THE NEWSLETTER OF THE MA THEMA TICAL ASSOCIATION OF AMERICA The Geometry Center Reaches Out Volume IS, Number 1 A major mission of the NSF-sponsored University of Minnesota Geometry Center is to support, develop, and promote the communication of mathematics at all levels. Last year, the center increased its efforts to reach and to educate diffe rent and dive rse groups ofpeople about the beauty and utility of mathematics. Center members Harvey Keynes and Frederick J. Wicklin describe In this Issue some recent efforts to reach the general public, professional mathematicians, high school teach• ers, talented youth, and underrepresented groups in mathematics. 3 MAA President's Column Museum Mathematics Just a few years ago, a trip to the local 7 Mathematics science museum resembled a visit to Awareness Week a taxidermy shop. The halls of the science museum displayed birds of 8 Highlights from prey, bears, cougars, and moose-all stiff, stuffed, mounted on pedestals, the Joint and accompanied by "Don't Touch" Mathematics signs. The exhibits conveyed to all Meetings visitors that science was rigid, bor• ing, and hardly accessible to the gen• 14 NewGRE eral public. Mathematical Fortunately times have changed. To• Reasoning Test day even small science museums lit• erally snap, crackle, and pop with The graphical interface to a museum exhibit that allows visitors to interactive demonstrations of the explore regular polyhedra and symmetries. 20 Letters to the physics ofelectricity, light, and sound. Editor Visitors are encouraged to pedal, pump, and very young children to adults, so it is accessible push their way through the exhibit hall. The to the casual browser yet has deep mathemati• 23 Career Kit for message conveyed to today's museum visitor cal content for the more serious investigator. Advisors is equally clear: science is vibrantly alive, is Computer graphics allow users to continuously fun, requires participation, and can be reward• deform a Platonic solid (that is, a regular poly• ing. hedron) into related polyhedra that are invari• 27 Contributed ant under the same symmetry group as the Papers in But where are the mathematics exhibits? One original Platonic solid. In particular, given a Burlington could argue that math is behind the scenes in Platonic solid (e.g., the cube) the museum visi• almost every exhibit; but if math is behind the tors can investigate which Archimedean solids 30 Institute in the scenes, then it is not in the public's eye. In an have the same symmetries. They can also dis• effort to make mathematics more visible in History of cover that within the family of polyhedra, there science museums, the Geometry Center is another Platonic solid (e.g., the octahedron) Mathematics teamed up with the Science Museum of Min• that is "dual" to the first Platonic solid, meaning nesota to develop an interactive mathematics that the faces and vertices of the first are trans• 36 Employment exhibit for the general public that uses com• formed into the vertices and faces of the second. Opportunities puter graphics to illustrate geometric ideas Users may also change the symmetry group in such as symmetry groups, tHings, regular order to explore a different family ofpolyhedra, polyhedra, and spherical geometry. or to explore planar tilings. The Mathematical The exhibit was designed for the science mu• Association of America seum environment with visitors ranging from See Geometry Center on page 4 1529 Eighteenth Street, NW Washington, DC 20036 Highlights from the Joint Mathematics Meetings on page 8 FOCUS February 1995 FOCUS Editorial FOCUS is published by The Mathematical Proof Beyond Reasonable Doubt Association of America, 1529 Eighteenth Street Northwest, Washington, DC 20036- Three events in the news during 1994 brought the issue of proof into the spotlight. One 1385, six times a year: February, April, June, was the O. J. Simpson trial taking place in Los Angeles. The second was the news that August, October, and December. Andrew Wiles had corrected the proof of Fermat's Last Theorem he had announced in Editor and Chair of the MAA Newsletter June 1993, but which was subsequently found to be flawed. The third was the discovery Editorial Committee: Keith J. Devlin, that the computer chip that is the computational heart of millions of widely used personal Saint Mary's College of California computers (Intel's Pentium chip) has a significant mathematical flaw. Associate Editor: Donald J. Albers, MAA The standard of proof required by law is well known: proof beyond reasonable doubt is Associate Executive Director and Director the all-important phrase. Of course, like most concepts in everyday life, once you start of Publications and Programs to examine it, it is not at all clear just what this definition amounts to. What is reasonable Managing Editor: Harry Waldman, MAA to one person can seem quite unreasonable to the next. I have heard it said-though I do Production Specialist: Amy Stephenson• not know if it is true-that prosecution lawyers have been known to object to the pres• Fabbri, MAA ence of mathematicians on juries, on the grounds that such persons will be able to see 'reasonable doubt' in any legal argument, and are thus very reluctant to convict. Proofreader: Meredith Zimmerman, MAA In the OJ. Simpson case, whatever the final outcome, neither the prosecution nor the Copy Editor: Nancy Wilson, Saint Mary's defense are likely to produce a proof that meets the totally precise standards of math• College of California ematics. From the time of its introduction in ancient Greece by Thales in the sixth century B.C., the formal proof has been the bedrock of mathematics, what Cambridge (England) Letters to the editor should be addressed to: mathematician Michael Atiyah has referred to as "the glue that binds mathematics to• Keith Devlin, Saint Mary's College of California, P.O. Box 3517, Moraga, CA gether." It is the absolute certainty provided by rigorous proofs that sets mathematical 94575; e-mail: [email protected] knowledge apart from all other kinds of knowledge, including all other sciences. Un• doubtedly, it is a lack of a proper appreciation of the nature of mathematical