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FOCUS November 2005

FOCUS is published by theMathematical Association of America inJanuary, February, March, April, May/June,August/September, October, November, andDecember.

Editor: Fernando Gouvêa, Colby College;fqgouvea@colby.edu

Managing Editor: Carol Baxter, MAAcbaxter@maa.org

Senior Writer: Harry Waldman, MAAhwaldman@maa.org

Please address advertising inquiries to:Rebecca Hall RHall@MarketingGeneral.com

President: Carl C. Cowen

First Vice-President: Barbara T. Faires,Second Vice-President: Jean Bee Chan,Secretary: Martha J. Siegel, AssociateSecretary: James J. Tattersall, Treasurer: JohnW. Kenelly

Executive Director: Tina H. Straley

Associate Executive Director and Directorof Publications: Donald J. Albers

FOCUS Editorial Board: Rob Bradley; J.Kevin Colligan; Sharon Cutler Ross; JoeGallian; Jackie Giles; Maeve McCarthy; ColmMulcahy; Peter Renz; Annie Selden;Hortensia Soto-Johnson; Ravi Vakil.

Letters to the editor should be addressed toFernando Gouvêa, Colby College, Dept. ofMathematics, Waterville, ME 04901, or byemail to fqgouvea@colby.edu.

Subscription and membership questionsshould be directed to the MAA CustomerService Center, 800-331-1622; e-mail:maahq@maa.org; (301) 617-7800 (outsideU.S. and Canada); fax: (301) 206-9789. MAAHeadquarters: (202) 387-5200.

Copyright © 2005 by the MathematicalAssociation of America (Incorporated).Educational institutions may reproducearticles for their own use, but not for sale,provided that the following citation is used:“Reprinted with permission of FOCUS, thenewsletter of the Mathematical Associationof America (Incorporated).”

Periodicals postage paid at Washington, DCand additional mailing offices. Postmaster:Send address changes to FOCUS,Mathematical Association of America, P.O.Box 90973, Washington, DC 20090-0973.

ISSN: 0731-2040; Printed in the United Statesof America.

FOCUS

FOCUS Deadlines

January February MarchEditorial Copy November 15 December 15 January 15Display Ads November 23 December 20 January 28Employment Ads November 11 December 9 January 15

InsideVolume 25 Issue 8

4 Don Albers Will Step Down as Director of PublicationsBy Gerald L. Alexanderson

6 Archives of American Mathematics Spotlight: The Walter Feit PapersBy Nikki Thomas

8 MAA and AMAYTC Welcome 2005-2006 Project ACCCESS Fellows

10 Change in MAA Bylaws to be Voted on at January Meeting

12 MAA Features Dickinson College Workshop Mathematics Projectat CNSF Exhibit

13 The Land of the MayaBy Jacqueline Dewar

14 Bob Witte, Champion of Project NExT, Dies

15 In Memoriam

16 Call for AMC Volunteers

18 Proof: Three Reviews

22 Letters to the Editor

24 MAA Prizes and Awards Announced at MathFest 2005

26 MAA Business at MathFest 2005By Martha Siegel

28 MathFest 2005 in Pictures

32 Supporting Assessment in Undergraduate Mathematics

On the cover: The MAA tour group contemplates the Temple of the Inscriptions in Palenque,Mexico. Photograph by Lisa Kolbe. See article on page 13.

FOCUS

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November 2005

Jacqueline Dewar, Keith Stroyan, andJudy Leavitt Walker are this year’s recipi-ents of the MAA’s most prestigious awardfor teaching, the Deborah and FranklinTepper Haimo Award for DistinguishedCollege or University Teaching of Math-ematics. They will receive their awardsat the Joint Mathematics Meetings, wherethey have also been invited to speakabout “the secrets of their success.” Thesepresentations will happen on Saturday,January 14, from 2:30 to 4:00 p.m.

During Jacqueline Dewar’s almost 32years at Loyola Marymount University inLos Angeles, “her enthusiasm, extraordi-nary energy, and clarity of thought haveleft a deep imprint on students, col-leagues, her campus, and a much largermathematical community.” (Here andbelow, quotations are from the award ci-tation.) Dewar has been deeply involvedin teacher education initiatives and re-search on mathematics teaching andlearning. She has put new ideas into prac-tice with great success. At LMU she hasintroduced several “laboratory” courses,ranging from Mathematics for Elemen-tary Teachers to a workshop course forfirst-year mathematics majors that helpsintroduce students to mathematics.Dewar is now involved in a new projectcalled Science Education for New CivicEngagements and Responsibilities.

Dewar’s talk at the Joint Meetings, Math-ematics is ———, will take definitionsof mathematics given by laypersons andexperts, and move on to examine the dif-ferences between them. Dewar will dis-cuss why and how we should attempt tomake students’ views agree a little betterwith those of the “experts,” and describean attempt to do this in a course for fu-ture teachers.

Keith Stroyan has taught at the Univer-sity of Iowa for over 30 years. He was apioneer in the introduction of technol-ogy and computers into calculus courses,but he does not use these tools merely todraw pretty pictures or to allow studentsto avoid learning to compute. Instead, hefocuses on a concrete, experiential ap-

proach, “using computer projects to en-gage teams of students in investigatingconcrete applications of mathematics.”One of his projects, for example, asks stu-dents to investigate the question: “Whydid we eradicate polio by vaccination, butnot measles?” Since he uses teaching as-sistants in this course, Stroyan has alsotrained many graduate and undergradu-ate students in this style of teaching.Stroyan was deeply involved in the “cal-culus reform” movement, received grantsto develop materials, and wrote text-books, all of which emphasize the roleof mathematics as “the language of sci-ence.”

Stroyan’s talk at the Joint Meetings iscalled Small Opportunities at a Big Uni-versity, and will focus on teaching el-ementary courses at a large research uni-versity. His abstract notes that “there aremany disadvantages teaching elementarycourses at our university.” As a state uni-versity, the University of Iowa has a re-sponsibility to serve students with a widespectrum of backgrounds, abilities, andgoals. Stroyan will describe how theseand other opportunities play a role in hisnew Calculus 2 course, Engineering Math2.

Judy Walker joined the faculty at theUniversity of Nebraska–Lincoln in 1996;and has already had significant impacton both her institution and the largermathematical community. Students “tes-tify that her courses are among the mostdemanding they ever had, yet consis-tently praise her ability to guide the di-

rection of a class through questions.” Shecreated a first-year seminar for non-ma-jors, The Joy of Numbers: Search for BigPrimes. Walker has also been deeply in-volved in the ALL GIRLS/ALL MATH program,has worked with undergraduate womenmathematics students, and is currentlyworking on a project focused onmentoring graduate students throughcritical transitions.

At the Joint Meetings, Walker will speakon The Joy of Teaching The Joy of Num-bers, which she describes as “one of myfavorite courses to teach.” Her abstractsays that “though all the students in thiscourse are honors students, they are notnecessarily mathematically inclined. Andyet these students, some of whom cringeat elementary algebra, learn to writeproofs as they work their way from thedefinition of what it means for one inte-ger to divide another, through card shuf-fling and identification numbers, to theRSA scheme for public key cryptogra-phy.”

The Mathematical Association ofAmerica first instituted Awards for Dis-tinguished College or University Teach-ing of Mathematics in 1991, with the goalof honoring college or university teach-ers who have been widely recognized asextraordinarily successful and whoseteaching effectiveness has been shown tohave had influence beyond their own in-stitutions. In 1993 the MAA Board ofGovernors renamed the award to honorDeborah and Franklin Tepper Haimo.

Dewar, Stroyan, and Walker Receive Haimo Awards

By Fernando Q. Gouvêa and the Haimo Committee

Judy Leavitt Walker Jacqueline Dewar Keith Stroyan

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FOCUS November 2005

No one is indispensable, we are told.One exception might be Don Albers inhis role as acquisitions editor for the vari-ous book series of the MAA. The ques-tion arose recently when Don informedExecutive Director Tina Straley that hewill be serving out his current term asAssociate Executive Director and Direc-tor of Publications but will not be avail-able for another. Needless to say thisprompted some quick thinking on thepart of Officers and the Executive Direc-tor.

Friends of Don and his wife, Geri, havelong known that Don and Geri wereawaiting the day when they could moveback to their California home. Both hav-ing been raised in the Minnesota-NorthDakota area, perhaps more than somethey have an appreciation for the North-ern California climate. Combining thiswith a shared love of hiking and camp-ing in the Western mountains made thedecision inevitable. It was only a ques-tion of when and some of us were sur-prised the decision had not come beforenow.

Fortunately the current plan is to replaceDon as Director of Publications withsomeone who could take on overseeingthe publications program in the Wash-ington headquarters, and to convincehim to continue as Acquisitions Editorfrom his new base in California, some-thing that should be easily possible in thisage of electronic communication. Donhas built the publications program to thepoint where having an Acquisitions Edi-tor working with the Director of Publi-cations is essential.

It is difficult to imagine our findingsomeone to replace him in this latter rolewho would have his experience, his trackrecord, his knowledge of the wide math-ematical community, and his extraordi-nary ability to wheedle manuscripts outof potential authors who otherwisemight not have thought about writing abook. Never content to wait for manu-scripts to come in through the mail, Dongets an idea for a book and goes out and

finds a suitable person to produce it. Un-der Don’s stewardship, MAA book saleshave quadrupled since 1991, bringing in$1.6 million last year. Instead of the oc-casional Carus Monograph that some ofus remember as the MAA’s book pro-gram, roughly twenty new books nowappear every year.

As hard as it is to believe at this point,Don did not start out life as an editor andadministrator. Born, raised and educatedin those cold states in the Northern tierof the country, he came to California in1968 and joined the faculty of MenloCollege, where he served as departmentchair for 21 years. Don and I met at thattime through our having a mutual friend,George Pólya. While at Menlo Don builta superb department of teachers andscholars, a group now sadly dispersed asthe result of a decision by the adminis-tration there to take the College in a dif-ferent direction. While at Menlo, though,Don became involved with the MAA asa member of various committees, Chairand later Section Governor of the North-ern California Section, Second VicePresident, and Editor of the (then) Two-Year College Mathematics Journal. WhenDon became editor, he changed andbroadened the publication so much thatby the end of his term as editor in 1984

the words “Two-Year” were droppedfrom the title. He imaginatively intro-duced all sorts of new features to theJournal—in his first issue, an interviewwith Pólya, a column called “The LighterSide,” a section on “Mathematical Gems”by Ross A. Honsberger, “Classics Revis-ited,” “The Signpost: News & Commen-tary,” as well as some unusually lively andwitty covers. These were precursors ofthings to come — and the MAA publi-cations would never be quite the same.

Don not only acquires good material forthe MAA publications program, he haspublished widely himself. He had theidea for the interviews that were a staplein the CMJ for years and became the coreof two volumes of the MathematicalPeople series, an idea now emulated invarious publications including the No-tices of the AMS. He continues to do in-terviews, the most recent that with TinaStraley in the September 2005 issue ofFOCUS.

Don arrived at the MAA headquarters in1991 with the title Director of Publica-tions and Programs, overseeing journalpublications in addition to the book se-ries, electronic publishing, book produc-tion, sales and marketing of publications.His Programs responsibilities includedthe Placement Tests Program, StudentChapters, Career Information, and MAAReps. Two years later as technologychanged so did his title to Director ofPublications and Electronic Services. Inthat new role he supervised developmentof MAA Online. Over the years Don alsohas served as the principal investigatoron grants related to electronic servicesand acting as liaison with the Dolciani-Halloran Foundation in their support forProject NExT, with the Educational Ad-vancement Foundation on the AmericanArchives of Mathematics, and withJSTOR. During periods of transition inthe Washington office he has taken onadditional tasks. He also introduced fournew book series (Spectrum, ClassroomResource Materials, Outlooks, and theMAA Problem Books) and was thefounding editor of Math Horizons for

Don Albers Will Step Down as Director of Publications

By Gerald L. Alexanderson

Don Albers

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November 2005

students. Forming such a student maga-zine was Don’s idea and the magazine hasbeen highly successful.

Probably what he enjoys most, though,is finding good manuscripts. And heknows a good one when he sees one—he seemingly reads everything he can gethis hands on. He’s always well ahead ofmost of us in keeping up on what hasappeared from just about any Englishlanguage publisher and many that are inother languages as well. Not satisfied withstaying abreast of what’s happening inmathematics, he regularly returns to hislongtime interest in astronomy, andother sciences as well. And then there arethe nonscientific fields. I would not liketo be paying the bill for shipping his vastcollection of books in Washington whenthey have to be sent to join his librarystill in California. Perhaps we shouldthink of his move as a consolidation oflibraries.

The MAA is lucky indeed that arrange-ments have been worked out to keep thisman of letters at the MAA to continuehis role as Acquisitions Editor. We canlook forward to many new and worth-while books that will be appearing in theMAA catalogue in the years ahead.

Gerald Alexanderson is Professor of Math-ematics at Santa Clara University andformer president of the MAA.

Eighth Annual

February 3 - 5, 2006A national showcase for research

projects of undergraduate women

in the mathematical sciences.

Main ProgramTalks by undergraduate women

about their own research

Plenary SpeakersLenore Blum,

Carnegie Mellon University

Krystyna Kuperberg,

Auburn University

For undergraduate participants, most local expenses

are covered and travel support is available.

For more information, to register,

apply for funding, or sign up to give a talk,

visit us on the web at

www.math.unl.edu/ ncuwmor write to us at

ncuwm@math.unl.edu

Department of Mathematics

University of Nebraska-Lincoln

203 Avery Hall

Lincoln, NE 68588-0130

Deadline for registrationJanuary 20, 2006

University of Nebraska-LincolnAn equal opportunity educator and employer with a comprehensive plan for diversity

Note from the Editor

Readers will have noticed that this is-sue of FOCUS is a little heftier than usual.This is due to the inclusion of a specialreport on the MAA’s SAUM program.This special report, edited by Bernard L.Madison and Lynn A. Steen, begins onpage 34 of this issue.

We are always looking for contributionsto FOCUS. At this point, we are particu-larly interested in more articles aboutmathematics, either expository or reflec-tive. But of course, we are also interestedin news: about mathematics, aboutmathematics teaching, about MAA ac-tivities. See the guidelines on how towrite and submit articles for FOCUSonline at:http://www.maa.org/pubs/focussubmission.html.

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FOCUS November 2005

The Archives of American Mathemat-ics at the Center for American Historyhas recently processed the papers of Aus-trian-born mathematician Walter Feit, apure mathematician who contributed toalgebra, geometry, topology, numbertheory, and logic and who co-authoredone of the most influential papers everwritten on finite group theory. SidnieFeit, Walter’s widow, donated the papersand provided a detailed initial inventory.

Feit was born in Vienna in 1930. At theage of nine, Feit’s parents placed him onthe last KinderTransport train allowed toevacuate Jewish children from Austria.The train left just two days before the warbroke out in 1939. Feit attended highschool in England and became passion-ate about mathematics. In 1946, he leftEngland for Florida and lived with hisaunt and uncle. The fall after his arrivalin the United States, Feit entered the Uni-versity of Chicago, and earned both hisBachelor’s and Master’s degrees in 1951.In 1955, he received his Ph.D. from theUniversity of Michigan under the super-vision of Robert Thrall, although Rich-ard Brauer was Feit’s true mentor.

Feit began his career in mathematics in1953, joining the mathematics faculty atCornell. There he worked on his 1963paper with John G. Thompson, “Solv-ability of Groups of Odd Order,” whichis widely regarded as the most influen-tial paper ever written on finite grouptheory. In 1964 Feit made the move toYale where he remained for 40 years un-til his retirement in 2003. While his mostfamous result is the proof of the Feit-Th-ompson theorem, he wrote and pub-lished extensively over the years, work-ing in finite group theory and modularcharacter theory. Feit passed away in2004.

Walter Feit was awarded the Cole Prizeby the American Mathematical Societyin 1965 for his work with Thompson andwas elected to the National Academy of

Sciences and the American Academy ofArts and Sciences. He also served as Vice-President of the International Math-ematical Union and editor for severaljournals, including 15 years as manag-ing editor for the Journal of Algebra. Hissixtieth and sixty-fifth birthdays were cel-ebrated with conferences in his honor, aswas his retirement in 2003.

A digital version of Walter Feit’s Memo-rial Service, including many photographsand information about his life, can befound online at the URL: http://

Archives of American Mathematics Spotlight

The Walter Feit Papers

By Nikki Thomas

www.math.ya le .edu/publ ic_html /WalterFeit/WalterFeit.html.

The Walter Feit Papers consist of corre-spondence and other printed material,including preprints, reprints, and manu-scripts. The collection contains a wealthof correspondence between Feit andJohn G. Thompson, primarily math-ematical in nature, but also coveringmore personal topics. Papers, preprints,and reprints comprise a substantial partof the collection, although most reprintsare included in duplicate, when available.

Walter Feit, 1988. From the Walter Feit Papers, Archives of Ameri-can Mathematics, Center for American History, The University ofTexas at Austin.

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November 2005

Early correspondence from J. G. Thompson to Walter Feit regarding their cooperative paper “Solvability of Groups of Odd Order,” ca.1960. From the Walter Feit Papers, Archives of American Mathematics, Center for American History, The University of Texas atAustin.

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FOCUS November 2005

Informal note between Walter Feit and John G. Thompson demonstrating their continued friendshipwell after the Feit-Thompson theorem was published, November 17, 1982. From the Walter Feit Pa-pers, Archives of American Mathematics, Center for American History, The University of Texas atAustin.

Of particular biographical relevance arethe materials related to Walter Feit’s me-morial service, as they include memoirsof his life in Vienna and England.

The finding aid for the Walter Feit Pa-pers is available online at: http://

www.lib.utexas.edu/taro/utcah/00433/cah-00433.html.

The Archives of American Mathematicsis located at the Research and Collectionsdivision of the Center for American His-tory on the University of Texas at Austincampus. Persons interested in conduct-

ing research or donating materials orwho have general questions about theArchives of American Mathematicsshould contact Kristy Sorensen, Archi-vist, k.sorensen@mail.utexas.edu, (512)495-4539. The Archives web page can befound at: http://www.cah.utexas.edu/collectioncomponents/math.html.

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November 2005

The MAA plans to award grants forprojects designed to encourage collegeand university women or high school andmiddle school girls to study mathemat-ics. The Tensor Foundation, workingthrough the MAA, is soliciting college,university, and secondary mathematicsfaculty (in conjunction with college oruniversity faculty) and their departmentsand institutions to submit proposals.

Grants will be up to $5,000 and will bemade to the institution of the projectdirector to be spent within the year. Col-lege and university mathematics faculty,or secondary school or middle schoolmathematics faculty working in conjunc-tion with college or university faculty, areeligible for TENSOR grants. Proposalswill be due in early February, 2006. Com-plete details are available through theMAA website at http://www.maa.org/projects/tensor_solic.html.

2006 MAA TENSOR

Grants for Women and

Mathematics Projects

_tâÇv{|Çzá

from theCUPM Curriculum Guide

A series of monthly columns by David Bressoud now discussesthe first six recommendations from this set of guidelines forthe undergraduate curriculum:

Introduction (February)

I. Who are we teaching? (March)

II. Teaching Students to Think (April)III. Only Connect! (May)IV. Math & Bio 2010 (June)V. Computational Science in the MathematicsCurriculum (July)

VI. On sustaining curricular innovation and renewal

(August)

Find the latest column at http://www.maa.org. All columns arearchived at http://www.maa.org/news/launchings.html.

Project ACCCESS (Advancing Com-munity College Careers: Education,Scholarship, and Service) has recentlychosen 32 two-year college faculty as Fel-lows for the 2005-06 academic year.Project ACCCESS is a joint project ofMAA and AMATYC (American Math-ematical Association of Two-Year Col-leges) for the mentoring and professionaldevelopment of new two-year collegemathematics faculty. Funding for theproject comes from the ExxonMobilFoundation. With the majority of lower-division mathematics students enrolledat two-year colleges, it is only natural thatthe two professional organizations findways to work together.

The first group of 28 ACCCESS Fellowswill complete their official year as Fel-lows at the AMATYC Conference in SanDiego in November. Here, they will be

joined by the new Fellows. Both groupswill participate in workshops and discus-sions relating to their teaching and otherprofessional commitments. Topics to beaddressed this year include creating ac-tive classrooms, assessment, demograph-ics of two-year colleges, and classroomresearch. Also, at the San Diego meeting,the 2004-05 Fellows will report on indi-vidual projects they carried out this pastacademic year at their home institutions.Fellows have created new course materi-als, evaluated innovative pedagogy, andconducted research on placement andassessment methods.

An important feature of ProjectACCCESS is Fellows’ attendance at anMAA Section meeting. One goal for theproject is that Fellows become activemembers of both MAA and AMATYC.Sections with a Section NExT program

MAA and AMATYC Welcome 2005-06 Project ACCCESS Fellows

are asked to include ACCCESS Fellowsin these activities. Fellows were over-whelmingly positive about their experi-ences at section meetings during this pastyear, but indicated that they would liketo see more talks or sessions focused onthe first two years of the undergraduatecurriculum.

Not every Section has an ACCCESS Fel-low in these first two cohorts, but theproject funding includes money for thecreation or support of a Section NExTas well as for other types of Section-basedactivities for two-year college faculty.Information about applying for fundingcan be found at: http://www.maa.org/projectacccess. Also at the website is acomplete list of both the 2004-05 and2005-06 ACCCESS Fellows.

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FOCUS November 2005

At the MAA MathFest in Albuquerque, the MAA Board of Governors approved a proposal to a change in the MAA Bylaws. Thischange will be voted on at the next Business Meeting of the Association, which will be held in January 2006 at the Joint Math-ematics Meetings in San Antonio, TX. It appears here in accordance with the rules governing changes in the Bylaws, which arecontained in the current MAA Bylaws, Article XI — Amendments to the Articles of the Association and Bylaws. The changeaffects Article VI, Section 2; which currently simply says:

VI.2. Each member of the Association residing in the United States, Canada or their possessions shall belong to one and onlyone Section.

It is proposed that this shall be changed to read as follows:

VI.2. Each member of the Association residing in the United States, its possessions, or Canada, shall belong to one and onlyone section. A member shall belong to the section determined by his or her MAA mailing address unless he or she has beenreassigned after requesting that the national office grant membership in another section.

The full text of the MAA Bylaws can be found online at http://www.maa.org (choose the “About the MAA” drop-down menu). Allmembers are encouraged to attend the Business Meeting at the January Joint Meetings in San Antonio to discuss and vote on thisamendment.

