linear algebra in the matical sciences." sophomore year and require one or two semesters of Since department goals must match institutional abstract algebra; virtually all require some upper missions, it would not be right for any commit• division work in analysis-the "theory of calculus." tee to recommend uniform goals for individual de• Nowadays, most require some computer work (pro• partments. We can, however, urge increased atten• gramming, computer use, or computer science) as tion to an important and distinctive feature of un• well as some applied work (statistics, differential dergraduate mathematics: National need requires equations, etc.) among the electives. This restores greater encouragement for students to continue a de facto six-course core to the major, typically their study of mathematics beyond the bachelor's half the total. level-whether as preparation for school teaching, The rise, fall, and restoration of a core cur• for university careers, or for employment in gov• riculum in the mathematics major paralleled simi• ernment and industry. In some institutions this lar patterns in other arts and sciences. Whereas encouragement may arise from a thriving pre• the CUPM recommendations of the late 1960s graduate program; in others it may evolve from an may have had too narrow a focus, the subsequent emphasis on liberal education. In all cases, depart• curricular chaos of the mid-1970s may have been mental objectives must be realistically matched too unstructured. Mathematicians began worrying 6 AAC-MAA TASK FORCE REPORT then-as AAC is now-about whether the typical ics majors to take some advanced sequence without student's experience with the mathematics major specifying which particular sequence it should be. may lack appropriate coherence and depth. Too Thus most mathematics majors today take a sub• often, it seemed to many, the mathematics major stantial sequence of courses, but they no longer all had become just an accumulation of courses with• take the same sequence of core courses. This is a out sufficient structure to ensure a common core of wise policy for undergraduate mathematics in to• learning. day's diverse climate: Each student who majors in In most colleges courses in the mathematical sci• mathematics should experience the power of deep ences are taught in many different departments. mathematics by taking some upper-division course Even upper division mathematics courses are com• sequence that builds on lower-division prerequisites. monly taught in departments of application: a re• It is neither necessary nor wise, however, to re• cent survey [28] shows that as many students study quire that all mathematics majors take precisely the post-calculus mathematics outside departments of same sequence. mathematics as do so within traditional mathe• Flexibility with rigor can be administered in a matics course offerings. Examples include discrete variety of ways. In one institution with a flourish• mathematics taught in departments of computer ing mathematics major, the mechanism is a per• science; methods of mathematical physics taught sonal "contract" developed to suit each student's in departments of physics; logic and model theory own objectives: taught in departments of philosophy; optimization Students arrange their major sequence according to a contract system. Potential majors meet with and operations research taught in departments of a member of the department and prepare a list of economics; and mathematical foundations of lin• courses and activities that will constitute the ma• guistics taught in departments of linguistics. The jor. This allows the student to arrange his or her program to suit special needs. The faculty mem• practice of cross-listing such courses or counting ber judges the appropriateness of the student pro• them as electives in the major varies enormously posal in terms of post-graduate plans, other stud• (and arbitrarily) from campus to campus. ies, and general departmental guidelines. This contract system has two distinct advantages: it Whether for good or ill, the diffusion of mathe• serves the personal needs of students, and the pro• matics courses both within departments of math• cess itself enhances students' education. The pro• cess of developing the contract provides an oppor• ematical sciences and into other departments has tunity for the student to work closely with a fac• moved the mathematics major away from a strict ulty member, to understand the variety of math• linear vertical pattern towards a more horizontal ematical options in a personal framework, and to see how a program ensuring depth and breadth of structure typical of the humanities or social sci• study can be achieved. ence major. Today's major, however, retains a dis• tinctive strength of mathematics: sequenced learn• Emphasizing Breadth ing. By its very nature, mathematics builds on it• At the same time as we stress the value of se• self and reinforces links among related fields. All quential courses to study in depth, we must also mathematics courses build on appropriate prereq• emphasize the essential contribution of breadth uisites. A student who progresses from calculus to building mathematical insight and maturity. to probability to operations research sees just as Whereas course sequences demonstrate depth by many connections as does one who moves through building in expected fashion on prior experience, the more traditional sequence of advanced calcu• the links that emerge among very different courses lus and real analysis. Although the focus of each (tying geometry to calculus, group theory to com• student's work is different, the contributions made puter science, number theory to analysis) re• by each track to the general objectives of study in veal depth by indirection: such links point to depth are comparable, and equally valuable. deeper common principles that lie beyond the Moreover, it is common for advanced courses to student's present understanding but are within be offered in sequences (e.g., Abstract Algebra I, grasp with further study. They show the moun• II; Real Analysis I, II; Probability and Statistics tain yet to be climbed-to shift metaphors from I, II) that begin with a three or four course chain depth to height-and offer hints of the explana• of prerequisites. Many departments, following the tory panorama to be revealed by some future and 1981 CUPM recommendations, require mathemat- more profound principles. CHALLENGES FOR COLLEGE MATHEMATICS 7

There are other good reasons to recommend Many measures can be used to monitor effective• breadth as an important objective of an under• ness of a mathematics major. Indicators of num• graduate mathematics major. Students who are bers of majors, of employability, of graduate school introduced to a variety of areas will more readily admissions, of eventual Ph.D.s, or of placement in discern the power of connected ideas in mathemat• teaching jobs are used by different departments ac• ics: unexpected links discovered in different areas cording to their self-determined missions. Many provide more convincing examples of a deep logical mathematics departments work hard to improve unity than do the expected relationships in tightly their effectiveness in one or another of these dif• sequenced courses. ferent dimensions. Exploration, experimentation, For the many majors who will teach (either in and innovation-along with occasional failures• high school or college), it is vitally important that are the hallmarks of a department that is commit• their undergraduate experience provide a broad ted to effective education. view of the discipline-since further study gener• Mathematics programs that work can be found ally is more narrow and specialized. For those in all strata of higher education, from small pri• seeking their niche in the world of mathematics, vate colleges to large state universities, from av• a broad introduction to many different yet inter• erage to highly selective campuses. The variety connected subjects, styles, and techniques helps of mathematics programs that work reveal what pique interest and attract majors. And for the can be achieved when circumstance and commit• many students who may never make professional ment permit it. When faculty resolve is backed by use of mathematics, depth through breadth offers strong administrative support, most mathematics a strong base for appreciating the true power and departments can easily adopt strategies to build scope of the mathematical sciences. Graduates of vigorous majors even while meeting other service programs that emphasize breadth will become ef• obligations. fective ambassadors for mathematics. One department that has had great success in Every student who majors in mathematics attracting students to major in mathematics bases should study a broad variety of advanced courses in its work on two "articles of faith:" order to comprehend both the breadth of the mathe• • We believe that faculty should relate to their stu• matical sciences and the powerful explanatory value dents in such a way that each student in the de• of deep principles. Such breadth can sometimes be partment will know that someone is personally in• terested in him and his work. achieved with courses offered by the department • We believe that careful and sensitive teaching that of mathematics, but more often than not it would helps students develop confidence and self-esteem be educationally advantageous for students to also is far more important than curriculum or teaching select a few mathematically-based courses offered technique. by other departments. Another department builds strength on a founda• tion of excellent introductory instruction: Effective Programs We put our best teachers in the introductory Departments of mathematics in colleges and uni• courses. We put the most interesting material versities exhibit enormous variety in goals and ef• in the introductory courses. We try to make the fectiveness. In various universities, the percentage statements of problems fun, not dry. We work very hard to motivate all topics, drawing on ap• of bachelor's degrees awarded to students with ma• plications in other disciplines and in the working jors in mathematics ranges from well under one• world. We are less interested in providing answers half of 1 percent to over 20 percent. In some than in motivating students to ask the right ques• departments the major is designed primarily to tions. prepare students for graduate school. Other de• Effective mathematics programs reflect sound prin• partments focus much of their major on preparing ciples of psychology as much as important topics students to teach high school mathematics, or on in mathematics: preparing students for employment in business and We try to make students proud of their efforts industry. Most departments fail to attract or retain in mathematical problem-solving, and especially many Afro-American, Hispanic, or Native Ameri• proud of their partial solutions-what some might call mistakes. We look at how much is right in an can students, whereas a few succeed in this very answer and teach how to detect and correct the difficult arena. parts that are wrong. 8 AAC-MAA TASK FORCE REPORT

Regular, formal recognition of student achieve• • The social support problem: To enhance stu• ment at different stages of the major serves to build dents' personal motivation and enthusiasm for students' confidence and helps attract students to studying mathematics. major in mathematics. Students know mathemat• These challenges have more to do with the suc• ics' reputation for being challenging, so recognition cess of a mathematics program than any curricu• of honest accomplishment can provide a tremen• lar structure. In the diverse landscape of Amer• dous boost to a student's fragile self-esteem. Effec• ican higher education, successful programs differ tive programs teach students, not just mathematics. enormously in curricular detail, but they all have in common effective responses to many of these Challenges for the 1990s broader challenges. The agenda for undergradu• Changes in the practice of mathematics and in ate mathematics in the 1990s must focus at least the context of learning pose immense challenges as much on these issues of context, attitude, and for college mathematics. Many of those issues that methodology as on traditional themes such as cur• pertain directly to course content, curricular re• ricula, syllabi, and content. quirements, and styles of instruction are under re• view by committees of the mathematical commu• Learning nity. We focus here on challenges that transcend One principal goal of the undergraduate math• particular details of courses and curriculum: ematical experience is to prepare students for life• • The learning problem: To help students learn to long learning in a sequence of jobs that will re• learn mathematics. quire new mathematical skills. Departments of • The teaching problem: To adopt more effective mathematics often interpret that goal as calling styles of instruction. for breadth of study. But another interpretation is • The technology problem: To enhance mathemat• just as important: because mathematics changes so ics courses with modern computer methods. rapidly, undergraduates must become independent • The foundation problem: To provide intellectu• learners of mathematics, able to continue their own ally stimulating introductory courses. mathematical education once they graduate. • The connections problem: To help students con• Most college students don't know how to learn nect areas of mathematics and areas of applica• mathematics, and most college faculty don't know tion. how students do learn mathematics. It is a tribute • The variety problem: To offer students a suf• to the efforts of individual students and teachers ficient variety of approaches to a mathematics that any learning takes place at all. Effective pro• major to match the enormous variety of student grams pay as much attention to learning as they career goals. do to teaching. • The self-esteem problem: To help build students' First-year students need special attention to confidence in their mathematical abilities. launch their college career on a suitable course. • The access problem: To encourage women and Typically, they carry with them a high school tra• minorities to pursue advanced mathematical dition of passive learning which emphasizes bite• study. sized problems to be solved by techniques provided • The communication problem: To help students by the textbook section in which the problem ap• learn to read, write, listen, and speak mathemat• pears. Unfortunately, by maintaining this tradi• ically. tional teaching format which perpetuates the myth • The transition problem: To aid students in mak• of passive mathematics learners, college calculus ing smooth transitions between major stages in teachers typically contribute more to the problem mathematics education. than to its solution. • The research problem: To define and encourage For example, calculus, the common entry point appropriate opportunities for undergraduate re• for potential mathematics and science majors, of• search and independent projects. ten fails to come alive intellectually as it should • The context problem: To ensure student atten• or as it is now at many institutions where calcu• tion to historical and contemporary contexts in lus reform efforts are underway. One school has which mathematics is practiced. found that new goals for calculus can significantly CHALLENGES FOR COLLEGE MATHEMATICS 9 enhance the entree of students into the study of They should come to recognize that it is common college mathematics: even for mathematicians to hear lectures or read The larger goals of the major are reflected in the material that they cannot grasp, and they must calculus sequence, which is founded on three prin• learn how to pick up clues from such experiences ciples: context, collaboration, and communica• that will fit into their personal mathematical puz• tion. "Context" means that we focus on the mean• ing and significance of calculus in the world. "Col• zles only some time later. They should learn the laboration" means that students work in groups value of persistence and the strategic value of go• and support each other. "Communication" means ing away and coming back. These "metacognitive" the recognition that calculus is first of all a lan• guage, not only for scientists, but for economists skills to control one's own learning are virtually and social scientists. Our goal is fluency. never learned in high school mathematics, so they Another institution uses calculus as a vehicle to must be planned into the early stages of the college broaden radically the view of mathematics that curriculum. students bring with them from high school: As students progress through their mathemat• ical study, they need to learn the value of li• Calculus should give students a solid base for ad• brary and electronic resources as tools for learn• vanced study. It is our opinion that our calculus courses were the weakest part of our program. We ing. Mathematics students rarely use the library or had, in effect, allowed the high schools to set the other sources of information, concentrating instead tone for our entire program. Our new course is so on mastering material in course texts. They need radically different from traditional calculus that our students are forced to confront the transition specific assignments that focus on the big map of from school to college mathematics. It carries sev• mathematics in order to gain perspective on their eral important messages, e.g., mathematics is cru• brief undergraduate tour. Undergraduate students cial for understanding science; mathematics has a strongly experimental side; mathematics is some• should not only learn the subject of mathematics, thing we all are capable of understanding deeply; but also learn how to learn mathematics. The ma• and mathematics is the most powerful of all the jor in mathematics should become more than the sciences. sum of its courses. By conscious effort to help stu• Some institutions offer special freshman semi• dents negotiate in unfamiliar terrain, instructors nars as a way to encapsulate the ideal of liberal can provide them with the tools of inquiry neces• education in an intimate setting that permits stu• sary to approach the literature and learn whatever dents to identify with faculty mentors. However, they need to know. in mathematics the massive tradition of calculus often stands in the way, so very few mathematics Teaching majors can trace the origin of their college major to The purpose of teaching, and its ultimate mea• a freshman seminar. Ideally calculus itself would sure, is student learning. So in some sense one be seen by colleges as the intellectual equivalent cannot discuss one without the other. However, of a freshman seminar in which students learn to as students must learn to learn, so teachers must speak a new language. If that analogy were to be learn to teach. In mathematics more than in most accepted, mathematics departments would teach other subjects, the role of teaching assistants and calculus only in a context that placed a great deal part-time instructors is particularly important, es• of emphasis on one-to-one communication between pecially in the first year [12]. Although there is student and teacher. Unfortunately, in too many no formula for successful teaching, there is con• institutions calculus is taught in large impersonal siderable evidence that separates certain practices settings that make meaningful person-to-person di• that have proven successful from those that are alogue unrealistic. Many efforts are now under• generally ineffective [55]. Teachers who study this way to reform the teaching of calculus [64]; most evidence can learn much from the experiences of of these experiments emphasize student motivation others. and styles of learning as a primary factor in reshap• Despite the general reputation of mathematics ing the course. as one of the most desirable environments for devel• One way or another, students should learn early oping rigorous habits of mind, criticism of under• in their college years how to study and learn math• graduate mathematics has been mounting in recent ematics. They should learn psychological as well years for failure in this, mathematics' distinctive as mathematical strategies for solving problems. area of strength. Those who study cognitive de- 10 AAC-MAA TASK FORCE REPORT velopment criticize standard teaching practices for tance of a supportive environment: constructive failing to develop fully students' power to apply teamwork in a context of challenging problems in their mathematical knowledge in unfamiliar ter• which instructors and students know each other rain. These critics conclude that present teaching personally builds mathematical self-esteem and, as practice in undergraduate mathematics does not a consequence, leads to greatly improved learn• do as much as it should to develop students' intel• ing [5, 30, 63]. Very different but equally strik• lectual power. ing lessons emerge from experiences of students The evidence of failure is persuasive, both lo• who study calculus in a technology-intensive en• cally and globally [57]. Data on the inability of vironment: by forcing students (and instructors) the profession to attract and retain the best and to focus on the behavior of mathematical objects brightest college graduates is confirmed by case (functions, algorithms, operators) rather than on studies of students who cannot make effective use their formalism, and by integrating visual, numer• of what they have learned. Although some very ical, and symbolic clues into the mathematical en• good students use a mathematics major as a plat• vironment, computers reveal to students and fac• form for substantial accomplishment, the majority ulty both avenues for insight and common sources of those who major in mathematics never move of misconception [36]. much beyond technical skills with standard text• A third example, but by no means the last that book problems. Passive teaching and passive learn• could be cited, can be found in evidence of im• ing results from an unconscious conspiracy of mini• proved student motivation and self-reliance that mal expectations among students and faculty, both occurs in those contexts where research-like experi• of whom find advantages in a system that avoids ences are used to enrich traditional classroom and the challenges of active learning that fully engages textbook experiences: students whose minds and both students and teacher. Both the curriculum eyes become engaged in the challenge of true dis• and teaching practices must respond to this chal• covery are frequently transformed by the experi• lenge of intellectual malnutrition that is all too ence [56]. common in today's major. The evidence from such diverse but non-tradi• tional instructional environments shows clearly Much of the research that bears on how students the effectiveness of instruction that builds self• learn college mathematics has been conducted in confidence on the foundation of significant accom• the setting either of high school mathematics or plishment in a context that is meaningful to the college physics. The results from these efforts are student. Here is an especially dramatic example: often surprising, yet not well known among uni• versity mathematicians. They show, among other In 1986 we began a critical evaluation of our program, course offerings, and teaching methods. things, that formal learning by itself rarely influ• This examination led to profound changes in our ences behavior outside the artificial classroom con• understanding of the teacher-student relationship, text in which the concept was learned [53, 54]. and of our role in the educational process. We found, for example, that we had not engaged our Students who know how to solve differential equa• students sufficiently to assume an active role in tions of motion often have no better insight into their. learning of mathematics. So we deliberately the behavior of physical phenomena described by modified our courses and attitude to experiment with active student participation in doing math• these equations than do others who never studied ematics problems and theory both in class and the equations; students who have learned course• outside class. based tests of statistical significance frequently do Results were strikingly positive, and we largely not recognize statistical explanations for events in discontinued the typical mathematics lecture for• mat, since lecturing kept students in a passive the world around them [47]. role. With an active participation method stu• Additional evidence of how young adults learn dents studied the text and worked problems be• fore class; faculty and students discussed difficult mathematics-or more often, why they fail to points in class. Students presented problems and learn-has accumulated in recent years as a re• results on the board in class with encouragement sult of many innovations in teaching tried on dif• and guidance from the instructor. We found that students became actively and enthusiastically in• ferent campuses. For example, intervention pro• volved in their learning of mathematics, with the grams designed to improve the mathematical per• instructor acting as a coach. formance of minority students show the impor- As a consequence of these changes, our faculty CHALLENGES FOR COLLEGE MATHEMATICS 11