proof that The FOCUS subscription price to individual results in continued attempts of amateur mathematicians to square the circle or to trisect members of the Association is $6.00, included in the annual dues. (Annual dues the angle, long after mathematicians have proved, conclusively, that such feats are for regular members, exclusive of annual impossible. SUbscription prices for MAA journals, are The proof that no ruler and compass construction can trisect a general angle is very $68.00. Student and unemployed members receive a 66 percent discount; emeritus different from any kind of proof O. J. Simpson's defense team might offer to prove that members recei ve a 50 percent discount; new it was impossible for their client to have commited the crime of which he stands accused. members receive a 40 percent discount for Even if the defense produces an eye witness who claims that O. J. was not at the murder the first two membership years.) scene, that will still not amount to mathematical proof; there always remains the possi• Copyright © 1995 by The Mathematical bility, however remote, of mistaken identity, hallucination, or plain dishonesty. In legal Association of America (Incorporated). cases, the judge and the jury must decide whether such doubts are 'reasonable' or not. Educational institutions may reproduce In mathematics, the standard of proof is far higher. The mathematician demands com• articles for their own use, but not for sale, plete, 100% certainty. provided that the following citation is used: "Reprinted with permission of FOCUS, the At least that is what our propaganda says. But does the product live up to the image? Has newsletter ofThe Mathematical Association Andrew Wiles proved Fermat's Last Theorem or hasn't he? What will it take to convince of America (Incorporated)." you? What will it take to convince Mazur or Ribet or Faltings, or any of the other handful Second-class postage paid at Washington, of number theorists who are best able to understand Wiles' long and difficult reasoning? DC and additional mailing offices. Would you be convinced if these experts agreed that the proof was correct? Won't there Postmaster: Send address changes to the remain a 'reasonable doubt' in your mind that these experts could be wrong? After all, Membership and Subscriptions De• Wiles himself is surely the greatest expert on his argument, and he was wrong initially. partment, The Mathematical Association of It took him well over a year to come up with a second version of the argument that America, 1529 Eighteenth Street Northwest, corrects (he believes) his earlier error. Washington, DC 20036-1385. ISSN: 0731-2040 With textbook examples such as the infinitude of the primes or the irrationality of the Printed in the United States of America. square root of 2, the issue seems fairly clear. At least it seems clear to mathematicians. Printed on recycled paper. Whether or not a typical trial jury made up of twelve ordinary citizens would be able to rule on the correctness or otherwise of Euclid's proof that there are infinitely many prime numbers is a question to which I do not know the answer. But, for the sake or argument, since these comments are written by a mathematician for consumption 'in the math• ematical family,' let's assume the absolute correctness of these classical proofs. After all, they are all sufficiently short and of a sufficiently simple nature that anyone with a basic mathematical education can follow them. 2 February 1995 FOCUS
of free airline its relations with other mathematics and President's Column mileage. A science organizations through various more serious bodies with acronyms like IPBM, It is a unique privilege and challenge to answer is that CBAMN,andCSSP.Inaddition,thePresi• serve as President of the MAA. I am lucky the President dent plays a major role in the Mathemati• to have extremely able predecessors who presides, rep• cal Olympiad ceremonies and represents serve as models, though I must confess resents, man• the national MAA at as many section meet• that my immediate predecessor, Don ages, and leads ings as possible. Kreider, is an especially hard act to fol• the MAA. low. On balance, though, I regard myself The President is the elected leader of the These are MAA. Fortunately, the MAA is run by an as very fortunate to be surrounded by many listed in ap• amiable and helpful colleagues. extremely able and professional staff in proximate reverse order of their impor• Washington headed by Marcia P. Sward, Several people have asked me, "What does tance. The President presides over and we have an excellent working rela• the MAA President do?" An outside ob• meetings of the Board of Governors and tionship. Among other things, her staff server would say that he or she reads reams the subcommittees that do much of the supports the extensive volunteer work in of journals, documents, reports, and pro• week-to-week work, namely the Execu• the organization. Moreover, there's a great posals, engages in lots of correspondence tive Committee and the Finance Commit• (especially via e-mail), and collects lots tee. The President represents the MAA in See President's Column on page 4
But what about proofs that stretch over darned convincing, but it is not quite the entific Knowledge, Popper argued that hundreds of pages, proofs that depend on comforting picture (for mathematicians, scientific theories are never proved; they other hundred-page-proof theorems ob• at least) provided by the notion of formal are at best accepted until shot down. tained by other mathematicians, and which proof that was so nicely captured by the This is not to say that mathematics is the use mathematical concepts and techniques mathematical logicians of the late nine• same as the natural sciences. There are that perhaps no single person fully under• teenth century. important differences. Mathematical stands? Wiles' proof definitely falls into The fact is, when you forget the rhetoric know ledge has a certainty far greater than this category. and look instead at actual mathematical the knowledge offered by any scientific Or what about proofs such as Appel and praxis, when you look at what