Travel to the Land of Cathay and Explore Its Ancient and Modern Culture

Journey to CHINA

the MAA’s 4th Annual Mathematical Study Tour

Contact Information:Lisa Kolbe Development Managerlkolbe@maa.org202-293-1170

June 6 - June 21, 2006

Full details, itinerary, and registration form will be available September 1, 2005 on MAA Online www.maa.org

Change in MAA Bylaws to be Voted on at January Meeting

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November 2005

The MAA Study Tour to the Landof the Maya began on May 24, 2005with a double-decker bus tour of thesites and neighborhoods of Merida,followed by free time for lunch andshopping. Later in the afternoon ourguides Chris, Alfonso, and Alonso,archeologists with connections to theMaya Exploration Center, presentedtalks about Mayan history and geom-etry. We were intrigued to hear thattraditional building methods stillemployed by indigenous peoples ofYucatan and Chiapas today seem tosupport archeologists’ measurementsand claims that the Mayan’s sacredgeometry employed ratios of 1 to

1 2 3 4 5, , , , , and φ in their

building designs. Then we turned toMayan numeration, examining, withmanipulatives, plausible algorithms forthe arithmetic operations with their nu-merals. Later in the trip lectures exam-ined the mathematics of the Maya cal-endars and aspects of archaeo-as-tronomy.

The next day, May 25, the physical chal-lenge began as we bussed a short distanceto Chichen Itza, a site that dates to the7th century. A path leading through asmall scrub forest suddenly opened toreveal an imposing view of our first pyra-mid, “El Castillo,” looming in the dis-tance. The 91 steps of this square-basedpyramid rise approximately 79 ft. to asplendid view of the amazingly flatYucatan horizon. (Such a horizon is idealfor observing various astronomicalevents.) Following a careful descent aidedonly by a rope stretched down the centerof the stairway, we queued up to climb61 slimy steps inside the pyramid singlefile to view an older inner temple con-taining a jaguar throne. It was commonpractice among the Mayans to build overearlier structures.

After a stroll through the ball court wearrived at the Group of a Thousand Col-umns adjacent to the Temple of the War-riors precisely at noon (well actually 1

PM because daylight savings time is ob-served in Mexico) on the day of the ze-nith at that latitude! Thus we were ableto experience and photograph an eventthat never happens in the continentalUnited States: the sun was directly over-head — objects cast no shadow. At thatvery moment little thermometers dan-gling from two of our group members’backpacks recorded the temperature as105 degrees.

Our group of 26 quickly bonded as wehelped and encouraged one another tonegotiate the uneven, large, and steeplyinclined stairs at the various sites. Someof us were challenged by fear of heights,others by claustrophobia, and all by theheat. But we all triumphed! Over the next10 days we visited sites at Dzbilchaltun,Uxmal, Labna, Edzna, Bonampak, andPalenque (where the photo on the coverof this issue was taken), accompanied byour marvelous guides who enthusiasti-cally shared their knowledge and passionfor the history and culture of the Mayaspast and present. Indeed, they were re-sponsible for a major find at Palenquejust a few years before.

Among the other amazing sights and ex-periences afforded by the trip were wait-ers and waitresses dancing while balanc-

ing bottles of beer ontheir heads, iguanastaking on various cam-ouflage hues accordingto their background,dung beetles workingcooperatively in pairsto move balls of you-know-what along thesacbe (raised roadbedsof ground limestone),bats hanging above usin Lolton cave, deli-cious food and Mexi-can beers and icecream, impressive cen-otes which are water-filled limestone sink-holes that provided a

stable supply of water for the ancientMaya people, a chance to splash in theGulf of Mexico, many opportunities topurchase various souvenirs, a wonderfulpicnic and waterfall with an icy pool forswimming at Golondrinas, and Olmecstone sculptures at La Venta park. Con-versations on our bus rides revolvedaround celestial events, the vagaries ofthe motion of the shadow of a gnomonat various latitudes, and how we mightincorporate what we had learned in ourcourses. As with the previous two MAAStudy Tours, wonderful Lisa Kolbe ex-pertly facilitated the various details andarrangements.

A number of us departed fromVillahermosa on the same plane for vari-ous connections in Houston. The coverstory on airplane travel magazine wastitled “Beijing Beckons.” A startling andpleasing coincidence: the destination forthe 2006 MAA Study is none other thanChina.

Jacqueline Dewar is Chair of the Depart-ment of Mathematics at LoyolaMarymount University in Los Angeles,California, and one of the recipients of thisyear’s Haimo Awards.

The Land of the Maya

By Jacqueline Dewar

Jacqueline Dewar at Uxmal.

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FOCUS November 2005

MAA joined mathematical, scientific,and engineering societies and universi-ties across the country in the Coalitionfor National Science Funding (CNSF)11th annual Science at Work exhibitionon May 10–11 in the Rayburn Building,the large House of Representatives Of-fice Building on Capital Hill . The pur-pose of the event is to exhibit to mem-bers of Congress and their staff the ben-eficial programs that the National Sci-ence Foundation supports. Each year, theMAA hosts a project funded by NSF thatfits the mission of MAA. During the dayof the exhibit, the exhibitors visit the of-fices of members of their state’s Congres-sional delegations. The program takesplace at 5 PM to 7:30 PM, after normalwork hours. A good number of the staff-ers and some of the Congressmen fromboth the House and the Senate stop bythe exhibition and reception.We askedthe three visitors to share their reactionsto the experience.

Nancy Baxter Hastings

I was excited and honored when Tinaasked me to spend a day with her “on theHill,” talking to congressional represen-tatives and staffers about the impact ofNSF funding in general on undergradu-ate mathematics education and in par-ticular on Dickinson College’s Workshop

Mathematics Project. Tina sug-gested that I bring along some stu-dents who were familiar with theproject, so I invited Jeff Goldsmith‘07 from Texas, and Carley Moore‘06 from Pennsylvania, who hadworked with me as TAs in Work-shop Calculus.

In preparation for the big day,Joanne Weissman, who has servedas the Workshop Mathematicsproject manager for the last twelveyears, and I worked with our pub-lic relations office to design aspiffy, professional-looking posterand color-coordinated brochure,and we filled small glass jars thathad the Dickinson seal on thefront with Hershey ChocolateKisses (a PA delicacy) to leave as thank-you gifts.

We were all … well, maybe not Tina … alittle nervous at first, but we quicklysettled down into a routine. Tina intro-duced us and talked about the value ofNSF funding. I think many of the peopleto whom we spoke were expecting Tinato say, “We need more!” Instead shethanked them for supporting funding inthe past and emphasized the value offunding undergraduate education. I wenton to mention two major impacts that

NSF funding has had on Dickinson Col-lege, namely providing students with ac-cess to state-of-the-art research equip-ment and supporting the developmentof innovative curricular materials thatsupport student learning. The main fo-cus, however, was on the students, Jeffand Carley. They spoke enthusiasticallyand articulately about their experiencesand observations in the workshop class-room. When we met with Rep. RalphHall (R-TX), who represents Jeff ’s dis-trict, Jeff and Carley sat in over-sized,blue leather wing chairs in front of the

MAA Features Dickinson College Workshop Mathematics Project

at CNSF Exhibit

Jeff Goldsmith, Nancy Baxter Hastings, andCarley Moore in front of the CNSF exhibit.

Carley Moore, a Congressional staffer, Jeff Goldsmith, and NancyBaxter Hastings

Nancy Baxter Hastings, Carley Moore, Abigail Wilson, andJeff Goldsmith.

FOCUS

13

November 2005

congressman’s desk and chatted amica-bly with him about the importance ofgetting a good education, while Tina andI contributed to the conversation from acomfy sofa off to the side. When Sen.Rick Santorum (R-PA) stopped to chatwith us in his outer office, he greeted uswarmly, asked Tina about our visit, andthen turned his attention to the students.

At the end of each meeting, we wouldleave our mementos — a Dickinsoncandy jar and a MAA stress ball (or two)— and then scurry, some times throughthe tunnels connecting the congressionalbuildings, to our next appointment.

The day culminated with a gala recep-tion for congressional representativesand their staffers. We talked to folks whodropped by, we posed for pictures, wenibbled on hors d’oeuvres, and then thefour of us collapsed in a cab and headedback to MAA Headquarters. It had beena terrific day. We could not have beenmore pleased with how things had gone,but we were all too tired to talk, so werode in silence enjoying the drivethrough downtown Washington and theend of a gorgeous day in D.C.

Jeff Goldsmith

My trip to D.C. for the CNSF exhibit wasa great experience. I’d been to Washing-ton before, but only to do sight-seeing,so this was a great opportunity for me totake part in the more business-orientedaspects of both mathematics and govern-ment funding. Professor Hastings, CarleyMoore and I had almost no idea what we

were getting ourselves into, but after acouple of meetings and the help of TinaStraley from the MAA, everyone founda comfortable flow.

We started the day with only two meet-ings, but thanks to the hard work of Kim-berly Niono, we soon had a very busy af-ternoon ahead of

us. I was particularly happy to meet withRalph Hall, the congressional represen-tative from my home district. He wasvery interested in how funding for theNSF had helped Professor Hastings’project, and how her project in turn af-fected students at Dickinson College.

I was very fortunate to have this oppor-tunity to learn how funding for programslike the NSF is handled by the govern-ment. I also now have a deep apprecia-tion for the hard work that goes into se-curing funds for the NSF and other pro-grams that have a huge impact on thelives of many people.

Carley Moore

My trip to Washington, D.C. for theCNSF exhibit was definitely a once in alifetime opportunity. Before this trip, myknowledge of the inner workings of theUS government was limited to what Ilearned from my 5th grade AmericanGovernment class and episodes of the“West Wing.” This experience not onlyintroduced me to a side of DC rarely seenby average tourists, but it also gave me agreater appreciation for the role of gov-ernment funding for science research and

the unique ability I have, as an Ameri-can, to voice my support of this funding.

In the beginning, I was not sure what Iwas getting myself into, but I was stillthoroughly honored to be asked to at-tend with Professor Hastings and an-other Dickinson student, Jeff Goldsmith.

Despite being excited and nervous aboutwhat the day would hold, I was eager toshare my perspective of the MAA-fundedWorkshop Mathematics Project atDickinson College. The whole day turnedout to be one pleasant surprise after an-other. At the start of the day, ProfessorHastings, Jeff, Tina Straley, and myselfhad two appointments (less than I hadexpected, though I was ready to do mybest at these meetings). By the end of theday we had visited with representativesfrom the offices of four senators and twocongressmen! While that in and of itselfwould have been considered a highlyproductive day, we also had the excellentopportunity to meet and speak with bothSenator Santorum and CongressmenRalph Hall in person, both of whom werevery receptive to what we shared.

That evening at the reception, I wasamazed to see the number of people thatturned out to see all the exhibits. It was agreat feeling to have a steady stream ofvisitors coming to our table and askingreal questions about the project. I am soglad that I have had this unique oppor-tunity, and I know that because of it, Ihave a new desire to learn more abouthow my government works and what myrole is within it.

Jeff Goldsmith speaks with a Congresssional aide at theCNSF reception.

Jeff Goldsmith outside an office in the RayburnBuilding.

14

FOCUS November 2005

Robert F. (Bob) Witte, age 66, a greatfriend of the MAA died October 3, 2005,following complications related to can-cer. Bob was born in Lowden, Iowa, andgraduated with a mechanical en-gineering degree from Iowa State Univer-sity, where he met his wife Pat, and sub-sequently received an MBA from theHarvard Business School. He also servedas a Captain in the United States AirForce.

During his nearly 40-year career withExxon Corporation, he spent 22 years inHouston and Baytown, where their chil-dren grew up. He transferred to Dallasin 1992 as Senior Program Officer of TheExxon Education Foundation (EEF, nowthe ExxonMobil Foundation) until hisretirement in 2000.

While Witte was Senior Program Officer,EEF’s support extended over severalMAA programs and activities: Math Ho-rizons, SUMMA, the Student Chaptersprogram, publication and disseminationof A Call for Change, and the Distin-guished Teaching Awards. His personalcommitment and drive to improvemathematics education continued oncehe retired. He served on numerousboards and committees dedicated tothis important objective.

One of the projects that Bob helped todevelop and nurture is Project NExT(New Experiences in Teaching), a pro-gram for new faculty. Project NExT isnow in its twelfth year, and counts morethan 800 Fellows to its credit. To say thathe was enthusiastic about Project NExTwould be an understatement.

In Bob’s Leitzel Lecture at MathFest 2001,What I Have Learned from the Mathemat-ics Community, he made clear his feel-ings about NExT when he said: “NExTis not just a new technique, a new cur-riculum, a new insight from research, al-though such resources are important.NExT demands sustaining the courageto question the status quo and the will-

ingness to change the ways in which youfulfill, indeed, live out, your responsibili-ties. And in the most crucial ways NExTis the network you have created, the pow-erfully supporting network of fellows,mentor consultants, leaders, and direc-tors.”

In his lecture, he offered the mathemat-ics community a challenging list of tasksfor the improvement of education, lacedwith such wry observations as Witte’sFirst Law of Productivity, which states that“If you work on the wrong thing, nomatter how hard you work, you willnever get the right answer.”

From 1994 until his retirement in 2000,he keenly observed the progress ofProject NExT. Fellows from those yearsremember the enthusiasm with whichBob participated in workshop sessionsand the eagerness with which he con-versed with them about their activitiesand aspirations. Without making himselfthe center of attention, he was always avital and welcome presence at ProjectNExT events.

According to Professor Christine Stevensof St. Louis University, Director ofProject NExT, “Even after his retirement,Bob remained deeply interested inProject NExT. He continued to read themessages on the Project NExT listserv,and, from time to time, he would sendme a message. Sometimes it would be anobservation about an issue in mathemat-ics education, and sometimes it wouldannounce the birth of a new grandchild.Often he would comment on someFellow’s contribution to the list and thenadd a comment like this: “I am sure he isa wonderful teacher. Makes me wish Icould be in his class. Bob truly believedthat the Fellows could change the face ofmathematics education.”

Bob’s family was fully aware of his com-mitment to Project NExT, and asks thatin lieu of flowers memorial donations inhis honor be used to sponsor Robert F.

Witte Project NExT Fellows. The MAAwill acknowledge Witte’s long-time sup-port by designating one or more RobertF. Witte Project NExT Fellows each year.Plans to establish an endowment for thispurpose are underway. Contributionsshould be made to the MAA, with a noteindicating that the donation is to sup-port a Witte Fellow, at 1529 EighteenthSt NW, Washington D.C. 20036.

He is survived by his wife of 45 years, Pat,their three children and four grandchil-dren: Beth, Frank and Lauren Apollo ofBellaire, Texas; Rob, Diane, Kyle and TyWitte of Irving, Texas; and Alison, Gary,and Karoline Beazley of GrandPrairie, Texas. He is also survived by hisbrother David Witte of Denver, Colo-rado, and an uncle, Ambrose Hoeger ofDubuque, Iowa.

Bob Witte, Champion of Project NExT, Dies

Robert F. Witte

FOCUS

15

November 2005

GET A JOB!Show your students the impact that

math has beyond the classroom.

Our new career brochure features eleven profiles that illus-

trate the emerging role that mathematics plays across all

disciplines, and is a great way to show your students the

great opportunities available to math majors.

We Do Math! Careers in the Mathematical Sciences is now

available online at the MAA Bookstore and through our

service center at 800-331-1622.

$30.00/100 copies plus shipping and handling

756 32

While an undergraduate at Texas Southern University

I was a part of the Louis Stokes Alliance for Minority

Participation Scholarship Program (LSAMP). This program

focused on minorities majoring in any of the major sciences

such as mathematics, computer science, physics, and

chemistry.

Having had great exposure to each one of these areas in

high school as well as college I decided to link my passion

for mathematics with something a little different, Business

Administration. This combination of mathematics and

business has created a framework that enables me to

apply analytical and problem solving skills in resolving

business management problems and issues, as well as

employ effective communication skills, orally and in

writing, consistent with the business and professional

environment. I have also developed the innovative

leadership and team – management skills necessary

for success in a diverse and changing workplace.

Since graduation in May 2004, I currently hold a position

in the Department of Defense as an Industrial Property

Management Specialist. I have a very big responsibility

dealing with issues in government property. I perform

audits, award surveys, conduct contract negotiations,

approve government transactions, investigate reports

of loss, damage, and destruction of property, create risk

analysis of property, and much more!

Careersin the Mathematical Sciences

Rochelle Johnson

THE MATHEMATICAL ASSOCIATION OF AMERICA

We Do Math!

B.S. Mathematics, Minor- Business

Administration

Industrial Property Management Specialist

Department of Defense

Defense Contract Management Agency

Serge Lang, well known number theo-rist and author of many important math-ematics books, passed away on Septem-ber 12 at the age of 78. Lang received hisPhD under Emil Artin, then taught atColumbia and Yale. He received theAMS’s Cole Prize in Algebra and alsotheir Leroy P. Steele Prize for mathemati-cal exposition. Lang wrote about all sortsof mathematics, from calculus to thefrontiers of research in number theory,often being the first one out of the gatewith a text on a new field, a service forwhich many graduate students have feltgrateful to him.

James J. Kaput, professor of mathemat-ics at the University of Massachsetts atDartmouth, died at age 63 after a jog-ging accident on July 31, 2005. Kaput waswell known for his work on mathemat-ics education. According to the obitiuaryin the Boston Globe, Kaput “was con-vinced that things like arcade games andhand-held devices were the keys to break-ing down mathematical concepts so any-one could understand them.” He taughtat UMass Dartmouth for 25 years, re-ceived many NSF grants, and helpedfound SimCalc Technologies. Kaput wasa member of the MAA for 28 years.

In Memoriam

George and Esther Szekeres both died onAugust 28, 2005. George was 94, andEsther was 95. They were part of the bril-liant group of Hungarian mathemati-cians of the 1930s which included PaulTurán and Paul Erdös. In the 1940s theymoved to Australia and helped invigo-rate the Australian mathematics researchcommunity. George and Esther madesignificant contributions to numbertheory and combinatorics, and Georgealso wrote on group theory and relativ-ity.

16

FOCUS November 2005

MAA Announces Search for Director of Publications

The Mathematical Association of America (MAA) is seeking a highly qualified person for the position of Director of Publica-tions. A candidate should have a significant record of work in publications in the mathematical sciences; a Ph.D. degree in amathematical science or mathematics education is preferred. The position requires successful experiences in all or most of thefollowing areas: book publishing; journal production; administration including financial management; editorial/reviewing ex-perience; mathematical writing not limited to research publications; and, electronic publications. Interest and experience withthe use of the internet in publications, grants, personnel management, and marketing experience are desirable. A successfulcandidate should have a vision for the future of MAA publications. Appointments may be made for two or three years, with theoption of renewal for multiple years.

The Director will oversee a staff of six located in the headquarters office and numerous editors. S/he oversees publication ofthree journals, three magazines, nine book series, and a variety of columns and articles. Electronic publications include all ofthese types of materials as well as the MAA Mathematical Sciences Digital Library (MathDL). The Director’s duties includepersonnel management, financial administration, acquisitions, production, inventory control, grant proposal writing and projectmanagement, and marketing. The Director reports to the Executive Director. S/he is a key member of the MAA’s staff leadershipteam, and will work closely with the Executive Director and other staff members, national officers, section officers, committeechairs, and others in strategic planning and program development.

The MAA, with nearly 30,000 members, is the largest association in the world devoted to college level mathematics. Member-ship includes college and university faculty and students, high school teachers, individuals from business, industry, and govern-ment, and others who enjoy mathematics. The Director of Publications is responsible for ensuring that publications encompassthe interests of all major constituencies of the MAA, embrace all areas of mathematics, and are easily available to all our mem-bers and the larger community who are interested in mathematics, especially expository mathematics and materials for facultyand students.

The deadline for submission of applications is January 21, 2006. Interviews will be held during the months of January andFebruary. It is expected that the new Director will begin work by July 2006, earlier if possible. The position is located at thenational headquarters of the MAA in Washington, DC. Salary will be based upon the candidate’s credentials or current salaryfor a reassignment position. The MAA offers a generous benefits package.

Candidates should send a resume and letter of interest to:

Ms. Julie KramanMathematical Association of America

1529 18th Street, NWWashington, DC 20036.

Applications may be submitted electronically to jkraman@maa.org. References will be requested after review of applications.Applications from individuals from underrepresented groups are encouraged. Additional information about the MAA and itspublication programs may be found on MAA’s website: http://www.maa.org . AA/EOE.

Call for AMC Volunteers

Interested in the American Mathemat-ics Competitions (AMC)? Would you liketo try your hand at anthropological field-work? Do you see yourself more as LaraCroft or Indiana Jones?

The AMC strategic planning workinggroup of the MAA seeks volunteers tohelp with a study of high school teach-ers’ attitudes about the American Math-ematics Competitions (AMC). Instead ofconducting a dry survey, we have hiredan expert ethnographer/anthropologist

to help us gather qualitative informationabout AMC participation.

Here is what you would be expected todo:

1. Attend a training session at the JointMeetings in San Antonio. This ismscheduled for Thursday, January 12,2006 from 5 pm to 7 pm.

2. Interview about five high schoolteachers by telephone (or in person ifconvenient).

3. Record your findings, using a con-venient web interface to transcribe yourfield notes.

If you would like to help, please contactFrank Farris at ffarris@scu.edu. Unfortu-nately, we cannot offer travel support, butwe believe that this interesting projectwill be well worth your time.

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17

November 2005

This book looks at the wide variety of ways in which math, statistics, and math education teach-ers have incorporated service-learning into their courses. These projects are not just stand-alone community service initiatives, but rather they specifically target the improvement ofmathematics skills and insights of the college students in the courses with which they areassociated. In some cases, the projects are the major focus of the courses. In others, theymay range from an essential component to one of several options. The book also specu-lates about heretofore untapped possibilities for service-learning, even including coursesin pure mathematics. College faculty often may not fully appreciate the wide range of sup-port mechanisms for such ventures even within their own institutions, so the bookincludes a lengthy chapter on the details of converting a rough idea to a solid action plan,sometimes even picking up financial support and other often unexpected benefits alongthe way. Creative teachers rarely implement a project in exactly the same way as a col-league might have, so the emphasis here is to display a wide range of successful projectsin order to encourage readers to develop some of their own.

Mathematics in Service to the Community:Concepts and models for service-learning in the mathematical sciencesCharles Hadlock, Editor

MAA Notes • NTE-66 • 277 pp., Paperbound, 2005 • ISBN: 0-88385-176-8List: $45.50 • MAA Member: $36.50

This book describes innovative approaches that have been used successfully by a variety ofinstructors in the undergraduate mathematics courses that follow calculus. Theseapproaches are designed to make upper division mathematics courses more interesting,more attractive, and more beneficial to our students. The authors of the articles in thisvolume show how this can be done while still teaching mathematics courses. Theseapproaches range from various classroom techniques to novel presentations of materi-al to discussing topics not normally encountered in the typical mathematics curriculum.