and students have become a community of learn• does not. To improve our understanding of the in• ers and scholars. Students now do mathematics tellectual development of college mathematics stu• in groups outside class, and more graduating se• niors are seeking advanced degrees in mathemat• dents, mathematicians should increase their efforts ics. The number of mathematics majors rose from to conduct research on how college students learn 69 in 1986 to 104 in 1988; the Mathematics De• mathematics. partment is now the largest unit in the School of Natural Sciences. Finally, and perhaps most im• We need to experiment with new ways to eval• portant, faculty affirm the belief that many more uate teaching. One key factor in good teaching students can realize their mathematical abilities. is how much students learn; other factors include such issues as how many students decide to major Several barriers separate educational studies and in mathematics, to go on to graduate school, or to experiments from the larger community of college work in mathematical careers. These are measures and university mathematicians. First, there are of the quality of teaching done both by an indi• very few individuals who conduct formal research vidual and by a department. They look not only dealing directly with college mathematics. Second, to indicators such as demonstrable knowledge, but mathematicians tend to distrust educational re• also to motivation, attitude, and enthusiasm for search. Third, and perhaps more important, math• the discipline. ematicians follow habit more than evidence in their Evaluation of teaching must involve teaching styles: even well-documented reports of robust indicators that reflect the broad purposes of better methods are insufficient to influence math• mathematics education. ematicians to change their teaching habits. (This Technology is not really too surprising, since neither do con• Computing has changed profoundly-and per• vincing classroom explanations of effective math• manently-the practice of mathematics at every ematical methods suffice to eradicate deep-seated level of use. College mathematics departments, misconceptions among students.) however, often lag behind other sciences in adapt• Too often mathematicians assume with little re• ing their curricula to computing, although con• flection that what was good for their education is siderable momentum is now building within the good enough for their students, not realizing that community for greater use of computing. The most of their students, not being inclined to be• delay in response may have been due in part to come mathematicians, have very different styles conservatism of mathematicians, but at least as of learning. College faculty must begin to recog• important is the simple fact of computer power: nize the proven value of various styles of instruc• only in the last few years have desk-top machines tion that engage students more directly in their achieved sufficient power to provide a legitimate own learning. Those who teach colleqe mathemat• aid to undergraduate (and research) mathematics. ics must seek ways to incorporate into their own As a consequence, scientific computation is becom• teaching styles the findings of research on teaching ing a third paradigm of scientific investigation• and learning. alongside experimental and theoretical science• Studies of metacognition and problem solving and the role of experiment in the practice of math• have yielded some insights that could be useful in ematics itself is increasing [52]. pedagogy, but they have also been frustrated by Computing can enhance undergraduate study in barriers that confront all teachers of mathematics many ways. It provides natural motivation for (for example, the difficulty of assessing just what many students, and helps link the study of mathe• has been learned, and the great length of time re• matics to study in other fields. It offers a tool with quired to develop effective problem-solving heuris• which mathematics influences the modern world tics). Such studies may yield insights that will and a means of putting mathematical ideas into change for the better the way teachers teach and action. It alters the priorities of courses, rendering students learn. But so far, college-level evidence certain favorite topics obsolete and making oth• is sufficiently slim to make the case unconvincing ers, formerly inaccessible, now feasible and neces• to those who most need to be persuaded. We re• sary [34, 68]. Computers facilitate earlier introduc• ally don't know how to induce most students to tion of more sophisticated models, thus making in• rise to the challenge of mathematical thinking; we struction both more interesting and more realistic. have much to learn about what works and what The penetration of computing into undergraduate 12 AAC-MAA TASK FORCE REPORT mathematics is probably the only force with suffi• mathematics change from carrying out prescribed cient power to overcome the rigidity of standard• procedures to formulating problems and interpret• ized textbooks [59, 66, 22]. ing results, it will become more important than The power of technology serves also an epistemo• ever for faculty to communicate clearly to students logical function by forcing mathematicians to ask the goals of the curriculum and how they might dif• anew what it means to know mathematics. Those fer from what students have been led to believe by who explore the impact of technology on educa• their prior study of school mathematics. tion indict introductory mathematics courses for One institution reports that computers have imparting to students mostly skills that machines changed the context of education in significant and can now do more accurately and more efficiently. It unexpected dimensions: is certainly true that typical indicators of student We constructed a strong computer-experimental component at all levels. Besides the obvious ad• performance document primarily that mathemat• vantages for building experience, context, and in• ics students can carry out prescribed algorithms• tuition, there are less obvious payoffs. For exam• just what computers (or calculators) can do. Col• ple, laboratories are places where students spend lots of time and which become, in reality, their lege faculty can no longer avoid the deep challenge habitats outside of their dormitory rooms. Stu• posed by computers for undergraduate mathemat• dents form allegiances and friendships in labora• ics: once calculations are automated, what is left tories. that can be taught effectively to average students? Different types of surprises were revealed on an• Responses to this challenge are taking shape in other campus that has pioneered use of computers experimental programs in many departments of in advanced courses: mathematics. It is, therefore, too early to describe The use of computer software made possible the introduction of topics previously reserved for grad• the impact computing will have on the mathemat• uate students. Examples include the use of MAC• ics major. Certainly in those courses and tracks SYMA, REDUCE, and MACAULEY in commu• devoted to applied mathematics, computing must tative algebra and algebraic geometry. For exam• ple, a 1989 honors thesis gave us strong evidence exert a major influence on the shape of the curricu• of the advances possible in learning mathematics lum. In this age it would be unconscionable to offer with the help of computational aids. a major in applied mathematics, statistics, or oper• Computers change not only how mathematics ations research without substantial and fully inte• is practiced, but also how mathematicians think. grated use of computer methods. Change will come Both changes are unsettling, yet ripe with oppor• more slowly in core subjects such as topology, anal• tunity for effective education. Indeed, in the realm ysis, and algebra. In each of these subjects there of computing, students and faculty must grope to• are impressive computer-based applications (e.g., gether towards a new balance of power among the fractals, coding theory, dynamical systems), yet many components of undergraduate mathematics. none of these applications has been central to the The transition of mathematics from a purely cer• traditional methodology of the subject as taught ebral paper-and-pencil (or chalk-and-blackboard) in introductory courses. Despite differences in the discipline to a high technology laboratory science is pace of change, however, there is no turning back: not inexpensive. Space must be expanded for labo• computers have dramatically altered the practice ratories; classrooms and offices need to be equipped of mathematics. To ensure an effective curriculum with computers and display devices; support staff for the twenty-first century, undergraduate math• must be hired to maintain both hardware and soft• ematics should change-both in objectives and in ware; faculty must be given time to learn to use pedagogy-to reflect the impact of computers on the computers, to learn to teach with computers, and practice of mathematics. to redesign courses and entire curricula to reflect Early experiments that make significant use of the impact of computing. Institutions must plan computing in undergraduate mathematics courses not only for an expensive transition, but also for show that as the balance of student work shifts continued operation at a higher plateau compara• from computation-which machines do better than ble to the traditional laboratory sciences. Colleges humans-to thought, the course becomes more dif• must recognize in budgets, staffing, and space the ficult, more unsettling, and less closely attuned to fact that undergraduate mathematics is rapidly be• student expectations [58]. As the ground rules of coming a laboratory discipline. CHALLENGES FOR COLLEGE MATHEMATICS 13