mathema• theory (let alone the 'facts' established in Haken's 1976 proof of the Four Color ticians actually do, not what they often a court oflaw). But at the same time, there Theorem, which involved an extensive say they do, then what passes for 'proof' are similarities that those of us in the computer calculation that no human can is far closer to the lawyers' notion than we mathematics profession often overlook. check by hand? The recent discovery of care to admit. A definition of proof that This state of affairs is undoubtedly less the flaw in the Pentium chip reminds us in most accurately reflects real, live math• comforting than the secure picture offered a graphic fashion that computers are fal• ematical proofs, is to be found not in math• by formal logic. (Though when was the lible-they too can make mistakes. Given ematical logic but in the sociology of last time you saw a proof of a real theorem the complexity of any modern computer science. In terms of the arguments that that followed the strict rules of logic? system, it is surely reasonable to assume determine what mathematical statements Remember, it took Whitehead and Russell that the computer system used in the proof we believe and what we do not, aproofis 362 pages of formal deduction in their of the Four Color Theorem had faults; often just an argument that (i) has been mammoth work Principia Mathematica indeed, it would be unreasonable to as• accepted by a number of mathematicians before they were able to prove 1+1 = 2.) sume otherwise. This leaves the proof of whom the community at large feels it can But, like it or not, for all its lack of com• the Four Color Theorem dependent on our trust on such matters, and (ii) has not yet fort, such is the world of mathematics that assumption that the errors in the system, been shown to be false. we actually inhabit. whatever they were, did not adversely If that is too uncertain for you, then you Personally, though I am attracted to the affect the calculations on which the proof will probably never know whether or not precision of cold logic, I find the socio• depended. Given that Appel and Haken's Fermat's Last Theorem has been proved logical aspect-the element of 'uncer• proof was checked by others, using differ• or not. Or whether the classification of tainty' -a pleasingly human aspect of the ent computer systems running different finite simple groups has been completed. subject. How about you? programs, such an assumption is surely Or any of hundreds of other 'facts' of reasonable. But in acknowledging that -Keith Devlin modern mathematics. fact, we pass from the idealistic world of The above are the opinions ofthe FOCUS 'absolute mathematical certainty' to the If the definition of a proof offered above editor, and do not necessarily reflect the less secure world of 'beyond reasonable looks familiar, it should. It is more or less official views of the MAA. The FOCUS doubt' -admittedly not the very messy the same thing that the late philosopher editor says he accepts the proofofthe Four version of the legal world, rather the far Karl Popper said about science praxis. In Color Theorem, and will accept the proof more certain 'beyond reasonable math• his highly influential 1963 book Conjec• ofFermat 's Last Theorem, once it has been ematical doubt.' This latter notion is pretty tures and Refutations: The Growth ofSci- certified by the appropriate experts.
3 FOCUS February 1995
President's Column from page 3 our students get the best education pos• em age, instant communication and the flood of information have increased the deal of wisdom and talent on the Execu• sible, rather than try to kill the reform pace of MAA business. The challenge is tive and Finance Committees, so the Presi• movement with anecdotes. to keep everyone involved and maintain dent never has to make uninformed People have also asked me what I hope to good communication. A new task force decisions. The sorts of issues that come up accomplish as MAA President. In addi• has been created by the Board to look into include decisions concerning appropriate tion to causing no harm, I have the follow• exactly these issues. computer support, electronic services for ing high priorities. I am very concerned the membership, and various financial about the job situation, though I have no Finally, as mentioned above, I support the issues. solutions. However, communication be• reform in mathematics and science edu• tween established mathematicians and cation in this incredibly fast-moving world My first rule ofleadership is "Do no harm!" "young" mathematicians has been sub• where quality education for all is essen• This is especially appropriate when so stantially improved over the past year tial. We need to reverse the widening gap many people are already doing so many thanks to the creation ofYMN, the Young between the haves and have-nots, while things well. For example, the MAA pub• Mathematicians' Network. Its weekly e• recognizing that this societal problem lications program and its numerous edu• mail newsletter has provided an invalu• extends far beyond the education estab• cation initiatives are in extremely able able forum for sharing frustration and lishment. hands. Let me say right off that I feel that information. At section meetings, I have the reforms taking place in science and In future columns, I will report on what is been trying to increase sensitivity among mathematics education are fundamentally currently going on in the MAA. Also, as I those who do the hiring. The MAA has sound even though there are potential pit• am wading through those reams of paper, already taken an important step in provid• falls. It is, of course, essential that we I will be on the lookout for references and ing professional development for new fac• continue to motivate and fully prepare the quotes that I believe readers will find in• ulty through the NExT program. More most academic students in serious solid teresting. Comments on these columns and needs to be done by the MAA for profes• mathematics and science. I urge all mem• other aspects of the MAA are welcome. sional development at all levels. bers of our community to work