One overriding goal of all of these articles is to encourage students to stretch their math-ematical boundaries. This stretching can be done in a variety of ways but there is one

common theme; students expand their horizons not merely by sitting and listening to lec-tures but by doing mathematics.

Innovative Approaches to Undergraduate Mathematics CoursesBeyond CalculusRichard J. Maher, Editor

MAA Notes • NTE-67 • 200 pp., Paperbound, 2005 • ISBN: 0-88385-177-6List: $48.95 • MAA Member: $39.50

Using the history of mathematics enhances the teaching and learning of mathematics. Todate, much of the literature prepared on the topic of integrating mathematics history inundergraduate teaching contains, predominantly, ideas from the 18th century and earlier.This volume focuses on 19th and 20th century mathematics, building on the earlier effortsbut emphasizing recent history in the teaching of mathematics, computer science, andrelated disciplines.

From Calculus to Computers is a resource for undergraduate teachers that provideideas and materials for immediate adoption in the classroom and proven examples tomotivate innovation by the reader. Contributions to this volume are from historians of math-ematics and college mathematics instructors with years of experience and expertise in thesesubjects.

From Calculus to ComputersUsing the Last 200 Years of Mathematics History in the ClassroomAmy Shell-Gellasch and Dick Jardine, Editors

MAA Notes • NTE-68 • 200 pp., Paperbound, 2005 • ISBN: 0-88385-178-4List: $48.95 • MAA Member: $39.50

Order Now!1.800.331.1622 • www.maa.org

New Notes from the

Mathematical Association of America®

18

FOCUS November 2005

Proof, a movie based on David Auburn’sPulitzer Prize winning play, was releasedin September. Featuring GwynethPaltrow, Anthony Hopkins, and JakeGyllenhaal in the main roles, the moviehad a mixed reception by the critics.Given the important role of mathemat-ics in the story, we decided to ask threepeople to have a look and tell us whatthey think. Harry Waldman is HeadWriter for FOCUS. He works at MAAheadquarters in the Publications Depart-ment, but in his spare time Harry has astrong interest in the cinema and haswritten several books about films.Melanie Wood is a graduate student atPrinceton and a winner of the MorganPrize for undergraduate research. JackieGiles is a member of the FOCUS edito-rial board who teaches at Central Col-lege, part of the Houston CommunityCollege System.

Harry Waldman

At a key point in the new film Proof, thecinematic adaptation of David Auburn’saward-winning play, Robert Llewelyn(Anthony Hopkins), in his early 60s, aformer distinguished University of Chi-cago mathematics professor, says, “Allcylinders are firing… the machinery isworking.” The “machinery” he’s referringto, as his daughter Catherine (GwynethPaltrow) knows, is his mind. A world-renowned figure in mathematics, he’sbeen absent form the field for years, butnow he’s evidently inspired and hard atwork on a proof. Catherine, a fine mathstudent, has cared for him as dementia— his long night of the living dead —has taken hold. She hopes beyond all rea-son that her revered father, who hadmade his name in Britain at the age of22, is back. That would set her free.

The film, which also features the excel-lent British character actor Roshan Sethas Professor Bhandari, was co-scripted byRebecca Miller — Arthur’s daughter.Completed in late 2004 and one of themain entries at the recent Venice andToronto film festivals, it tries to bring thearena of mathematics and its inhabitants

to life. Set at the University of Chicago,it talks a good game, but doesn’t showenough of mathematicians in action.

Proof isn’t just a movie about math-ematics; it’s a mathematical movie.The scenes may as well have been laidout by diagram: Let’s put a touchingfather-daughter moment here, a star-tling revelation there, and use inter-stitial flashbacks to cube the total emo-tional-resonance quotient.

Stephanie ZacharekSalon

But the film is not really about math-ematics anyway. That’s just a backdrop,window dressing, so to speak. The film-makers have chosen mathematics as themetaphor through which to make theirpoints, including a few about the fielditself. There’s the impression that math-ematics is dominated by a few geniuses.That real mathematicians must showtheir talent early before they flame out.And that mathematics is too incompre-hensible to show on the screen. Thus allwe get is a bit of clichéd “mathematics.”

We do, however, see a lot of mathemati-cal books and journals — some of themfrom the Mathematical Association ofAmerica! Who are the people involvedwith this stuff? They are young wizards,we’re informed, striving for greatness.The world of rarefied mathematics, ap-parently, requires genius and its substan-tiation. If you can’t maintain the loftymathematics — exemplified by a proofso original that it causes your colleagues’heads to spin — you might as well headfor the hills. The young mathematicianshave only a bit of time: the clock is tick-ing. There are those in the mathematicscommunity who believe this but others,like Catherine, disagree. So implies thenew film from director John Madden.

Then again, Robert may be the exceptionto this rule. Although he’s suffering fromdementia, he just might have come upwith a major mathematical proof late inlife that will set the math world ablazeand restore him to his former glory. Oris this mysterious “proof,” whichCatherine has uncovered, not really his?Has she perhaps inherited her father’sgenius? Does the future hold promise orperil for her? Is she her father’s daughterin more ways than one?

Proof: Three Reviews

Photo provided by Miramax Films; used with permission

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19

November 2005

Self-centered young graduate mathemat-ics student Hal (Jake Gyllenhaal) repre-sents the young mathematicians. Heteaches, he coaches young hockey play-ers, he plays drums in a band. He’s no

geek. When we first meet Hal, he’s wear-ing glasses. That’s the last time he can seeclearly, so to speak. After that, sans spec-tacles, he grows stubble and is in it forthe glory. Getting close to Catherine, heoffers her a way out of her emotionalbind. He’s fond of saying, “There’s noth-ing wrong with you.”

So the film comes down to proofs of loveand trust in other ways. Catherine’s proofis of her commitment and love for herfather, the sacrifices she makes for himas his life ends — and of her careerchoices. It’s also about proof of Hal’strust and love for her, and hers for him.However, her father’s final years havebeen a nightmare for her. She had tomake sacrifices socially and education-ally. She couldn’t pursue her work be-cause her father couldn’t manage with-out her.

We do see that Catherine is depressedand in mourning even while her fatheris alive but in decline. Her sister Claire(Hope Davis), is a rather clear-headedNew Yorker, although she is portrayed asmaterialistic and hyperorganized. Hav-ing perhaps inherited these characteris-tics of her father, she has already givenup on him. There’s never a scene betweenher and Robert. Claire arrives in Chicagoto try to get her sister, who appears toher to be in need of psychiatric help, outof there. Catherine, a masochist of a sort,will have none of Claire or her ideas.

What’s perhaps most notable in the 100-minute film for MAA members is thatthe MAA contributed several hundredjournals and books to the producers of

this film so that they could recreate theambience of a real mathematician’s of-fice. Look, therefore, for some of the realmathematics in the film: the AmericanMathematical Monthly, the College Math-

ematics Journal, and MathematicsMagazine.

Melanie Wood

You should see the movie Proof — butnot just because you are a mathemati-cian. Feel free to take along non-math-ematician friends or recommend it toyour relatives. Proof is an emotionallycompelling drama, which manages to

present a fresh story without resortingto the laughingly preposterous premisesthat are the recent formula for an origi-nal plot. Gywneth Paltrow drives theshow with her convincing portrayal ofCatherine, the daughter of mathemati-cian Robert (Anthony Hopkins) who hasgone crazy in his later years and requiresthe constant attention of his daughter. Atthe start of the movie, Robert has justdied, and his other daughter Claire(Hope Davis) flies in for the funeral andto take care of Catherine. Claire’s over-bearing manner is not the only annoy-ance Catherine also faces. Hal (JakeGyllenhaal), a former student of Robert’s,wants to go through over a hundrednotebooks left by Robert to see if thereis any new mathematics in them.Catherine is grieving and struggling tocope with her father’s death and what thismeans for her own life.

Against considerable odds and despitea shaky start, Proof proves itself in ev-ery area. Thanks largely to Paltrow’sbeautifully unadorned performance,an exceptional portrait of psychologi-cal fragility that is honest, direct anddevastating, this is a film that reallyhas to be seen.

Kenneth TuranThe Los Angeles Times

At first, the movie deals with the issuesof grieving, insanity, caring for ill par-ents, and though it is engaging in theseaspects, it might leave you wonderingwhen they are going to get to the math(and might leave everyone else in theaudience relieved!). A few early math-

ematical jokes will allow you to identifyyour compatriots in the audience, butmostly the mathematical issues come latein the movie. A proof is discovered in anotebook in Robert’s desk and raises aquestion of the authorship. Was theproof done by Robert when everyonethought he was insane and incapable ofwork? Or is it possible that his daughterCatherine, a talented mathematics stu-dent, but not even finished with her edu-cation, could be the author? (Warning:the rest of review gives the answer.)

The drama of the sisters and Hal andthe past-and-present father drags a bitbecause some of the time it seems tobe not much more than elevated bick-ering. All in all, this is the sort of workthat gets credit for more gravity thanit genuinely possesses, merely be-cause of its subject.

Stanley KauffmannThe New Republic

Before it was a movie, Proof was a PulitzerPrize play, which ran on Broadway andwon Tony Awards for Best Play, Best Ac-tress, and Best Director. Of course, themovie medium changes a lot in the pre-sentation, but there were two big changesin substance from the play that were bothdisappointing. In the play, it is made clearearly on that Robert did not do lucidmathematics in his relapse from insan-ity, and in particular, that the proof mustbe Catherine’s. Knowing that the workis Catherine’s, we in the audience sym-pathize with this woman not only be-cause her sister and new boyfriend don’tseem to trust her, but also because hermathematical work is not recognized.

The movie takes a different approach,presumably to maximize suspense or setup a big plot twist. We are led to believethat Robert was able to start workingduring his relapse, and moreover that hehad a program for a proof of an impor-tant result. Catherine also starts her ownmathematical work, and it is possiblethey are working together. In a break-down, Catherine even seems to claim thatshe has stolen the proof from her dad.

The movie does in the end make it al-most clear that the proof is Catherine’s,

Because it takes itself so seriously,some viewers might come away con-vinced that Proof is a substantial workof art. You couldn’t prove it by me.

Colin CovertThe Star Tribune (Minneapolis)

20

FOCUS November 2005

except that one very mysterious scene isleft unexplained. While watching amovie, Catherine tells her father of herprogress on some result, and he makes asuggestion — apparently a good one,judging by her reaction. This scene notonly contradictsthe impressionthat Robert wasnot able to domathematics afterhis mental prob-lems began, butalso raises ques-tions. Even ifCatherine workedout the details, ifher father gaveher suggestions,was it really herproof? Could hehave given her re-ally significants u g g e s t i o n s ?Could he haveoutlined thewhole thing? Thisleads the audienceto disbelieveCatherine, and thus makes it harder forthem to feel sympathy for her plight.

This change also means that the movieloses the chance to unequivocally portraya young woman as an excellent math-ematician who thinks in new and creativeways.

The second major change from the playis that Catherine is portrayed as actuallyhaving mental problems (the play makesthat questionable at best). She appearsto be more disturbed than just by nor-mal grief. The movie uses flashing col-ors and scenes and lights going by quicklyin one scene to give an impression of in-sanity. While it is exciting for mathemat-ics that there have been two recent ma-jor motion pictures portraying math-ematicians, it is unfortunate that theyboth feature mathematicians with sig-nificant psychological problems.

One of the repeated themes of Proof isthat mathematicians do their best workby the time they are 26, and by 27 it is alldownhill. Many of my colleagues were alittle depressed by the movie for this rea-

son. The movie’s end underscores thisidea since the important proof was in factdone by Catherine (before she turned27), and not by her older father. Anotherinteresting “observation” made is math-ematicians’ inclination for serious par-

tying and use of speed. I am not surewhat that will do for recruitment into ourfield.

Jacqueline Brannon Giles

As I viewed the movie Proof I tried tostrip myself of the tendency to compareand contrast it with other movies aboutmathematicians. I wanted to study themovie as a single unit of entertainment.The dedication and compassion of ayoung female mathematician(Catherine) who cared enough about anailing, yet renowned older mathemati-cian, was touching and heart warming.It was the intergenerational connectingand caring that impressed me.

The point lies in a darker place — atthe intersection of insanity and genius,and the inheritance of both. It asks ifpathology can be separated from gift.It wonders if a wunderkind will fadepast his 20s, or if the child of a mad-man is doomed to crack. And it talksabout math like music.

Amy BiancolliThe Houston Chronicle

Catherine’s birthday, possibly markingthe beginning of her decline in capabil-ity, was also the day of her father’s burial.This highlighted both transition andcontinuity, for in the daughter’s struggleto aid her father she was inspired to do

her best work. Thedeath of the fatherspawned the birthof the daughter.

Catherine’s inti-mate interactionand constant con-tact with her fa-ther, who was alsoher mentor andher teacher,spurred on the ge-nius in her. Never-theless, she depos-ited her work in adrawer only to beshared after her af-fection for theyoung man (Hal)was acted on. Onceshe felt free toemote and share

with Hal, she was willing to share thetreasured mathematical thoughts in hernotebook. It suggested to me that her de-velopment as a female preceded her fulldevelopment as a mathematician.

A schism occurred because of distrustamong the characters, yet there was a sat-isfying resolution at the end. The daugh-ter, together with her father’s former stu-dent, went on to affirm and improve themathematical work inspired by her fa-ther, who sincerely desired that hisdaughter would “do” mathematics. Whilethe movie leaves this somewhat unde-cided, I believe that the father’s hope wasindeed actualized in the decision of thedaughter to continue the legacy of “writ-ing and doing mathematics.” This couldeasily be a visionary statement for thesenior mathematicians in our profes-sional community; Out of the declineand death of our great mathematiciansshould come the birth and rise of thenext generation of mathematicians.

Photo provided by Miramax Films; used with permission

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November 2005

Where Mathematics, History and Teaching Interact

®

Convergence: Where Mathematics, History and Teaching Interact, is the MAA's newonline magazine in the history of mathematics and its use in teaching. Convergence is a resource andforum for mathematics teachers of high school classes and lower-division college classes who want to usethe history of mathematics to engage and motivate their students and help them better understand themathematical ideas. The editors, Victor J. Katz, from the University of the District of Columbia, and FrankSwetz, from Penn State University, Harrisburg, welcome all mathematics teachers to visit the Convergencewebsite (http://convergence.mathdl.org) and see what the magazine has to offer.

The magazine includes:

o Expository articles dealing with the history of various topics in the mathematics curriculum, each accompanied by an online discussion group.

o Translations of original sources, with commentary showing the context of the works.

o Reviews of current and past books, articles, teaching aids, and websites on the history of mathematics.

o Classroom suggestions.

o Historical problems.

o What Happened Today in History?

o Quotation of the day.

o An up-to-date calendar of meetings dealing with the history of mathematics and its use in teaching.

The magazine is currently free to all, although registration is required. We encourage you to log on, to usethe material in your classes, to participate in the discussion groups, and to contribute new articles basedon your own experiences.

Convergence: http://convergence.mathdl.org

The Mathematical Association of America

:

22

FOCUS November 2005

Found Math: Mathematics and Football

Tsk, tsk. Both Ms Bazemore and the WSJflubbed this one. (FOCUS, August/Sep-tember 2005, “Found Math” on page 43.)

Bazemore, with any background in math,should merely have stated that the expla-nation of how this could have happenedis left as an exercise for the reader.

The WSJ editors, having no significantsports staff, could have pulled in any kidoff the street who would have pointedout that on fourth down the team threwa “Hail Mary” pass which was intercepted30 yards downfield, run back 23 yardswhere the ball was fumbled and recov-ered by the passing team. It could just aseasily have been a fumbled punt return.

I’m sure you’ll get hundreds of similarletters, mostly from mathematicians.

Phil Stein

As you might expect the Best of the Webcolumn also got many such responses, allof which pointed out interesting scenarios.You can see these responses at the OpinionJournal site: the original item is at http://w w w. o p i n i o n j o u r n a l . c o m / b e s t /?id=110006727 (last item), and the com-ments on the replies at http://w w w. o p i n i o n j o u r n a l . c o m / b e s t /?id=110006731 (look for “Figuring Foot-ball.” But of course the point was not thatsuch scenarios did not exist, but rather thatMs. Bazemore’s answer reinforces the no-tion that mathematicians are all too ca-sual about the connection between math-ematical problems and the real world.

Sample Mound Master Question RaisesDoubts

The third “Sample Mound Master Ques-tion” in the article “Math Youth Days atthe Ballpark” in the August/Septemberissue of FOCUS caught my attention:

“The Sky Sox play at the AAA level, onestep below the major leagues. Over theyears, 75% of Sky Sox players have goneon to the majors. If the Sky Sox currently

have 24 men on their roster, how many ofthe current players would we expect to goon to the majors?”

We don’t have enough information tosolve this problem, but I expect the trueanswer is significantly less than 75% of24, or 18.

For a concrete model, let’s assume that aplayer who gets promoted to the majorsplays for the Sky Sox for 2 years first,while players who don’t get promotedstick around for 6 years before giving up.Suppose further that each year 2 playersgive up and retire, while 6 players arepromoted to the majors. These 8 playersare replaced by 8 new players, of whom6 will eventually get promoted in 2 yearsand 2 will retire after 6 years. Clearly 75%of all players go on to the majors. Buteach season we have 6 first year playerswho will eventually be promoted, 6 sec-ond year players who will be promoted,and 2 players each in years 1, 2, 3, 4, 5,and 6 who will never be promoted. Thusin any given season we have 12 playerswho will eventually be promoted to themajors and 12 who will not.

My model is clearly overly simple, andmy numbers might not be realistic, butit clearly illustrates the fact that the per-centage of all players who eventually goon to the majors is not at all the samething as the percentage of players at anyone time who will eventually get pro-moted to the majors.

Paul K. StockmeyerCollege of William and Mary

Gene Abrams replies:

It is indeed the case that, as indicatedquite correctly in the comment by meansof a particular model, that “the percent-age of all players who eventually go onto the majors” is not at all the same thingas “the percentage of players at any onetime who will eventually get promotedto the majors.” I thank Prof. Stockmeyerfor pointing this out.

Here would be a better (at least a math-ematically precise) way to phrase a ques-tion of the type I am trying to convey:

“Historically, a large percentage of playerswho play for the Sky Sox eventually go onto play in the major leagues. The Sky Soxcurrently have 24 players on their roster.Suppose that 75% of the players on thecurrent roster eventually go on to play inthe majors. How many future major leagu-ers are there on the current roster?”

Or indeed the mathematical format ofthe question could be changed to:

“Historically, a large percentage of playerswho play for the Sky Sox eventually go onto play in the major leagues. The Sky Soxcurrently have 24 players on their roster.Suppose that 18 of the players on the cur-rent roster eventually go on to play in themajors. What percentage of the currentroster consists of future major leaguers?”

I will definitely change the wording ofthis type of question in the future.

What is now somewhat interesting to meis the following. The Sky Sox mediapeople are fond of using the followingstatistic in their advertising: “73% of SkySox players go on to play in the majors.”In the context of the current discussion,I guess there could be more than one in-terpretation of that phrase!

Intermountain Section Award Winner

In the August/September 2005 FOCUS,the winner of the Intermoutain SectionAward is correctly reported to be AfshinGhoreishi. However, that report fails tosay that Professor Ghoreishi teaches atWeber State University. For every otherwinner the institutional affiliation isgiven.

Lee BadgerWeber State UniversityChair of the Intermountain Section

Chicken Nuggets Puzzler

Thought you might like to know that ourmath club worked at its first meeting onthe puzzler in the August/September is-sue of FOCUS (page 45). It was perfectfor that group and we had fun figuring itout.

Letters to the Editor

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November 2005

I also wonder whether any readers cameup with the same outside-the-box idea Idid. See, you go into McDs and ordernine 9-piece boxes, and also ask for fourempty 20-piece cartons. Take the food toyour table, and repack 80 of the 81 piecesinto the 20 piece cartons, then go backto the counter and ask for a refund onfour 20 piece containers. That leaves youwith 1 piece, and so allows you to fill anyorder. If you can talk McDs into buyingback chicken pieces, you convert theproblem into one permitting any inte-ger coefficients, and so you can fill anyorder of size a multiple of the GCD ofthe box sizes.

Dan KalmanAmerican University

More on Nuggets

I was vastly amused by seeing that prob-lem in the most recent issue of FOCUS.I believe I am the source, although I guesssomeone else could have come up withit independently.

It happened 20 years ago, when my sonhad a summer job at the McGill com-puting centre and we walked togetherdowntown. My office was in the samebuilding as he worked. About 3/4 of amile from our house, we pass aMcDonald’s and he mentioned as wepassed by that quite frequently he andhis friends would go in and buy a load ofMcNuggets. He then told me that theycome in boxes of 6 and 9 and buckets of20. One of us (I don’t recall which), thencame up with the question. We answeredit and continued on our way. Then afterhe graduated and was elected to the en-gineering honors society, we started re-ceiving their official magazine, The Bent,which features a puzzle page every issue.Although my son was living elsewhere,the magazine came to our house and Inoticed it.

Eventually I thought to send in thispuzzle and it got duly printed. Since myson graduated in 1988, this was probablyin 1990 or 91. I forgot about it then untilmy other son got a master’s from MITand started receiving their alumni maga-zine Technology Today and saw the exact

same problem in their problem section(which seems to appear in alternate is-sues). It was probably about three yearsago that it appeared. It is not a greatstretch to suppose that some MIT alum-nus had been a member of the honorsociety and seen it The Bent, but forgot-ten its provenance by then. Then I sawin the sci.math newsgroup one day andmade some attempt to determine if The

Bent appearance was the earliest, but noone who responded had ever even seenthat appearance. So, while it would takesome research to find out if it all goesback to my contribution, I can choose tobelieve it.

Michael BarrMcGill University

Correction

In the October issue of FOCUS on page 26 we listed the NAMReception and Banquet as taking place on Friday, January 13, 2006from 6:00 p.m. to 9:30 p.m. The NAM Reception and Banquetwill be held on Saturday January 14, 2006 from 6:00 p.m. to 9:30p.m. On page 27 we listed the AMS Reception and Banquet asbeing held on Saturday, January 14, 2006 from 6:30 p.m. to 10:30p.m. The AMS Reception and Banquet will actually be held onSunday, January 15, 2006 from 6:30 p.m. to 10:30 p.m. We aresorry for the errors.

2006 MAA Membership Renewals

Check the mail for your 2006

MAA Membership renewal notice.

We appreciate your continued

support of the MAA.

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FOCUS November 2005

Merten M. Hasse Prize

Carl B. Allendoerfer AwardFor articles published in Mathematics Magazine

Robert B. Eggleton and William P. Galvin (posthumously)for “Upper Bounds on the Sum of Principal Divisors ofan Integer”Mathematics Magazine, v. 77June 2004, pp. 190-200

Professor Eggleton was unable to attend.