Foundations rare to find such arrangements related specifically Because of the highly sequenced nature of the to mathematics departments. What is now rare mathematics curriculum, no student can com• should become common: To ensure equal opportu• plete an undergraduate mathematics major with• nity for access to undergraduate mathematics ma• out having secured a proper foundation of calculus, jors, mathematics departments should work with linear algebra, and computing in the first two years nearby two-year colleges to maintain close artic• of college. For many students, half of the credits ulation of programs. required for the major are taken in the first two Connections years. So the nature of mathematical learning in these years is of crucial importance both for indi• Recent studies of the mathematical sciences [7, vidual success in completing a strong mathematics 8] point to two special features that have charac• major, and for programmatic success in building a terized twentieth-century research: the extensive critical mass of upper class mathematics majors. growth in areas of application (no longer just lim• One-third of the first and second year college ited to physics and engineering) and the impressive students in the United States are enrolled in two• unity of mathematical theories (revealed by the fre• year colleges, including over two-thirds of Afro• quent use of methods from one specialty to solve American, Hispanic, and Native American stu• problems in another specialty). Connectedness, dents. It is clear from these figures that any ef• therefore, is inherent in mathematics. It is what fort to strengthen the undergraduate mathematics gives mathematics its power, what establishes its major, especially to recruit more majors among mi• truth, and what reveals its beauty. nority students, must be carried out in a manner Mathematics is widely recognized as the lan• that includes two-year colleges as a full partner in guage of science. Its enabling role in the develop• preparing the foundation for study in depth. ment of the physical sciences formed the paradigm The tradition of common texts and relatively of the scientific method. Today it is beginning to standard syllabi for standard mathematics courses playa similar role in the biological sciences, where in the first two years has facilitated transfer of mathematical tools as diverse as knot theory, non• students and credits during these years even as linear dynamics, and mathematical logic are being it has mitigated against the open intellectual en• applied to model the structure of DNA, the flow of vironment many believe to be essential for ef• blood, and the organization of the brain. fective learning. Now, however, as momentum Similar connections have emerged in the human, builds for reform of courses in the first two years, social, and decision sciences. Statistical models un• and as departments experiment in an effort to dergird virtually every study of human behavior; reshape the entire mathematics major, there is axiomatic studies have helped establish a rigor• some risk that students from lower socio-economic ous theory of social choice; and multi-dimensional backgrounds-the predominant clientele of the mathematical analysis is employed widely to model two-year colleges-will find themselves pursuing a the multitudinous attributes of economic, psycho• course of study that is inconsistent with the efforts logical, or social behavior. Today mathematics is of four-year colleges to improve the undergraduate truly the language of all science-physical, biolog• mathematics major. ical, social, behavioral, and economic. Some four-year institutions that are engaged in Even as the connections multiply between ab• curricular reform are extending the scope of their stract ideas of mathematics and concrete embodi• mathematics program to include informal consor• ments in the world, so too have the internal con• tia with other nearby institutions. One private nections within the mathematical sciences prolifer• Eastern liberal arts college is building just such ated. Key theorems and deep problems that link arrangements into its mathematics program: separate mathematical specialties have provided a force for vast growth of interdisciplinary research. Plans are underway to create a partnership with a local ~ommunity college and a public school sys• Examples abound, including such areas as stochas• tem to mterest students, especially minority stu• tic differential equations at the interface of prob• dents, in mathematics. ability theory and analysis; combinatorial geome• Many institutions maintain regular ties with lo• try joining arithmetical methods of discrete math• cal high schools or community colleges, but it is ematics to problems of space, shape, and position; 14 AAC-MAA TASK FORCE REPORT and control theory that employs tools from analy• is the Holy Grail of undergraduate mathematics sis, linear algebra, statistics, and computer science [60]. Maturity is one objective of study in depth, to formulate effective mechanisms of control for au• but its meaning must be derived from the context tomated processes. of a student's level and goals. Depth itself is a If the undergraduate major does not reveal con• metaphor for many things. To a mathematician nections, it has not revealed mathematics. Most it signals knowledge, insight, complexity, abstrac• mathematics courses and most mathematics ma• tion, and proficiency; to some others it connotes jors do make substantial contributions to this ob• such elusive concepts as ownership, empowerment, jective. Indeed, it is not uncommon for sophomores and control. Although most colleges equate study to select mathematics as a major instead of chem• in depth with the major-a circumstance reflected istry or biology precisely because in their mathe• also in this report-it is important to recognize matics courses they can see more clearly the logical that for some students the major may not achieve connections among different parts: in mathematics the objectives that many have for study in depth. they can "figure things out" rather than just mem• For these students, curricular structures other than orizing results. (Of course, many students make the traditional major may better approach their the opposite choice, but usually for other reasons.) goals for study in depth. At its best, mathematics overflows with connec• Many college students study mathematics as an tions, both internal and external. But one must be important adjunct to another field which is their honest: undergraduate courses do not always show primary interest (e.g., economics, education, biol• mathematics at its best. At their worst, especially ogy). Some colleges offer joint majors that com• in lower-division courses through which both ma• bine study of mathematics with study in a related jors and non-majors must pass, they reveal math• field, usually tied together with some type of joint ematics as a bag of isolated tricks: problems in project. The ever-present danger in such options elementary courses are often solved more by recog• is that they merely combine two shallow minors nition of which section of the text they come from without ever achieving the depth traditionally re• than by any real understanding of fundamental quired in a major. Notwithstanding this risk, one principles. Dealing with open-ended problem sit• must acknowledge that some objectives of study in uations should be one of the highest priorities of depth are well within the range of an effective joint undergraduate mathematics. For example: major, say, in mathematics and biology where se• • Mathematics teachers could bring in outside nior students employ mathematical models based ("real-world") examples to illustrate applica• primarily on lower division mathematics to model tions of material being studied in regular course• a biological phenomenon and then test and modify work. the model based on laboratory data. • Student projects could emphasize connections, Teacher education poses a special case of partic• either to fields that use mathematics or from one ular significance, since mathematics is one of the part of mathematics to another. few disciplines taught throughout all twelve grades • Greater emphasis on multi-step problems amen• of school. It is obviously important for our nation able to a variety of approaches would wean stu• that school teachers be both competent and en• dents away from the school tradition of bite• thusiastic about mathematics. Special committees sized, self-contained problems. recommend standards for preparation of mathe• • Problem-oriented seminars provide wonderful matics teachers [14, 18], and these recommenda• opportunities to explore links between various tions provide one particular perspective on study branches of mathematics. in depth. Such problems would be pregnant with ambiguity, Prospective secondary school teachers of math• ripe with subtle connections, and overflowing with ematics generally pursue an undergraduate de• opportunities for multi-faceted analyses. gree that includes a major in mathematics, of• ten constrained in special ways to ensure breadth Variety appropriate to the responsibilities of high school Mathematicians are fond of talking about an elu• mathematics teachers. However, the appropriate sive concept called "mathematical maturity" that mathematical preparation of prospective elemen- CHALLENGES FOR COLLEGE MATHEMATICS 15 tary and middle school teachers-who commonly portunity believe that the widespread display of teach several subjects, and sometimes teach the "geniusism" as a measure of worth in mathematics whole curriculum-is subject to much debate these is in part a mask for sexism-an unconscious em• days. Many national studies have recommended phasis on behavior intended to preserve the status that prospective elementary school teachers, like quo regarding access to leadership in teaching and secondary school teachers, major in a liberal art research. or science rather than in the discipline of educa• Fortunately there is a growing recognition in the tion. However, the traditional mathematics major mathematical community that old traditions must is generally inappropriate for teachers at this level, be replaced with new approaches better suited to and today there appears to be virtually no exam• the demographic realities of our age. We need to ple of a viable alternative. Even more vexing is the recognize that individuals bring very different but question of achieving depth in mathematics appro• equally valuable strengths to the study of mathe• priate to an elementary school teacher within a ma• matics. A multiplicity of approaches that encour• jor in some other field. Some interesting ideas can age student growth in many different dimensions is be found in the "new liberal arts" initiative spon• far more effective than a single-minded focus lead• sored by the Alfred P. Sloan Foundation which has ing to a linear ranking in one narrow dimension attempted to infuse quantitative methods in tradi• of "brightness." Not every value in mathemati• tionalliberal arts subjects [38]. cal talent can be measured well by timed tests or Most of the issues, guidelines, and recommenda• intercollegiate competitions; the "Putnam power• tions in this study focus on the traditional math• house" is not the only standard by which under• ematics major, which is where most students who graduate majors should be judged. (The William study mathematics in depth are to be found. How• Lowell Putnam Examination, a national contest for ever, study in depth can be done at any level undergraduates, is the Nobel competition of colle• and in many contexts. Mathematics departments giate mathematics. It stumps even faculty with should take seriously the need to provide appropri• questions so hard that the median national score ate mathematical depth for students who wish to for undergraduates is frequently 0.) concentrate in mathematics without pursuing a tra• Specific efforts to focus the mathematics curricu• ditional major. lum on the interests and abilities of all students can bring dramatic results, as this campus report Self-Esteem shows: One of the greatest impediments to student At the time of the first registration for first-year achievement in mathematics is the widespread rep• students, fewer than 20 individuals in the entire utation of mathematics as a discipline for geniuses. freshman class indicate that mathematics is a pos• Many facets of school and college practice con• sible major. A year later, the number is in the 50's, and by the junior year the number is over spire to portray mathematics in "macho" terms: 100. One reason for this impressive increase in only those who are bright, aggressive, and inclined student interest in a mathematics major is the towards arrogance are likely to succeed. Those departmental position that mathematics is for ev• eryone, not just the gifted. We attempt to demon• who do not instantly understand-including many strate the power and applicability of mathematics thoughtful, reflective, creative students-are made by emphasizing breadth of study during the sec• to feel "deeply dumb," like outsiders who don't get ond and third years. the point of an in-joke. Self-confidence increases when students succeed, It is hard to overstate the power of intimidation and decreases when they fail. "What students need to erode students' self-confidence. Many calculus to build self-confidence are genuine small successes teachers recognize the problem: bright freshman of their own" [67]. Initial successes come from rou• "show-offs"-usually white males-whose ques• tine homework, but these are insufficient to the tions are designed not so much to elicit answers task. More effective are instructional strategies and build understanding as to demonstrate their that engage the student in active learning: open• superior intelligence to their classmates. The rit• ended problems, team work that builds diverse ual is not unlike the bluffing maneuvers that male problem-solving skills; undergraduate research ex• animals employ to claim dominant status in a herd. periences; independent study. Building students' Many who are concerned about equality of op- well-founded self-confidence should be a major pri- 16 AAC-MAA TASK FORCE REPORT ority for all undergraduate mathematics instruc• not all learn mathematics in the same way. Class• tion. room methods must fit both the goals of the ma• jor (e.g., to help students to learn to communicate Access mathematically) and the learning styles of individ• Data from many sources [3, 48, 65] show that ual students (e.g., need for peer support and posi• women and members of certain minority groups tive feedback). These same principles apply to all often discontinue their study of mathematics pre• students, not just to students of color. To pro• maturely, before they are prepared appropriately vide effective opportunities for all students to learn for jobs or further school. Afro-American and His• mathematics, colleges must offer a broader spec• panic students drop out of mathematics at very trum of instructional practice that is better attuned high rates throughout high school and college; only to the variety of students seeking higher education. a tiny fraction complete an undergraduate math• ematics major [46]. In college, women major in Communication mathematics almost as often as men do, but they College graduates with majors in mathematical• persist in graduate studies at much lower rates. ly-based disciplines are often perceived by society (Interestingly, mathematics comes closer to achiev• as being verbally inept: the stereotype of the com• ing an even balance of men and women among its puter hacker who cannot communicate except with undergraduate majors than virtually any other dis• a computer has permeated the business world, and cipline; this record of equality disappears, however, tainted mathematics graduates with the same rep• in graduate school.) utation. Recognizing the legitimate basis for this Evidence from various intervention programs concern in the incomprehensible writing of their shows that the high drop-out rate among minority own upper-division students, many mathematics students can be reduced [4, 5, 30, 63]. Appropri• departments are beginning to emphasize writing ate expectations that provide challenges without in mathematics courses at all levels. the stigma of "remediation" together with assign• The forms of writing employed in mathemat• ments and study environments that reinforce group ics courses include the standard genres used in learning have proved successful on many campuses. other disciplines (expository essays, personal jour• Mentoring programs of various types open doors nals, laboratory reports, library papers, research of opportunity to women and minorities who have reports) as well as some that are particularly rele• traditionally been under-represented in mathemat• vant to mathematics (proofs, computer programs, ically based fields. What becomes clear from these solutions to problems). Many students and pro• programs is that the tradition of isolated, compet• fessors are uneasy about what writing means in a itive individual effort that dominates much math• mathematics class, about how to grade it, and how ematics instruction does not provide a supportive to improve it. Few mathematicians know how to learning environment for all students. teach students to improve their writing or speak• Assignments that stress teamwork on problems ing, although there is increasing professional inter• chosen to relate to student interests can help many est in this issue [31, 37, 61]. students succeed in mathematics. The experiences One department focuses on communication of students who work in teams to solve large com• throughout the major, and stresses writing and puter science projects and of those who participate speaking mathematics in a required senior collo• in science research groups show clearly the benefit quium: of incentives for careful work that is created by the team atmosphere. Mathematicians must learn that The conclusion of the major features the collo• quium course "Mathematical Dialogues." The the teaching strategies they recall as being success• emphasis here, as in earlier courses, IS on commu• ful in their own education-and in the education nication, as well as on the connections among the of a mostly white male professional class-do not different branches of mathematics. Mathematical Dialogues consists of lectures from invited schol• necessarily work as well for those who grow up in ars discussions, and independent work. Students vastly different cultures within the American mo• are' expected to read papers and write reviews, to saic. listen to talks and to deliver them. Programs that work for minority students are In industry, one of the most important tasks built on the self-evident premise that students do for a mathematician is to communicate to non- CHALLENGES FOR COLLEGE MATHEMATICS 17 mathematicians in writing and orally the math• these transitions; many drop out of the mathemat• ematical formulation and solution of problems. ics pipeline as a consequence, often to the detri• Each student's growth in mathematical maturity ment of their future study in many disciplines. depends in essential ways on continual growth in Mathematics education at all levels, from grade the ability to communicate in the language of school through graduate school, should take as a mathematics: to read and write, to listen and goal to smooth out the roughness caused by these speak. Students must learn the idioms of the dis• difficult transitions. College mathematics depart• cipline, and the relation of mathematical symbols ments should, in particular, seek to streamline the to English words. They need to learn how to inter• transition of students to college, to upper-division pret mathematical ideas arising in many different mathematics, and to graduate school. sources, and how to suit their own expression of In college, students often experience a different mathematics to different audiences. Mathematics type of transitional problem that applies in vir• majors should be offered extensive opportunities to tually all courses: to understand the relation be• read, write, listen, and speak mathematical ideas at tween theory and applications. This is probably each stage of their undergraduate study. Indeed, the most common complaint that students and fac• writing and speaking is the preferred test of com• ulty in collateral disciplines raise about undergrad• prehension for most of the broad goals of study in uate mathematics courses: they are often perceived depth. as being too theoretical and insufficiently applied. Although in some cases this perception may be Transitions well justified, in many other instances the problem As students grow in mathematical maturity rests more with insufficient effort to demonstrate from early childhood experiences to adult employ• the value of theory to application than with an ac• ment, they face a series of difficult transitions tual excess of theory. The problem is not that the where the nature of mathematics seems to change transition from application to theory is inappro• abruptly. These "fault lines" that cross the terrain priate, but that it is often taken without sufficient of mathematics education appear at predictable effort to build appropriate motivation or connec• stages: tions. Smooth curricular transitions improve stu• dent learning and help maintain momentum for the • Between arithmetic and algebra, when letter study of mathematics. symbols, variables, and relationships become im• portant. Research • Between algebra and geometry, when logical The role of so-called "capstone experiences" proof replaces calculation as the methodology of such as undergraduate research, theses, or senior mathematics. projects is one of the more controversial ingredi• • Between high school and college, when the ex• ents in discussions of the mathematics major. Typ• pectation for learning on one's own increases sig• ically, such requirements are common in the hu• nificantly. manities and the sciences, especially in more selec• • Between elementary and upper-division college tive institutions. In the humanities they are viewed mathematics, when the focus shifts from tech• as opportunities for integration; in the sciences, as niques to theory, from solving problems to writ• opportunities for research. In both science and hu• ing proofs. manities, capstone requirements offer apprentice• • Between college and graduate school, when the ships in the investigative methods of the field. level of abstraction accelerates at a phenomenal In mathematics, however, there has been lit• rate. tle consensus about objectives, feasibility, or ben• • Between graduate school and college teaching, efits of this type of requirement. Very few in• when the realities of how others learn must take stitutions heeded the 1981 CUPM call for a re• precedence. quired course in mathematical modelling for all • Between graduate school and research, when the majors. Many mathematicians believe in coverage new Ph.D. must not just solve a serious problem, as more crucial to understanding: standard theo• but learn to find good problems as well. rems, paradigms of proof, and significant counter• Students experience real trauma in crossing examples in all major areas must be covered before 18 AAC-MAA TASK FORCE REPORT a student is ready to advance to the next stage Students preparing to teach mathematics in high of mathematical maturity. In this view, learning school also have open an enormous range of appro• what is already known is a prerequisite to discov• priate projects to translate interesting newer math• ering the unknown. Moreover, special capstone ematics into curriculum appropriate to the schools. courses appear superfluous since each course pro• In some cases students may want to undertake re• vides its own capstone-the fundamental theorem search into how people learn mathematics, to ex• of calculus, the central limit theorem in statistics, plore for themselves the effectiveness of various in• the fundamental theorem of algebra-which ties structional strategies and the impact of computers together a long chain of prior study. When forced on development of mathematical understanding. to choose between a capstone experience or another Internships in industry, co-op programs that mix advanced course, advocates of coverage will unhesi• study with work, and summer research opportuni• tatingly choose the latter. ties in industrial or government laboratories pro• Because of mathematics' austere definition of vide rich environments for breaking down the ar• "research"-a definition which, incidentally, rules tificial barriers of courses and classrooms. They out the professional work of more than half the enable students to integrate mathematics learned nation's mathematics faculty-many mathemati• in several different courses; to experience the role of cians believe that except in very rare cases, un• mathematical models; to extend their mathemat• dergraduates cannot do research in mathemat• ical repertoire beyond just what has been taught; ics. Moreover, in most areas of mathematics, stu• and to establish mathematical concepts in a con• dents cannot even assist in faculty research, as text of varied use, applications, and connections. they do quite commonly in the laboratory sci• Experiences of departments with long-standing ences. The exceptions in mathematics are prin• traditions of undergraduate research or senior cipally where computer investigation-the math• projects confirm both the value of such work and ematician's laboratory-can aid the research ef• the effort required for success: fort. As a consequence, many mathematicians be• While these projects require a great deal of time and effort on the part of students and faculty, we lieve that further coursework would better serve generally feel that it is well worth it. Most of the the goals of integration (because the higher one students report that they had worried about the progresses in mathematics, the more internal links senior project for their first three years, but had ultimately found it to be a very worthwhile and one can see) and at the same time help advance stimulating part of their college experience. All the student towards better preparation for further recommended that this important aspect of the study or application of mathematics. undergraduate experience be retained. Others feel that any encounter with a substan• One department in an institution whose academic tial problem that a student does not know how calendar permits extended blocks for full-time to solve can provide a legitimate and rewarding study in one subject requires all majors to com• research experience. Indeed, many colleges have plete a major project in the senior year: used summer experiences with undergraduate re• The final project in the major field should demon• strate application of the skills, methods, and search as an effective strategy to recruit students knowledge of the discipline to the solution of a to careers in the mathematical sciences [26, 56], problem that would be representative of the type and the National Science Foundation is actively to be encountered in one's career. Project activ• ities encompass research, development and appli• supporting such programs. There are now many cation, involve analysis or synthesis, are experi• diverse programs offering research experiences for mental or theoretical, emphasize a particular sub• undergraduate mathematics majors. area of the major or combine aspects of several subareas. In applied areas-especially in statistics, com• Another department uses summers to provide op• puting, and operations research-it is easier to de• portunities for research experiences: student par• velop projects that are sufficiently rich and varied ticipants range from freshmen to seniors, and en• so that students can make progress along various gage in a wide variety of mathematical investiga• lines of investigation. Computers now are mak• tions: ing inroads in theoretical areas of mathematics, We are convinced that everyone working in math• permitting exploration of conjectures that hereto• ematics can find problems appropriate for un• fore were beyond the range of any undergraduate. dergraduates. Many problems can be attacked CHALLENGES FOR COLLEGE MATHEMATICS 19

without any knowledge of the complex machin• more than superficial attention to the historical, ery which generated them. Mathematicians know cultural, or contemporary context in which math• how exciting mathematical research can be. The best way to generate interest in mathematics is to ematics is practiced. Today, however, as math• provide undergraduates with the chance to expe• ematical models are used increasingly for policy rience that excitement. and operational purposes of immense consequence, Since hard work by itself is insufficient to en• it is vitally important that students of mathemat• sure reasonable progress on a mathematical prob• ics learn to think through these issues even as they lem, there is ever-present danger that undergradu• learn the details of mathematics itself. ates confronted with difficult theoretical problems Examples abound of mathematical activity that will flounder and become discouraged. Strong fac• leads directly to decisions of great human import. ulty intervention can prevent disaster, but exces• Software written for the Strategic Defense Initia• sive supervision undermines the independence that tive depends on mathematical theories of orbital is supposed to result from the project. Effective dynamics for its performance, and on the ability undergraduate research experiences require careful of logicians and computer scientists to verify that planning and steady, unobtrusive leadership. One complex untestable programs will perform cor• must carefully choose problems to be suggested to rectly under any possible situation. Debates about undergraduates for the research experiences: they the relation of carbon dioxide buildup to global must be tailored to the individual undergraduate. warming and consequent implications for govern• Effective programs provide stepping stones to mental and industrial policies center in large part help students progress from routine homework to on different interpretations of statistical and math• independent investigation. For example, one in• ematical projections. Computer-controlled trading stitution plans a progression leading to the senior of stocks, epidemiological studies of AIDS, and im• project: plications of various voting rules offer other exam• Mathematics majors enroll in a Junior Seminar ples where mathematics really matters in impor• where they are asked to read critically two se• tant decisions affecting daily life. nior projects from earlier years to describe the strengths and weaknesses of these papers, and to Students of mathematics should be encouraged suggest how they would improve on these papers had they written them. This Seminar also helps to see mathematics as a human subject whose the• acquaint these students with appropriate stan• ories often begin in ambiguity and controversy. dards of exposition in mathematics. It takes decades, sometimes centuries, for schol• The range of opportunities for independent in• ars to sculpt and polish the precise theories that vestigation is so broad and the evidence of ben• are expounded in today's textbooks. Historical efit so persuasive as to make unmistakably clear analogs provide useful yardsticks to students (and that research-like experiences should be part of ev• faculty) who seek to understand the limits of what ery mathematics student's program. Undergradu• mathematics can contribute to public policy. As ate research and senior projects should be encour• society comes to rely increasingly on mathemati• aged wherever there is sufficient faculty to provide cal analyses-often well-disguised-of social, eco• appropriate supervision. Effective programs must nomic, or political issues, mathematics majors be tailored to the needs and interests of individual must confront the social and ethical implications students; no single mode of independent investiga• of such activity. All such issues can be enlightened tion can lay claim to absolute priority over others. by appropriate historical case studies, and moti• Flexibility of implementation is crucial to ensure vated by compelling debates of our age. All math• that all experience the exhilaration of discovery ematics students should engage in serious study of which accompanies involvement with mathemati• the historical context and contemporary impact of cal research. mathematics. One possible strategy to achieve both this ob• Context jective and several others as well is to adapt a Mathematics courses-especially those taken by modelling project or course to problems of signif• majors-have traditionally been taught as purely icant societal impact. In such a setting students utilitarian courses in techniques, theory, and ap• could undertake original investigation, gain expe• plications of mathematics. Most courses pay no rience in reading, writing, listening, and speaking 20 AAC-MAA TASK FORCE REPORT about mathematically rich material, explore his• Mechanisms for Renewal torical antecedents and contemporary debates, and Constant vigilance is needed to maintain qual• gain experience in team work to address complex, ity in any academic department. This is espe• open-ended problems. For many students a cap• cially true in mathematics, where the subject is stone project on a public policy issue would be a continually evolving, where external departments fitting way to relate their mathematics major to impose their own often-conflicting demands, where liberal education. so much teaching effort is devoted to remedial, el• ementary, and lower division work, and where the Social Support very ability of the discipline to attract sufficient The abstract, austere nature of mathematics numbers of students to careers in the mathemati• provides relatively few intrinsic rewards for the cal sciences is now in serious doubt. We focus here typical undergraduate who is trying to pursue a on five mechanisms of renewal: field of study and at the same time learning to es• • DIALOGUE: To talk with students and col• tablish and maintain personal friendships. In this leagues. context the social support provided by departmen• • ASSESSMENT: To measure what is happening. tal activities can be decisive in tipping the bal• • FACULTY DEVELOPMENT: To improve intellec• ance either for or against a mathematics major. tual vitality. Peer group support helps build mathematical self• • DEPARTMENTAL REVIEW: To listen to col• confidence and enhances the intrinsic rewards that leagues and clients. come from mathematical achievement. • GRADUATE EDUCATION: To provide leadership Virtually all successful mathematics depart• for improvement. ments instigate and support a variety of extra• curricular activities. Examples include mathemat• The key ingredient is listening--to one's students, ics clubs, student chapters of the Mathematical As• to one's discipline, to one's colleagues, to one's sociation of America, or chapters of the mathemat• friends, and to one's critics. Departments that ics honorary society Pi Mu Epsilon. Another com• listen-and learn-will thrive. mon feature of successful departments is informal faculty-led sessions to help students solve problems Dialogue posed in collegiate periodicals or to prepare for na• Departments often know very little about their tional contests such as the Putnam Examination or students' views of the undergraduate mathematics the Mathematical Modelling Contest. Banquets, major. That different students pursue mathemat• picnics, and barbecues lend a light touch that help ics for very different reasons is clear. Most de• students become acquainted with each other and partments must accommodate students with quite with the faculty of the entire department. different purposes, although certain departments Other activities can enrich students' experiences tend to focus their programs on one or another ob• with their courses by providing links to the world jective (for example, preparation for jobs, prepara• beyond the campus. Undergraduate colloquia with tion for teaching, preparation for graduate school). visiting mathematicians from industry or universi• Many departments, especially small departments, ties is one common mechanism. Alumni involve• find it impossible to sustain several different pro• ment through career nights or other activities can grams of equal high quality. help students imagine what they too could do Mathematicians also frequently know almost with their major. Current students will be in• nothing about the expectations held by their col• spired when departments make visible the variety leagues in cognate disciplines for the mathemati• of accomplishments of their graduates-not only cal preparation of students with other majors. It those who have become mathematicians but also is not uncommon for the three interested parties• the majority who have used their undergraduate mathematics professors, science faculty advisors, mathematics for other ends. Mathematics depart• and students-never to discuss goals or objectives, ments should exert active leadership in promoting but only credit hour requirements. It should come extracurricular activities that enhance peer group as no surprise that in the absence of good commu• support among mathematics majors. nication, misunderstandings flourish. CHALLENGES FOR COLLEGE MATHEMATICS 21