with the My e-mail address is ross@math. reform movement, make the improve• A second major concern of mine is to uoregon.edu. ments necessary, and make sure that all of improve Board effectiveness. In the mod- Ken Ross Geometry Center from page I The Science Museum of Minnesota esti• oping future exhibits with the Science mation to another without having to worry mates that about three thousand people use Museum of Minnesota. We hope that in (or even know!) about where the informa• the exhibit during a typical week. Geom• the near future, people visiting science tion is coming from. etry Center staff member Tamara Munzner, museums will also have the impression We at the Geometry Center are using the one of the authors of the current museum that mathematics, like the sciences, is alive, Web as part of our effort to make math• exhibit, is currently talking to other sci• accessible, and fun! ematics more accessible and understand• ence museums who may be interested in able to non-mathematicians. Documents this exhibit, and is also looking into devel- Mathematics on the Internet are written in hypertext, which allows users Every year, millions of people go to sci• to "click" on certain words, phrases, and ence museums. We are rapidly approach• pictures in order to obtain additional in• ing an age in which a comparable number formation. Our hypertext documents are of people have access to the Internet. The designed to present mathematics to the past year has seen a tremendous surge in masses, using a combination of exposi• the number of Internet users; a partial tory prose, computer-generated graphics, explanation for this is the popUlarity of and animations. Furthermore, we have so-called WorldWide Web (WWW) used the unique capabilities of the Web to browsers such as NCSA Mosaic. (See the produce interactive documents that allow August 1994 FOCUS for more informa• users to explore and discover mathemat• tion on using NCSA Mosaic as a tool for ics for themselves. We have a very popu• Internet access.) You might say that the lar Gallery of Interactive On-Line Internet has become a "Web of Dreams" Geometry in which Internet explorers in• that operates under the philosophy "Build vestigate and visualize the mathematics it, they will come." Using a WWW of quasiperiodic tilings of the plane, wall• browser, it is relatively painless (in fact, paper groups, geodesics on manifolds, it's a lot of fun!) to electronically obtain Riemann surfaces, and symmetries of the A view ofthe Geometry Center s interactive Gallery information from all over the world. The hyperbolic plane. The Geometry Center's ofOn-Line Geometry, as it appears using the NCSA WWW browsers use "hyperlinks" to al• Mosaic WWWbrowser. Each picture and underlined WWW documents are accessible from our word is actually a "hyperlink" that brings up addi• low the user to pass from one site of infor- so-called "home page." Readers who have tional information.
4 February 1995 FOCUS access to the WorldWide Web may view ing with this new immersion; there are WWW documents, but we cannot evalu• our home page using the Uniform Re• movies of the rotating surface and of a ate who these people are, nor determine source Locator (URL) given by family of level sets, and there are interac- their backgrounds. Are we reaching teach• ers from high schools and commu• http://www.geom.umn.edu nity colleges? Are we reaching In fact, this article is available on-line populations that are underrepresented as well as part of the Geometry Center's in mathematics? Are we making a Online Multimedia Document Library! difference in the way that ordinary people think about mathematics? Are The Geometry Center is also looking at we affecting the way that mathemati• how to use the WorldWide Web in order cians think about presenting math• to disseminate research-level math• ematics? We have some (very ematics to professional mathematicians. positive) comments sent to us by us• Already we live in an age in which a ers, but it is difficult to extrapolate large portion of research mathematics from this data. We need to understand is distributed in preprint form, thanks how to assess our outreach through to TEX, for formatting equations, elec• the Internet better. tronic drawing programs for producing figures, and laser printers for putting it Summer Enrichment A documentfrom Davide Cervone smultimedia preprint. The docu- all on paper. Color printers and copiers ment not only presents a new mathematical result, but also illus• Through the WorldWide Web, the Ge• now allow mathematicians to include trates one way to present mathematics using current technologies. ometry Center distributes information color figures in preprints, thus increas• to anyone who wants it. But the Ge• ing our ability to convey information about tive pictures that the reader may manipu• ometry Center also runs hands-on, in• geometric structures. Yet our pre prints and late in order to see the surface from a de• house, summer enrichment programs in journal articles are fundamentally limited sired viewpoint. We are currently which we target certain populations. This to static images. No paper-based preprint considering how to use this powerful past summer our target populations were or journal allows authors to publish an medium to communicate mathematics in sixth through eighth grade females and animation showing a rotating surface. curricular material for undergraduates in underrepresented minorities, talented under• None allow the reader to alter parameters mathematics, science, and engineering. graduates, and high school math teachers. interactively in a model and subsequently generate a new figure that perhaps even We believe that our presentation of math• The four-week summer program for sixth the author has never seen. However, these ematics on the Web is successfully reach• through eighth graders was a joint venture abilities are available in hypertext docu• ing thousands of people from all walks of with the Office of Special Projects in the ments, and the Geometry Center has be• life, but it is very difficult to assess the University of Minnesota School of Math• gun to experiment with these