MAA Prizes and Awards Announced at MathFest 2005

Robert L. Devaneyfor “Chaos Rules!”Math HorizonsNovember 2004, pp. 11-14.

George Pólya AwardFor articles published in the College Mathematics JournalThere were two Pólya Awards this year:

Stephen M. Walk

for “Mind Your ∀ ∃s and s ”

College Mathematics Journal, v. 35November 2004, pp. 362-369.

Brian Hopkins andRobin J. Wilsonfor “The Truth About Königsberg”College Mathematics Journal, v. 35May 2004, pp. 198-207.

Brian Hopkins

The awarding of prizes for expository writing has long beenone of the high points of MathFest. The Alder Award, foryounger teachers, has added a prize for distinguished teachingto the annual MathFest awards ceremony. These pages list theawards offered at MathFest; more details, including full cita-tions and responses, can be found online at http://www.maa.org/news/080105mfwa.html.

MAA President Carl Cowen presides over the awardsceremony at MathFest.

Maureen Carroll and Stephen Doughertyfor “Tic-Tac-Toe on a Finite Plane”Mathematics Magazine, v. 77, no.4October 2004, pp. 260-274.

Trevor Evans AwardFor articles published in Math Horizons.

For a noteworthy expository paper appearing in an MAA pub-lication, at least one of whose authors is a younger mathemati-cian

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November 2005

Henry L. Alder Award for Distinguished Teachingby a Beginning Collegeor University Mathematics Faculty MemberThe Alder Awards honor beginning college or university fac-ulty whose teaching has been extraordinarily successful andwhose work is known to have had influence beyond their ownclassrooms. This years winners are:

Steven Finch and John Wetzelfor “Lost in a Forest”The American Mathematical Monthly, v. 111, no.8October 2004, pp. 645-654.

Professors Finch and Wetzel were unable to attend.

Judith Grabinerfor “Newton, Maclaurin, and the Authority of Mathematics”The American Mathematical Monthly, v. 111, no.10December 2004, pp. 841-852.

Professor Grabiner was unable to attend.

Chauvenet Prize for Expository WritingThe Chauvenet Prize is given for an outstanding expositoryarticle on a mathematical topic by a member of the Associa-tion.

Matthew DeLongTaylor University

Sarah GreenwaldAppalachian State

University

Laura TaalmanJames Madison

University

Lester R. Ford AwardFor articles published in The American Mathematical MonthlyThere were five Ford Awards this year:

Tom Apostol and MamikonMnatsakanianfor a series of three articles:

“Isoperimetric and IsoparametricProblems,” The American Math-ematical Monthly, v. 111, no.2,February 2004, pp. 118-136.

“A Fresh Look at the Method ofArchimedes,” The AmericanMathematical Monthly, v. 111, no.6June/July 2004, pp. 496-508.

“Figures Circumscribing Circles,”The American Mathematical Monthlyv. 111, no.10, December 2004, pp.853-863.

Henry Cohnfor “Projective Geometry over 1and theGaussian Binomial Coefficients” TheAmerican Mathematical Monthly, v. 111,no.6, June/July 2004, pp. 487-495.

Mamikon Mnatsakanian

Alan Edelman

Alan Edelman and Gilbert Strangfor “Pascal Matrices”The American Mathematical Monthlyv. 111, no.3, March 2004, pp. 361-385.

John Stillwellfor “The Story of the 120-Cell”Notices of the AMS, January 2001,pp. 17-24.

26

FOCUS November 2005

MathFest in Albuquerque marked thebeginning of a new era for the MAA. Forthe first time, our in-house meetings staffmanaged the entire summer meeting.Associate Secretary Jim Tattersall; Direc-tor of Membership, Marketing and Meet-ings Jim Gandorf; Director of Programsand Services Michael Pearson, and theirstaff did a great job. Knoxville MathFestin 2006 will be even better!

The MAA and all of its members shouldcongratulate the newly elected officers ofthe MAA. They will take office after theJanuary 2006 Joint Mathematics Meet-ings: Joe Gallian, President-Elect, CarlPomerance, First Vice-President, andDeanna Haunsperger, Second Vice-Presi-dent.

The Association is currently engaged ina strategic planning process. First Vice-President Barbara Faires heads the plan-

ning effort. We are in the midst of exam-ining three important areas: Revenue, theAmerican Mathematics Competitions,and Professional Development. WorkingGroups, appointed by the President andchaired by Barbara Faires, Frank Farris,

and Nancy Hagelgans, respectively, aremeeting with our staff with a timetablethat requires final reports next Novem-ber. At its meeting on August 3, theBoard of Governors voted that the nextthree areas of study should be Mem-bership, Students, and Governance.Working Groups will be formedshortly..In other Board action, the Board ap-proved a revision of the short versionof the mission statement to be postedonline and in other MAA documents.The old statement said: The Math-ematical Association of America is thelargest professional society that fo-cuses on undergraduate mathematicseducation. The new statement says: TheMathematical Association of Americais the largest professional society thatfocuses on mathematics accessible atthe undergraduate level.

The Board welcomed new SectionGovernors: John Bukowski of the Allegh-eny Mountain Section, Richard Gillmanof the Indiana Section, Dan Curtin of theKentucky Section, John A. Winn, Jr. ofthe Metro New York Section, Christo-pher Masters of the Nebraska–SE SouthDakota Section, Kathleen Hamm of theNorthern California Section, FrederickWorth of the Oklahoma–Arkansas Sec-tion, Jane Arledge of the Rocky Moun-tain Section, John Koker of the Wiscon-sin Section. It also thanked the outgoinggovernors: Donald M. Platte, Roger B.Nelson, Rodger Hammons, Raymond N.Greenwell, John Fuelberth, Leonard F.Klosinski, Lisa Mantini, Hortensia Soto-Johnson, and Richard Poss, respectively.

Governors-at-Large were elected by theBoard to begin a three-year term begin-ning after the January 2006 Joint Math-ematics Meetings: Jeremy Kilpatrick ofthe University of Georgia was electedGovernor-at-Large Representing TeacherEducation, replacing Joan Ferrini-Mundy; Peter Stanek, Lockheed (ret.),was elected Governor-at-Large Repre-

senting Mathematicians OutsideAcademia, replacing Peter DeLong.

Future meetings of the Association, ap-proved by the Board of Governors canbe found on the MAA web site. TheBoard approved two additional onespending successful contract negotiations:the 2012 Joint Mathematics Meetings, tobe held in Boston, Massachusetts, onJanuary 4–7 (Wednesday-Saturday), andthe 2013 Joint Mathematics Meetings, tobe held in San Diego, California, Janu-ary 9–12 (Wednesday-Saturday).

Daniel Maki was reelected to a four-yearterm on the Audit and Budget Commit-tees. Lowell Beineke, Editor of The Col-lege Mathematics Journal, was elected toserve a two-year term on the ExecutiveCommittee representing publications.Nancy Hagelgans was reelected to athree-year term as Chair of the Commit-tee on Sections. As provided in the By-laws, she will serve another three-yearterm on the Board of Governors and on

MAA Business at MathFest 2005

By Martha Siegel, Secretary of the Association

Joe Gallian, MAA President-Elect

Deanna Haunsperger newly elected SecondVice-President.

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November 2005

the Executive Committee of the Associa-tion.

Allen Schwenk of Western MichiganUniversity was elected by the Board toserve as Editor-Elect of MathematicsMagazine from January 1, 2005 to De-cember 31, 2005. He will assume the re-sponsibilities of Editor on January 1,2006. The Board approved the incom-ing editorial board of the Magazine.

The Board elected Daniel Velleman ofAmherst College as Editor-Elect of TheAmerican Mathematical Monthly, to suc-ceed its current Editor, Bruce Palka. TheEditor-Elect begins his term on January1, 2006 and will become editor in 2007,for a term of five years.

The Board approved Meritorious ServiceAwards for six sections and selected threeoutstanding teachers as winners of the2006 Deborah and Franklin TepperHaimo Awards. The Board also approvedthe winner of the Gung-Hu Award forDistinguished Service to Mathematics.These honors will be presented at theJoint Prize Session in January 2006. Seepage 3 for more on the Haimo Awardwinners.

Francis Edward Su of Harvey Mudd Col-lege was elected James R. C. Leitzel Lec-turer for MathFest 2006, and Tim Gowersof Cambridge University was electedEarle Raymond Hedrick Lecturer forMathFest 2006. Bernd Sturmfels of Uni-versity of California Berkeley was electedPólya Lecturer for 2005–06 and 2006–07.President Cowen thanked the outgoingPólya Lecturer, Martin Isaacs of the Uni-versity of Wisconsin Madison.

The Board approved the regulations forthe MAA’s David P. Robbins Prize in Al-gebra, Combinatorics, and DiscreteMathematics. The Board also revised eli-gibility guidelines for the Henry L. Alderand the Deborah and Franklin TepperHaimo Awards. As soon as final revisionshave been made, the regulations will beposted on the MAA web site.

In addition, the Board approved changesin the section bylaws of the Allegheny,Iowa, and Kansas Sections, and approved

a change in the MAA Bylaws Article VI,Section 2 and discussed a correspondingchange in the “model” Section Bylaws. Asrequired by the Bylaws themselves, theBylaws change is presented in detail inthis issue of FOCUS (on page 12) andwill be voted on at the Business Meetingof the MAA to be held during the Janu-ary Joint Meetings.

MAA grants and program activities con-tinue to increase. Ideas come from ouractive member-volunteers who serve oncommittees and from the very able staff.We have more than 10 projects currentlyactive, including our PREP workshopsand are being funded for many programsand activities including a large assess-ment project and undergraduate re-search centers and conferences.

Our bottom line this year is an operat-ing surplus of almost $600,000. The com-pleted 2004 audit was very positive, ac-knowledging the great improvement inAssociation finances and financial pro-cedures in recent years. The total incomefrom publications in 2005 seems to beon target, thanks to the efforts of ourPublications staff and our fine group ofmember-Editors. We cannot rest on theselaurels, though. External funding is likelyto be more competitive, especially in the

collegiate mathematics area, so we con-tinue to try to increase revenue and con-trol spending.

The Carriage House Conference Center,funded with a generous gift from Pauland Virginia Halmos, is progressingwell. The renovation of the historicbuilding behind 1529 Eighteenth Streetis now in the hands of our architect and,pending permits, we are about to beginthe construction. Active ProgrammingCommittees for the Center are planningfor the future now.

One of the most ambitious projects ofthe MAA now in the implementationphase is the CUPM study of the under-graduate mathematics curriculum origi-nally led by chair Harriet Pollatsek.David Bressoud, the new chair of CUPMhas arranged many opportunities forMAA members to hear about the imple-mentation of the recommendations,

which have Board approval. Reference tothe CUPM page will lead you to the Il-lustrative Resources that are being devel-oped by Susanna Epp and her commit-tee to accompany this effort.

The Third MAA Mathematical Studytour was a great success in its visit toMexico this spring. Check MAA Onlineon September 1 for the announcementof the 2006 trip — June 6- 21 to China.

The array of programs, publications, ser-vices, and activity in the MAA is mind-boggling. We thank the staff and themany member-volunteers who make itall happen! Hope to see everyone in SanAntonio in January.

JimTattersall, MAA Associate Secretary

Have You Moved?

The MAA makes it easy to change youraddress. Please inform the MAA ServiceCenter about your change of address byusing the electronic combined mem-bership list at MAA Online (http://www.maa.org) or call (800) 331-1622,fax (301) 206-9789, email:maaservice@maa.org, or mail to theMAA, PO Box 90973, Washington, DC20090.

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FOCUS November 2005

MathFest 2005 in Pictures

Carl Cowen welcomes all at theOpening Banquet.

Statues near one of the entrances tothe hotel.

Bob Devaney talked about iteratingthe exponential map.

Main entrance of the AlbuquerqueConvention Center.

MAA staff at the Los Amigos Roundup:Jurgita Schwan, Jim Gandorf, RichHamilton, Lisa Kolbe, and TinaStraley.

MAA President Carl Cowen intro-duces the Hedrick Lectures.

Packed escalator in the Conven-tion Center, with rainbows pro-vided by the suspended prisms.

Don Albers, Richard Guy, and Jack Graverbefore the Silver and Gold Banquet.

Suspended prisms in the Convention Center.

Lowell Beineke buys someMAA gear for his grand-daughter.

Joe Gallian lectures at the OpeningBanquet.

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November 2005

President-elect Joe Gallian and ColmMulcahy.

The activities table: fun for ev-eryone.

A pensive Joe Gallian at theMAA Business Meeting.

Harold Reiter and Richard Anderson at thePresident’s Reception at the NationalAtomic Museum.

MAA Associate Secretaries, past andpresent: Jim Tattersall, Ken Ross, and DonVan Osdol.

Hedrick Lecturer Jeff Lagarias does his thing.

Jeff Lagarias’ co-authors and their fates afterthe dot-com bubble.

After hours: Bev Ruedi, Lisa Kolbe, andElaine Pedreira of the MAA headquar-ters staff.

The Sudoku craze had arrived in Al-buquerque.

Carol Baxter, managing editor ofFOCUS, did her part at the regis-tration desk.

Nathaniel Dean introduces BlackwellLecturer Leona H. Clark.

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FOCUS November 2005

Lisa Kolbe sells MAA gear.

Twenty-five year members of the MAA atthe Silver and Gold Banquet.

Math and Science T-shirts on sale.

Wood Mobius exhibited Möbius strips; seehttp://woodmobius.com.

A particularly important message.

Ken and Ruth Ross preparing to lift off.

Three generations ofBeinekes: Lowell, Jennifer,and Anna Rose.

Gizem Karaali (UCSB) and MichaelMcJilton (College of the Desert).

The Man in Charge:Jim Gandorf, MAADirector of Member-ship, Marketing, andMeetings.

Todd Shayler, Carl Erickson, andNathaniel Watson, all participants inthis year’s SMALL program at Will-iams College.

Did she attend any of the talks?

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November 2005

El Editor

The eInstruction exhibit displayed inno-vative educational software.

The mathematicians arethataway.

At the Board of Governors meeting, some of the Powerful Ones:Second Vice-President Jean Bee Chan, Secretary Martha Siegel,President Carl Cowen, Executive Director Tina Straley, and Parlia-mentarian Wayne Roberts.

Fifty-year members of the MAA at the Silver and GoldBanquet.

Water was essential foreveryone.

Madilyn Magno, daughter of MAAstaffer Gretchen Magno, models MAAbabywear.

Don Albers and Beverly Ruedi ofthe MAA Publications Depart-ment.

Checking email could be done inside the exhibit area.

Bob Devaney explains about topologists.

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FOCUS November 2005

Supporting Assessment in Undergraduate Mathematics

A Special Report

Driven by quests for more coherencein the curriculum and more accountabil-ity for learning produced, during the lastquarter century assessment has becomemuch more prominent in U.S. highereducation. Most of the initial pressurefor assessment came from sources exter-nal to the faculty — from accreditingagencies, governing boards, and univer-sity administrations. Thus early assess-ment work by disciplinary faculties waslargely motivated by the need to respondto these external pressures. Nonetheless,the underlying goal of improving coursesand academic programs so that studentlearning increased is of paramount im-portance. This has been the guidingprinciple of MAA’s support of assessmentin undergraduate mathematics over thepast fifteen years.

MAA’s support of assessment has con-sisted of three major components,roughly in five-year intervals during thefifteen years, 1990-2005. This special re-port concerns the third of those compo-nents, which was largely underwritten bya 2001 grant from National ScienceFoundation to support a project entitledSupporting Assessment in Undergradu-ate Mathematics (SAUM).

MAA’s work in assessment began in 1990with the appointment of the CUPM Sub-committee on Assessment. In 1995 theSubcommittee issued guidelines for de-partments to use in establishing a cycleof assessment aimed at program im-provement (CUPM, 1995). The secondcomponent included use of these guide-lines in mini-courses and other aware-ness activities, culminating in 1999 withpublication of Assessment Practices inUndergraduate Mathematics, MAA Notes#49 (Gold, et al.,1999). This volume con-tains over seventy case studies of assess-ment in undergraduate mathematics andincludes CUPM’s 1995 assessment guide-lines in an appendix. Prefaces to thisvolume offer additional context forMAA’s support of assessment through-out the 1990s.

In the initial phase of the SAUM project,copies of Assessment Practices in Under-graduate Mathematics were sent to everycollege and university mathematics de-partment in the United States. In subse-quent years project personnel led forumson assessment at meetings of MAA’s re-gional sections and arranged four seriesof multi-session workshops to supportassessment efforts on individual cam-puses. These workshops brought to-gether teams of faculty to develop theirown assessment plans and to share bothprogress and impediments with col-leagues from other institutions. Casestudies emerging from these workshops,reviewed and edited by project staff, havenow been published by the MAA bothon-line and as a printed volume (Steen,et al., 2005). Other activities of the SAUMproject included a mini-courses, postersessions, invited and contributed papersessions, an “assessment reception,” anda website (http://www.maa.org/saum/)which MAA will maintain as a perma-nent Internet home for support of assess-ment in undergraduate mathematics.

This special section of FOCUS containsadaptations of three essays that introducethe SAUM case studies. Pull-quotes con-tain observations derived by project lead-ers from the case studies themselves.

ReferencesCommittee on the Undergraduate Pro-

gram in Mathematics. (1995) “Assess-ment of Student Learning for Improv-ing the Undergraduate Major in Math-ematics,” FOCUS 15(3) (June, 1995) 24-28. Reprinted in Assessment Practices inUndergraduate Mathematics, BonnieGold, et al., eds. Washington, DC: Math-ematical Association of America, 1999, p.279-284. http://www.maa.org/saum/maanotes49/279.html

Gold, Bonnie, Sandra Z. Keith, and Will-iam A. Marion (1999). Assessment Prac-tices in Undergraduate Mathematics.MAA Notes No. 49. Washington, DC:Mathematical Assoc. of America.www.maa.org/saum/maanotes49/index.html

Steen, Lynn Arthur, et al. (2005) Support-ing Assessment in Undergraduate Math-ematics: Case Studies from a NationalProject. Washington, DC: MathematicalAssociation of America.

SAUM PersonnelThe SAUM project was directed by Ber-nard L. Madison, Professor of Mathemat-ics at the University of Arkansas with theassistance of Bonnie Gold (MonmouthUniversity), William E. Haver (VirginiaCommonwealth University), LaurieHopkins (Columbia College), RichardJardine (Keene State College), Sandra Z.Keith (St. Cloud State University), WilliamA. Marion, Jr. (Valparaiso University), andLynn A. Steen (St. Olaf College). MAA As-sociate Executive Directors Thomas W.Rishel and Michael Pearson managed theproject, while Peter Ewell, vice presidentof the National Center for Higher Educa-tion Management Systems (NCHEMS)served as project evaluator.

A complimentary copy of the new SAUMreport is available to the first 1000 depart-ments submitting requests through theproject website http://www.maa.org/saum,where you can also access the full text of thereport.

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November 2005

Tensions and TethersAssessing Learning in Undergraduate Mathematics

By Bernard L. Madison

In 2001, after a decade of encouragingand supporting comprehensive assess-ment of learning in undergraduatemathematics, the Mathematical Associa-tion of America (MAA) was well posi-tioned to seize an opportunity for fund-ing from the National Science Founda-tion (NSF) to intensify and extend thissupport. As a result, NSF awarded MAAa half-million dollars for a three-yearproject “Supporting Assessment in Un-dergraduate Mathematics” (SAUM) thatprovided a much-needed stimulus for as-sessment at the departmental level. The

need for such a program is rooted in thevarious and often conflicting views of as-sessment stemming from worry aboutuses of the results, difficulties and com-plexities of the work, and possible con-flicts with traditional practices. Facultynavigating through these views to de-velop effective assessment programs en-counter numerous tensions between al-ternative routes and limiting tethers thatrestrict options. Against this backgroundthe MAA launched SAUM in January2002.

The goal of SAUM was to encourage andsupport faculty in designing and imple-menting effective programs of assess-ment of student learning in some cur-ricular block of undergraduate math-ematics. SAUM leaders were reasonablysure that many faculty would welcomehelp with assessment because many col-leges and universities were under man-dates to develop and implement pro-grams to assess student learning—man-dates originating in most cases from ex-ternal entities such as regional accredit-

Case Studies

Developmental, Quantitative Literacy,and Precalculus Programs

Allegheny College. Assessing IntroductoryCalculus and Precalculus Courses.

Arapahoe Community College. Math-ematics Assessment in the First Two Years.

Arizona Western College. Using Assess-ment to Troubleshoot and Improve Devel-opmental Mathematics.

Cloud County Community College.Questions About College Algebra.

Mount Mary College. Assessing the Gen-eral Education Mathematics Courses at aLiberal Arts College for Women.

Portland State University. Assessment ofQuantitative Reasoning in Applied Psy-chology.

Portland State University. AssessingQuantitative Literacy Needs Across theUniversity.

San Jose State University. Precalculus inTransition: A Preliminary Report.

Virginia Commonwealth University. AnAssessment of General Education Math-ematics Courses’ Contribution to Quan-titative Literacy.

Mathematics-Intensive Programs.

North Dakota State University. Devel-oping a Departmental Assessment Pro-gram.

United States Military Academy. Assess-ing the Use of Technology and Using Tech-nology to Assess.

Virginia Tech. A Comprehensive Assess-ment Program—Three Years Later.

Weill Cornell Medical College, Qatar.Assessment of a New American Programin the Middle East.

Mathematics Programs to Prepare Fu-ture Teachers.

Monmouth University. Assessment in aMiddle School Mathematics TeacherPreparation Program.

University of Texas at Brownsville andTexas Southernmost College. Using Prac-tice Tests in Assessment of Teacher Prepa-ration Programs.

The Undergraduate Major in Math-ematics.

American University. Learning OutcomesAssessment: Stimulating Faculty Involve-ment Rather Than Dismay.

Colorado School of Mines. The Develop-ment, Implementation, and Revision of aDepartmental Assessment Plan.

Keene State College. Assessing StudentOral Presentation of Mathematics.

Point Loma Nazerene University. Keep-ing Assessment Simple.

Portland State University. SurveyingMajors in Developing a Capstone Course.

Saint Mary’s University of Minnesota.Assessing Written and Oral Communica-tion of Senior Projects.

Saint Peter’s College. Assessing the Math-ematics Major: A Multifaceted Approach.

South Dakota State University. Assess-ing the Mathematics Major Through aSenior Seminar.

University of Arkansas, Little Rock. As-sessment of Bachelors’ Degree Programs.

University of Nevada, Reno. Assessmentof the Undergraduate Major Without Fac-ulty Buy-in.

Washburn University. Assessing theMathematics Major With a Bottom-UpApproach.

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ing bodies. Our expectations were accu-rate. We found many faculty willing totackle assessment but unenthusiastic andeven skeptical about the work.