Undergraduate mathematics shares many bor• they can, upon graduation, get jobs that pay more ders with other subjects and institutions: verti• than their professors earn. Professors, in contrast, cally with high schools below and graduate schools expect students who are eager to take on challeng• above; horizontally with science, business, and en• ing problems and who will learn on their own what• gineering. Each border is a potential impediment ever they need to make progress. Students, in this to the smooth flow of ideas and students. Mathe• exaggerated portrait, feel responsible only for what matics departments must work hard to maintain ef• they have been taught, whereas faculty judge as fective articulation across these many boundaries: truly significant only those things students can do • With high schools whose curriculum is also which they have not been taught. changing and whose students will arrive at col• It is important for mathematics departments to lege with new expectations. help faculty and students recognize their own per• • With departments in the physical sciences and spectives on mathematics and understand the per• engineering whose students use advanced math• spectives of others. Doing this is not the same as ematics. covering a syllabus of mathematical topics; it in• volves instead various strategies to enable faculty • With graduate schools in the mathematical sci• ences, which attract and retain far too few U.S. and students to discuss mathematics in informal ways. Such discussions are an important part of students. the process by which students grow from the lim• • With employers of bachelors degree graduates ited school perspective to the self-directed stance who expect employees who can function effec• of a professional. tively in a work environment. Announcing or publishing department goals is Regular discussion is essential to maintain effec• not sufficient to achieve this important objective. tive policies that will satisfy these many boundary What is required is a process that engages all stu• conditions. dents in significant and repeated discussion of indi• To the extent that resources permit, depart• vidual goals throughout their undergraduate study ments should seek to determine and then accom• of mathematics. In particular, careful and indi• modate different student career interests. This vidualized advising is crucial to students' success. means that even small departments should provide Effective advising builds an atmosphere of mutual mechanisms (e.g., independent study, special sem• respect among faculty and students. Courses, ca• inars) to allow students of diverse interests to re• reer objectives, motivations, fears, celebrations are ceive a major suitable to their career objectives. all part of advising, and of special importance in Mathematics is too diverse and student purposes the long, slow process of building students' self• too different for any single set of eight to ten confidence. courses to meet all needs equally well. Students too must recognize that the practice Assessment of mathematics is quite different from the text• Many would argue that goals for study in depth book image they usually bring with them from high can be effective only if supported by a plan for school. Often students expect of college math• assessment that persuasively relates the work on ematics merely advanced topics in the spirit of which students are graded to the objectives of school mathematics: a succession of techniques, ex• their education. Assessment in courses and of the ercises, and test problems, each explained by the major as a whole should be aligned with appro• instructor with sufficient clarity that what remains priate objectives, not just with the technical de• for the student is only the requirement of prac• tails of solving equations or doing proofs. Many tice and memorization. Such expectations do little specific objectives can flow from the broad goals to foster creativity, independence, criticism, and of study in depth, including solving open-ended perspective-the more important goals of liberal problems; communicating mathematics effectively; education. close reading of technically-based material; pro• The different perspectives of mathematics stu• ductive techniques for contributing to group ef• dent and mathematics professor often approach forts; recognizing and expressing mathematical caricature. Eager students expect of college classes ideas embedded in other contexts. Open-ended directed instruction in tools of the trade with which goals require open-ended assessment mechanisms; 22 AAC-MAA TASK FORCE REPORT although difficult to use and interpret, such devices der thinking and habits of mind suitable for effec• yield valuable insight into how students think. tive problem solving, then these are the things that Relatively few mathematics departments now should be tested. Moreover, assessment should be require a formal summative evaluation of each stu• an integral part of the process of instruction: it dent's major. The few that do often use the should arise in large measure out of learning envi• Graduate Record Examination (or an undergrad• ronments in which the instructor can observe how uate counterpart) as an objective test, together students think as well as whether they can find with a local requirement for a paper, project, or right answers. Assessment of undergraduate ma• presentation on some special topic. Many insti• jors should be aligned with broad goals of the ma• tutions, frequently pressured by mandates from jor: tests should stress what is most important, not on high, are developing comprehensive plans for just what is easiest to test. assessing student outcomes; a few are exploring innovative means of assessment based on portfo• Faculty Development lios, outside examiners, or undergraduate research The relation of research and scholarship to fac• projects. Here's one example that blends a cap• ulty vitality is one of the most difficult issues facing stone course with a senior evaluation: many departments of mathematics, especially in smaller institutions. Professional activity is crucial The Senior Evaluation has two major compo• nents to be completed during the fall and spring to inspired teaching and essential to avoid faculty semesters of the senior year. During the fall burn-out. Mathematical research in its traditional semester the students are required to read twelve sense plays only a small role in the mechanisms re• carefully selected articles and to write summaries of ten of them. (Faculty-written summaries of quired to maintain intellectual vitality of a math• two articles are provided as examples.) This work ematics department: only about one in five full• comprises half the grade on the senior evaluation. time faculty in departments of mathematics pub• During the fall semester each student chooses one lish regularly in research journals, and fewer than article as a topic for presentation at a seminar. During the spring semester the department ar• half of those have any financial support for their ranges a seminar whose initial talks are presented research. Clearly the community needs to encour• by members of the department as samples for the age and support a broader standard as a basis for students. At subsequent meetings, the students present their talks. Participation in the seminar maintaining faculty leadership both in curriculum comprises the other half of the grade for the Senior and in scholarship. Evaluation. The first step is to expand the definition of pro• Because of the considerable variety of goals of fessional activity from "research" to "scholarship," an undergraduate mathematics major, it is widely more in a manner akin to that currently recognized acknowledged that ordinary paper-and-pencil tests in some other academic disciplines. Applied con• cannot by themselves constitute a valid assessment sulting work, software development, problem solv• of the major. Although some important skills and ing, software and book reviews, expository writ• knowledge can be measured by such tests, other ing, and curriculum development are examples of objectives (e.g., oral and written communication; activities that serve many of the same purposes as contributions to team work) require other meth• research: they advance the field in particular direc• ods. Some departments are beginning to explore tions, they engage faculty in active original work, portfolio systems in which a student submits sam• they serve as models for students of how mathe• ples of a variety of work to represent just what matics is actually practiced, and they provide op• he or she is capable of. A portfolio system al• portunities for student projects. lows students the chance to put forth their best Teaching in new areas is also a form of schol• work, rather than judging them primarily on areas arship in mathematics. Unlike many other disci• of weakness. plines where faculty rarely teach outside their own The recommendations [41J from the National areas of specialty, mathematicians are generally ex• Council of Teachers of Mathematics for evalua• pected to teach a wide variety of courses. Learn• tion and assessment of school mathematics convey ing and then teaching a course far outside one's much wisdom that is applicable to college math• zone of comfort is an effective way to build inter• ematics. Assessment must be aligned with goals nal connections which then spill over in all courses of instruction. If one wants to promote higher or- one teaches. A teacher who is still an active learner CHALLENGES FOR COLLEGE MATHEMATICS 23 sets a fine example for students concerning the true regularly, not just when some crisis threatens the meaning of scholarship. status quo. The second step is to insist on greater commu• Client disciplines expect from mathematics de• nication about professional activity in mathemat• partments an amazing repertoire of support ser• ics so that it becomes public. Only the bright vices for students who will major in other fields light of public scrutiny by colleagues in various [44]. Some demand a magic bullet-s-a perfect in• institutions-i-not only on one's own campus-s-can fusion of just those mathematical methods (and affirm the quality and value of professional work. no more) needed in the other field; others expect "Public" need not mean merely publication; lec• a rigorous filter that will pass on only those stu• tures, workshops, demonstrations, reports of vari• dents who are sufficiently bright to function ably in ous sorts can serve the same objective. What mat• upper-division work in other fields. Occasionally, ters is that the result become part of the profession, but all too rarely, an external discipline will require and be evaluated by the profession. To ensure con• mathematics primarily to enable students to bene• tinued vitality of undergraduate mathematics pro• fit from the intrinsic values of mathematics: logic, grams, all mathematics faculty should engage in rigor, analysis, symbol-sense, etc. Since the expec• public professional activity, broadly defined. tations of other departments are often not clearly conveyed by the list of mathematics courses that Department Review they recommend or require, regular reviews pro• vide a good mechanism-s-but not the only one-i-to More than any other academic discipline, math• ensure that different departments at least under• ematics is constrained to serve many masters: stand their differing perspectives and objectives. the many sciences that depend on mathematical Virtually all departments receive informal feed• methods; the demand of quantitative literacy that back from graduates, employers, and graduate undergirds general education; the need to edu• schools. Speaker programs that bring students cate teachers for our nation's schools; the need and faculty into contact with users of mathemat• of business and industry for mathematically lit• ics serve both to inform students about the broad erate employees; the expectation of mathemati• world of mathematics beyond their classroom walls cal proficiency by faculty and students in natu• and to provide informal feedback to help regulate ral science, business, engineering, and social sci• the curriculum and keep it properly tuned to the ence; the professional standards of employers for needs of graduates. All such informal means of entry-level technical personnel; and the require• feedback are valuable and must be encouraged. ments of the mathematical sciences themselves for However, they are no substitute for formal, reg• well-prepared graduate students. It is an enormous ular, external review. Both external reviews and challenge for a department of mathematics, one informal feedback are needed to assure quality in that very few are able to fulfill with distinction departments of mathematics. in every dimension. There are many advantages to a regular program Because of these diverse demands, it is especially of external reviews that should form the basis of all important that departments of mathematics un• reviews: dergo regular review, with both external and inter• nal mechanisms to provide evaluation and advice. •A broad-based review provides a strategic op• External requirements mandate periodic review of portunity to document the accomplishments of all departments in many colleges and universities, a department. Well-structured reviews can ef• especially in public institutions. But in other in• fectively counter external (political) demands for stitutions, department goals are defined implicitly narrow or inappropriate instruments of assess• without self-reflection or benefit of external per• ment such as a multiple-choice examination of spectives. At worst, the goals of such depart• all graduates. ments are defined by coverage of standard text• • Reviews provide a structured and neutral forum books. Often it takes a crisis-i-such as when the for mathematicians to discuss with those who engineering or business school complains about cer• use mathematics both the mathematical needs tain courses-i-for departments to step back and ex• of client disciplines and the common issues that amine their objectives. Reviews should take place both mathematics and the client discipline face 24 AAC-MAA TASK FORCE REPORT

in accommodating changes that are underway in uate education in mathematics is not now serving the mathematics curriculum. U.S. interests especially well: • By involving members of the faculty outside the • Too few U.S. mathematics majors choose to en• department in the review-especially those from ter graduate school in a mathematical science. fields that are served by mathematics and those • U.S. mathematics students do less well in gradu• involved in faculty curriculum committees-a ate school-and drop out more often-than for• department of mathematics can help educate eign nationals. colleagues across the campus about the special • Many students finish graduate school ill-equip• opportunities and challenges of teaching math• ped for the breadth of teaching duties typically ematics. Ignorance can usually be turned to expected of undergraduate mathematics teach• understanding through discussions prompted by ers. the occasion of a regular review. • Relatively few who finish doctoral degrees in • By including non-academic reviewers such as in• mathematics actually go on to effective research dustrial executives, scientists, professionals, and careers in mathematics. community leaders, the department can gain valuable insight into the qualities that will be In the 1970s, as the number of U.S. students expected of graduates who enter the work force. applying to graduate school in mathematics be• • Regular reviews encourage faculty members to gan to decline, the graduate schools responded by think about the department's program as a increasing the number of international students, whole, rather than only about the courses they most of whom had completed a more intense and teach. Such discussions make it more likely that specialized education in mathematics than is typi• the curriculum will remain responsive to stu• cal of American undergraduates. Hence the level of dent needs, and to the changing demands of the mathematics expected of beginning graduate stu• mathematical sciences. Reviews provide an ideal dents gradually shifted upward to an international mechanism for the department to assert control standard that is well above current U.S. under• over its own program. graduate curricula. Consequently, the failure or The Mathematical Association of America can pro• drop-out rate of U.S. students increased, creating vide advice to departments both about the struc• pressure for more international students and even ture of effective reviews and about appropriate con• higher entrance expectations. sultants or reviewers. It is time to break this negative feedback loop by encouraging better articulation of programs Graduate Education and standards between U.S. undergraduate col• Even though relatively few mathematics majors leges and U.S. graduate schools. Such cooperation go on to receive a graduate degree in the mathe• is needed both to enhance the success of U.S. stu• matical sciences, the health of college mathemat• dents and to enable the graduate schools to bet• ics is inextricably linked with the status of gradu• ter match their programs with the needs of the ate education. As the sole agent for advanced de• colleges and universities who employ a majority grees, graduate schools bear alone the responsibil• of those who receive advanced degrees. Renewal ity for preparing college mathematics teachers; as of undergraduate mathematics will require commit• the primary locus of mathematical research, grad• ment, leadership, and support of graduate schools. uate schools shape the nature of the discipline, and One good mechanism for such cooperation hence of the curriculum. Much of the responsibil• would be an exchange of visitors between under• ity for renewing undergraduate mathematics rests graduate and graduate institutions so that each with the graduate schools, since it is they who pro• can learn about the needs of the other. Espe• vide the primary professional education of those cially as change occurs in the content and nature who are responsible for undergraduate mathemat• of the undergraduate major, it is very important ics: college faculty. that graduate schools maintain programs of study Indicators from many sources [12, 35, 44] suggest and research that are appropriately linked to the that the match between undergraduate and grad- undergraduate program in mathematics. CHALLENGES FOR COLLEGE MATHEMATICS 25

Summary • Mathematicians should increase their efforts to understand better how college students learn Without becoming entangled in specific curricu• mathematics. lum and course recommendations-which are the • Evaluation of teaching must involve robust indi• proper province of other committees of mathemat• cators that reflect the broad purposes of mathe• ics professional organizations-we can nevertheless matics education. enumerate several broad principles implied by our study of the undergraduate mathematics major: ACCESS AND ENCOURAGEMENT • Effective programs teach students, not just GOALS AND OBJECTIVES mathematics. • The primary goal of a mathematical sciences ma• • National need requires greater encouragement jor should be to develop a student's capacity to for students to continue their study of mathe• undertake intellectually demanding mathemati• matics beyond the bachelor's degree. cal reasoning. • To provide effective opportunities for all stu• • The undergraduate mathematics curriculum dents to learn mathematics, colleges should of• should be designed for all students with an in• fer a broader spectrum of instructional practice terest in mathematics. that is better attuned to the variety of students • Applications should motivate theory so that the• seeking higher education. ory is seen by students as useful and enlighten• • To ensure for all students equal access to higher ing. mathematics education, mathematics depart• • Mathematics majors should be offered extensive ments should work with nearby two-year colleges opportunities to read, write, listen, and speak to maintain close articulation of programs. mathematical ideas at each stage of their under• • Smooth curricular transitions improve student graduate study. learning and help maintain momentum for the study of mathematics. BREADTH AND DEPTH • All students who major in mathematics should USING COMPUTERS study some sequence of upper division courses • The mathematics curriculum should change to that shows the power of study in depth. reflect in appropriate ways the impact of com• • Every student who majors in mathematics puters on the practice of mathematics. should study a broad variety of advanced • Colleges must recognize in budgets, staffing, and courses. space the fact that undergraduate mathematics • Mathematics departments should take seriously is rapidly becoming a laboratory discipline. the need to provide appropriate mathematical DOING MATHEMATICS depth to students who wish to concentrate in • Dealing with open-ended problem situations mathematics without pursuing a traditional ma• should be one of the highest priorities of under• jor. graduate mathematics. • Mathematics majors should complete a minor in • All undergraduate mathematics students should a discipline that makes significant use of mathe• undertake open-ended projects whose scope ex• matics. tends well beyond typical textbook problems. LEARNING AND TEACHING • Undergraduate research and senior projects should be encouraged wherever there is sufficient • Instruction should encourage students to explore faculty to provide appropriate supervision. mathematical ideas on their own. • Students majoring in mathematics should un• • Undergraduate students should not only learn dertake some real-world mathematical modelling the subject of mathematics, but also learn how project. to learn mathematics. • Those who teach college mathematics should STUDENTS seek ways to incorporate into their own teaching • Building students' well-founded self-confidence styles the findings of research on teaching and should be a major priority for all undergradu• learning. ate mathematics instruction. 26 AAC-MAA TASK FORCE REPORT