possibilities ematics. The program was the culmina• through its Web documents. tion of a year-long effort to provide exciting and enriching mathematics to stu• As an example, we highlight a recent dents who are in the process of making mathematical result by Center postdoc decisions about whether (or how much) Davide Cervone. Cervone discovered a mathematics will playa part in their fu• certain immersion of the real projective ture. The curriculum for the summer course plane with one handle into three-space that was developed by postdocs at the Geom• settles a long-standing conjecture in sim• etry Center and in the School of Math• plicial geometry. The result is being sub• ematics, and featured units on planar mitted to a traditional journal, but because Sum of the angles isometries, symmetry groups, and the Pla• the fundamental result concerns exhibit• of a triangle tonic and Archimedean solids. The last two ing a surface with special properties, weeks of the summer institute focused on Cervone chose to write a preprint using geometry and visualization, and made ex• hypertext as his medium. Cervone's docu• tensive use of the Geometry Center. The ment, obtainable by using the URL Spberical Euclidean students met at the Geometry Center to http://www.geom.umn.edu/docs/dpvc/ use the Center's facilities and Center-pro• RP2.html duced software. As a closing group project of the solids unit, the students constructed begins with a highly accessible introduc• A sample frame from Lori Holthus' video that intro• what we believe to be the world's largest tion to the problem (complete with duces high school teachers and students to non• Euclidean geometry. paper model of the MAA symbol-a six• hyperlinks to basic definitions !), and then foot-high icosahedron. proceeds to present and prove his new impact of our efforts quantitatively. We result. He then provides the reader with have records of hundreds of thousands of This construction was motivated by a multiple ways of visualizing and interact- connections to the Geometry Center FOCUS article (February 1994) that
5 FOCUS February 1995 showed a group of elementary school stu• A much longer and more intensive sum• dents who had built the world's largest mer program is the Geometry Center's rhombicosidodecahedron. In that polyhe• research experience for undergradu• dron, each side had a length ofthirteen inches. ates. We invite twenty undergraduates Rather than try to better that record, we chose to spend ten weeks at the Geometry to construct an icosahedron of roughly the Center conducting research in topics same size; this resulted in our icosahedron chosen from mathematics, computer having edges that measure thirty-seven science, and math education. They inches. The event attracted the attention of work under the supervision of their the local television stations and newspapers "coach" Tony Phillips from SUNY who not only covered the event but also in• Stony Brook, and collaborate on The summer enrichment class poses with their creation: terviewed the beaming students. Weeks later, projects with Centerpostdocs, Univer- an icosahedron of staggering proportions! the icosahedron was prominently displayed sity math professors, and representa• at the MAA booth at MathFest, the joint tives from industry. This year many ofthe teachers discussed and debated applica• summer meeting of the MAA and theAMS. students presented their results at a poster tions of statistics to DNA fingerprinting We hope to encourage more students to ex• session during MathFest, and several will and questioned whether vitamin supple• plore symmetry and solid geometry by build• also be traveling to an upcoming AMS ments have any significant impact on ing models; a teacher's guide to creating a meeting to present their work. The stu• health. The course participants learned that large icosahedron as a class project is avail• dents tackled difficult problems in celes• seemingly significant events (such as win• able on-line from the Geometry Center's tial mechanics, dynamical systems, ning and losing streaks in sports) may WWWserver Riemannian manifolds, and mathematical actually be the natural outcomes of ran• biology. The products of their labors in• dom events. Information about the http://www.geom.umn.edu/docs/fjw/ clude software and videos to illustrate the CHANCE project, including a complete Icosahedron.html problems and solutions. One student, who set of handouts from the summer course, began teaching high school mathematics is available from the Geometry Center's Calculus: A Student this fall, even produced a computer-ani• WWW server at the URL mated video designed to introduce high http://www.geom. umn .edu/docs/snell/ Centered Approach school students and teachers to non-Eu• chance/welcome.html clidean geometry! She completed this Ithaca College will host a National Sci• sophisticated video in just ten weeks Conclusions ence Foundation funded five-day work• (thanks to the support of the Geometry shop, June 25th-30th, on the use of Center staff) and the result was so impres• In all of these varied activities among di• classroom activities and group projects sive that a publisher wants to market her verse popUlations, there is one common to teach calculus. The workshop will pro• video. thread: the Geometry Center strives to vide an overview of the calculus reform present mathematics visually and interac• movement and the experience of work• In addition to our summer program for tively. We use tools such as computer ing with the materials developed at Ithaca undergraduates, the Geometry Center also graphics, animations, hypertext, videos, College. A principal aim is to enable a annually runs an intensive two-week resi• and world-record icosahedra to achieve wide range of participants from colleges dential summer course for high school our goals. And although we do not charge and universities, two-year colleges, and teachers of mathematics. The programs admission to visitors, we are like a mod• secondary schools to become involved in introduce new mathematics to the teach• ern science museum: we are trying to con• these new pedagogical approaches. ers and encourages them to develop