During the three years of SAUMwe promoted assessment to hun-dreds of faculty in professionalforums and worked directly with68 teams of faculty from 66 col-leges or universities in SAUMworkshops. The final SAUMworkshop—restricted to assess-ing learning in the major—con-cludes in January 2006. Most ofthe 68 teams had two or threemembers, with two usually at-tending the workshop sessions.As these teams worked face-to-face at the workshop sessions, asthey continued their work backhome, and as we promoted as-sessment to the larger audiencesin professional forums, skepti-cism was evident in lack of enthusiasmand inevitably brought forth argumentsagainst assessment as we were advocat-ing it.

The arguments were basically of twotypes: tensions and tethers. I use tensionshere as forces that mitigate against mean-ingful and effective assessment, pullingtoward easier and less effective models.A common example is the tension be-tween doing assessment that is effectivein plumbing the depths of student un-derstanding and doing assessment thatis practical and more superficial. Mosttethers are ties to past and present prac-tices that are likely to continue and pos-sibly prevent or restrict developing effec-tive assessment. For example, many in-structional programs are tied to tradi-tional in-course testing and have noplans to change, placing significant lim-its on assessment.

Below, I describe some of these assess-ment tensions and tethers, along withsome ways SAUM tried to ease the ten-sions and untie the tethers. First, how-ever, I will explore SAUM retrospectivelyand describe how it evolved from a de-cade of assessment activity by the MAA.

From Awareness to Ownership

The SAUM proposal to NSF was basedon an unarticulated progression of stepsnecessary to get college and university

faculty fully committed to meaningfuland effective assessment of student learn-ing. The first step is awareness, the sec-ond, acceptance; next comes engagement,and finally ownership.

First, we aimed to make faculty aware ofthe nature and value of assessment bystimulating thought and discussion. Sec-ond, we encouraged acceptance throughknowledgeable and respected plenaryspeakers at workshops, and collegial in-teraction with others interested in andsometimes experienced in assessment.Examples of the plenary presentations,documented on the SAUM website (http://www.maa.org/saum/) are presentationsand writings by Lynn Steen and PeterEwell. Their combined overview of howassessment is positioned in the largerarena of federal, state, and universitypolicies and practices can be surmisedfrom their article The Four A’s: Account-ability, Accreditation, Assessment, and Ar-ticulation (Ewell & Steen, 2003). This ar-ticle is based on a presentation by PeterEwell at the joint session of Workshops#1 and #2 at Towson University in Janu-ary 2003.

Peter Ewell was an unexpected and valu-able resource at workshops, giving ple-nary presentations and generously agree-ing to consult with individual teams. His

broad historical perspective, vast expe-rience in consulting with and advisingcolleges and universities, and intimateknowledge of policies of accrediting bod-ies gave teams both encouragement and

helpful advice. Further,Peter’s view as a non-math-ematician was helpful bothfor his questioning and hisknowledge of other disci-plines. Peter’s expertise wasnicely complemented by LynnSteen’s wide experience withmathematics, mathematicseducation, and mathematicsand science policy issues.

Third, we urged workshopparticipants to engage in de-signing and implementing anassessment program at theirhome institutions. Face-to-face workshop sessions re-quired exit tickets that were

plans for actions until the next face-to-face session. Teams presented these plansto their workshop colleagues and then re-ported at the next session on what hadbeen done. As noted by Peter Ewell inhis evaluator’s report (see below), thisstrategy provided strong incentive forparticipants to make progress at theirown institutions so that they would havesomething to report at the next sessionof the workshop.

Finally, we promoted ownership by re-quiring that each team write a case studydescribing its assessment program orpresent a paper or poster at a professionalmeeting. Paper sessions were sponsoredby SAUM at MathFest 2003 in Boulder,Colorado and at the 2004 Joint Math-ematics Meetings in Phoenix, Arizona.SAUM also sponsored a poster session atPhoenix. The case studies have been pub-lished in the new MAA report Support-ing Assessment in Undergraduate Math-ematics: Case Studies from a NationalProject (Steen et al., 2005). An additionalpaper session is scheduled for the 2006Joint Mathematics Meetings in San An-tonio, Texas.

Background for SAUM

SAUM’s background goes back to anMAA long-range planning meeting in

SAUM Workshop: January was a good time to be in Phoenix.

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the late 1980s. At that meeting I askedwhat the MAA was going to do regard-ing the growing movement on assess-ment that had entered the US highereducation scene about a decade earlier.Indicative of the fact that no plans hadbeen made by MAA, I soon found my-self as chair of the 12-member Subcom-mittee on Assessment of MAA’s Commit-

tee on the Undergraduate Program inMathematics (CUPM). We were chargedwith advising MAA members on policiesand procedures for assessment of learn-ing in the undergraduate major for thepurpose of program improvement. Veryfew of the subcommittee members hadany experience in or knowledge of thekind of assessment we would eventuallyunderstand that we needed, and westruggled with the multiple meaningsand connotations of the vocabulary sur-rounding the assessment movement.Nevertheless, we plowed into our workat the summer meeting in Columbus,Ohio, in 1990.

In retrospect, our work developed inthree distinct phases: (1) understandingthe assessment landscape that includedoutspoken opposition to assessment; (2)developing guidelines for assessment;and (3) compiling case studies of assess-ment programs in mathematics depart-ments. A fourth phase, seen in retro-spect, was the extensive faculty awarenessand professional development made pos-sible by SAUM.

Two vehicles proved very helpful in Phase1. First, in 1991, I moderated an e-maildiscussion on assessment among four-

teen academics (twelve mathematiciansand two non-mathematicians) that in-cluded four members of the AssessmentSubcommittee. Some of the discussantswere opposed to assessment as it wasthen evolving; their worries ranged fromoperational issues like extra work to fun-damental issues like academic freedom.E-mail was neither user-friendly norregularly read in 1991, and managing theinformation flow and compiling it intoa coherent report was quite challenging.Nonetheless, a report was written andpublished in Heeding the Call for Change(Madison, 1992), edited by Lynn Steen,who had been both helpful and encour-aging on my involvement with assess-ment.

Appended to the report of the 1991 e-mail discussion is a reprint of a seminalarticle by Grant Wiggins consisting of thetext of his 1990 keynote address to theassessment conference of the AmericanAssociation for Higher Education(Wiggins, 1992). This annual conferencebegan in 1985 and over the past two de-cades has been the premier conveningevent on assessment in higher education.Between 1990 and 1995, I attended theseconferences, learned about assessmentoutside mathematics, and eventuallymastered the language. Plenary speak-ers such as Wiggins, Patricia Cross, andPeter Ewell were impressive in their ar-ticulate command of such a large aca-demic landscape.

Phase 2 of the work of the AssessmentSubcommittee consisted of producing adocument on assessment that wouldboth encourage assessment and guidedepartment faculties in their efforts todesign and implement assessment pro-grams. Grounded largely in the e-maildiscussion and a couple of AAHE assess-ment conferences, I forged a first draftof guidelines that was based on assess-ment as a cycle that eventually wouldhave five stages before it repeated. By1993 the Subcommittee had a draft readyto circulate for comment. Aside from be-ing viewed as simplistic by some becauseof inattention to research on learning, theguidelines were well received and CUPMapproved them in January 1995 (CUPM,1995).

Further plans of the Subcommittee in-cluded gathering case studies as examplesto guide others in developing assessmentprograms. The small number of contri-butions to two contributed paper ses-sions that the Subcommittee had spon-sored did not bode well for collectingcase studies, especially on assessment oflearning in the major. However, stronginterest and enrollment in mini-courseson assessment indicated that case stud-ies might soon be available. One of theSubcommittee members, WilliamMarion, had expressed interest in team-ing up with Bonnie Gold and SandraKeith to gather and edit case studies onmore general assessment of learning inundergraduate mathematics. By agree-ing to help these three, I saw the work ofthe Subcommittee as essentially finishedand recommended that we be dis-charged. The Subcommittee was dis-solved, and in 1999 Assessment Practicesin Undergraduate Mathematics contain-ing seventy-two case studies was pub-lished as MAA Notes No. 49, with Gold,Keith, and Marion as editors (Gold, et al.,1999).

Two years later, in 2001, NSF announcedthe first solicitation of proposals in thenew Assessment of Student Achievementprogram. During two weeks in May 2001while I was serving as Visiting Mathema-tician at MAA, with help and encourage-

ment from Thomas Rishel, and with theencouragement and advice from mem-bers of CUPM, most notably WilliamHaver, I wrote the proposal for SAUM. Iwas fortunate to gather together a teamfor SAUM that included the principals

Although most programs ex-pect to collect data that con-firms their current practice,even excellent programs havethe potential to improve. Bydesigning assessments mostlikely to identify areas whereimprovement is appropriate,programs can continue to getbetter. Students ultimately ben-efit from this approach.

—Laurie Hopkins

Increasingly, accrediting agenciessuch as ABET (engineering) andNCATE (teacher education) are lessinterested in the mathematicscourses students have taken or thecontent they have studied than instudents’ demonstration of math-ematical performance. “Demon-strated proficiency” is rapidly re-placing named courses.

—Dick Jardine

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in MAA’s decade of work on assessment:Bonnie Gold, Sandra Keith, WilliamMarion, Lynn Steen, and myself. Goodfortune continued when William Haveragreed to direct the SAUM workshopsand Peter Ewell agreed to serve asSAUM’s evaluator.

In August 2001 I learned that the NSFwas likely to fund SAUM for the re-quested period, January 1, 2002, to De-cember 31, 2004, at the requested bud-get of $500,000. Because we were rea-sonably sure of an award, we were ableto begin work early, in effect extendingthe period of the project by severalmonths. The award was made official(DUE 0127694) in fall 2001.

The 1995 CUPM Guidelines on Assess-ment are reprinted as an appendix toAssessment Practices in UndergraduateMathematics (CUPM, 1995) and an ac-count of MAA’s work on assessment isin the foreword (Madison, 1999). Alighter account of my views on encoun-tering and understanding assessment,“Assessment: The Burden of a Name,” canbe found on the website of Project Ka-leidoscope (Madison, 2002).

Tensions and Tethers

As noted above, throughout SAUM andthe MAA’s assessment work that pre-ceded SAUM, various tensions and teth-ers slowed progress and prompted long

discussions, some of which were helpful.Some faculty teams were able to ease orcircumvent the tensions while others stillstruggle with the opposing forces. Like-

wise, some were able to free their pro-gram of the restraints of certain tethers,while others developed programs withinthe range allowed.

A major obstacle to negotiating thesetensions and tethers is the lack of docu-mented success stories for assessmentprograms. Very few programs have gonethrough the assessment cycle multipletimes and used the results to make

changes that result in increased studentlearning. This absence of success storiesrequires that faculty work on assessmentbe based either on faith or on a sense ofduty to satisfy a mandate. In theory, theassessment cycle makes sense, but imple-mentation is fraught with possibilities fordifficulties and minimal returns. Thereis, thus, considerable appeal to yield totensions—to do less work or to workonly within the bounds determined by atether to traditional practice.

Easing Tensions

The most prominent tension in assess-ment is between what is practical andwhat is effective in judging student per-formance and understanding. There areseveral reasons for this, some of whichinvolve other tensions. Multiple-choice,machine-scored tests are practical butnot effective in probing the edges anddepths of student understanding or fordisplaying thought processes or miscon-ceptions. Student interviews and open-ended free-response items appear to bemore effective in this probing, but are notpractical with large numbers of students.We know too little about what is effec-

tive and what the practical methods mea-sure, but we believe that getting studentsto “think aloud” is revealing of how theylearn. Unable to see evidence of value inthe hard work of effective assessment, wevery often rely on the results of practicalmethods—believing that we are measur-ing similar or highly correlated con-structs.

To ease this tension between the practi-cal and the “impractical,” we recom-mended that faculty start small and groweffective methods slowly. Interviewing arepresentative sample of students is re-vealing; comparing the results of theseinterviews with the results of practicalmethods can provide valuable informa-tion. Knowing how students learn caninform assessment in an essential andpowerful way. We know too little abouthow mathematical concepts are learned,especially in a developmental fashion,and we know too little about how assess-ment influences instruction. This is bothan impediment to doing assessment anda challenging reason for doing it. Onecan use it as an excuse for waiting untilwe know more about learning, or one canmove ahead guided by experience butalert to evidence of how learning is oc-curring and how learning and assessmentare interconnected.

Many assessment programs are the re-sult of requirements by accrediting agen-cies or associations. Often these require-ments boil down to applying three orfour tools to measure student learningoutcomes for majors and for general edu-cation. For example, the tools for a ma-jor could be a capstone course, exit in-

Goals for courses below calculus of-ten include some broad skills (e.g.to read a newspaper critically) andsome effective outcomes (e.g. to feelless mathophobic). Partner disci-plines and committees on generaleducation can provide useful inputas mathematics departments definethese course goals. Concrete learn-ing objectives can be developedeven for effective goals.

—Bonnie Gold

One lesson learned by SAUM par-ticipants is to not overwhelm the as-sessment program with too manyprocedures or data. Keep assess-ment simple. To do that it is oftenprudent to proceed in stages. For ex-ample, choose just one goal to pi-lot test the assessment process. Ana-lyzing results in one area can serveas a prototype for more comprehen-sive assessment in the future.

—William Marion

One theme that arises repeatedlyin assessment programs is theneed to account for the cultureof the host institution. Assessmentcan be conducted more readilyif it is part of institutional culture.Failing to account for the culturalenvironment is a common causeof reduced effectiveness of as-sessment efforts.

—Dick Jardine

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terviews, and an end-of-program com-prehensive examination. Consequently,discipline faculties can meet the require-ments by doing minimal work—desig-nating a capstone course, interviewinggraduating seniors, and selecting an off-the-shelf major field achievement test—and getting minimal benefits.

Reflecting on the results of the assess-ment and considering responses such asprogram or advising changes requiresmore work and raises questions aboutpast practices. This tension between get-ting by with minimal work for minimalpayoff and probing deeply to expose possi-bly intractable problems does offer pauseto faculty whose time is easily allocatedto other valued work.

The tension pulling toward meetingmandated assessment requirementsminimally is reinforced by the bad repu-tation that assessment has among facul-ties. This reputation derives both fromworries about uses of assessment resultsfor accountability decisions and fromnumerous reports of badly designed andpoorly implemented large-scale highstakes assessments. Unfortunately, fewpeople understand the broad assessmentlandscape well enough to help facultyunderstand that their assessment workhas educational value that is largely in-dependent of the public issues that areoften used to discredit assessment. For-tunately, in SAUM we did have peoplewho understood this landscape andcould communicate it to mathematicsfaculty.

Mathematicians are confident of theirdisciplinary knowledge and generallyagree on the validity of research results.However, their research paradigm of rea-soning logically from a set of axioms andprior research results is not the empiri-cal methodology of educational practicewhere assessment resides. This tensionbetween ways of knowing in very differ-ent disciplines often generates disagree-ments that prompt further evidencegathering and caution in drawing infer-ences from assessment evidence. Even-tually, though, decisions have to be madewithout airtight proof.

This tension is amplified by the complex-ity of the whole assessment landscape.For example, the so-called three pillarsof assessment—observation, interpreta-tion, and cognition—encompass wholedisciplines such as psychometrics andlearning theory (NRC, 2001).

Assessment of learning in a coherentblock of courses often provides informa-tion that can be used to compare learn-ing in individual courses or in sectionsof a single course, and hence to judgecourse and instructor effectiveness. Suchcomparisons and judgments create ten-sion between individual faculty member’sacademic freedom and the larger interestof programs. Indeed, learning goals for ablock of courses do place restrictions onthe content of courses within the block.

Mathematics faculty members are accus-tomed to formulating learning goals interms of mathematical knowledge ratherthan in terms of student performance inusing mathematics. This creates tensionbetween testing what students know andtesting for what students can do. Sincejudging student performance is usuallyfar more complex than testing for spe-cific content knowledge, this tension isclosely related to that between practicalversus effective tension discussed above.

Partly because of the nature of math-ematical knowledge, many instructionalprograms have not gathered empirical

evidence of what affects student learn-ing. Rather, anecdotal information—of-ten based on many years of experiencewith hundreds of students—holds sway,indicative of the tension between a cul-ture of evidence and a culture of anecdotalexperience. Since empirical evidence isoften inconclusive, intuition and expe-rience will be valuable, even more sowhen bolstered by evidence.

Untying Tethers

Mathematics programs in colleges anduniversities are very tradition-bound,and many of these traditions workagainst effective assessment of studentlearning. Sometimes, these tethers canbe untied or loosened; sometimes theycannot. The tethers we encountered inSAUM include:

Tethers to traditional practices in programevaluations. We are accustomed to evalu-ating programs by the quantity of re-sources attracted to the program—in-puts—as opposed to quality of learningoutcomes. One reason for this traditionalpractice is the lack of evidence aboutlearning outcomes, or even an articula-tion of what they are.

Tethers to traditional faculty rewards sys-tem. Traditionally, mathematics facultyrewards are based on accomplishmentsthat do not include educational or em-pirical research results much less amor-

Laurie Hopkins leads a session at a SAUM workshop.

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phous scholarship on assessment. Evenif scholarship on assessment is recog-nized and rewarded, the outlets for suchwork are very limited. Unlike the situa-tion in mathematics research, standardsfor judging empirical assessment workare not widely agreed to and, conse-quently, are inconsistent.

Tethers to traditional lecture-style teach-ing. Especially with large classes, lecture-style teaching severely limits assessmentoptions, especially for formative assess-ment. Some electronic feedback systemsallow lecturers to receive informationquickly about student understanding ofconcepts, but probing for the edges ofunderstanding or for misconceptionsrequires some other scheme such as in-terviewing a sample of the students.

Tethers to a traditional curriculum. Thetraditional college mathematics curricu-lum is based largely in content, so assess-ment of learning (includinglearning goals) has beencouched in terms of thiscontent. Standardized test-ing has centered on thiscontent. Students and fac-ulty expect assessmentitems to address knowledgeof this content. Conse-quently, there is resistanceto less specific assessmentitems, for example, open-ended ill-posed questions.

Tethers to traditional in-course testing. This tetherwas very apparent in thework of SAUM workshopteams. Going beyond as-sessing learning in a singlecourse to assessing learningin a block of courses was amajor step for many faculty teams. Thisstep involved a range of issues from de-veloping learning goals for the block tologistical arrangements of when andwhere to test. Even when learning goalswere agreed to, assessing areas such asgeneral education or quantitative literacyoffered special challenges.

Recognizing this tether, Grant Wigginshas compared assessment of quantitativeliteracy to performance of sports. One

can practice and even master all the in-dividual skills of basketball, but the as-sessment of basketball players is based onperformances in actual games. Wigginsconcludes that assessment for quantita-tive literacy threatens all mainstreamtesting and grading in all disciplines, es-pecially mathematics (Wiggins, 2003).

Components of SAUM

SAUM had five components that wereaimed at encouraging faculty to design,develop, and implement meaningful as-sessment programs. The plan, as out-lined earlier, was to move faculty in de-partments of mathematics from aware-ness of assessment, to acceptance, to en-gagement, and finally to ownership.

Component 1

The initial component was aimed atstimulating thought and discussion,

thereby raising awareness about assess-ment and why it could be a valuable partof an instructional program. There werethree principal vehicles:

• Panels at national and regional profes-sional meetings.

• Ninety-minute forums at meetings ofMAA Sections. Forums were held atseventeen of the twenty-nine sections.

• Distributing Assessment Practices inUndergraduate Mathematics (Gold et

al., 1999) to the chairs of each of the3000 plus departments of mathemat-ics in two- and four-year colleges oruniversities in the United States.

Component 2

A chief project goal was to publish a newvolume containing new case studies to-gether with updated case studies fromAssessment Practices in UndergraduateMathematics. For reasons that are un-clear, few of the original case studies wereupdated. The project had more successin gathering new case studies, mainlybecause the workshops provided natu-ral vehicles for generating them.

SAUM originally planned to support sixareas of assessment: (i) the undergradu-ate mathematics major, including thosefor prospective secondary school math-ematics teachers; (ii) general educationcourses in mathematics and statistics, in-

cluding those in-tended to achievequantitative literacy;(iii) blocks of math-ematics courses forprospective elemen-tary or middle schoolteachers; (iv) place-ment programs or de-velopmental math-ematics courses; (v)reform courses orother innovations; and(iv) classroom assess-ment of learning.

However, as SAUMdeveloped and work-shop teams enrolled,this original list of sixareas evolved into five:the major, general

education, mathematics for teachers,pre-calculus mathematics, and math-ematics in mathematics-intensive ma-jors. Well over half of the sixty-eightSAUM teams worked in just one of theseareas—assessment of the major.

Component 3

Development and delivery of four fac-ulty development workshops plus a self-paced online workshop was the central

SAUM Workshop: Are you sure that will work?

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component of SAUM. As noted above,the workshop teams provided almost allthe new case studies and provided a criti-cal audience for selecting resources tosupport assessment. William Haver wasthe principal organizer and designer ofthe SAUM workshops.

Preliminary evidence indicates that theworkshops were successful in moving thefaculty teams to engagement with assess-ment and many to ownership. Workshopparticipants repeatedly told us that theinteraction among teams was important,and we relied heavily on this feature tomove participants from acceptance toengagement and ownership. Knowledgeand experience of workshop leaders andpresenters seemed to work for awarenessand acceptance but not much further.

We do not have evidence about the ef-fectiveness of the online workshop. Al-though the suggested readings in theonline workshop are selected to movefaculty through the awareness, accep-tance, engagement, and ownership se-quence, face-to-face support and colle-gial interaction may be an essential in-gredient that is missing from the onlineapproach.

Although not specified as a goal in theoriginal SAUM proposal, a significant ac-

complishment of SAUM was identifyingand developing leadership in assessmentof learning in undergraduate mathemat-ics. SAUM began with six leaders, noneof whom claimed broad expertise in as-sessment or in conducting workshops forfaculty on assessment. Since each work-shop session would require four or moreleaders or consultants, recruiting newleaders seemed essential.

We were fortunate that in the first andsecond workshops several leadersemerged. From these leaders we re-cruited Rick Vaughn (Paradise ValleyCommunity College), William Martin(North Dakota State University), LaurieHopkins (Columbia College), KathySafford-Ramus (St. Peter’s College), andDick Jardine (Keene State College).These new leaders provided experiencein assessment at various levels at a vari-ety of institutions and enriched our sub-sequent workshop sessions by sharingtheir experiences. Two of the five—Laurie Hopkins and Dick Jardine—as-sisted with editing the SAUM case stud-ies.

Component 4

Construction of the SAUM website(www.maa.org/saum) began at the out-

set of the project. The site, part of MAAOnline, has several major sections thatsupported SAUM activities and continueto provide resources for undergraduatemathematics assessment. These include:

• An annotated bibliography on assess-ment drawn from multiple sources.