• Careful and individualized advising is crucial to References students' success. • All mathematics students should engage in seri• ous study of the historical context and contem• [1] Albers, Donald J.; Anderson, Richard D.; Lofts• gaarden, Don O. Undergraduate Programs in the porary impact of mathematics. Mathematical and Computer Sciences: The 1985• • Mathematics departments should actively en• 1986 Survey. MAA Notes No.7. Washington, courage extracurricular programs that enhance D.C.: Mathematical Association of America, 1987. peer group support among mathematics majors. [2] Albers, Donald J.; Rodi, Stephen B.; Watkins, Ann RENEWAL E. (Eds.). New Directions in Two- Year College Mathematics. New York: Springer-Verlag, 1985. • It is important for mathematics departments to [3] American Council on Education. One Third of help faculty and students recognize their own a Nation. Report of the Commission on Minor• perspectives on mathematics and understand the ity Participation in Education and American Life. perspectives of others. Washington, D.C.: American Council on Education, • Assessment of undergraduate majors should be 1988. aligned with broad goals of the major; tests [4] Anderson, Beverly; Carl, Iris; Leinwand, Steven should stress what is most important, not just (Eds.). Making Mathematicians Work for Minori• what is easiest to test. ties. CBMS Issues in Mathematics Education, Vol• • To ensure continued vitality of undergraduate ume 2. Providence, R.I.: American Mathematical mathematics programs, all mathematics faculty Society (To Appear). should engage in public professional activity, [5] Asera, Rose. "The Math Workshop: A Descrip• broadly defined. tion." In Fisher, Naomi; Keynes, Harvey; Philip Wagreich (Eds.). Mathematicians and Education • Regular external reviews and informal feedback Reform. CBMS Issues in Mathematics Education, are needed to assure quality in departments of Volume 1. Providence, R.I.: American Mathemati• mathematics. cal Society, 1990, pp. 47-62. • Renewal of undergraduate mathematics will re• [6] Association of American Colleges. Integrity in the quire commitment, leadership, and support of College Curriculum. Washington, D.C.: Associa• graduate schools. tion of American Colleges, 1985. In most respects both prevailing professional [7] Board on Mathematical Sciences. Mathematical wisdom and current practice for the mathematics Sciences: A Unifying and Dynamic Resource. Na• tional Research Council. Washington, D.C.: Na• major reflect well the major goals of AAC's In• tional Academy Press, 1986. tegrity. Discussion continues on many campuses [8] Board on Mathematical Sciences. Mathemati- about whether the major should focus inward to• cal Sciences: Some Research Trends. National wards advanced study in the mathematical sciences Research Council. Washington, D.C.: National or outward towards preparation for diverse careers Academy Press, 1988. in science and management. These discussions are [9] Board on Mathematical Sciences. Renewing U.S. more about strategies than long-term goals, how• Mathematics: A Plan for the 1990s. National ever, since either emphasis can advance the broad Research Council. Washington, D.C.: National AAC goals of coherence, connections, and intellec• Academy Press, 1991. . tual development. [10] Boyer, Ernest 1. College: The Undergraduate Expe• Liberal education provides a versatile back• rience. New York: Harper & Row, 1987. ground for a life of ever-changing challenges. [11] Case, Bettye Anne. Teaching Assistants and Part• Among the many majors from which students can time Instructors. Washington, D.C.: Mathematical choose, mathematics can help ensure versatility for Association of America, 1988. the future. Habits of mind nurtured in an under• [12] Case, Bettye Anne. "How Should Mathemati- cians Prepare for College Teaching?" Notices of graduate mathematics major are profoundly use• the American Mathematical Society, 36 (December ful in an enormous variety of professions. The 1989) 1344-1346. challenge for college mathematicians is to ensure [13] College Entrance Examination Board. Academic that the major provides-and is seen by students Preparation in Mathematics: Teaching for Transi• as providing-not just technical facility, but broad tion From High School To College. New York: The empowerment in the language of our age. College Board, 1985. CHALLENGES FOR COLLEGE MATHEMATICS 27

[14] Committee on the Mathematical Education of 1990. Teachers. A Call for Change: Recommendations for [28] Garfunkel, Solomon and Young, Gail. "A Study of the Mathematical Preparation of Teachers of Math• Mathematical Offerings in Diverse Departments." ematics. Washington, D.C.: Mathematical Associ• In Mathematical Sciences: Servant to Other Disci• ation of America, 1991. plines. Committee on Mathematical Sciences in the [15] Committee on the Undergraduate Program in year 2000, National Research Council. Washington, Mathematics. A General Curriculum in Mathemat• D.C.: National Academy Press, (To Appear). ics for Colleges. Washington, D.C.: Mathematical [29] Gilfeather, Frank. "University Support for Mathe• Association of America, 1965. matical Research." Notices of the American Math• [16] Committee on the Undergraduate Program in ematical Society, 34 (November 1987) 1067-1070. Mathematics. Recommendations for a General [30] Gillman, Leonard. "Teaching Programs That Mathematical Sciences Program. Washington, D.C.: Work." Focus, 10:1 (1990) 7-10. Mathematical Association of America, 1981. [31] Gopen, George D. and Smith, David A. "What's an [17] Committee on the Undergraduate Program in Assignment Like You Doing in a Course Like This?: Mathematics. Reshaping College Mathematics. Writing to Learn Mathematics." In Connolly, Paul MAA Notes No. 13. Washington, D.C.: Mathe• and Yilardi, Teresa (Eds.). Writing to Learn Math• matical Association of America, 1989. ematics and Science. New York: Teachers College [18] Committee on the Undergraduate Program in Press, 1989; reprinted in The College Mathematics Mathematics Panel on Teacher Training. Rec• Journal, 21 (1990) 2-19. ommendations on the Mathematical Preparation of [32] Graham, William R. "Challenges to the Mathemat• Teachers. MAA Notes No.2. Washington, D.C.: ics Community." Notices of the American Mathe• Mathematical Association of America, 1983. matical Society, 34 (February 1987) 245-250. [19] Conference Board of the Mathematical Sciences. [33] Howson, Geoffrey; Kahane, J.-P.; Lauginie, P.; de New Goals for Mathematical Sciences Education. Turckheim, E. (Eds.). Mathematics as a Service Washington, D.C.: Conference Board of the Math• Subject. International Commission on Mathemat• ematical Sciences, 1984. ical Instruction Study Series. Cambridge: Cam• [20] Connors, Edward A. "A Decline in Mathematics bridge University Press, 1988. Threatens Science-and the U.S." The Scientist, 2 [34] Howson, Geoffrey and Kahane, J.-P. (Eds.). The (November 28, 1988) 9, 12. Influence of Computers and Informatics on Mathe• [21] Connors, Edward A. "1989 Annual AMS-MAA Sur• matics and its Teaching. International Commission vey." Notices of the American Mathematical Soci• on Mathematical Instruction Study Series. Cam• ety, 36 (November 1989) 1155-1188. bridge: Cambridge University Press, 1986. [22] Cunningham, Steven and Zimmerman, Walter [35] Jackson, Allyn B. "Graduate Education in Mathe• (Eds.). Visualization in Teaching and Learning matics: Is it Working?" Notices of the American Mathematics. MAA Notes. Washington, D.C.: Mathematical Society, 37 (1990) 266-268. Mathematical Association of America (To Appear). [36] Karian, Zaven. Report of the CUPM Subcommit• [23] David, Edward E. (Ed.) Renewing U.S. Mathemat• tee on Symbolic Computation. Washington, D.C.: ics: Critical Resource for the Future. Committee Mathematical Association of America (To Appear). on Resources for the Mathematical Sciences, Na• [37] Knuth, Donald E.; Larrabee, Tracy; Roberts, Paul tional Research Council. Washington, D.C.: Na• M. Mathematical Writing. MAA Notes No. 14. tional Academy Press, 1984. Washington, D.C.: Mathematical Association of [24] David, Edward E. "Renewing U.S. Mathematics: America, 1989. An Agenda to Begin the Second Century." Notices [38] Koerner, James D. (Ed.). The New Liberal Arts: of the American Mathematical Society, 35 (October An Exchange of Views. New York: Alfred P. Sloan 1988) 1119-1123. Foundation, 1981. [25] Davis, Philip J. "Applied Mathematics as Social [39] Madison, Bernard L. and Hart, Therese A. A Chal• Contract." Mathematics Magazine, 61 (1988) 139• lenge of Numbers: People in the Mathematical Sci• 147. ences. Committee on Mathematical Sciences in the [26] Davis-Van Atta, David, et al. Educating America's Year 2000, National Research Council. Washington, Scientists: The Role of the Research College. Ober• D.C.: National Academy Press, 1990. lin, Ohio: Oberlin College, 1985. [40] McKnight, Curtis C., et al. The Underachieving [27] Fisher, Naomi; Keynes, Harvey; Philip Wagreich Curriculum: Assessing U.S. School Mathematics (Eds.). Mathematicians and Education Reform. from an International Perspective. Champaign, Ill.: CBMS Issues in Mathematics Education, Volume 1. Stipes Publishing Company, 1987. Providence, R.I.: American Mathematical Society, [41] National Council of Teachers of Mathematics. Cur- 28 AAC-MAA TASK FORCE REPORT

riculum and Evaluation Standards for School Math• College Mathematics Teaching. Committee on the ematics. Reston, Virg.: National Council of Teach• Teaching of Undergraduate Mathematics. Washing• ers of Mathematics, 1989. ton, D.C.: Mathematical Association of America, [42] National Institute of Education. Involvement in 1990. Learning: Realizing the Potential of American [56] Senechal, Lester. Models for Undergraduate Re• Higher Education. Washington, D.C.: National In• search in Mathematics. MAA Notes No. 18. Wash• stitute of Education, 1984. ington, D.C.: Mathematical Association of Amer• [43] National Research Council. Everybody Counts: A ica, 1990. Report to the Nation on the Future of Mathematics [57] Simon, Barry (Ed.). Report of the Committee Education. Washington, D.C.: National Academy on American Graduate Mathematics Enrollments. Press, 1989. Washington, D.C.: Conference Board of the Math• [44] National Research Council. Moving Beyond Myths: ematical Sciences, 1987; Notices of the American Revitalizing Undergraduate Mathematics. Commit• Mathematical Society, 34 (August, 1987) 748-750. tee on Mathematical Sciences in the Year 2000, Na• [58] Smith, David A.; Porter, Gerald J.; Leinbach, L. tional Research Council. Washington, D.C.: Na• Carl; Wenger, Ronald H. (Eds.). Computers and tional Academy Press, (To Appear). Mathematics: The Use of Computers in Undergrad• [45] National Science Board Task Committee on Under• uate Instruction. MAA Notes No.9. Washington, graduate Science and Engineering Education. Un• D.C.: Mathematical Association of America, 1988. dergraduate Science, Mathematics and Engineering [59] Steen, Lynn Arthur (Ed.). Calculus for a New Cen• Education, Vols. I, II. Washington, D.C.: National tury: A Pump, Not a Filter. MAA Notes No. Science Foundation, 1986, 1987. 8. Washington, D.C.: Mathematical Association of [46] National Science Foundation. Women and Minori• America, 1988. ties in Science and Engineering. Washington, D.C.: [60] Steen, Lynn Arthur. "Developing Mathematical National Science Foundation, 1988. Maturity." In Ralston, Anthony and Young, Gail [47] Nisbett, Richard E., et al. "Teaching Reasoning." S. (Eds.). The Future of College Mathematics. New Science, 238 (October 30, 1987) 625-631. York: Springer-Verlag, 1983, pp. 99-110. [48] Oaxaca, Jaime and Reynolds, Ann W. (Eds.). [61] Sterrett, Andrew (Ed.). Using Writing to Teach Changing America: The New Face of Science and Mathematics. MAA Notes No. 16. Washington, Engineering, Final Report. Task Force on Women, D.C.: Mathematical Association of America, 1990. Minorities, and the Handicapped in Science and [62] Thurston, William P. "Mathematical Education." Technology, January 1990. Notices of the American Mathematical Society, 37 [49] Office of Technology Assessment. Educating Scien• (September 1990) 844-850. tists and Engineers, Grade School to Grad School. [63] Treisman, Philip Uri. "A Study of the Mathemat• Washington, D.C.: Office of Technology Assess• ics Performance of Black Students at the University ment, 1988. of California, Berkeley." In Fisher, Naomi; Keynes, [50] Ralston, Anthony and Young, Gail S. The Future of Harvey; Philip Wagreich (Eds.). Mathematicians College Mathematics. New York: Springer-Verlag, and Education Reform. CBMS Issues in Mathemat• 1983. ics Education, Volume 1. Providence, R.I.: Ameri• [51] Resnick, Lauren B. Education and Learning to can Mathematical Society, 1990, pp. 33-46. Think. Committee on Mathematics, Science, and [64] Tucker, Thomas (Ed.). Priming the Calculus Pump: Technology Education, Commission on Behavioral Innovations and Resources. MAA Notes No. 17. and Social Sciences and Education, National Re• Washington, D.C.: Mathematical Association of search Council. Washington, D.C.: National America, 1990. Academy Press, 1987. [65] Widnall, Sheila E. "AAAS Presidential Lecture: [52] Rheinboldt, Werner C. Future Directions in Com• putational Mathematics, Algorithms, and Scientific Voices from the Pipeline." Science, 241 (Septem• ber 30, 1988) 1740-1745. Software. Philadelphia, Penn.: Society for Indus• trial and Applied Mathematics, 1985. [66] Wilf, Herbert S. "The Disk with the College Educa• [53] Romberg, Thomas A. "Policy Implications of the tion." American Mathematical Monthly, 89 (1982) Three R's of Mathematics Education: Revolution, 4-8. Reform, and Research." Paper presented at annual [67] Wilf, Herbert S. "Self-esteem in Mathematicians." meeting of American Educational Research Associ• The College Mathematics Journal 21:4 (September ation, 1988. 1990) 274-277. [54] Schoenfeld, Alan H. Mathematical Problem Solving. [68] Zorn, Paul. "Computing in Undergraduate Math• New York: Academic Press, 1985. ematics." Notices of the American Mathematical [55] Schoenfeld, Alan H. (Ed.) A Source Book for Society, 34 (October 1987) 917-923. November-December 1990 FOCUS 9

Mathematics thor developed his theme with selections from the lives and writ• Magazine ings of familiar figures in the scientific community of the nineteenth century." When asked what motivated him to explore this topic, Archibald responded that "The relationship between connectivity Martha J. Siegel, Editor and smoke-rings which I described in my paper seemed to me to illustrate ... rather well ... sources of abstract mathematical ideas in the concrete world."

LESTER R. FORD AWARD Jacob Eli Goodman of City College, City University of New York, and Janos Pach and Chee K. Yap, both of The aim of Mathematics Maga• the Courant Institute of Mathematical Sciences, New York Univer• zine is to provide lively mathe• sity accepted this award for "Mountain Climbing, Ladder Moving, matical exposition in both long Ring Width of a Polygon," in The American Mathematical Monthly, articles and short notes. This 96 (1989): 494-510. Their paper realizes "a unified approach journal, published five times a to three elegant results that generally confirm one's intuition about year, strives to provide material of movement problems along polygonal paths in the plane. For exam• interest to collegiate mathemati• ple, the authors show that under modest hypotheses on the path, cians-both faculty and students. Of particular appeal to the Ed• a ladder of fixed length can be carried along a polygonal path by two people holding the ladder at its ends. Intuition in this area itorial Board are manuscripts that stress the relationships among can be deceiving." Pack explained that the challege of "an open several mathematical areas as well as the use of mathematics in problem posed by Chee Yap on a computational geometry day at other disciplines. the Courant Institute" piqued his interest. Moreover, he expressed We welcome papers on historical topics and look for the kind of writ• delight with the "beauty and simplicity of the mathematical tools ing that students in advanced undergraduate and beginning gradu• required for the solution." Yap described the essay as an attempt ate courses would read and enjoy. Although all papers are refereed, to solve two conjectures which he had posed in his earlier paper, "How to Move a Chair through a Door." The award winning essay the Magazine is not a research journal. There is a hope, however, "partially solves" one of these two conjectures. that the articles and other features will stimulate research. Every is• sue includes a lively "Reviews" section (featuring articles and books The Ford Committee also recognized Doron Zeilberger of Temple about mathematics but outside the mainstream of mathematics lit• University for "Kathy O'Hara's Constructive Proof of the Unimodal• erature), "Problems," "Letters," and "Notes." In three issues each ity of the Gaussian Polynomials" in The American Mathematical year, we present problems and solutions for major national and Monthly, 96 (1989): 590-602. "Often in mathematics, one has international mathematical contests. The new Editorial Board has to use techniques quite distant from the field in which a problem agreed to publish short classroom gems: clever and innovative is found in order to give a proof. Thus, combinatorialists have for mathematics suitable for presentation in the classroom. Student many years regretted that all proofs of the unimodality of Gaus• members of the MAA and their advisors should find the Magazine sian polynomials, which occupy a central position in combinatorics, particularly suited to their interests. use analysis or algebra. In 1987, Kathy O'Hara offered a brilliant, purely combinatorial, but very difficult proof. In presenting this dif• ficult proof to Monthly readers, Zeilberger offers several simplifi• cations but modestly downplays his own contributions so that the focus remains on O'Hara's resplendent triumph." GEORGE POLYA AWARD In 1990, two authors accepted this award 1990 MAA Journal Awards for their outstanding mathematical exposition: Richard Dean Nei• dinger of Davidson College for "Automatic Differentiation and APL," Each year the MAA's three journal awards committees honor in the College Mathematics Journal, 20 (1989): 238-251; and Is• the most accomplished articles from each of the Association's rael Kleiner of York University for "Evolution of the Function Con• periodicals-the Carl B. Allendoerfer Award for papers in Math• cept: A Brief Survey," in the College Mathematics Journal, 20 ematics Magazine; the Lester R. Ford Award for papers in The (1989): 282-300. Neidinger's essay is a "beautifully written arti• American Mathematical Monthly; and the George P61ya Award for cle that fulfills its stated goal while at the same time teaching the papers in The College Mathematics Journal. Winners received a reader a programming language (APL). The recursive techniques check, a certificate, and, most importantly, the respect and admira• are interesting since they can be implemented so easily in APL tion of their colleagues. The mathematical community recognizes and, conversely, the reader is prompted to learn APL because it these authors for their exceptionally skillful mathematics exposition is perfect for the job at hand. The level is appropriate for a calcu• and for generously sharing their insights with us. lus student who could be drawn into looking in a new way at the familiar, and sometimes boring, rules for differentiation." Neidinger CARL B. ALLENDOERFER AWARD Fan R. K. Chung, Martin Gard• reports that an article in Mathematics Magazine inspired him to ner, and Ronald L. Graham received this award for "Steiner Trees teach students about the computer language APL-he regarded on a Checkerboard," in Mathematics Magazine, 62 (1989): 83• this article as a "natural means to explain the idea and practical 96. Gardner has written several books published by the MAA and, significance of automatic differentiation." for twenty-five years, authored Scientific American's "Mathemati• cal Games" column. Chung and Graham are associated with Bell In his paper, Israel Kleiner, the second P6lya Award recipient, Communications Research and AT&T Bell Laboratories, respec• "makes a point that is particularly important for students and teach• tively. Their "presentation is accessible and engaging and includes ers to understand these days when graphing calculators have problems and examples to involve the reader at various levels." further reinforced the naive idea of functions as graphs. At a time when discoveries about dynamical systems are making clear The Allendoerfer Committee also honored Thomas W. Archibald the ubiquity of everywhere-discontinuous functions, continuous of Arcadia University for "Green's Second Identity in Its First Fifty but nowhere differentiable functions, and other functions formerly Years," in Mathematics Magazine, 62 (1989): 219-232. In his thought to be pathological, it is a mistake to limit students to a sev• historical account of Green's second identity, Archibald illustrates enteenth century viewpoint by identifying the function concept with "the vital role of interplay between mathematics and physics in the the geometric notion of a graph. Kleiner's article is well-written, evolution of some fundamental ideas of both disciplines. The au- with apt quotations, and an excellent bibliography." 10 FOCUS November-December 1990