ac• vey to students, teachers, and the general tivities suitable to use in their classrooms. public that mathematics is alive, requires Grant funds will provide for room and This year the program on probability and participation, and can be extremely re• board. Faculty who teach significant num• statistics was called CHANCE and ana• warding. We continue to seek new oppor• bers of students from underrepresented lyzed current events as reported in news• tunities and vehicles to spread this groups are particularly encouraged to ap• papers and journals. For example, the important message. ply. Applicants will be expected to dem• onstrate a commitment to use some of the materials in a calculus classroom during Pr~liminary Call for Student Papers the following academic year. There will be a variety of follow-up activities for Eighth Annual MAA Undergraduate Student Paper Sessions The eighth MAA participants. Undergraduate Student Paper Sessions will take place at the Joint MAA-AMS summer For additional information and an appli• meeting in Burlington, Vermont, August 6-8, 1995. For further information and/or to cation form, contact Diane Driscoll submit nominations for fifteen-minute papers along with brief abstracts, contact Ron Schwartz, Department of Mathematics Barnes, ComputerlMath Sciences, University of Houston-Downtown, 1 Main St., and Computer Science, Ithaca College, Houston, TX 77002~ e-mail: [email protected] number of awards to support 1212 Williams Hall, Ithaca NY 14850- student travel have been made available through a grant from the Exxon Education 72S4~ e-mail: [email protected]. Foundation. Deadline for submission of abstracts is June 26th. 6 February 1995 FOCUS Mathematics Awareness Week-April 23-29, 1995 Mathematics and Symmetry Mathematics Awareness Week is held each year to increase public understanding of and appreciation for mathematics. This year's theme is "Mathematics and Symmetry." More information can be obtained from the Joint Policy Board for Mathematics( (202)234- 9570) and on the MAA Gopher gopher gopher.maa.org or on the WorldWide Web at: URL''http://www.maa.org''. Plans are underway to make MAW visuals available via the WorldWideWeb at the University of Minnesota's Geometry Center; information will be posted on the MAA Gopher. Activities for Mathematics Awareness Week are generally organized by college and university departments, institutional public information offices, student groups, and related associations and interest groups. They include a wide variety of workshops, competitions, exhibits, festivals, lectures, and symposia. The number and breadth of activities increases annually. For example, one college has now sponsored a high school mathematics day for several years to encourage women to continue their studies in mathematics. At one university, two departments-mathematics and architecture-cooperated to plan and produce an interactive traveling exhibit that provides hands-on experience for such topics as codes, tilings, chaos, geometry, graphs, and computer science. Coordination and support for Mathematics Awareness Week is provided by the Joint Policy Board for Mathematics, which includes the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Math• ematics. Additional financial support comes from the United States Army Research Office, Oxford University Press, and Springer• Verlag. MATHEMATICS AWARENESS WEEK VISUALS ORDER FORM Name ______Address ______City/State/Zip ______Phone ______
1995 MAW "Mathematics & Symmetry" Poster (color) @ $7.00 = $ ______1995 MAW "Mathematics & Symmetry" Postcards (color) @ $5.00 = $ ______(set of 10, 5 each of 2 designs) Order both the poster and postcards for only $10.00 @$IO.00= $ ___
**Reduced Prices on Past Visuals**
1 EACH OF 4 POSTERS FROM PAST MAWS* 1994 MAW "Mathematics & Medicine" Poster (color) 1993 MAW "Mathematics & Manufacturing" Poster (color) 1992 MAW "Mathematics & the Environment" Poster (color) 1991 MAW "Mathematics: It's Fundamental" Poster (color) A $20 value for only $14.00 ___ @$14.00=$ ___
1 EACH OF 4 SETS OF POSTCARDS FROM PAST MAWS 1994 MAW "Mathematics & Medicine" Postcards (set of 10, 5 each of 2 designs) 1993 MAW "Mathematics & Manufacturing" Postcards (set of 3, 1 each of 3 designs) 1992 MAW "Mathematics & the Environment" Postcards (set of 4, 1 design) 1991 MAW "Mathematics Applications" Postcards (set of 4, 1 each of 4 designs) A $10.50 value for only $6.50 __ @$6.50 = $ ___
TOTALAMOUNTDUE$ ______
All orders must be prepaid. All posters are rolled and mailed in a tube. Prices include third-class postage. Please make check payable to AMS and send this firm to: JPBM Visuals, 1529 Eighteenth Street, NW, Washington, DC 20036. * The JPBM may substitute another set of posters if supplies are depleted.
7 FOCUS February 1995 $~TIC8~ Joint Mathematics Meetings g ~ January 4-7,1995 o a':l ~ 00 San Francisco ~ ~ ~~" ~ J'to""" ,,' , ~ ...... ~ ~_'_'~ b( Cisco • J an~~~1
Prizes Awarded in San Francisco
Prize winning members of the MAA, AMS, and AWM were honored at a joint prize session held at the January meetings in San Francisco. The Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Mathematics was presented to Dr. Anneli Lax of the Courant Institute of New York University. The award is the most presti• gious one given by the Association, and Donald Saari Barbara T. Faires consists of a cash prize of $4000, a cita• Anneli Lax tion, and recognition by the American mathematics community. Dr. Lax has been editor of and indeed the entire driving force behind the MAA's New Mathematical Library, one of the most distinguished and successful series since its founding in 1961. The complete citation for Dr. Lax's award, written by Ivan Niven, will appear in the February issue of the American Mathematical Monthly. In receiving the award, Dr. Lax observed that she was being rewarded for an activ• ity that had been its own reward, and ended her remarks with some advice to all future mathematical writers: "Let us practice what we preach, read and write carefully, avoid trendy slogans, and go beyond math• Left to right: David Moore. Lisa Mantini. Robert Devaney ematical correctness, syntactic correct• ness, and