• A communication center for SAUMworkshops, sessions at national meet-ings, and section forums.

• Links to dozens of sites that have in-formation on assessment relevant tothe goals of SAUM.

• A frequently asked questions (FAQ)section containing brief answers to 32common questions about assessment.

• The online assessment workshop.• Online copies of case studies and other

papers in Assessment Practices in Un-dergraduate Mathematics (Gold, et al.,1999).

• Draft case studies including exhibitsand supporting documents.

• The contents of the SAUM case stud-ies volume, Supporting Assessment inUndergraduate Mathematics (Steen,2005).

Component 5

Dissemination of SAUM employs threemedia—print, electronic, and personal.Products include the SAUM case studiesvolume and this special report in FO-CUS; the SAUM website enumeratedabove; and presentations at nationalmeetings, so far including 24 contributedpapers and 18 poster exhibits.

Beyond SAUM

Through the SAUM workshops, nearly200 mathematics faculty members par-ticipated in the development and imple-mentation of programs of assessment in66 college and university mathematicsdepartments. In addition, several hun-dred other faculty became more awareof the challenges and benefits of assess-ment through other SAUM activities.The SAUM web site (www.maa.org/saum) and the associated volume of casestudies (Steen et al., 2005) constitutevaluable resources for others interestedin assessment.

Nonetheless, the accomplishments ofSAUM are probably insufficient to pro-

SAUM workshop director Bill Haver chats with SAUM director Bernie Madi-son at the SAUM Poster Session at the 2003 Joint Mathematics Meetings inPhoenix.

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vide a critical mass of experience and un-derstanding necessary for assessment tobecome a natural part of instructionalprograms in most mathematics depart-ments. Because assessment is largelyalien to beliefs of many mathematics fac-ulty and to traditions in most mathemat-ics departments, much further work willbe needed to overcome the tensions anduntie the tethers discussed here.

Increased external demands for account-ability for student learning will keep fac-ulty attentive but still unenthusiasticabout assessment. Only success storiesthat are documented to the satisfactionof skeptical mathematicians will breakthrough the tacit resistance and causefaculty to take ownership of and workdiligently on assessment programs. Per-haps some of the SAUM-inspired pro-grams will provide these stories. In ad-dition, support from MAA for assess-ment in undergraduate mathematics willlikely be needed for years to come.

References

National Research Council. (2001).Knowing What Students Know. Com-mittee on the Foundations of Assess-ment. James W. Pellegrino, NaomiChudowsky, and Robert Glaser, eds.Washington, DC: National AcademiesPress, 2001

Committee on the Undergraduate Pro-gram in Mathematics. (1995). “Assess-ment of Student Learning for Improv-ing the Undergraduate Major in Math-ematics,” FOCUS 15(3) (June) pp. 24-28. Reprinted in Assessment Practicesin Undergraduate Mathematics, BonnieGold, et al., eds. Washington, DC:Mathematical Association of America,1999, p. 279-284. www.maa.org/saum/maanotes49/279.html

Gold, Bonnie, Sandra Z. Keith, and Wil-liam A. Marion. (1999). AssessmentPractices in Undergraduate Mathemat-ics. MAA Notes No. 49. Washington,DC: Mathematical Assoc. of America.www.maa.org/saum/maanotes49/index.html

Madison, Bernard L. (1992). “Assessmentof Undergraduate Mathematics.” InHeeding the Call for Change: Suggestionsfor Curricular Action, Lynn Arthur

Steen, editor. Washington, DC: Math-ematical Association of America, pp.137-149. www.maa.org/saum/articles/assess_undergrad_math.html

Madison, Bernard L. (1999). “Assess-ment and the MAA.” Foreword to As-sessment Practices in UndergraduateMathematics, Bonnie Gold, et al., eds.Washington, DC: Mathematical Asso-ciation of America, p. 7-8.www.maa.org/saum/maanotes49/7.html

Madison, Bernard L. (2002). “Assess-ment: The Burden of a Name.” Wash-ington, DC: Project Kaleidoscope.w w w. p k a l . o r g / t e m p l a t e 2 . c f m?c_id=360 . Also at: http://w w w. m a a . o r g / s a u m / a r t i c l e s /assessmenttheburdenofaname.html.

Steen, Lynn Arthur, et al. (2005) Sup-porting Assessment in UndergraduateMathematics: Case Studies from a Na-tional Project. Washington, DC: Math-ematical Association of America.

Wiggins, Grant. (2003). “‘Get Real!’: As-sessing for Quantitative Literacy.” InQuantitative Literacy: Why NumeracyMatters for Schools and Colleges, Ber-nard L. Madison and Lynn ArthurSteen, editors. Princeton, NJ: NationalCouncil on Education and the Disci-plines, pp.121-143. www.maa.org/saum/articles/wigginsbiotwocol.htm

Wiggins, Grant. (1992).”Toward Assess-ment Worthy of the Liberal Arts.” InHeeding the Call for Change: Suggestions

for Curricular Action, Lynn A. Steen,editor. Washington, DC: MathematicalAssociation of America, pp. 150-162.w w w. m a a . o r g / s a u m / a r t i c l e s /wiggins_appendix.html

Bernard Madison, project director, is Pro-fessor of Mathematics at the University ofArkansas. E-mail: bmadison@uark.edu.

SAUM Poster Session: This is what we did, and here is what we found out.

A complimentary copy of the newSAUM report is available to the first1000 departments submitting requeststhrough the project website http://www.maa.org/saum, where you canalso access the full text of the report.

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Asking the Right Questions

By Lynn Arthur Steen

Assessment is about asking and an-swering questions. For students, “howam I doing?” is the focus of so-called “for-mative” assessment, while “what’s mygrade?” often seems to be the only goalof “summative” assessment. For faculty,“how’s it going?” is the hallmark ofwithin-course assessment using instru-ments such as ten-minute quizzes or one-minute responses on 3 x 5 cards at theend of each class period. Departments,administrations, trustees, and legislatorstypically ask questions about more ag-gregated levels: they want to know notabout individual students but aboutcourses, programs, departments, andentire institutions.

The conduct of an assessment dependsimportantly on who does the asking andwho does the answering. Faculty are ac-customed to setting the questions and as-sessing answers in a context where out-comes count for something. When as-sessments are set by someone other thanfaculty, skepticism and resistance oftenfollow. And when tests are administeredfor purposes that don’t “count,” (for ex-ample, sampling to assess general edu-cation or to compare different pro-grams), student effort declines and re-sults lose credibility.

The assessment industry devotes consid-erable effort to addressing a variety ofsimilar contextual complications, suchas:

• different purposes (diagnostic, forma-tive, summative, evaluative, self-assess-ment, ranking);

• different audiences (students, teachers,parents, administrators, legislators,voters);

• different units of analysis (individual,class, subject, department, college, uni-versity, state, nation);

• different types of tests (multiplechoice, open ended, comprehension,performance-based, timed oruntimed, calculator permitted, indi-

vidual or group, seen or unseen, ex-ternal, written or oral);

• different means of scoring (norm-ref-erenced, criterion referenced, stan-dards-based, curriculum-based);

• different components (quizzes, exams,homework, journals, projects, presen-tations, class participation);

• different standards of quality (consis-tency, validity, reliability, alignment);

• different styles of research (hypoth-esis-driven, ethnographic, compara-tive, double-blind, epidemiological).

Distinguishing among these variablesprovides psychometricians with severallifetimes’ agenda of study and research.All the while, these complexities cloudthe relation of answers to questions andweaken inferences drawn from resultinganalyses.

These complications notwithstanding,questions are the foundation on whichassessment rests. The assessment cyclebegins with and returns to goals and ob-jectives (CUPM, 1995). Translating goalsinto operational questions is the mostimportant step in achieving goals sincewithout asking the right questions wewill never know how we are doing.

Two Examples

In recent years two examples of this tru-ism have been in the headlines. The morevisible—because it affects more people—is the new federal education law knownas No Child Left Behind (NCLB). Thislaw seeks to ensure that every child is re-ceiving a sound basic education. With

this goal, it requires assessment data tobe disaggregated into dozens of differ-ent ethnic and economic categories in-stead of typical analyses that report onlysingle averages. NCLB changes the ques-tion that school districts need to answerfrom “What is your average score?” to“What are the averages of every sub-group?” Theoretically, to achieve its titu-lar purpose, this law would require dis-

tricts to monitor every childaccording to federal stan-dards. The legislated re-quirement of multiple sub-groups is a political and sta-tistical compromise be-tween theory and reality.But even that much hasstirred up passionate debatein communities across theland.

A related issue that con-cerns higher education hasbeen simmering in Con-gress as it considers reau-thorizing the law that,

among other things, authorizes federalgrants and loans for postsecondary edu-cation. In the past, in exchange for thesegrants and loans, Congress asked collegesand universities only to demonstrate thatthey were exercising proper stewardshipof these funds. Postsecondary institu-tions and their accrediting agencies pro-vided this assurance through financialaudits to ensure lack of fraud and bykeeping default rates on student loans toan acceptably low level.

But now Congress is beginning to ask adifferent question. If we give you moneyto educate students, they say, can youshow us that you really are educatingyour students? This is a new question forCongress to ask, although it is one thatdeans, presidents, and trustees should askall the time. The complexities of assess-ment immediately jump to the fore-ground. How do you measure the edu-cational outcomes of a college education?As important, what kinds of assessmentswould work effectively and fairly for allof the 6,600 very different kinds ofpostsecondary institutions in the UnitedStates, ranging from 200-student beau-tician schools to 40,000-student researchuniversities? Indicators most often dis-

SAUM Workshop: This plan must be finished before wego home.

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cussed include the rates at which studentscomplete their degrees or the rates atwhich graduates secure professional li-censing or certification. In sharp con-trast, higher education mythology stillembraces James Garfield’s celebratedview of education as a student on oneend of a log with Mark Hopkins on theother end. In today’s climate of publicaccountability, colleges and universitiesneed to “make peace” with citizens’ de-mand for candor and openness anchoredin data (Ekman, 2004).

I cite these examples to make two points.First, the ivory tower no longer shelterseducation from external demands for ac-countability. Whether faculty like it ornot, the public is coming to expect ofeducation the same kind of transparencythat it is also beginning to demand ofgovernment and big business. Especiallywhen public money is involved—as it isin virtually every educational institu-tion—public questions will follow.

Second, questions posed by those outsideacademe are often different from thoseposed by educators, and often quite re-freshing. After all these years in whichschool districts reported and compared

test score averages, someone in power fi-nally said “but what about the variance?”Are those at the bottom within strikingdistance of the average, or are they hope-lessly behind with marks cancelled outby accelerated students at the top? Andafter all these years of collecting tuitionand giving grades, someone in power hasfinally asked colleges and universitieswhether students are receiving the edu-cation they and the public paid for. Ask-ing the right questions can be a power-ful lever for change, and a real challengeto assessment.

Mathematics

One can argue that mathematics is thediscipline most in need of being askedthe right new questions. At least untilvery recently, in comparison with otherschool subjects mathematics has changedleast in curriculum, pedagogy, and as-sessment. The core of the curriculum ingrades 10-14 is a century-old enterprisecentered on algebra and calculus, em-broidered with some old geometry andnew statistics. Recently, calculus passedthrough the gauntlet of reform andemerged only slightly refurbished. Al-gebra—at least that part known incon-

gruously as “College Algebra”—is now inline for its turn at the reform carwash.Statistics is rapidly gaining a presence inthe lineup of courses taught in grades 10-14, although geometry appears to havelost a bit of the curricular status that wasprovided by Euclid for over two millen-nia.

When confronted with the need to de-velop an assessment plan, mathematicsdepartments generally take this tradi-tional curriculum for granted and focusinstead on how to help students throughit. However, when they ask for advicefrom other departments, mathemati-cians are often confronted with ratherdifferent questions (Ganter & Barker,2004):

• Do students in introductory mathemat-ics courses learn a balanced sample ofimportant mathematical tools?

• Do these students gain the kind of expe-rience in modeling and communicationskills needed to succeed in other disci-plines?

• Do they develop the kind of balance be-tween computational skills and concep-tual understanding appropriate for theirlong-term needs?

Greece, 250 BCE

If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields ofthe Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow andthe last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the whitebulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourthpart of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, thedappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of thecows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal tothe fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now thedappled in four parts were equal in number to a fifth part, and a sixth of the yellow herd. Finally the yellow were in number equalto a sixth part, and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, givingseparately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be calledunskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.

But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number withthe black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with theirmultitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that theirnumber, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours intheir midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in yourmind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in thisspecies of wisdom.

—Archimedes, Counting the Cattle of the Sun

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• Why can’t more mathematics problemsemploy units and realistic measure-ments that reflect typical contexts?

These kinds of questions from math-ematics’ client disciplines strongly sug-gest the need for multi-disciplinary par-ticipation in mathematics departments’assessment activities.

Similar issues arise in relation to peda-gogy, although here the momentum ofvarious “reform” movements of the lasttwo decades (in using technology, inteaching calculus, in setting K-12 stan-dards) has energized considerable changein mathematics instruction. Althoughlectures, problem sets, hour tests, and fi-nal exams remain the norm for math-ematics teaching, innovations involvingcalculators, computer packages, groupprojects, journals, and various mentoringsystems have enriched the repertoire ofpostsecondary mathematical pedagogy.Many assessment projects seek to com-pare these new methods with traditionalapproaches. But client disciplines andothers in higher education press evenfurther:

• Do students learn to use mathematicsin interdisciplinary or “real-world” set-tings?

• Are students encouraged (better still, re-quired) to engage mathematics activelyin ways other than through routineproblem sets?

• Do mathematics courses leave studentsfeeling empowered, informed, and re-sponsible for using mathematics as a toolin their lives? (Ramaley, 2003)

Prodded by persistent questions, math-ematicians have begun to think afreshabout content and pedagogy. In assess-ment however, mathematics still seemsfirmly anchored in hoary traditions.More than most disciplines, mathemat-ics is defined by its problems and exami-nations, many with histories that are de-cades or even centuries old. National andinternational mathematical Olympiads,the William Lowell Putnam undergradu-ate exam, the Cambridge Universitymathematics Tripos, not to mentionpopular problems sections in most math-ematics education periodicals attest to

the importance of problems in definingthe subject and identifying its star pu-pils. The correlation is far from perfect:not every great mathematician is a greatproblemist, and many avid problemistsare only average mathematicians. Some,indeed, are amateurs for whom problemsolving is their only link to a past schoollove. Nonetheless, for virtually everyoneassociated with mathematics education,assessing mathematics means asking stu-dents to solve problems.

Mathematical Problems

Problems on mathematics exams havedistinctive characteristics that are foundnowhere else in life. They are stated with

precision intended to ensure unambigu-ous interpretation. Many are about ab-stract mathematical objects—numbers,equations, geometric figures—with noexternal context. Others provide arche-type contexts that are not only artificialin setting (e.g., rowing boats across riv-ers) but often fraudulent in data (in-vented numbers, fantasy equations). Incomparison with problems people en-counter in their work and daily lives,most problems offered in mathematicsclass, like shadows in Plato’s allegoricalcave, convey the illusion but not the sub-stance of reality.

Little has changed over the decades orcenturies. Problems just like those oftoday’s texts (only harder) appear inmanuscripts from ancient Greece, India,and China (see sidebars). In looking atundergraduate mathematics exams from100 or 150 years ago, one finds few sur-prises. Older exams typically includemore physics than do exams of today,since in earlier years these curricula wereclosely linked. Mathematics course ex-ams from the turn of the twentieth cen-tury required greater virtuosity in accu-rate lengthy calculations. They were, af-ter all, set for only 5% of the population,not the 50% of today. But the centralsubstance of the mathematics tested andthe distinctive rhetorical nature of prob-lems are no different from typical prob-lems found in today’s textbooks andmainstream exams.

Questions suitable for a mathematicsexam are designed to be unambiguous,to have just one correct answer (whichmay consist of multiple parts), and toavoid irrelevant distractions such as con-fusing units or complicated numbers.Canonical problems contain enough in-formation and not an iota more thanwhat is needed to determine a solution.Typical tests are time-constrained andinclude few problems that students havenot seen before; most tests have a highproportion of template problems whosetypes students have repeatedly practiced.Mathematician and assessment expertKen Houston of the University of Ulsternotes that these types of mathematicstests are a “rite of passage” for studentsaround the world, a rite, he adds, that is

China, 100 CE

• A good runner can go 100 paces while apoor runner covers 60 paces. The poorrunner has covered a distance of 100paces before the good runner sets off inpursuit. How many paces does it take thegood runner before he catches up to thepoor runner?

• A cistern is filled through five canals.Open the first canal and the cistern fillsin 1/3 day; with second, it fills in 1 day;with the third, in 2 1/2 days; with thefourth, in 3 days, and with the fifth in 5days. If all the canals are opened, howlong will it take to fill the cistern?

• There is a square town of unknown di-mensions. There is a gate in the middleof each side. Twenty paces outside theNorth Gate is a tree. If one leaves the townby the South Gate, walks 14 paces duesouth, then walks due west for 1775 paces,the tree will just come into view. Whatare the dimensions of the town?

• There are two piles, one containing 9gold coins and the other 11 silver coins.The two piles of coins weigh the same.One coin is taken from each pile and putinto the other. It is now found that thepile of mainly gold coins weighs 13 unitsless than the pile of mainly silver coins.Find the weight of a silver coin and of agold coin.—Nine Chapters on the Mathematical Art.

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“never to be performed again” once stu-dents leave university. Unfortunately,Houston writes, “learning mathematicsfor the principal purpose of passing ex-aminations often leads to surface learn-ing, to memory learning alone, to learn-ing that can only see small parts and notthe whole of a subject, to learningwherein many of the skills and much ofthe knowledge required to be a workingmathematician are overlooked” (Hous-ton, 2001).

All of which suggests a real need to as-sess mathematics assessment. Some is-sues are institutional:

• Do institutions include mathematical orquantitative proficiency among theireducational goals?

• Do institutions assess the mathematicalproficiency of all students, or only ofmathematics students?

• Others are more specifically mathemati-cal:

• Can mathematics tests assess the kindsof mathematical skills that society needsand values?

• What kinds of problems would best re-flect the mathematical needs of the av-erage educated citizen?

• Can mathematics faculty fairly assessthe practice of mathematics in other dis-ciplines? Should they?

Issues and Impediments

Assessment has had a tenuous impact inhigher education, especially amongmathematicians who are trained to de-mand rigorous inferences that are rarelyattainable in educational assessment.Some mathematicians are unrelentinglycritical of any educational research thatdoes not closely approach medicine’sgold standard of randomized, doubleblind, controlled, hypothesis-drivenstudies. Their fears are not unwarranted.For example, a recent federal projectaimed at identifying high quality educa-tional studies found that only one of 70studies of middle school mathematicscurricula met the highest standards forevidence (What Works, 2005). Virtuallyall assessment studies undertaken bymathematics departments fall far shortof mathematically rigorous standardsand are beset by problems such as con-

founding factors and attrition. Evidencedrawn entirely from common observa-tional studies can never do more thansuggest an hypothesis worth testingthrough some more rigorous means.

Notwithstanding skepticism from math-ematicians, many colleges have investedheavily in assessment; some have evenmade it a core campus philosophy. Insome cases this special focus has led theseinstitutions to enhanced reputations andimproved financial circumstances.Nonetheless, evidence of the relation be-tween formal assessment programs andquality education is hard to find. Listsof colleges that are known for their com-mitment to formal assessment programsand those in demand for the quality oftheir undergraduate education are virtu-ally disjoint.

Institutions and states that attempt toassess their own standards rigorously of-ten discover large gaps between rhetoricand reality. Both in secondary andpostsecondary education, many studentsfail to achieve the rhetorical demands ofhigh standards. But since it is not politi-cally or emotionally desirable to brandso many students as failures, institutionsfind ways to undermine or evade evi-dence from the assessments. For ex-ample, a recent study shows that on av-erage, high stakes secondary school exitexams are pegged at the 8th and 9thgrade level to avoid excessive failure rates(Achieve, 2004). Higher education typi-cally solves its parallel problem either bynot assessing major goals or by doing soin a way that is not a requirement forgraduation.

• How, if at all, are the mathematical,logical, and quantitative aspects of aninstitution’s general education goals as-sessed?

• How can the goals of comprehendingand communicating mathematics beassessed?

When mathematicians and test expertsdo work together to develop meaningfulassessment instruments, they confrontmajor intellectual and technical hurdles.First are issues about the harmony ofeducational and public purposes:

• Can a student’s mathematical proficiencybe fairly measured along a single dimen-sion?

• What good is served by mapping a mul-tifaceted profile of strengths and weak-nesses into a single score?

Clearly there are such goods, but theymust not be oversold. They include fa-cilitating the allocation of scarce educa-tional resources, enhancing the align-ment of graduates with careers, and —

with care—providing data required toproperly manage educational programs.They do not (and thus should not) in-clude firm determination of a student’sfuture educational or career choices. Toguard against misuse, we need always toask and answer:

• Who benefits from the assessment?• Who are the stakeholders?• Who, indeed, owns mathematics?

Mathematical performance embracesmany different cognitive activities thatare entirely independent of content. Ifcontent such as algebra and calculus rep-resents the nouns—the “things” of math-ematics—cognitive activities are theverbs: know, calculate, investigate, in-vent, strategize, critique, reason, prove,communicate, apply, generalize. Thisvaried landscape of performance expec-tations opens many questions about the

India, 400 CE

• One person possesses seven asavahorses, another nine haya horses, andanother ten camels. Each gives two ani-mals, one to each of the others. Theyare then equally well off. Find the priceof each animal and the total value ofthe animals possessed by each person.

• Two page-boys are attendants of aking. For their service one gets 13/6dinaras a day and the other 3/2. The firstowes the second 10 dinaras. Calculateand tell me when they have equalamounts.

—The Bakhsali Manuscript

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purpose and potential of mathematicsexaminations. For example:

• Should mathematics exams assess pri-marily students’ ability to perform pro-cedures they have practiced or their abil-ity to solve problems they have not seenbefore?

• Can ability to use mathematics in di-verse and novel situations be inferredfrom mastery of template procedures?

• If learned procedures dominate concep-tual reasoning on tests, is it mathemat-ics or memory that is really being as-sessed?

Reliability and Validity

A widely recognized genius of Americanhigher education is its diversity of insti-tutions: students’ goals vary, institutionalpurposes vary, and performance stan-dards vary. Mathematics, on the otherhand, is widely recognized as universal;more than any other subject, its content,practices, and standards are the same ev-erywhere. This contrast between insti-tutional diversity and discipline univer-sality triggers a variety of conflicts re-garding assessment of undergraduatemathematics.