FOCUS Employment FOCUS offers a 15% discount for the same adver• DEPARTMENT OF MATHEMATICS tisement in three or more consecutive issues. Advertisements University of Alberta The MAA will invoice advertisers after the first oc• Applications are invited for a tenure-track position currence specified in insertion orders. All invoices The Mathematical Association of America's more (File GP-1) at the assistant professor level, beqin• include a tear sheet as proof of contract fulfillment. than 30.000 members all receive FOCUS and its ning July 1, 1991. The possibility of an appointment "Employment Advertisements" as a standard memo ADVERTISING COPY DEADLINES at the associate professor level is not precluded. bership benefit. Most of these FOCUS readers Requirements are a PhD and proven ability or describe themselves as mathematicians teaching The Association publishes FOCUS six times per demonstrated potential for research and teaching. in secondary schools. colleges and universities. or year. Advertisement copy deadlines include: Salary for assistant professor is currently $36,910• working in business. industry. and government. $53.374; for associate professor currently $45,694• • January-February 1991 issue $67.658. Send vita and arrange for three letters NOTE TO ADVERTISERS: FOCUS will post Its Friday, 30 November 1990 of reference to be sent to: Professor R. Bercov, January-February 1991 Issue durtng the third • March-April 1991 issue Chairman, Department of Mathematics, University week of December 1990. "you wish to advise of Alberta, Edmonton, Alberta, Canada T6G 2G1. Monday, 28 January 1991 potential employment candidates that your Instl· In accordance with Canadian Immigration require• tutlon will Interview In January durtng the Joint After these deadlines. we advise potential aover• ments. priority will be given to Canadian citizens Meetings In Ssn Francisco, California, please do tisers to telephone MAA headquarters to inquire and permanent residents of Canada. Closing date so with confidence-the January-February 1991 about advertising space availability in these issues. for applications is January 15, 1992. Please quote Issue will raach rssders at lesst two weeks be• The Association will accept postdeadline edveruse• file number when responding to this advertisement. fore the meetings commence. ments on a discretionary basis only. The University of Alberta is committed to the princi• ple of equity in employment. The University encour• Rates for both classified and display FOCUS Ern• Anyone wishing to place an employment advertise• ages applications from aboriginal persons, disabled ployment Advertisements: ment in FOCUS should contact: persons, members of visible minorities, and women. • 50 words or less: $50.00 Siobnan B. Chamberlin • More than 50 words: $55.00 per column inch FOCUS Employment Advertisements DEPARTMENT OF MATHEMATICS The Mathematical Association of America University of Alberta Each advertising column measures 14.33 picas or 1529 Eighteenth Street Northwest 2 7/16 inches wide. Advertisements spanning two Washington, DC 20036-1385 Applications are invited for a tenure-track position columns measure 29.66 picas or 4 '¥'6 inches wide. (202) 387-5200 in Approximation Theory (File Ap-1) at the assis• Advertisements spanning three columns measure FAX: (202) 265-2384 tant professor level, beginning July 1, 1991. The 45 picas or 7 1/2 inches wide. e-mail: [email protected] possibility of an appointment at the associate pro• fessor level is not precluded. Requirements are a PhD and proven ability or demonstrated poten• tial for research and teaching. Salary for assistant professor is currently $36,910-$53,374; for asso• ciate professor currently $45,694-$67,658. Send vita and arrange for three letters of reference to be SOUTHWESTERN UNIVERSITY ASSISTANT PROFESSOR sent to: Professor R. Bercov, Chairman, Depart• OF MATHEMATICS ment of Mathematics, University of Alberta, Edmon• Georgetown, Texas ton, Alberta, Canada T6G 2G1. In accordance with Roanoke College Southwestern University invites applications for a Canadian Immigration requirements, priority will be tenure-track position in computer science. beqin• Tenure-track appointment at the assistant professor given to Canadian citizens and permanent residents ning August 1991. Responsibilities include teaching level beginning August 1991. PhD in mathemat· of Canada. Closing date for applications is January nine credit hours per semester, continuing scholarly ics required. Salary commensurate with qualifica• 15, 1992. Please quote file number when respond• activity, and assisting in the design of a new cur• tions. Excellent teaching emphasized, active schol• ing to this advertisement. The University of Alberta riculum emphasizing formal mathematical methods arship encouraged. Commitment to liberal learning is committed to the principle of equity in employ• such as derivation and verification of programs. A expected. Roanoke College is a private. liberal arts ment. The University encourages applications from PhD in computer science or in mathematics with college affiliated with the Lutheran Church and 10' aboriginal persons. disabled persons, members of strong supporting work in computing is required. cated in the Roanoke Valley of Virginia. Position is visible minorities. and women. Rank and salary are commensurate with experience open until filled. Send vita, graduate transcript, and and expertise. three letters of recommendation to: Dr. W. D. Er• DEPARTMENT OF MATHEMATICS gle, Department of Math, Computer Science, and Southwestern University is a selective. undergrad• University of Alberta Physics, Roanoke College, Salem, VA 24153: (703) uate institution committed to a broad-based liberal 375-2449. AAlEOE. Applications are invited for a tenure-track position arts and sciences education. Affiliated with the in Numerical Analysis (File NA-1) at the assistant United Methodist Church, it has over 1.200 students professor level. beginning July 1, 1991. The possi• and a history of stable enrollment. Southwestern's CHAIR bility of an appointment at the associate professor endowment of more than $126 million ranks among Department of Mathematics & Statistics level is not precluded. Requirements are a PhD the highest per student of undergraduate institutions Saint Mary's College of Minnesota and proven ability or demonstrated potential for re• in the country. The University has a strong commit• search and teaching. Salary for assistant protes• ment to faculty support, with faculty salaries at the Applications are invited for the position of depart• sor is currently $36,910-$53,374; for associate pro• 92nd percentile for liB institutions, and an endow• ment chair. Nine-month, tenure-track position to fessor currently $45,694-$67.658. Send vita and ment in excess of $1 million for faculty development. lead a seven-member department, starting 8/91. The University is located in Georgetown, Texas. 28 PhD in mathematics or statistics required, as well as arrange for three letters of reference to be sent miles north of Austin. the state capital. and site of demonstrated excellence in teaching and communi• to: Professor R. Bercov, Chairman, Department the University of Texas. cation at the undergraduate level, and experience of Mathematics, University of Alberta, Edmonton, or potential for departmental administration. Alberta, Canada T6G 2G1. In accordance with Applicants should send a letter of application, vita, Canadian Immigration requirements, priority will be and three current letters of recommendation to: Review begins December 1 and continues until the given to Canadian citizens and permanent residents Search Committee, Department of Mathematics and position is filled. Send vita and names & addresses of Canada. Closing date for applications is January Computer Science, Box 6399 SU Station, George· of three references to: Louis A. Guillow, Campus 15, 1992. Please quote file number when respond• town, TX 78626. Review of applications will begin Box 4, Saint Mary's College of Minnesota, Winona, ing to this advertisement. The University of Alberta on January 8, 1991. Women and minority candi· MN 55987. is committed to the principle of equity in employ• dates are encouraged to apply. Southwestern Uni• Saint Mary's College of Minnesota is committed to ment. The University encourages applications from versity is an Affirmative Action. Equal Employment affirmative action and encourages applications from aboriginal persons, disabled persons, members of Opportunity Employer. minorities and women. visible minorities, and women. November-December 1990 FOCUS 11

HOBART AND NORTHERN KENTUCKY UNIVERSITY EMORY & HENRY COLLEGE WILLIAM SMITH COLLEGES Department of Mathematics Tenure-track position beginning August 1991. PhD Department of Mathematics and Computer Science or near completion. Rank and salary open. Able and Computer Science The Department of Mathematics and Computer Sci• to teach wide range of undergraduate mathematics Two assistant professor, tenure-track positions start• ence anticipates having two tenure-track positions. and beginning computer science courses. Commit• ing in September 1991. Salary is competitive. The positions are pending funding and would be• ment to excellent teaching essential. Apply with iet• gin in August 1991. A doctorate in mathematics ter, cv, names of three references to: Richard Ptau, For the first position applicants should have a PhD in or a mathematical science is required. Statisticians Dean, Emory & Henry College, Emory, VA 24327. computer science or a PhD in mathematics and ex• are encouraged to apply. Responsibilities include AAlEOE. perience in computer science. Duties include teach• undergraduate teaching (12 hours/semester), schol• ing undergraduate computer science, participating arly activity, and service. The department's empha• in the Colleges' Interdisciplinary General Curricu• sis is on quality teaching. Good oral and written lum, and the possibility of teaching some mathe• communication skills in English are mandatory. matics (depending on interests and qualifications). Review of applicants will begin February 1, 1991. For the second position applicants should have a Apply to: Professor Frank Dietrich, Chair of Search PhD in mathematics; specialty open, but preference Committee, Department of Mathematics and Com• COLBY may be given to algebraists, applied mathemati• puter Science, Northern Kentucky University, High• cians, or those with demonstrated computer science land Heights, KY 41706. expertise. Duties include teaching undergraduate mathematics, participating in the Colleges' Interdis• NKU is an Affirmative Action/Equal Opportunity Em• ciplinary General Curriculum, and the possibility of ployer and actively seeks the candidacy of minori• teaching some computer science (depending on in• ties and women. terests and qualifications). Department of For both positions a strong commitment to teach• WESTERN ing and promise of continued scholarly activity is NEW ENGLAND COLLEGE Mathematics required. Teaching load: two courses per trimester. Department of Mathematics/Computer Science Colby anticipates having two regular Hobart College for men and William Smith College (i.e., tenure track) faculty openings (at Assistant/associate professor of mathematics, ten• for women are coordinate, four-year, liberal arts col• any level) in mathematics/computer ure-track position(s) commencing September 1991. leges committed to teaching and interdisciplinary science, commencing September 1991. Candidate will instruct 12 credit hours per semester. Our aim is to fill one vacancy in algebra study with a combined enrollment of 1,900 students. Applicants must have a PhD in mathematics or and one in computer science (exact Within an hour's drive are three major universities: mathematical sciences with excellence in and a interests not important). A strong Cornell, Rochester, and Syracuse. strong commitment to teaching. Scholarly activity research program plus a firm commit• Send detailed resume, three letters of recommenda• is expected and supported through released time, ment to undergraduate teaching are tion (at least one including comments on teaching), summer research grants, and professional travel essential for these positions. The PhD and undergraduate and graduate transcripts (pho• funds. There are no restrictions as to area of spe• degree is required. tocopies acceptable) to: Prof. Kevin Mitchell, Fac• cialization. Salary is competitive and commensurate We particularly welcome applications ulty Box 75, Department of Mathematics and Com• with credentials and experience. Western New Eng• from women, and well-qualified female puter Science, Hobart and William Smith Colleges, land College is an independent, nondenominational, applicants at the level of Assistant or Geneva, NY 14456. Evaluation of applications will coeducational institution with Schools of Arts and Associate Professor will in addition be begin December 15, 1990 and will continue until Sciences, Business, Engineering, and Law, with a considered for one of the college's Clare the position is filled. Women and minorities are total day-evening enrollment of over 5,000 students. Boothe Luce Chairs for women scientists. strongly encouraged to apply. An Equal Opportu• The department has 11 full-time faculty members Colby is a highly selective, private, nity/Affirmative Action Employer. and over 70 undergraduate majors. Send cover liberal arts college with an enrollment of letter (indicate if you plan to attend AMS/MAA an• 1700students. The Mathematics nual meeting in San Francisco), curriculum vitae, Department has 9 full-time and one DARTMOUTH COLLEGE and three letters of recommendation to: Dennis part-time faculty, and teaches bachelor's John Wesley Young M. Luciano, Chairman, Department of Mathemat• courses in mathematics and computer Research Instructorship ics and Computer Science, Western New England science. The college provides generous College, Springfield, Massachusetts 01119. Appli• support for faculty research. The John Wesley Young Research Instructorship is cations will be screened on a rolling basis until the a two-year, postdoctoral appointment for promising The college offers a rich computing positions are filled. Western New England College new or recent PhDs whose research interests over• environment based on VAX mainframes is an Equal Opportunity/Affirmative Action Employer lap a department member's. Current departmental and Apple Macintosh workstations (IIfx, and encourages applications from women and mi• Hexand SE3O). The Mathematics interests include areas in algebra, analysis, alge• nority candidates. Department in-house teaching/research braic geometry, combinatorics, computer science, laboratory comprises 18 Macintosh II1Hcx differential geometry, logic and set theory, num• computers, a central fileserver, and ber theory, probability, and topology. Teaching du• MIAMI UNIVERSITY associated visual-display apparatus. An ties of four ten-week courses spread over two or Oxford, Ohio Internet connection provides access to three quarters typically include at least one course Mathematics Education Position supercomputers and other networked in the instructor's speciality and include elemen• Department of Mathematics and Statistics resources. Colby is an acknowledged tary, advanced, and (at instructor's option) gradu• national leader in the development of ate courses. Nine-month salary of $32,500, supple• Tenure-track assistant professorship beginning Au• computer-aided instruction in mented by summer (resident) research stipend of gust 1991 in the area of mathematics education. mathematics. $7,150 (two-ninths). Send letter of application, re• Duties include teaching 8-9 hours per semester, continuing scholarship, and service. Applicants Please send a letter of application and a sume, graduate transcript, thesis abstract, descrip• resume, and the names of three referees, should have (by 8/91) a doctorate in mathematics tion of other research activities and interests if ap• to: Professor Keith Devlin, Chair, propriate, and 3 or preferably 4 letters of recom• education or a doctorate in mathematics with exper• Department of Mathematics, Colby mendation (at least one should discuss teaching) tise in mathematics education. Please send vita, College, Waterville, ME 04901. Review to: Phyllis A. Bellmore, Department of Math and transcripts. and three reference letters to: John of applications will begin December 5, CS, Bradley Hall, Hanover, NH 03755. Applica• Skillings, Math Education Search, Department of 1990,and continue until the positions tions received by Jan. 15 receive first consideration; Mathematics and Statistics, Miami University, Ox• are filled. Colby is an AA/EO Employer applications will be accepted until position is filled. ford, Ohio 45056. Review of applications will com• and strongly encourages applications Dartmouth College is committed to affirmative ac• mence on January 15, 1991. Women and minorities from women and minorities. tion and strongly encourages applications from mi• are encouraged to apply. Miami provides equal op• norities and women. portunity in employment and education. 12 FOCUS November-December 1990