political correctness in serving 1990). The award consists of a cash prize Award for Distinguished College or Uni• our discipline in our individual ways." of $1000, a certificate, and the recogni• versity Teaching, consisting of a cash tion of the mathematical community. prize of $1000, a certificate, and the rec• The Chauvenet Prize for expository writ• ognition ofthe mathematical community. Responding to the award, Professor Saari ing on mathematics by a member of the The recipients were Professor Robert dedicated his acceptance to the two math• MAA was presented to Professor Donald Devaney of Boston University, Professor ematicians who most influenced his early G. Saari of Northwestern University, for Lisa Mantini of Oklahoma State Univer• career, Harry Pollard and Ralph Boas. his paper "A Visit to the Newtonian N• sity, and Professor David Moore of Purdue body problem via elementary complex Three MAA members received the University. variables," published in the American Deborah and Franklin Tepper Haimo Mathematical Monthly, vol. 97 (February 8 February 1995 FOCUS Who Invented Newton's Method? "Children, don't try this at home," warned Professor Dunham of Muhlenberg College Displaying the original writings ofall three Dr. William Dunham, referring to the task was the MAA Student Lecturer at the Joint mathematicians to support his claim, of solving the pair of simultaneous equa• Annual Meetings. 'Who invented Dunham showed how the method New• tions Newton's Method?' was the main theme ton himself used to find approximate roots of his talk. xx + yY= 1000 was entirely algebraic, involving no refer• After describing Newton's life, and that ence to calculus or fluxions, and how it xY + y= 100 of his great rival Leibniz, Dunham traced was only implicitly that Raphson used dif• This particular pair of equations was in the now-familiar approximation method ferentiation in his later version of Newton 's fact solved in the eighteenth century by from Newton's own work, through a later approach. Simpson, using a new method he had in• development of Raphson, and on to (In his solution to the equations above, vented, the method known nowadays as Simpson. It is only with Simpson, declared Simpson started with initial approxima• Newton's Method. Dunham, that we first see the method tions x = 4.5 and y = 2.5, and after two nowadays called Newton's Method. iterations obtained the solutionx=4.5519, y =2.4495.)
Students enjoy the annual Make-four-Own Sundae party after the Left to right: Bonnie L. Thomas (Sonoma State University), Joan student lecture. Chan (Sonoma State University), MAA President Ken Ross (Univer• sity o/Oregon), and his wife, Ruth.
Barbara Beechler Theresa Michnowicz Roben Eslinger Donald W. Robinson Howard Lewis Penn
Certificates of Meritorious Service to At the same ceremony, the AMS awarded The Norbert Wiener Prize in Applied the MAA were presented to Professors the Ruth Lyttle Satter Prize in Math• Mathematics, awarded jointly by the Barbara Beechler (Southern California ematics to Sun-Yung Alice Chang AMS and SIAM, went to Professors Section), Robert Eslinger (Oklahoma• (UCLA), the Frank Nelson Cole Prize in Hermann FIaschka (co-coordinating edi• Arkansas Section), Barbara Faires (Al• Algebra to Professors David Harbater tor of Physica D: Nonlinear Phenomena) legheny Mountain Section), Theresa (University of Pennsylvania) and Michael and Ciprian Foias (Indiana University). Michnowicz (New Jersey Section), Raynaud (University of Paris Sud Orsay), TheAWM's Louise Hay Award for Con• Howard Lewis Penn (Maryland-District and the Award for Distinguished Public tributions to Mathematics Education of Columbia-Virginia Section), and Service to Mathematics to Professor was made to Professor Etta Z. Falconer of Donald W. Robinson (Intermountain Sec• Donald J. Lewis (University of Michigan). Spelman College. tion). 9 FOCUS February 1995 Board of Governors How to Get Mathematics Stories into the Meeting Newspapers
The following are just a few of the many So you think you have just the right math• panelist at the session. "People are fasci• items of business that occupied the ematical news item to catch the attention nated by what motivates the mathemati• Association's Board of Governors for their of the local newspaper. What does it take cians to do what they do," Davidson seven-and-a-half-hour meeting in San to make that desire a reality? This was the observed, a point echoed by comments Francisco on January 3rd. main question addressed at a public infor• from K. C. Cole of the Los Angeles Times. • Former MAA President Len Gillman, mation session organized by the Joint For those mathematicians who want to who retired from the Board at the meet• Policy Board for Mathematics at the San write their own news stories or expository ing' observed that he had served on the Francisco meetings. articles, make sure you have your tenure Board for twenty-two consecutive years, Former Los Angeles Times science writer first, warned Paul W. Davis, an applied a figure he believed to be a record. Lee Dembart suggested that the secret was mathematician at Worcester Polytechnic Institute who writes articles for SIAM • Treasurer Jerry Porter announced that to ask the big, five o'clock question. Not News. For, as Davis reminded the audi• the cost of moving the meeting from its your big question, which is probably, ence, the mathematics profession tends to original venue, Denver, Colorado, which "What can this journalist do for me?" but view with considerable disdain anyone the MAA and the AMS had decided to the journalist's big question, the question who dares to venture into the world of ex• do in the interest of protecting its mem• that, according to Dembart, all journalists pository writing. The remark brought bers from a voter climate hostile to gays ask as their days' work is coming to an laughter, but it was laughter in agreement. and lesbians, was $22,000. He remarked end: "Did I make the front page?" The mathematical community clearly has that this loss may be more than made up "Try to see your story through the a long way to go in the public relations by the greater attendance at the San Fran• journalist's