Assessment of school mathematics issomewhat different from thepostsecondary situation. Partly becauseK-12 education is such a big enterpriseand partly because it involves many le-gal issues, major assessments of K-12education are subject to many layers oftechnical and scholarly review. Items arereviewed for, among other things, accu-racy, consistency, reliability, and (lack of)bias. Exams are reviewed for balance,validity, and alignment with prescribedsyllabi or standards. Scores are reviewedto align with expert expectations and de-sirable psychometric criteria. The resultsof regular assessments are themselves as-sessed to see if they are confirmed by sub-sequent student performance. Even abrief examination of the research armsof major test producers such as ETS, ACT,or McGraw Hill reveal that extensiveanalyses go into preparation of educa-tional tests.

In contrast, college mathematics assess-ments typically reflect instructors’ beliefs

about subject priorities more than anyexternal benchmarks or standards ofquality. This difference in methodologi-cal care between major K-12 assessmentsand those that students encounter inhigher education cannot be justified onthe grounds of differences in the “stakes”for students. Sponsors of the SAT andAP exams take great pains to ensure qual-ity control in part because the conse-quences of mistakes on students’ aca-demic careers are so great. The conse-quences for college students of unjusti-

fied placement procedures or unreliablefinal course exams are just as great.

• Are “do-it-yourself ” assessment instru-ments robust and reliable?

• Can externally written (“off the shelf ”)assessment instruments align appropri-ately with an institution’s distinctivegoals?

• Can locally written exams that have notbeen subjected to rigorous reviews forvalidity, reliability, and alignment pro-duce results that are valid, reliable, andaligned with goals?

Professional test developers go to con-siderable and circuitous lengths to scoreexams in a way that achieves certain de-sirable results. For example, by using amethod known as “item response theory”they can arrange the region of scores withlargest dispersion to surround the pass-ing (so-called “cut”) score. This mini-

mizes the chance of mistaken actionsbased on passing or failing at the expenseof decreased reliability, say, of the differ-ence between B+ and A- (or its numeri-cal equivalent).

• How are standards of performance—grades, cut-scores—set?

• Is the process of setting scores clear andtransparent to the test-takers?

• Is it reliable and valid?

Without the procedural checks and bal-ances of the commercial sector, under-graduate mathematics assessment israther more like the Wild West—a liber-tarian free-for-all with few rules and noestablished standards of accountability.In most institutions, faculty just make uptests based on a mixture of experienceand hunch, administer them without anyof the careful reviewing that is requiredfor development of commercial tests, andgrade them by simply adding and sub-tracting arbitrarily assigned points.These points translate into grades (forcourses) or enrollments (for placementexams) by methods that can most chari-tably be described as highly subjective.

Questions just pour out from anythoughtful analysis of test construction.Some are about the value of individualitems:

SAUM Workshop: Laurie Hopkins watches as small groups work.

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• Can multiple choice questions truly as-sess mathematical performance abilityor only some correlate? Does it matter?

• Can open response tasks be assessed withreliability sufficient for high-stakes tests?

• Can problems be ordered consistently bydifficulty?

• Is faculty judgment of problem difficultyconsistent with empirical evidence fromstudent performance?

• What can be learned from easy problemsthat are missed by good students?

Others are about the nature and balanceof tests that are used in important assess-ments:

• Is the sampling of content on an examtruly representative of curricular goals?

• Is an exam well balanced between nar-row items that focus on a single proce-dure or concept and broad items that cutacross domains of mathematics and re-quire integrated thinking?

• Does an assessment measure primarilywhat is most important to know and beable to do, or just what is easiest to test?

Interpreting test results

Public interest in educational assessmentfocuses on numbers and scores—percentpassing, percent proficient, percent

graduating. Often dismissed by educa-tors as an irrelevant “horse race,” publicnumbers that profile educational accom-plishment shape attitudes and, ulti-mately, financial support. K-12 is themajor focus of public attention, but aswe have noted, pressure to document theperformance of higher education is ris-ing rapidly.

Testing expert Gerald Bracey warnsabout common misinterpretations of testscores, misinterpretations to which poli-ticians and members of the public arehighly susceptible (Bracey, 2004). Onearises in comparative studies of differ-ent programs. Not infrequently, resultsfrom classes of different size are averagedto make overall comparisons. In suchcases, differences between approachesmay be entirely artificial, being merelyartifacts created by averaging classes ofdifferent sizes.

Comparisons are commonly made usingthe rank order of students on an assess-ment (for example, the proportion froma trial program who achieve a proficientlevel). However, if many students arebunched closely together, ranks can sig-nificantly magnify slight differences.Comparisons of this sort can truly makea mountain out of a molehill.

Another of Bracey’s cautions is of pri-mary importance for K-12 assessment,but worth noting here since higher edu-cation professionals play a big role indeveloping and assessing K-12 math-ematics curricula. It is also a topic sub-ject to frequent distortion in politicalcontests. The issue is the interpretationof nationally normed tests that reportpercentages of students who read or cal-culate “at grade level.” Since grade levelis defined to be the median of the groupused to norm the test, an average class(or school) will have half of its studentsfunctioning below grade level and halfabove. It follows that if 30% of a school’seighth grade students are below gradelevel on a state mathematics assessment,contrary to frequent newspaper innuen-dos, that may be a reason for cheer, notdespair.

Bracey’s observations extend readily tohigher education as well as to other as-

pects of assessment. They point to yetmore important questions:

• To what degree should results of programassessments be made public?

• Is the reporting of results appropriate tothe unit of analysis (student, course,department, college, state)?

• Are the consequences attached to differ-ent levels of performance appropriate tothe significance of the assessment?

Program Assessment

As assessment of student performanceshould align with course goals, so assess-ment of programs and departmentsshould align with program goals. Butjust as mathematics’ deep attachment totraditional problems and traditional tests

often undermines effective assessment ofcontemporary performance goals, so de-partments’ unwitting attachment to tra-ditional curriculum goals may under-mine the potential benefits of thorough,“gloves off” assessment. Asking “how canwe improve what we have been doing?”is better than not asking at all, but all toooften this typical question masks an as-sumed status quo for goals and objectives.Useful assessment needs to begin by ask-ing questions about goals.

Many relevant questions can be inferredfrom Curriculum Guide 2004, a reportprepared recently by MAA’s Committeeon the Undergraduate Program in Math-

In most institutions, courses belowcalculus—including developmen-tal, quantitative literacy, and precal-culus courses—constitute the majorpart of a mathematics department’sworkload. Nonetheless, they are notwhere faculty invest their greatest ef-forts. They are the least mathemati-cally interesting courses offered bythe department, are filled with un-enthusiastic students who dislikeand fear mathematics, and are of-ten the most difficult and frustrat-ing courses to teach. As a result,however, these are courses in whicheffective assessment can yield thegreatest improvement in both stu-dent learning and faculty workingconditions.

—Bonnie Gold

An effective assessment plan mustbe anchored in the department’smission statement. So departmentfaculty should first review and up-date (or if necessary, write) theirmission statement. Goals for stu-dent learning—broad descriptionsof competencies or skills studentsshould achieve—are based on thedepartment’s mission. After goalshave been articulated, learning ob-jectives—measurable outcomesthat tell when a goal has beenachieved (or not)—can be devel-oped.

—William Marion

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ematics (CUPM, 2005). Some ques-tions—the first and most important—are about students:

• What are the aspirations of students en-rolled in mathematics courses?

• Are the right students enrolled in math-ematics, and in the appropriate courses?

• What is the profile of mathematicalpreparation of students in mathematicscourses?

Others are about placement, advising,and support:

• Are students taking the best kind ofmathematics to support their careergoals?

• Are students who do not enroll in math-ematics doing so for appropriate rea-sons?

Still others are about curriculum:

• Do program offerings reveal the breadthand interconnections of the mathemati-cal sciences?

• Do introductory mathematics coursescontain tools and concepts that are im-portant for all students’ intended ma-jors?

• Can students who complete mathemat-ics courses use what they have learnedeffectively in other subjects?

• Do students learn to comprehend math-ematically-rich texts and to communi-cate clearly both in writing and orally?

A consistent focus of this report and itscompanion “voices of partner disci-plines” (mentioned above) is that the in-creased spread of mathematical methodsto fields well beyond physics and engi-neering requires that mathematics de-partments promote interdisciplinary co-operation both for faculty and students.Mathematics is far from the only disci-pline that relies on mathematical think-ing and logical reasoning.

How is mathematics used by other depart-ments?Are students learning how to use math-ematics in other subjects?Do students recognize similar mathemati-cal concepts and methods in different con-texts?

Creating a Culture of Assessment

Rarely does one find faculty begging ad-ministrators to support assessment pro-grams. For all the reasons cited above,and more, faculty generally believe intheir own judgments more than in theresults of external exams or structuredassessments. So the process by which as-sessment takes root on campus is moreoften more top down than bottom up.

A culture of assessment appears to growin stages (North Central Assoc., 2002).First is an articulated commitment in-volving an intention that is accepted by

both administrators and faculty. This isfollowed by a period of mutual explora-tion by faculty, students, and adminis-tration. Only then can institutional sup-port emerge conveying both resources(financial and human) and structuralchanges necessary to make assessmentroutine and automatic. Last shouldcome change brought about by insightsgleaned from the assessment. And thenthe cycle begins anew.

Faculty who become engaged in this pro-cess can readily interpret their work aspart of what Ernest Boyer called the“scholarship of teaching,” (Boyer, 1990)thereby avoiding the fate of what LeeShulman recently described as “drive-byteachers” (Shulman, 2004). Soon theyare asking some troubling questions:

• Do goals for student learning take intoaccount legitimate differences in educa-tional objectives ?

• Do faculty take responsibility for thequality of students’ learning?

• Is assessment being used for improve-ment or only for judgment?

Notwithstanding numerous impedi-ments, assessment is becoming a main-stream part of higher education pro-grams, scholarship, and literature. Incollegiate mathematics, however, assess-ment is still a minority culture beset byignorance, prejudice, and the power of adominant discipline backed by centuriesof tradition. Posing good questions is aneffective response, especially to math-ematicians who pride themselves on their

SAUM Workshop: That sounds like a good idea.

Courses in quantitative literacy arespecially well suited to alternative as-sessment methods such as portfolios,journals, projects, group work, es-says, and student-created problems.However, to use these alternativemethods efficiently, faculty must es-tablish rubrics to enable readers tosummarize diffuse information rap-idly and in ways that give useful in-formation and that can be comparedacross courses.

—Bonnie Gold

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ability to solve problems. The key to con-vincing mathematicians that assessmentis worthwhile is not to show that it hasall the answers but that it is capable ofasking the right questions.

References

Achieve, Inc. (2004). Do GraduationTests Measure Up? Washington, DC.

“Assessment of Student AcademicAchievement: Levels of Implementa-tion.” (2002). Handbook of Accredita-tion. Chicago, IL: North Central As-sociation Higher Learning Commis-sion.

Boyer, Ernest L. (1990). Scholarship Re-considered: Priorities of the Professori-ate. Princeton, NJ: Carnegie Founda-tion for the Advancement of Teaching.

Bracey, Gerald W. (2004). “Some Com-mon Errors in Interpreting Test Scores.”NCTM News Bulletin, April, p. 9.

Committee on the Undergraduate Pro-gram in Mathematics. (1995). “Assess-ment of Student Learning for Improv-ing the Undergraduate Major in Math-ematics,” FOCUS, 15:3 (June) 24-28.www.maa.org/saum/maanotes49/279.html

CUPM Curriculum Guide 2004. (2004).Washington, DC: Mathematical Asso-ciation of America.

Ekman, Richard. (2004). “Fear of Data.”University Business, May, p. 104.

Ganter, Susan and William Barker, eds.(2004). Curriculum FoundationsProject: Voices of the Partner Disciplines.Washington, DC: Mathematical Asso-ciation of America.

Houston, Ken. (2001). “Assessing Under-graduate Mathematics Students.” InThe Teaching and Learning of Math-ematics at University Level: An ICMIStudy. Derek Holton (Ed.). Dordrecht:Kluwer Academic, p. 407-422.

Ramaley, Judith. (2003). Greater Expec-tations. Washington, DC: Associationof American Colleges and Universities.

Shulman, Lee S. (2004). “A Different Wayto Think About Accountability: NoDrive-by Teachers.” Carnegie Perspec-tives. Palo Alto, CA: Carnegie Foun-dation for the Advancement of Learn-ing, June.

What Works Clearinghouse. (2005).www.whatworks.ed.gov.

Lynn Arthur Steen, project editor, is Pro-fessor of Mathematics at St. Olaf College.E-mail: steen@stolaf.edu.

Assessing AssessmentThe Evaluator’s Perspective

By Peter Ewell

The SAUM project took place within abroader context of assessment in Ameri-can higher education. Faculty teams inmathematics departments experiencedin microcosm what their colleagues inmany other disciplines were simulta-neously experiencing, and their actionswere shaped by larger forces of politicsand accountability affecting their insti-tutions. At the same time, their effortsto develop and document viable depart-ment-level approaches to assessment inmathematics helped inform the nationalassessment movement—a field badly inneed of concrete, discipline-level ex-amples of good practice. Evaluation ofSAUM helped bridge these two worlds.

In my personal role as project evaluator,I continued to participate in nationalconversations about assessment’s pur-poses and prospects throughout thethree-year grant period. But watchingSAUM participants struggle with theday-to-day reality of crafting workableassessment approaches in their own de-partments helped keep me honest about

what could and could not be accom-plished. Similarly, the participant expe-riences that were revealed through theevaluation information we compiled of-ten mirrored what was happening toother “early adopters” elsewhere.

The first section of this essay sets thewider stage for SAUM by locating theproject in a national context of assess-ment. A second section reflects on myrole as project evaluator, and describesthe kinds of evaluative information wecollected to examine the project’s activi-ties and impact. A third section presentssome of what we learned—focused pri-marily on what participants told us abouthow they experienced the project and thechallenges they faced in implementingassessment initiatives back home.

SAUM in a National Context

The so-called “assessment movement” inhigher education began in the mid-1980swith the confluence of two major forces.One originated inside the academy,

prompted by growing concerns aboutcurricular coherence and the convictionthat concrete information about how andhow well students were learning could becollectively used by faculty to improveteaching and learning (NIE, 1984). Thisversion of “assessment” was low-stakes,incremental, faculty-owned, and guidedby a metaphor of scholarship. The otherdriving force for assessment originatedoutside the academy prompted bypolicymakers’ growing concerns aboutthe productivity and effectiveness of col-leges and universities (NGA, 1986). Thisversion of “assessment” was high-stakes,publicly visible, accountability-oriented,and infused with the urgency of K 12 re-form embodied in A Nation at Risk(USDOE, 1983).

Although fundamentally contradictory,both these forces were needed to launchand sustain a national movement. Ex-ternal authorities—first in the guise ofstates and later in the guise of regionalaccrediting organizations—served toconstantly keep assessment at the fore-

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front of institutional attention. But be-cause these external requirements wereat first fairly benign—and because aca-demic leaders quickly saw the need toprotect the academy’s autonomy by de-veloping locally-owned processes thatmight actually be useful—internal pref-erences for diverse evidence-gatheringapproaches aimed at institutional im-provement served to discharge account-ability as well for many years (Ewell,1987).

The environment within which theSAUM project was launched was shapedby fifteen years of growing in-stitutional experience withsteering between these contra-dictory poles of assessment. By2000, virtually every institutioncould claim that it “did assess-ment,” at least in the sense thatit had developed learning out-comes goals for general educa-tion and that it periodically sur-veyed its students and gradu-ates. Most could also point tothe beginnings of an institu-tion-level organizational infra-structure for assessment—a co-ordinator operating out of theacademic affairs office perhaps,or a faculty-staffed institutionalassessment committee. Abouta third could lay claim to moresophisticated efforts includingtesting programs in general education,portfolios assembled by students and or-ganized around general learning out-comes like “effective communication” or“critical thinking,” or specially-designedassignments intended to both grade stu-dents individually and provide facultywith broader information about patternsof student strength and weakness in vari-ous abilities. Indeed, as revealed by theprograms at such gatherings as the an-nual Assessment Forum hosted by theAmerican Association for Higher Edu-cation (AAHE), there was a steady in-crease in the sophistication of institu-tional assessment efforts with respect tomethod and approach throughout thisperiod, and equally steady progress infaculty acceptance of the fact that assess-ment was a part of what colleges anduniversities, for whatever reason, had todo.

By 2000, moreover, the primary reasonwhy institutions had to “do assessment”had become regional accreditation. Statemandates for assessment in public insti-tutions, instituted in the wake of theNational Governors Association’s Timefor Results report in the mid-1980s, hadlost a lot of steam in the recession thatappeared about 1990. States had otherthings to worry about and there were fewresources to pursue existing mandates inany case. Accreditors, meanwhile, wereunder mounting pressure by federal au-thorities to increase their focus on stu-

dent learning outcomes. Regional ac-crediting organizations must be “recog-nized” by the U.S. Department of Edu-cation in order for accredited status toserve as a gatekeeper for receipt of fed-eral funds. The federal recognition pro-cess involves a regular review of accredi-tation standards and practices against es-tablished guidelines. And since 1989these guidelines have emphasized the as-sessment of learning outcomes moreforcefully each time they have been re-vised by the Department. Accreditors arestill accorded the leeway to allow insti-tutions to develop their own learningoutcomes and to assess them in their ownways. But by 2002, when SAUM waslaunched, it was apparent thataccreditors could no longer afford to al-low institutions to get by with little orno assessment—which had up to thenessentially been the case—if they hoped

to maintain their recognized status. Theresult was growing pressure on institu-tions to get moving on assessment, to-gether with growing awareness amonginstitutional academic leaders that a re-sponse was imperative.

But even at this late date, assessment re-mained something distant and faintly“administrative” for the vast majority ofcollege faculty. It was rarely an activitydepartments engaged in regularly out-side professional fields like engineering,education, business, or the health pro-fessions where specialized accreditation

requi rementsmade assess-ment manda-tory. And evenin these cases,the fact thatdeans and otheracademic ad-m i n i s t r a t o r swere front andcenter in theprocess, com-plete with therequisite guide-lines, memos,schedules, andr e p o r t s — a l lwritten in pas-sive prose—made it likelythat faculty in

departments like mathematics wouldkeep their distance. At the same time,despite their growing methodologicalprowess, few institutions were able to ef-fectively “close the loop” by using assess-ment results in decision-making or toimprove instruction. Periodic assess-ment reports were distributed, to be sure,but most of them ended up on shelves tobe ritually retrieved when external visi-tors inquiring about the topic arrived oncampus. Much of the reason for this phe-nomenon, in hindsight, is apparent. As-sessment findings tend to be fine-grainedand focused, while institutional decisionsremain big and messy. Real applicationrequired smaller settings, located muchcloser to the teaching and learning situ-ations that assessment could actually in-form.

SAUM Poster Session: The United States Military Academy’s poster is in goodhands.

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In this context, the notion of groundingassessment in the individual disciplineswhere faculty professionally lived andworked made a great deal of sense. Forone thing, assessment practitioners haddiscovered that methods and approachesought appropriately to vary substantiallyacross fields and that such purportedly“generic” academic abilities as “criticalthinking” and “communication” weremanifest (and thus had to be assessed)very differently in different disciplinarycontexts. At the disciplinary level, more-over, learning outcomes were generallymuch more easily specified than at theinstitutional level where of necessity theyhad to be so broadly cast that they oftenlost their meaning. More importantly,faculty tended to listen to one anothermore carefully in disciplinary commu-nities bound by common languages andfamiliar hierarchies of respect. Evenwhen assessment leaders on campus werefaculty instead of administrators, theirobvious background in methods derivedfrom education and the social sciencesoften distanced them from colleagues inthe sciences, humanities, fine and per-forming arts—as well as mathematics.For all these reasons, anchoring assess-ment in individual disciplinary commu-nities was critical if it was to become ameaningful activity for faculty.

But why mathematics? In my view,mathematics became an “early adopter”of assessment for at least three reasons.First, the discipline is embedded in mul-tiple aspects of teaching and learningbeyond its own major at most institu-tions. Like colleagues in writing—butunlike those in physics, philosophy, andFrench—mathematics faculty had tostaff basic skills courses in general edu-cation. As a result, both their course de-signs and pedagogies in such offerings ascalculus and statistics must be closelyaligned with a range of client disciplinesincluding the sciences, engineering, busi-ness, and the social sciences. As a “basicskill,” moreover, mathematics is generallyassessed already at most institutions inthe form of placement examinations, soat least some members of every math-ematics department have experiencewith test construction and use. Wheredevelopmental mathematics courses areoffered, moreover, they are often evalu-

ated directly because the question of ef-fectiveness is of broad institutional in-terest—a condition not enjoyed by, say,a course in Chaucer. All these factorsmeant that at least some members of anyinstitution’s mathematics departmenthave at least some familiarity with

broader issues of testing, evaluation, andpedagogy.

Second, mathematics has more explicitconnections than most other disciplineswith the preparation of elementary andsecondary school teachers. Even if math-ematics faculty are not explicitly locatedin mathematics education programs,many at smaller colleges and universitiesthat produce large numbers of teachersare aware of pedagogical and assessmentissues through this connection, and thisknowledge and inclination can translatequickly to the postsecondary level.

Finally, at least at the undergraduatelevel, learning outcomes in mathematicsare somewhat more easily specified thanin many other disciplines. Although Ihave learned through the SAUM project

that mathematicians are as apt to dis-agree about the nuances of certain as-pects of student performance as anyother body of faculty—what constitutes“elegance,” for instance, or an effectiveverbal representation of a mathematicalconcept—they can certainly come to clo-sure faster than their colleagues else-where on a substantial portion of whatundergraduate students ought to knowand be able to do in the discipline. Forall these reasons, mathematics was par-ticularly well positioned as a disciplinein 2002 to broaden and deepen conver-sations about assessment through aproject like SAUM.

Evaluating SAUM: Some Reflections

Serving as SAUM’s external evaluatorprovided me with a personally un-matched opportunity to explicitly testmy own beliefs and assumptions as anassessment practitioner. On the onehand, I have spent almost 25 years advo-cating for assessment, helping to developassessment methods and policies, andworking with individual campuses todesign assessment programs. One can-not do this and remain sane unless oneis at some level convinced of assessment’sefficacy and benefit. Yet evaluation is anempirical and unforgiving exercise.SAUM’s central premise was that it ispossible to create a practice-based infra-structure for assessment that depart-ments of mathematics could adapt andadopt for their own purposes, and thusimprove teaching and learning. On thelarger stage of institutional and publicpolicy, this premise has been the basis formy professional career. The opportunityto “assess assessment” as it was acted outby one important discipline —and toreflect on what I found—was both ex-citing and sobering.