JOHNS HOPKINS UNIVERSITY TRINITY COLLEGE EASTERN OREGON STATE COLLEGE Mathematics Applications are invited for a junior position in statis• The Department of Mathematics at Trinity College tics, to begin in fall 1991. Selection is based invites applications for a tenure-track position at the Applications are invited for two tenure-track posi• on demonstration and promise of excellence in re• rank of assistant professor, beginning in the aca• tions in mathematics at the assistant or associate search, teaching, innovative application. AAlEOE. demic year '91-'92. The normal teaching load is professor level beginning in the fall of 1991. The Applicants are asked to furnish a vita, transcripts, a five semester-courses per year ("3/2"). While we successful candidates will be committed to teach• letter describing professional interests and aspira• will be happy to receive applications from those with ing a wide range of undergraduate mathematics tions, and arrange for three letters of recommen• any specialty, we will be particularly interested in al• courses. An interest and background in geome• dation to be sent to: Prof. John D. Wierman, gebraists, logicians, and persons whose research try is preferred for one of the positions, but ap• Chairman, Mathematical Sciences Department, The interests might intersect with current department plications from all areas of mathematics are wel• Johns Hopkins University, Baltimore, MD 21218. members' areas: complex analysis, functional anal• come. A PhD in mathematics or a related disci• ysis, geometry, graph theory, combinatorics, and pline is required. Applications, including a vita and mathematical statistics. Requirements for the po• three current (1987--90) letters of recommendation, JOHNS HOPKINS UNIVERSITY sition: PhD in mathematics, evidence of teaching should be sent to: Mathematics Search Commit• The Mathematical Sciences Department invites ap• excellence at the undergraduate level, indications tee Chair, Badgley Hall of Science, Eastern Oregon plications for the 1991-92 of promise in research, and interest in curriculum State College, La Grande, OR 97850-2899. Appli• development. cation deadline: February 1, 1991, or until positions EliEZER NADDOR Applicants should send a cv, three letters of refer• are filled. Telephone inquiries: Professor Robert POSTDOCTORAL FELLOWSHIP. ence (at least one of which addresses teaching) and Brandon, (503) 962-3631. AAlEOE. a statement of teaching and research interests to: The Fellow is to be an outstanding graduating doc• toral student in mathematics, statistics, or opera• Search Committee Chair HAVERFORD COLLEGE tions research, who plans an academic research Dept. of Mathematics Mathematics career. The fellowship provides full support for 12 Trinity College Haverford, PA 10941 months of postdoctoral study at the department in Hartford, CT 06106 Haverford College announces a tenure-track open• an area of interest to some department faculty mem• No decision will be made prior 10 January 21, after ing for 1991-92 in the Department of Mathemat• ber, free from teaching and administrative duties. which the position may be filled at any time. ics, at the assistant (or possibly associate) profes• Selection is made without discrimination on the ba• sor level. Applications are invited from candidates sis of race, color, religion, sex, or national origin. Representatives of the Department will attend the employment register at the Joint Annual Meetings with research interests in any field of mathematics. Applicants should provide a current vita, a letter Candidates should demonstrate a strong commit• in San Francisco in January 1991. describing career aspirations and a research plan ment to teaching a broad spectrum of undergradu• for the fellowship year, and transcripts, and should Trinity College is an Equal Opportunity/Affirmative ate courses, and to research. Send curriculum vi• arrange for three letters of recommendation to be Action employer. Women and members of minority tae, statement of research and teaching interests, sent, by January 15, 1991, to: groups are especially encouraged to apply. and three letters of recommendation to: Curtis Professor John C. Wierman, Chairman Green, Chair, Department of Mathematics, Haver• Mathematical Sciences Department FURMAN UNIVERSITY ford College. Haverford, PA 19041. Haverford Col• 220 Maryland Hall lege is an EO/AA Employer. Women and minority The Johns Hopkins University GreenVille, South Carolina 29613 candidates are encouraged to apply. Baltimore, Maryland 21218 The Department of Mathematics at Furman Univer• Deadline for applications: December 7, 1990 Applicants for positions in algebra, analysis, dif• sity, an undergraduate, liberal arts college, invites ferenlial equations, geometry, number theory, and applications for a tenure-track assistant professor• (Late applications may be considered until the posi• topology should contact the Mathematics Depart• ship beginning September 1, 1991. A PhD in a tion is filled, but this cannot be guaranteed.) ment instead of the Mathematical Sciences Depart• mathematical science is required. All areas of spe• ment. cialization are acceptable. Excellence in teaching FERRIS STATE UNIVERSITY and continued scholarly activity are expected of all faculty. A vita, graduate and undergraduate tran• Head, Department of Mathematics UNIVERSITY OF FLORIDA scripts, and three letters of recommendation should Ferris State University invites nominations and ap• Anticipated, two LECTURER openings, non-tenure• be send to: Robert Fray, Department of Mathe• plications for the position of Head of the Depart• accruing but continuing full-time positions, to teach matics. Application deadline: February 1, 1991 ment of Mathematics. The Department of Mathe• undergraduate statistics courses starting August EOE/AAE. matics currently has 22 faculty and is responsible 1991. Requires master's level training in statistics for undergraduate education in math and computer and strong commitment to quality teaching. Expe• HUMBOLDT STATE UNIVERSITY (HSU) science and for baccalaureate programs in applied rience desirable, but not necessary. Apply by Jan• mathematics and actuarial science. QUALIFICA• Arcata, CA 95521 uary 15, 1991. Women and minorities are encour• TIONS: An earned doctorate in mathematics or ap• aged to apply. Send curriculum vitae, graduate tran• Applications are invited for a tenure-track assistant plied mathematics, or an earned doctorate in math• scripts, three letters of recommendation to: Geoff or associate professor position for fall 1991. HSU, ematics education with a master's in mathematics; Vining, Dept. of Statistics, 483 Little Hall, Univ. of located on the California north coast, has an ac• professional development and teaching experience FL, Gainesville, FL 32611-2049. AAlEOE. tive mathematics faculty and a strong undergradu• appropriate to senior rank; familiarity with a broad ate major with options in applied mathematics, com• spectrum of math instruction; ability to work with WAKE FOREST UNIVERSITY puter science, and teacher preparation, as well as others in a broad array of disciplines; and personal a master's program in mathematics modeling of en• qualities of integrity, industriousness, organization, Department of Mathematics vironmental systems. Candidates must have a doc• leadership, and interpersonal skills. and Computer Science torate in a mathematical science or mathematics Ferris State University is a career-oriented, open• Applications are invited for two tenure-track posi• education. All qualified applicants with a commit• admission, state-funded institution in western Michi• tions in mathematics at the assistant professor level ment to teaching excellence and professional activ• gan with 12,000 students and over 120 degree pro• beginning August 1991. Duties include teaching ities will be considered. Preference will be given to grams. Its schools include Allied Health, Arts and mathematics at the undergraduale and graduate applicants with a teaching credential and/or teach• Sciences, Business, Education, Pharmacy, Technol• levels and continuing research. A PhD is required. ing experience in elementary or secondary schools. ogy, and the College of Optometry. The departmenl has 22 members and offers a BS Women and minorities are especially encouraged to and MA in mathematics and a BS in computer sci• apply. Applicants should send vita, transcripts, and Review of applications is in progress and will con• ence. Send letter of application and resume to: three letters of reference to: Search Committee, tinue until the position is filled Send letter of in• Richard D. Carmichael, Chairman, Department of Department of Mathematics, Humboldt State Uni• terest, curriculum vitae, 3 letters of reference, and Mathematics and Computer Science, Wake Forest versity, Arcata, CA 95521 by February 1 for full con• official transcripts to: George Wales, Search Com• University, Box 7311, Winston-Salem, NC 27109 sideration. HSU is an Equal Opportunity/Affirmative mittee Chair. Starr 120, Ferris State University, Big AAlEO Employer. Action Employer. Rapids, MI 49307. EO/AAE. November-December 1990 FOCUS 13

GANNON UNIVERSITY NATURAL SCIENCES AND CALIFORNIA STATE UNIVERSITY Mathematics Department MATHEMATICS DEPARTMENT Long Beach University Square, Erie, PA 16541 Trinity College of Vermont Tenure-trackposition Statistics, begin fall 1991: Re• Burlington, VT 05401 The Department of Mathematics invites applications quires PhD math, applied math, or statistics, strong for a tenure-track position beginning August 1991. Trinity College invites applications for a faCUltypo• coursework math, research interests stat; evidence Applicant must possess a PhD with strength in prob• sition In mathematics, August 1991. Trinity is effective teaching and strong research potential; ability and statistics, and demonstrate evidence of located in Burlington, VT, a progressive and in• knowledge actuarial math desirable. Teach 3-4 good undergraduate teaching with research poten• tellectually stimulating community. Trinity's mis• classes per semester undergrad math and stat, MA• tial. Send resume and three letters of recommenda• sion stresses academic excellence, the education of level stat; research in speciality. Asst. or as• tion to: Dr. Rafal Ablamowicz, Chairman. For full women, combining the liberal arts with career prepa• soc. prof. preferred. Send resume, transcripts, consideration, please apply by March 1,1991. Gan• ration, and supporting life-long learning. PhD in 3 references letters to: Samuel G. Councilman, non University is an Equal Opportunity/Affirmative math and demonstrated independent teaching abil• Chair, Math Dept., CSU Long Beach, CA 90840• Action Employer. ity required. Duties: 24 credit hours of instruction 4502. Position open until filled; selection begins per year, student advising, committee work. Contin• 12/1/90. CSULB is an Equal Opportunity/Affirmative TRANSYLVANIA UNIVERSITY ued faculty development expected and supported. ActionlTitie IX Employer. Rank and salary depend on qualifications and ex• Lexington, KY 40508 perience. Application: letter, vita, graduate tran• script, one letter of reference addressing teaching, MATHEMATICS DEPARTMENT Computer Science: assistant/associate profes• UNIVERSITY OF NORTH DAKOTA sor, Transylvania University. Full-time, tenure-track. names/addresses/phone numbers of 2-3 additional PhD in computer science, or master's in computer references. Submit to: Dr. John F. Heinbokel, Box 8162 University Station science with PhD in a closely related field. Rank and Chair, NSM Department, Trinity College, Burlington, Grand Forks, NO 58202-8162 VT 05401. Review of applications will continue until salary dependent upon background. Exceptionally Applications are invited for 2 or more tenure-track position filled. Applications from women and minori• well-qualified candidates may be considered for a positions at the assistant professor level starting Au• ties are encouraged. ANEOE. Bingham Award for Excellence in Teaching; smaller gust 16, 1991. Consideration will be given to all ar• "start-up" grants are available for less experienced eas of mathematics, as well as statistics and math faculty. This recognition provides a supplement of CLARION UNIVERSITY education. Applicant must possess a strong com• up to 50% of base salary for the position. Transyl• Clarion, Pennsylvania mitment to teaching and research and have com• vania is a private, liberal arts college with a strong pleted PhD requirements by starting date. Teaching commitment to excellence in undergraduate educa• Applications are invited for a tenure-track position loads are three courses/semester. Must be eligible tion. The program in computer science is of long• in mathematics at the assistant/associate professor to work in US. Salary and fringes competitive. Ap• standing and is recognized for its outstanding qual• level, beginning fall 1991. PhD (by August 1991) plication deadline: March 1, 1991. Send resume, ity. Please send letter of application, curriculum vi• in the mathematical sciences and evidence of suc• copy of transcripts, and three letters of reference to: tae, and names of three references to: Dr. Dwight cessful mathematics teaching are required, Faculty Selection Committee. UND is an ANEOE. W. Carpenter, Computer Science Program, Transyl• load is 12 hours per semester. Salary and bene• vania University, Lexington, KY 40508. An Equal fits are highly competitive. Located in northwest• Opportunity Employer. ern Pennsylvania, adjacent to Interstate 80, Clar• ion University is one of fourteen institutions in the • TRANSYLVANIA UNIVERSITY State System of Higher Education. With 6,100 stu• dents, it offers the bachelor's degree in mathemat• m Lexington, KY 40508 ics, with options in applied mathematics, actuarial WESTERN The Mathematics Program invites applications for a science, computer science, and in mathematics ed• KENTUCKY tenure-track position commencing in the fall of 1991 ucation. Send letter of application, personal vita, UNIVERSITY and a possible one-year sabbatical replacement for transcripts, and three current letters of reference 1991-92. Transylvania University is a private, lib• to: Main Campus Search Committee, Department eral arts college with a strong commitment to aca• of Mathematics, Clarion University, Clarion, Penn• sylvania 16214. Screening will begin December DEPARTMENT HEAD demic excellence. Applicants must have a PhD in DEPARTMENT OF MATHEMATICS mathematics and a commitment to undergraduate 1, 1990, but applications will be accepted until the teaching. Salary and rank will depend on qualifica• position is filled. Clarion University actively seeks The Department of Mathematics is seeking tions and experience. Exceptionally well-qualified minorities and women applicants, and is an affirma• an energetic and enthusiastic individual to tive action/equal opportunity employer. serve as Head of the Department. The candidates may be considered for a Bingham Ex• Department consists of 28 full-time faculty cellence in Teaching Award which supplements fac• and offers baccalaureate and masters ulty member's salary by up to 50%; smaller, "start• THE UNIVERSITY OF degree programs with approximately 270 up"grants are available for less experienced faculty. SOUTH CAROLINA undergraduate majors and minors. Send letter of application, resume, undergraduate Department of Mathematics Western has an enrollment of 15,000 and graduate transcripts, and three letters of ref• students and is located in Bowling Green, erences to: David L. Shannon, Mathematics Pro• Applications are invited for anticipated tenure-track KY (population approximately 50,000), two gram Director, Transylvania University, Lexington, faculty positions at all ranks. Applicants in all ar• hours south of Louisville and one hour north of Nashville, TN along interstate 1-65. KY 40508. The search will remain open until the eas of mathematics will be considered. The de• positions are filled. Transylvania University is an partment is building on existing research strengths The successful candidate will hold a Equal Opportunity Employer. and is increasing the scope of its program in ap• Doctorate in Mathematics and have at least five years of college teaching experience. plied mathematics. The PhD degree or its equiva• Applicants should demonstrate ITHACA COLLEGE lent is required, and all appointments will be con• administrative leadership skills and provide sistent with the department's commitment to excel• evidence of effective teaching, public The Department of Mathematics and Computer Sci• lence in research and teaching at the graduate and service, and research/scholarly activities. ence at Ithaca College has two tenure-eligible posi• undergraduate levels. A resume, containing a sum• Review of applications will begin February tions in mathematics available for the 1991-92 aca• mary of research accomplishments and goals, and 1, 1991, and will continue until the position demic year. Qualifications: PhD preferred, active four letters of recommendation should be sent to: is filled, with expected date of appointment ABDs considered. Rank: assistant professor or July 1, 1991. Send letter of application, vita, Dr. Colin Bennett, Chairman above. All successful candidates will be expected to and names, addresses and phone numbers Department of Mathematics teach a wide variety of mathematics courses at the of at least three references to: Office of University of South Carolina undergraduate level. Screening begins December Academic Affairs, Mathematics Department Columbia, South Carolina 29208 17. 1990. Send vita to: Dr. Eric Robinson, Chair, Head Search, Western Kentucky University, Bowling Green, KY 42101. Department of Mathematics and Computer Science, The closing date for applications is January 31, Ithaca College, Ithaca, NY 14850. An Affirmative 1991. The University of South Carolina is an Af• An Affirmative Action/Equal Opportunity Employer. Women and minorities are encouraged to apply. Action/Equal Opportunity Employer. firmative Action/Equal Opportunity employer. 14 FOCUS November-December 1990