On a personal note, I also came tostrongly value my role as “participant-observer” in the project and the oppor-tunities that it provided me (and I hopeto the project’s participants) to see be-yond customary professional bound-aries. For my own part, I was gratifiedto witness many of the lessons about howto go about assessment that I had beenpreaching to Provosts and Deans formany years confirmed in microcosm

The goal for developmentalcourses seems clear: to preparestudents for credit-bearing courses.However, judging by course con-tent, the goal often appears to beremediating what students haven’tlearned in grades high school.These two goals may be quite dif-ferent. High schools attempt to pre-pare students for all possible edu-cational futures. Once a student isin college, however, educationalaims may be much better defined.Many students, for example, maynot ever need to study calculus. Soremediating high school deficien-cies may not be necessary or ap-propriate. The most direct way toassess developmental courses isnot to sample student skills in highschool mathematics but to inves-tigate student success in subse-quent credit-bearing courses.

—Bonnie Gold

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among mathematics faculty at the de-partmental level. But I also saw (at timesto my chagrin) the many differences inperception and failures of communica-tion that can occur when such organiza-tional boundaries are crossed.

As one telling example, at one of theSAUM department-level workshops Iencountered a departmental team thatreported a particularly frustrating bu-reaucratized approach to assessment atits institution being undertaken in re-sponse to an upcoming accreditation re-view—an institution that I knew fromanother source was being cited as a“model” of flexible and creative assess-ment implementation by the accreditorin question. I like to think that such in-sights, and they occurred throughout theproject, helped keep me humble in thebalance of my work in assessment.

At the same time, I like to think that myboundary-spanning role helped partici-pants achieve some of the project’s ob-jectives. An instance here, as the previ-ous example suggests, was my consider-able ongoing work with accrediting or-ganizations, which allowed me to inter-

pret their motives and methods forSAUM participants, and perhaps set abroader context for their local assessmentefforts.

Like many large, multi-faceted projects,SAUM presented many evaluation di-lemmas. Certainly, it was perfectlystraightforward to conduct formative

data-collection efforts intended to guidethe future implementation of project ac-tivities. For example, we collected par-ticipant reactions from the sixteen SAUMworkshops conducted at MAA sectionmeetings and used them to focus and im-prove these sessions. Responses fromsection meeting participants early in theprocess stressed the need for concreteexamples from other mathematics de-partments that faculty could take homewith them. Participating faculty ob-served that they often learned as muchfrom interaction with other participantsas from the material presented. Theselessons were steadily incorporated intosessions at later section meetings (as wellas into the design of the SAUM depart-ment-level workshops that were begin-ning to take shape at that time) and par-ticipant reactions steadily improved.Similarly, we learned through a follow-up of SAUM department-level workshopparticipants that a three-meeting formatwas superior to a two-meeting formatand returned to the former in theproject’s last phase.

But determining SAUM’s effectiveness ina more substantive way posed significantchallenges. The most important of thesewas the fact that the bulk of the project’santicipated impact on mathematics fac-ulty and departments would occur (if itdid) well after the project was over. (Asgood an example of this dilemma as anyis the fact that the publication in whichthis essay appears is one of the project’sprincipal products; yet it reaches yourhands as a reader only after the conclu-sion of the formal project evaluation!)Determining SAUM’s effect on assess-ment practice in mathematics depart-ments thus had to be largely a matter offollowing the experiences of project par-ticipants—particularly those mathemat-ics faculty who attended the multi-ses-sion department-level workshops—when they returned to their home de-partments to apply what they learned.We did this primarily through e mail sur-veys given to participants ten months toa year after the conclusion of their work-shop experience. The multi-session for-mat of the department-level workshopsalso helped the evaluation because ateach of the workshop’s concluding ses-sions we were able to explicitly ask par-

ticipants about their experiences betweenworkshop sessions. What we learnedabout the experience of assessment at thedepartmental level is reported in the fol-lowing section.

We also set the stage for a more formalevaluation of SAUM’s impact by con-ducting an electronic survey of math-ematics departments early in the project’sinitial year. This was intended to pro-vide baseline information about existingdepartment-level assessment practices.

A similar survey of departmental assess-ment practices will be undertaken at theconclusion of the project in the fall of2005. The baseline survey was adminis-tered via MAA departmental liaisons toa sample of 200 mathematics depart-ments stratified by size, institutionaltype, and location. 112 responses werereceived after three email reminders sentby the MAA, yielding a response rate of56%. Questions on the electronic sur-vey were similar to questions that we alsoposed to 316 individuals who attendedSAUM workshop sessions at sectionmeetings, which constituted anothersource of baseline information.

Justifying the project’s potential impact,both sets of baseline data suggested thatin 2002 most mathematics departmentswere at the initial stages of developing asystematic assessment approach. About40% of department liaisons (and only20% of participants at section meetings)reported comprehensive efforts in whichassessment was done regularly in mul-tiple areas, and another 35% (and 31%of participants at section meetings) re-

Although all programs that certifyK-12 teachers have some assess-ment measures in place, these as-sessments have often been createdto satisfy an external requirementrather than to suggest changes inprogram practice. Program direc-tors typically do not seek data thatmay suggest that they change theirpractice. While measures so con-ceived are not useless, they rarelycreate productive conversationsthat lead to enriched student learn-ing.

—Laurie Hopkins

Accreditation agencies and univer-sity administrators generally focusassessment either on the major oron general education. Hence math-ematics departments have not feltmuch pressure to assess mathemat-ics-intensive programs such ascourses taken by future engineers,doctors, architects, economists, andother professionals. Thus these “ser-vice” courses are ripe for assess-ment.

—Dick Jardine

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ported that assessment was done “in afew areas.” About 10% percent of depart-ment liaisons (and 21% of participantsat section meetings) reported that assess-ment was “just getting started,” and 15%percent (and more than a quarter of par-ticipants at section meetings) reportedthat there was “no systematic effort.” Re-spondents from research universities re-ported somewhat lower levels of activitythan other types of institutions. Differ-ences in responses between departmentliaisons and the regular mathematics fac-ulty who presumably attend sectionmeetings are notable and reflect the pat-tern of reporting on institutional assess-ment activities typical of the late 1980s:in these surveys, administrators routinelyreported higher levels of institutionalengagement in assessment than was ap-parent to faculty at their own institutions(El-Khawas, 1987).

Baseline survey results also revealed thatthe mathematics major is the most popu-lar target for assessment activities, with

almost three quarters of responding de-partments indicating some activity here.About half of the departmental liaisonssurveyed indicated that assessment takesplace in either general service or reme-dial courses, and about a third reportedthat assessment takes place in courses forprospective teachers and in placementand advising. Not surprisingly, commu-nity colleges were somewhat more likelyto report assessment in remedial and de-velopmental courses, and less likely to re-

port assessment of the major. Mastersdegree granting universities were morelikely to be engaging in assessment ofgeneral service courses and courses forprospective teachers. Doctorate-grant-ing research universities were somewhatless likely than others to be undertakingassessment in any of these areas.

Survey results suggested that mathemat-ics departments are using a wide varietyof assessment methods. The most popu-lar method was faculty-designed exami-nations, which 62% of departments re-ported using. 53% of departments re-ported using standardized tests, which isnot surprising, but more than 40% ofdepartments reported employing so-called “authentic” approaches like worksamples, project presentations (oral andwritten), or capstone courses. About40% also reported using surveys of cur-rently-enrolled students and programgraduates. Standardized examinationstended to be used slightly more by de-partments engaging in assessment of re-medial and developmental courses,while projects and work samples weresomewhat more associated with assess-ing general service courses.

Finally, the departmental baseline sur-vey asked respondents about their fa-miliarity with Assessment Practices inUndergraduate Mathematics (Gold et al.1999), which had then been in print forseveral years. Some 19% reported thatthey had consulted or used the volume,while another 35% noted that they wereaware of it, but had not used it. Thebalance of 46% indicated that they werenot aware of the volume. As might beexpected, awareness and use were some-what related to how far along a depart-ment felt it was with respect to assess-ment activity. About 61% of respon-

dents from departments reporting com-prehensive assessment programs in placesaid they were at least aware of Assess-ment Practices and about 25% had ac-tively used it. Only about a third of thosereporting no systematic plans had evenheard of the volume and none had usedit. Certainly, these baseline results leaveplenty of room for growth and it will beinstructive to see if three years of SAUMhave helped move the numbers.

Emerging Impacts

Despite the fact that most of the SAUMproject’s impact will only be apparentafter the publication of this report, evalu-ation results to date suggest some emerg-ing impacts. The majority must be in-ferred from responses to the follow-upsurveys administered to department-level workshop participants about a yearafter they attended, focused on their con-tinuing efforts to implement assessmentprojects in their own departments. Many

of these results parallel what others havefound in the assessment literature aboutthe effective implementation of assess-ment at the institutional level, and fordisciplines beyond mathematics.

Colleagueship. Like any change effort inthe academy, implementing assessmentcan be a lonely business because its fac-ulty practitioners are dispersed acrossmany campuses with few local colleaguesto turn to for practical advice or support.Indeed, one of the most important earlyaccomplishments of the national assess-ment movement in higher education wasto establish visible and viable networksof institution-level assessment colleaguesthrough such mechanisms as the AAHEAssessment Forum (Ewell, 2002a).

Results of the evaluation to date indicatethat SAUM is clearly fulfilling this rolewithin the mathematics community. Afirst dimension here is simply the factthat SAUM is a network of mathemati-cians, not just “people doing assessment.”As one faculty member told us, “history,agriculture, and even physics have differ-

The biggest assessment challenge incollege algebra and precalculus isdeciding what the goals are. As longas the course is primarily being usedto prepare students for calculus, thegoals are fairly clear, as is an appro-priate assessment: see how well stu-dents do in calculus. On the otherhand, if the majority of students donot take calculus, the goals of thecourse are much less clear. In thiscase, determining why students takethe course and how they use it is anessential first step in any assessmentprocess.

—Bonnie Gold

Planning and assessment ofcourses intended primarily fornon-majors should extend beyondthe mathematics department. En-suring that the needs of other ma-jors are met, as well as those ofthe institution, can do wonders forthe reputation of the mathematicsdepartment on campus and canlead to support for its other initia-tives.

—Bonnie Gold

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ent flavors of assessment from math-ematics” and the opportunity to workwith other mathematicians on math-ematics topics in assessment was criticalin grounding effective departmental ef-forts. Another dimension is simply thereassurance for individual mathemati-cians who first get involved in localprojects that assessment is a going con-cern. Here it was useful for SAUM par-ticipants to learn that many mathemati-cians are already involved in assess-ment—more than many realized—andthat assessment is not a peripheral activ-ity that only a few mathematics depart-ments are involved in. These points wereseen by SAUM participants as particu-larly important in “selling” assessment toother faculty when they returned to theirhome departments.

The team basis for participation inSAUM workshops meant automaticcolleagueship and mutual support. Sim-ply being away together with colleagues,far from the pressures of everyday cam-pus work was also deemed helpful. Atthe same time, working with other cam-puses at the workshop in multiple en-counters helped build a feeling amongSAUM participants of being part of alarger “movement” that had momentum.This was especially important for facultywho felt, in the words of one, that theyhad “been thrust into a leadership posi-tion on assessment” with little real prepa-ration for this role. Knowing that otherswere in the same position and sharingapproaches about what to do about it wasseen as especially important.

The same was true of learning aboutmore specific assessment approacheswhere, as expected, SAUM departmentsborrowed liberally from one another.But by far the most important impact ofhaving colleagues was the stimulus theyprovided to keep participating depart-ments moving. The need to present de-partmental progress periodically andpublicly was important to this dynamic:teams at the workshops knew that theywere going to have to report to theirpeers, so worked hard to have somethingto say. As one representative noted, “ifwe had run out of time and didn’t ac-complish what we intended, it probablywouldn’t have had any consequences oncampus—we were already doing morethan most departments—but because wehad to have presentations ready at dif-

ferent workshops we were pushed to fol-low through on plans and to discuss andrevise our activities.” This “peer stimu-lation” effect was a particularly impor-tant dynamic in SAUM, and parallelssimilar lessons learned aboutcolleagueship in other assessment-re-lated change projects (e.g., Schapiro andLevine, 1999).

On a more sobering note, however, earlyevaluation results also suggest the diffi-culty of maintaining colleagueship ab-sent the explicit framework of a project

or a visible network to support it. Fewdepartmental representatives reportedcontacting other participants on theirown, largely due to pressures of time.Again confirming lessons of the assess-ment movement more generally, an in-frastructure for sustaining assessment inmathematics must be actively built; it willnot just happen as a result of peoples’good experiences. Learning By Doing.Another lesson of the assessment move-ment nationally was the importance ofearly hands-on experience and practice.Evaluation results to date suggest thatSAUM is strongly replicating this find-ing within mathematics departments.Rather than looking for the best “modelprogram” and planning implementationdown to the last detail, SAUM depart-ments, like their colleagues in many dis-ciplines, learned quickly that time in-vested in even a messy first efforttrumped similar investments in “perfect-ing” design. As one participant told us,“the most important lesson that I learnedwas to just get started doing something.”Another said, “begin with a manageableproject, but begin.” Small projects cannot only illustrate the assessment processin manageable ways with only limited in-vestments of resources but can alsoquickly provide tangible accomplish-ments to show doubting colleagues. Theimportance of this insight was reinforcedby the fact that most participants alsodiscovered that assessment was a gooddeal more time-consuming than theyhad first imagined, even for relativelysimple things.

At the same time, participating depart-ments learned the importance of find-ing their own way in their own way, andthat local variations in approach are bothlegitimate and effective. Like their col-leagues elsewhere in assessment, they alsolearned tactical lessons about implemen-tation that could only be learned by do-ing. As one department reported, “forus, designing an assessment programmeans finding a balance between gettinggood information ... and not increasingfaculty workload too much.” Additionalcomments stressed the importance ofknowing that “one size does not fit all”and that good assessment should be re-lated to local circumstances.

Assessment is a marathon, not asprint. Formerly, accreditation vis-its and program reviews were pe-riodic events that prompted fran-tic last-minute efforts to assembledata in support of vague claims ofdepartmental effectiveness. Nowcontinuous assessment is the goal,the way departments routinelyshould go about the business ofhelping students learn mathemat-ics.

—Dick Jardine

The final stage in an assessmentprogram is the feedback loop.What do the results say aboutwhether students are meeting thegoals the faculty has set, andwhether the departmental pro-gram is designed in such a waythat the students can meet thosegoals? In other words, if studentsare falling short, is it the studentswho are underperforming or is itthe program that is deficient? Ineither case, the faculty needs toreview the program to see whatimprovements can be made sothat students are successful in de-veloping the desired competen-cies.

—William Marion

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Finally, participants encountered aspectsof local departmental culture that couldnot be addressed through formulaicmethods. One of them summarized thiscondition nicely: “there are rules at myinstitution about how we have to do as-sessment even though those rules areunwritten, unarticulated (except whenviolated), and specific to my institutionand the larger community. I used tothink that these rules were to be foundsomewhere in the literature ... now Iknow that I’m dealing with the unknownand with rules that are likely being madeup as we go. This makes me much moreconfident in my own ideas instead ofbacking down when I am told that some-thing is ‘not allowed.’”

Growing Maturity. As their projectsevolved, most participating departmentsreported a growing maturity with assess-ment. Several departments doing pro-gram-level assessment, for example, hadreplicated their assessment models inanother related department or program(e.g., computer science), or had beenworking with other departments at theinstitution to help them develop an as-sessment approach. Others imple-mented or regularized activities that theyhad planned or experimented with at theworkshop. Most indicated that they hadexpanded their departmental assessmentefforts to become more systematic andcomprehensive—adding new assessmenttechniques and applying them to morecourses or involving more faculty. As oneparticipating department reported, “Webelieve one of our greatest accomplish-ments is to have engaged a significantproportion of the department (morethan half the faculty) in assessment inone way or another.”

Growing maturity is also apparent in or-ganizational and motivational dimen-sions. With regard to the former, severaldepartments reported that they had dis-covered the importance of having a de-partmental advocate or champion for as-sessment who could set timelines, en-force deadlines, and provide visibility. Afew also reported “regularizing” assess-ment activities—in one case allowing theoriginal project leader to hand off assess-ment activities to a newly interested fac-ulty member to coordinate or lead. As

one departmental representative put it,participating “got us off to a great startand developed a sense of confidence thatwe are in a better posture with assess-ment than other departments on ourcampus.”

Motivational shifts were more subtle, butreflected a shift toward internal insteadof external reasons for engaging in as-sessment. Mirroring experience in otherfields, many mathematicians first heardabout assessment through accreditationor their administration’s desire to “cre-ate a program.” But as their participa-tion in SAUM progressed, many also re-ported new attitudes toward assessment.As one faculty member put it, “Earlier[activities] were about responding tooutside pressure…[later activities] wereabout doing this for ourselves.” Anothernoted, “Most people [at my institution]are not as advanced in assessment as weare in the mathematics department. ...The task still seems to most people like anecessary activity conducted for externalreasons, rather than an activity that hasintrinsic value to improve their ownwork.” This shift requires time to accom-plish and findings from other fields em-phasizes the fact that outside pressuresor occasions are important to start thingsmoving on assessment at the institu-tional level (Ewell, 2002b). But SAUMparticipants began to recognize also thatsustaining assessment requires the kindof internal motivation that can only be

developed over time and through collec-tive action.

Changing Departmental Culture. Twentyyears after the emergence of assessmentas a recognizable phenomenon in highereducation, it has yet to become a “cul-ture of use” among faculty in disciplinesthat lack professional accreditation.Many reasons for this have been ad-vanced, ranging from alien language tolack of institutional incentives for en-gagement, but by far the most prominentis the imposition of assessment require-ments by external authorities (Ewell,2002a). Consistent with national expe-rience in other disciplines, SAUM work-shop participants thus returned to theirown departments determined to make adifference, but they faced an uphill battleto change their colleagues’ attitudesabout assessment and, in the longer term,to begin to transform their department’sculture.

A first milestone here was the fact thatparticipation in SAUM itself helped le-gitimize the work of developing assess-ment. Being part of a recognized, NSF-funded project was important in con-vincing others that the work was impor-tant. So was the clear commitment ofworkshop participants to working ontheir projects. As one faculty membertold us, “Because [the participants] weregenuinely interested, ... that interest andenthusiasm has been acknowledged byothers.” Several also mentioned the value

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of knowing the “justifications for assess-ment” in communicating with fellow fac-ulty members.

But SAUM participants also tended toend up being the “assessment people” intheir departments—accorded legitimacyfor their activities to be sure, but not yetjoined by significant numbers of col-leagues. As one wryly stated, “it has prob-ably made more work for me as when Ishare an idea of something that we cando, I usually get put in charge of doingit.” Another doubted that he and hisSAUM teammate had gained much stat-ure in the department because of theirparticipation, “but I guess at least morepeople recognize what we have done.”

As national experience suggests, more-over, wider impacts on departmentalculture with respect to assessment re-

quire time to develop—more than thethree years of engagement most SAUMparticipants have to date enjoyed. Mostindicated that their colleagues were ingeneral more informed about assessmentas a result of SAUM and were thereforemore willing to agree that it might bebeneficial for their departments or insti-tutions. So despite little groundswell ofenthusiasm, most did report slowprogress in changing departmental atti-tudes. One participant captured the typi-cal condition succinctly when he re-ported that his colleagues “remain largelyindifferent to assessment…they are infavor of improving programs as long as

it doesn’t bother them.” Another de-scribed this condition as follows: “I thinkthere is still a degree of skepticism aboutall of this, but at least we don’t run intooutright hostility or claims that this is alla great waste of time and effort.” Echo-ing these comments, a third reported that“the department has definitely becomemore open to the idea of assessment ...for one thing, they have finally realizedthat it is not going away ... for another, ifthere is someone willing to do the work,they will cooperate.” These are no smallachievements. But the overall pattern ofimpact to this point remains one of in-creased awareness and momentum forassessment among SAUM departmentswith only a few early signs of a changeddepartmental culture.

“Assessing assessment” through theSAUM evaluation remains an ongoingactivity. Like assessment itself in manyways, the task will never be finished. Butit is safe to conclude at this point thatmathematics has built a resourcethrough SAUM that if maintained, willbe of lasting value. On a personal note, Ihave learned much from my colleaguesin mathematics and have been gratefulfor the opportunity to work with themon a sustained basis. And from a nationalperspective, I can say without reservationthat they are making a difference.

References

El-Khawas, Elaine (1987). CampusTrends: Higher Education Panel Report#77. Washington, DC: AmericanCouncil on Education (ACE).

Ewell, Peter T. (1987). Assessment, Ac-countability, and Improvement: Manag-ing the Contradiction. Washington, DC:American Association of Higher Edu-cation (AAHE).

Ewell, Peter T. (2002a). An EmergingScholarship: A Brief History of Assess-ment. In T.W. Banta and Associates,Building a Scholarship of Assessment.San Francisco: Jossey Bass, 3-25.

Ewell, Peter T. (2002b). A Delicate Bal-ance: The Role of Education in Man-agement. Quality in Higher Education,8, 2, 159-172.

Gold, Bonnie, Sandra Z. Keith, and Wil-liam A. Marion, eds. (1999) Assessment

Practices in Undergraduate Mathemat-ics. MAA Notes, Vol. 49. Washington,DC: Mathematical Association ofAmerica. Web: http://www.maa.org/saum/maanotes49/index.html.

National Governors Association (1986).Time for Results: The Governors’ 1991Report on Education. Washington, DC:National Governors Association.

National Institute of Education, StudyGroup on the Conditions of Excellencein American Higher Education (1984).Involvement in Learning: Realizing thePotential of American Higher Education.Washington, DC: U.S. GovernmentPrinting Office.

Schapiro, Nancy S. and Levine, Jodi H.(1999). Creating Learning Communi-ties: A Practical Guide for Winning Sup-port, Organizing for Change, and Imple-menting Programs. San Francisco:Jossey-Bass.

U. S. Department of Education, NationalCommission on Excellence in Educa-tion (1983). A Nation at Risk: The Im-perative for Educational Reform. Wash-ington, DC: U.S. Government PrintingOffice.

Peter Ewell, project evaluator, is vice presi-dent of the National Center for HigherEducation Management Systems(NCHEMS). E-mail: peter@nchems.org.

Effective placement can be crucialto student success. Good placementprocesses generally involve multiplecomponents including results of aplacement test, high school rank,grade point average, last mathemat-ics course taken, SAT or ACT scores,and student self-descriptions of howgood they are at mathematics. Ex-amining how well each factor andthe overall system predicts successand adjusting the formula in re-sponse to this analysis is an impor-tant part of the assessment of theseintroductory courses.

—Bonnie Gold

A complimentary copy of the re-port Supporting Assessment in Un-dergraduate Mathematics is avail-able to the first 1000 departmentssubmitting requests through theproject website http://www.maa.org/saum.