WEST CHESTER UNIVERSITY EAST STROUDSBURG UNIVERSITY MILLERSVILLE UNIVERSITY Mathematics Education and East Stroudsburg, PA 18301 Millersville University of PA, Mathematics Depart• Computer Science Full-time, tenure-track position, Assistant/associate ment. Tenure-track assistant professor in math• West Chester University seeks applicants for a professor depending on qualifications. Master's ematics beginning August 1991. Duties include tenure-track assistant or associate professorship in plus 10 credits in mathematics required; doctorate teaching upper and lower division undergraduate mathematics education beginning September 1991 degree and discrete mathematics background pre• mathematics courses. Must have strong commit• to teach pre- and inservice mathematics content ferred, Primary duties in undergraduate instruction. ment to excellence in teaching and scholarship, and methods courses for elementary majors, as 24 hour per academic year load. Competitive salary PhD in mathematics or near completion. Candi• well as other mathematics service courses. Also, and excellent benefits. Send current resume, tran• dates with specialty in any field of mathematics may will be responsible for the recruitment and advise• scripts, and 3 letters of recommendation to: Prof. apply; some preference will be given to applica• ment of elementary education majors who concen• Edward Hogan, Search Committee Chairperson, tions received by 1/28/91, but consideration of ap• trate in mathematics. A doctorate in mathematics Mathematics Department, East Stroudsburg Univer• plications will continue until position filled. Submit resume, copies of transcripts, and three letters of education is preferred but a candidate at disser• sity, East Stroudsburg, PA 18301. Deadline for recommendation (at least two attesting to teach• tation level may be considered. Applicants must applications is February 1, 1991. A State System ing effectiveness) to: Dr. Dorothee Jane Blum, have experience teaching at the elementary school of Higher Education University. An Affirmative Ac• Search Committee/FOl190, Department of Math• level and/or teaching preservice mathematics for el• tion/Equal Opportunity Employer, ematics, MILLERSVILLE UNIVERSITY, Millersville, ementary majors. Preference will be given to appli• cants with a demonstrated interest in research and PA 17551. An Affirmative Action/Equal Opportunity grant proposal writing in K-8 mathematics educa• STATE UNIVERSITY OF Employer. tion. Please send transcripts, vita, and three let• NEW YORK AT PURCHASE ters of reference, postmarked by January 31, 1991, Four-year liberal arts college near New York City THE UNIVERSITY OF SCRANTON to: Prof. John Kerrigan, Mathematics Dept., WEST has available a tenure-track assistant professorship Mathematics Department CHESTER UNIVERSITY, West Chester, PA 19383. in mathematics starting September 1991. Respon• AAfEOE. Women and Minorities Are Encouraged to sibilities include teaching five courses per year, su• The University of Scranton is a Jesuit university with Apply. pervising senior theses, and conducting research. over 3,500 undergraduates. The Mathematics De• Must be able to teach elementary computer sci• partment has 15 full-time faculty and about 50 ma• jors, DEPARTMENT HEAD ence courses. Interest in teaching general educa• tion courses a plus. Send three letters of refer• One (possibly two) tenure-track position is available Mathematics and Computer Science at the Univer• ence and resume to: Dr. Martin Lewinter, Chair, for fall 1991 for faculty interested in a teaching en• sity of Northern Iowa, a department with a strong tra• Mathematics Search Committee, State University vironment where research is encouraged and sup• dition of excellence in teaching and teacher prepa• of New York, 735 Anderson Hill Road, Purchase, ported. Individuals with expertise in any area of ration and a growing program of scholarship and NY 10577-1400. Letters of reference should ad• mathematics will be considered. Preferred areas research in mathematics, computer science, and dress teaching and research potential. Application include applied mathematics, probability/statistics, mathematics education, is seeking a new depart• deadline is January 4, 1991 or until position is filled. actuarial mathematics, algebra, and analysis. Rank ment head, Responsibilities include budgeting; fac• Equal Opportunity/Affirmative Action Employer. and salary are open and competitive. ulty assignment, evaluation & development; exter• nal relations; and some teaching and scholarly ac• Submit a vita, transcripts, and three references to: tivity. Required characteristics include being ap• BOWDOIN COLLEGE Mathematics Faculty Search Committee, University pointable as a full professor in the department, lead• Brunswick, Maine 04011 of Scranton, Scranton, PA 18510 or phone (717) ership & academic administrative skills, and good 914-6113. Screening will begin at once and appli• communication skills, Mathematics Department: Two tenure-track assis• cations will be considered until all positions have tant professorships starting fall 1991. Initial appoint• been filled. An AAfEO Employer and Educator. Appointment will cover the academic year plus sum• ment for three years with renewal possible. PhD mer session and begin June or August 1991. Year required and strong research record or potential ex• salary will be near $70,000 plus excellent fringe pected. Field open, but a preference will be given to MURRAY STATE UNIVERSITY benefits. Application screening begins January 14, candidates in applied mathematics for one position. Department of Mathematics & Statistics 1991. For further information, contact: Philip East, Normal teaching load is two courses per semester. Mathematics and Computer Science, University of Candidates with record of effective undergraduate Applications are invited for tenure-track positions Northern Iowa, Cedar Falls, IA 50614-0506. teaching preferred. Review of candidates begins at the assistant/associate professor level beginning 15 January, but applications will be considered until August 1991. Preference will be given to applicants UNI is an affirmative action/equal opportunity both positions are filled. Women and minorities are in statistics and mathematics education. educator and employer. encouraged to apply, Send resume and 3 letters of Responsibilities include a maximum three course recommendation to: Wells Johnson, Chair, Depart• teaching load consisting of a wide variety of un• FROSTBURG STATE UNIVERSITY ment of Mathematics, Bowdoin College, Brunswick, dsrqraduate and graduate level courses, continuing Mathematics ME 04011. Bowdoin College is committed to Equal research/scholarly activities, and university and de• Opportunity through Affirmative Action, partmental service. A PhD in statistics or a PhD Full-time, tenure-track, instructor/assistant profes• in applied mathematics with a statistics background sor beginning fall 1991, to teach 12 credits of GRAND VALLEY STATE UNIVERSITY is required for the statistics position, The mathe• introductory level mathematics per semester and matics education position requires a PhD in math• share departmental responsibilities, Salary range Allendale, Michigan ematics education or a PhD in mathematics with a is $25,000-$35,000. Master's degree in mathemat• Tenure-track positions are open for fall 1991 in background in mathematics education. Salary will ics required, doctorate preferred in mathematics or mathematics, mathematics education, statistics, be competitive. Screening will begin January 14, mathematics education. Candidate must have a computer science, and information systems. Du• 1991 and continue until the positions are filled. strong commitment to undergraduate teaching and ties include teaching undergraduate and/or gradu• a continuing interest in mathematical development. Applicants who are not US citizens must provide ate courses, student advising, and professional de• Preferable attributes include teaching experiences, their visa status and any other information relevant velopment. Earned doctorate and strong teaching experience with applications of mattiematics, and to their ability to accept employment. recommendations required. interest in applications of technology to classroom Send letter of application with vita, graduate tran• teaching. Send letter of interest, resume, transcripts GVSU is located in greater Grand Rapids, the sec• scripts or a list of graduate courses taken, and direct and three letters of recommendation no later than ond largest metropolitan area in Michigan, and of• three letters of recommendation to: January 15, 1991 to: Mr. C. Douglas Schmidt, Di• fers numerous cultural and recreational opportuni• rector of Personnel Services, Frostburg State Uni• ties. Cost of living is moderate and quality of life Dr. Robert Pervine, Search Committee Chair versity, Frostburg, MD 21532. Questions may be is high. Applications accepted until positions are Department of Mathematics & Statistics directed to: Dr. Richard Weimer, Department Chair; filled. Send resume and names of three references Murray State University (301) 689-4377. Women and minorities are encour• to: Faculty Search Committee, Math & CS Dept., Murray, KY 42071 aged to apply. AAfEEO, GVSU, Allendale, MI 49401. An EO/AA Institution MSU is an EO/AA employer. November-December 1990 FOCUS 15

CENTRAL MICHIGAN UNIVERSITY BENTLEY COLLEGE EASTERN ILLINOIS UNIVERSITY Three tenure-track positions and one tentative ten• Waltham, Massachusetts Department of Mathematics Charleston, IL 61920 ure-track position. All are at assistant professor The Department of Mathematical Sciences antici• rank. Priorities for the three indicated positions are pates at least one opening for a tenure-track po• A tenure-track position starting fall semester 1991 1. Functional analysis/operator theory; 2. Combi• sition starting in fall 1991. A PhD in mathemat• is anticipated. Doctorate in mathematics or mathe• natorics/design theory; and 3. Mathematics educa• ics, statistics, quantitative methods, operations re• matical sciences (including appl. math., computing, tion. The tentative position is in statistics and its sta• search, or a related field is required. Located in math. ed., or stat.) is required. Excellence in teach• tus will be determined by January 1991. Statistics suburban Boston, Bentley College has long been ing is expected and potential for scholarly activities candidates may also be considered as a fourth prior• known for its leadership in the education of busi• is desired. Rank is open. Full consideration given ity for one of the three positions mentioned above. ness professionals. In recent years, the school has to applications received by January 1, 1991. Appli• Candidates for all positions should have a doctor• experienced dramatic growth and currently enrolls cations from minorities and women are encouraged. ate in the appropriate field of mathematics, show about 7,500 graduate and undergraduate students Contact: Ira Rosenholtz, Chairperson, at the above promise of excellence in teaching, and have demon• in both business and liberal arts programs. While address. strated research ability. Candidates in mathematics heavy emphasis continues to be placed on quality education should have teaching experience in K-12 teaching, research and other scholarly activities are and the ability to teach undergraduate mathemat• CALIFORNIA STATE UNIVERSITY also encouraged and expected. Send resume to: HAYWARD ics courses. Duties include teaching and research Prof. Charles. R. Hadlock, Chair, Department of with a normal teaching load of 9 semester hours. Mathematical Sciences, Bentley College, 175 For• Dept. of Mathematics & Computer Science Preference will be given to candidates who com• est Street, Waltham, MA 02154-4705. Bentley Col• The Department invites applications for entry-level, plement existing research interests in the depart• lege is an Equal Opportunity/Affirmative Action em• tenure-track assistant professor position in mathe• ment. Salaries are competitive and benefits include ployer. matics beginning fall 91. Applicants should hold university-paid TIAA, medical, dental, and group life. PhD degree in mathematics, be committed to ex• Send resume, transcripts, and three letters of rec• cellence in teaching, exhibit competence and po• ommendation to: R. J. Fleming, Dept. of Math• HOPE COLLEGE tential to engage in professional activities including ematics, Central Michigan Univ., Mt. Pleasant, MI Holland, Michigan 49423 research and publication. All areas of specialization 48859 by January 21, 1991. Late applications will will be considered, including mathematics educa• be received until the positions are filled. Central The Department of Mathematics at Hope College tion. The department enrolls more than 700 majors Michigan University is an affirmative action/equal invites applications for two tenure-track positions, beginning August 1991. Candidates should have in its four degree programs: BS in mathematics, opportunity employer. All persons including mem• BS in computer science, and MS in both disciplines. bers of minority groups, women, handicapped per• a PhD in mathematics (or one of the mathemati• cal sciences), a commitment to excellence in teach• Please send resume and 3 references to: Mathe• sons, disabled veterans, and veterans of the Viet• matics Faculty Search Committee, Dept. of Math• nam Era are encouraged to apply. ing, and evidence of continuing professional ac• tivity. Scholarship and research are encouraged ematics & Computer Science, California State Uni• and supported. Candidates should have a strong versity, Hayward, Hayward, CA 94542-3092. Ap• KINGS'S COLLEGE interest in research projects involving undergrad• plications received by Jan. 1,1991 will be assured full consideration. Applications will be accepted un• Wilkes-Barre, PA 18711 uates. A letter of application, vita, graduate and undergraduate transcripts, and at least three let• til position is filled. Position #91-92 MATH-TI-1. Math Position: assistant professor. PhD required. ters of recommendation should be sent to: Prof. CSUH is an Equal Opportunity/Affirmative Action ABO will be considered. Successful applicant will John Stoughton, Mathematics Search. Departmen• Employer. teach an advanced course for majors in addition tal representatives will interview at the San Fran• to introductory courses. Primary interest and com• cisco meetings. Hope College is a Christian, co• TEIKYO LOREnO mitment must be to quality undergraduate instruc• educational, residential, liberal arts college affiliated HEIGHTS UNIVERSITY tion and to formally training young mathematicians. with the Reformed Church in America. Hope Col• Denver, Colorado Public scholarship and active involvement in the lege complies with federal and state requirements mathematical community are also expected. Ap• for nondiscrimination in employment. Applications Two positions at the asst. professor level will be plications, consisting of teaching and research in• are strongly encouraged from women and minority available in August 1991 to teach college algebra terests, vita, transcripts, and three letters of recom• persons. and/or calculus. A PhD in mathematics, mathe• mendation should be submitted to: Dr. William matics education, or related discipline is required; Shergalis, Dean of the College of Arts & Sciences, teaching experience in a multicultural setting is de• UNIVERSITY OF TENNESSEE King's College, Wilkes-Barre, PA 18711. Re• sirable. The applicant should have an international AT CHAnANOOGA view of applicants will begin on December 1, 1990. perspective and be sensitive to the special needs KING'S COLLEGE IS AN EQUAL OPPORTUNITY Department Head of Japanese and other international students study• EMPLOYER AND SPECIFICALLY INVITES AND ing mathematics in English. Send letter of appli• The University of Tennessee at Chattanooga invites ENCOURAGES APPLICATIONS FROM WOMEN cation, cv, and three letters of reference to: Dr. applications for Head of the Department of Math• AND MINORITIES. W. Hodson, Mathematics Search Committee, Teikyo ematics. A PhD in a mathematical science and at Loretto Heights University, 3001 S. Federal Blvd., least five years of college mathematics teaching ex• Denver, CO 80236 by January 12, 1991. Inter• PURDUE UNIVERSITY perience are required. Applicants should provide views will be conducted at the January MAA meet• evidence of leadership in curriculum development, Tenure-track position in the area of quantitative• ing. Equal Opportunity Affirmative Action Employer. mathematics psychology beginning fall 1990. teaching, public service, and research/scholarly ac• tivities. In this primarily undergraduate institution, Strong research credentials or research potential the faculty is expected to exhibit excellence in teach• HAMILTON COLLEGE essential. Salaries and benefits highly competitive, ing while maintaining a strong commitment to re• Department of Mathematics excellent research facilities, and support available. search and public service. The mathematics de• and Computer Science Curriculum vitae, three letters of recommendation, partment has 21 faculty members including a Chair Clinton, NY 13323 and representative preprints and/or reprints should of Excellence in applied mathematics. Located in be sent to: Dr. Jerome Busemeyer, c/o Department a very scenic metropolitan area of 400,000, UTS Two-year, tenure-track position. PhD and prior of Psychological Sciences, Purdue University, West has a student enrollment of 7,800. Send appli• teaching experience desirable. Duties involve teach• Lafayette, Indiana 47907. Applications received cation with current vita to: Dr. Paul L. Gaston, ing five courses per year at a small, highly selec• prior to November 1 will be given fullest considera• Dean, College of Arts and Sciences, 119 Holt Hall, tive, 4-year liberal arts college. Excellence in teach• tion though applications will be accepted until filled. UTC, Chattanooga, TN 37403-2598. Applications ing required. To apply, send curriculum vitae and Women and minorities are especially encouraged to received by January 31, 1991 will be assured full three letters of reference (at least one about teach• apply. consideration. Women and minorities are encour• ing) to: Richard Bedient, Chair; (315) 859-4138. aged to apply. UTC is an Equal Opportunity Em• Women and members of minorities are encouraged Purdue University is an Equal Opportunity Affirma• ployment/Affirmative ActionlTitle IX Section 504 in• to apply; Hamilton College is an Equal Opportu• tive Action Employer. stitution. nity/Affirmative Action Employer. Calendar

National MAA Meetings Seaway State University of New York at Oneonta, Oneonta, New York: Spring 1991 16-19 January 1991 74th Annual Meeting, San Francisco, Cal- Southeastern University of South Alabama, Mobile, Alabama: ifornia (Board of Governors, 15 January 1991) 5 and 6 April 1991 8-11 August 1991 67th Summer Meeting, Orono, Maine (Board Southern California University of California at Irvine, Irvine, of Governors, 7 August 1991) California: 10 November 1990 8-11 January 1992 75th Annual Meeting, Baltimore, Maryland Southwestern New Mexico State University, Las Cruces, New (Board of Governors, 7 January 1992) Mexico: 5 and 6 April 1991 Texas Stephen F. Austin State University, Nacogdoches, Texas: Sectional MAA Meetings 4-6 April 1991 Wisconsin University of Wisconsin, Oshkosh, Wisconsin: 26 Allegheny Mountain West Virginia State College, Institute, West and 27 April 1991 Virginia: 12 and 13 April 1991 Eastern Pennsylvania and Delaware University of Delaware, Other Meetings Newark, Delaware: 10 November 1990 Florida Eckerd College, St. Petersburg, Florida: 1 and 2 March 9-10 November 1990 Fourth Annual Southeastern Small Col• 1991 lege Computing Conference, Lenoir-Rhyne College, Hickory, North, Illinois Eastern Illinois University, Charleston, Illinois: 26 and Carolina. For further information. contact: Frank Cheatham. 27 April 1991 Campbellsville College, 200 West College Street, Campbellsville, Indiana Anderson University, Anderson, Indiana: 23 March Kentucky 42718; (502) 465-8158. 1991 9-11 November 1990 Third Annual International Conference Intermountain Ricks College, Rexburg, Idaho: 12 and 13 April on Technology in Collegiate Mathematics, Ohio State University. 1991 Invited lectures, panel discussions, workshops, minicourses, and Iowa Drake University, Des Moines, Iowa: Spring 1991 contributed paper sessions on the use of technology in collegiate Kansas Southwestern College, Winfield, Kansas: 5 and 6 April mathematics teaching. Topics: precalculus, calculus, differential 1991 equations, fractals, statistics, linear algebra, and related issues. Kentucky Northern Kentucky University, Highland Heights, Ken- Send your early registration fee of $45.00 to: Franklin D. Demana tucky: 5 and 6 April 1991 and Bert K. Waits, 1990 Conference on Technology in Collegiate Louisiana and Mississippi University of Mississippi, Biloxi, Mis- Mathematics, Department of Mathematics, Ohio State University. sissippi: 1 and 2 March 1991 231 West Eighteenth Avenue, Columbus, Ohio 43210. Maryland-District of Columbia-Virginia Towson State Univer• 7-9 January 1991 SIAM Workshop on Automatic Differentiation sity, Towson, Maryland: 16 and 17 November 1990; Virginia Com• of Algorithms: Theory, Implementation, and Application, Hilton Ho• monwealth University, Richmond, Virginia: Spring 1991 tel, Breckenridge, Colorado. Organizer: Andreas Griewank of Ar• Metropolitan New York Columbia University, New York, New gonne National Laboratory. For further information. contact: SIAM York: 4 and 5 May 1991 (50th Anniversary) Conference Coordinator, Dept CC0900, 3600 University City Sci• Michigan Calvin College, Grand Rapids, Michigan: 10 and 11 ence Center, Philadelphia, Pennsylvania 19104-2688: (215) 382• May 1991 9800; [email protected]. Fax: (215) 386-7999. Missouri The University of Missouri at Rolla, Rolla, Missouri: 5 20 January 1991 An informal Workshop on the Teaching of Cal• and 6 April 1991 culus will immediately follow the MANs annual meeting in San Fran• Nebraska Nebraska Wesleyan University, Lincoln, Nebraska: cisco, California. No advance registration necessary. For further 26 and 27 April 1991 information, contact: Gilbert Strang, Room 2-240, MIT.Cambridge, New Jersey Seton Hall University, South Orange, New Jersey: Massachusetts 02139. 10 November 1990 28-30 January 1991 Second ACM-SIAM Symposium on Dis• Northeastern Framingham State College, Framingham, Mas- crete Algorithms, Cathedral Hill Hotel, San Francisco, California. sachusetts: 16 and 17 November 1990 Organizer: Alok Aggawal of the IBM T. J. Watson Research Cen• Northern California California State University at Hayward, ter. For further information, contact SIAM Conference Coordinator Hayward, California: February or March 1991 at address above (7-9 January 1991). Oklahoma and Arkansas Cameron University, Lawton, Okla- 11-15 February 1991 Twenty-Second Southeastern Interna• homa: 29 and 30 March 1991 tional Conference on Combinatorics, Graph Theory, and Com• Pacific Northwest Seattle Pacific University, Seattle, Washing- puting, Louisiana State University (LSU), Baton Rouge, Louisiana ton: 20-22 June 1991 70803. Invited speakers include: R. Brualdi, P. Erdos, C. Godsil, Rocky Mountain University of Northern Colorado, Greeley, Col- W. Pulleyblank, and R. Thomas. For further information, contact: orado: Spring 1991 J. G. Oxley of the Department of Mathematics at LSU.

FOCUS NOVEMBER-DECEMBER 1990 The Mathematical Association of America Second class postage paid 1529 Eighteenth Street, NW at Washington, DC and Washington, DC 20036 additional mailing offices.