eye," advised Dembart. "Ask arena. On the other hand, a positive sign cisco meeting than would have been in yourself why the journalist should want to that things might be changing was the large Denver, following the controversial spend time working on your story." attendance at the session, which was Colorado ballot measure passed in No• chaired by Hugo Rossi, the new editor of vember 1992 (see FOCUS, vol. 13,no.l, One way to increase your chances of suc• the AMS Notices, and also included Andy p.4). cess with a mathematics story is to talk about the psychological aspects, accord• Evangelista, who writes science articles • The governors unanimously endorsed the ing to Keay Davidson, the science writer in the publicity office of the University of nomination of Andrew Granville of the at the San Francisco Examiner, another California, San Francisco. University of Georgia for the Hasse Prize, in recognition of his article "Zaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle," published in Schattschneider Demonstrates New the April 1992 issue of American Math• ematical Monthly (vol. 99, no. 4, pp. 318- Packing Result 331). Dr. Doris Schattschneider. professor of mathematics at Moravian College and a vice• • The governors gave unanimous support president of the MAA. demonstrated that it was possible to pack some 1100 randomly to a resolution expressing appreciation shaped mathematicians into the large ballroom of the San Francisco Hilton on January to the Exxon Education Foundation for 6th, when she drew a more than capacity audience for her AMS-MAA invited address forty years of corporate support of edu• "Ingenious Mathematical Amateurs: M. C. Escher (artist) and Marjorie Rice (home• cation reform in U.S. schools and col• maker)." With all 970 seats occupied and about a hundred people sitting or standing leges, including generous and critical around the edges. many had to be turned away at the doors. support for a number ofMAA programs, In one of the most successful talks of the meeting, Dr. Schattschneider described how projects, and publications. both Escher and Rice, in the absence of any formal mathematical training, developed their own diagrammatic notations to study symmetry and tilings. For Escher, the key to the mathematical ideas he used was provided by a paper of George P61ya's, Schattschneider observed; for Rice it came from an article by Martin Gardner in Sci• entific American, followed later by correspondence with Schattschneider herself. The work of both demonstrated the power and the importance of visual information in mathematical reasoning. Schattschneider concluded, adding that their status as math• ematical "amateurs" proved the importance of good, popular exposition to draw in others whose special talents can further the development of mathematics and the influ• ence of mathematics on human culture.
10 February 1995 FOCUS MAA Book Exhibit a Big Success Expert Advice
As in previous years, the Books Exhibit was a major attraction at the Joint Annual "In God we trust, all others bring data!" Meetings, with most of the 3500 meeting attendants passing through the display at advised MAA Teaching Award winner least once. Sales of MAA books were particularly strong. Among the most popular David Moore in his address at the award MAA titles this year were Linear Algebra Problem Book, by Paul Halmos; Lion winners special session at the San Fran• Hunting and other Mathematical Pursuits, edited by Gerald L. Alexanderson and cisco meetings. His statistician's caution Dale and AU the Math that's Fit to PrInt, FOCUS editor Keith Devlin. was directed toward all who seek to im• prove the teaching of collegiate mathemat• ics. Moore advised that some of the crucial data we need in order to evaluate what we do is to be found in the hands of the edu• cational psychologists. "We should regard teaching as a craft," he said, "and that means that anyone can learn it, and all who are employed to teach mathematics have an obligation to learn it." Provide each of the students in your class with a list of names and phone numbers of the entire class to encourage them to feel part of a group and perhaps to help each other, was part of the advice offered by award winner Lisa Mantini, who entitled her presentation "To Challenge with Com• passion." Insist that each student comes to see you in your office at least once so they see you out of the classroom and realize you are approachable, was another piece of advice she offered. With the aid of interactive computer tech• nology, some very simple ideas can very quickly lead to deep and interesting math• ematical questions that can be investigated both computationally and analytically. This was the principal message conveyed by the presentation of awardee Robert Devaney, who demonstrated some dra• matic experiments on randomly generated iterative systems. Want to know more? During the course of the corning year, FOCUS plans to publish written versions of each of the presenta• tions given by the three Teaching Award winners for 1994.
11 FOCUS February 1995 Award Winners from the MathFest, Minneapolis, MN, August 1994
Bruce Berndt Leonard Gillman Charles W. Groetsch Joan P. Hutchinson The Lester R. FordAward The Lester R. Ford Award The George P6lya Award The Carl B. Allendoerfer "Ramanujan-For Low• "An Axiomatic Approach "Inverse Problems and Award brows," American to the Integral," American Torricelli's Law," College "Coloring Ordinary Maps, Mathematical Monthly, Vol. Mathematical Monthly, Mathematics Journal, Vol. Maps of Empires, and Maps 100, pp. 644-656. (Co• Vol. 100, pp. 16-25. 24, pp. 210-217. of the Moon," Mathematics authored with S. Bhargava.) Magazine, Vol. 66, pp. 211- 226.
Dan Kalman Edgar Lorch Joseph H. Silverman Dan Velleman The George P6lya Award The Lester R. Ford Award The Lester R. FordAward The Lester R. Ford Award "Six Ways to Sum a Se• "Szeged in 1934," Ameri• "Taxicabs and Sums of Two "Versatile Coins," Ameri• ries," College Mathematics can Mathematical Monthly, Cubes," American Math• can Mathematical Monthly, Journal, Vol. 24, pp. 402- Vol. 100, pp. 219-230. ematical Monthly, Vol. 100, Vol. 100, pp. 26-33. (Co• 421. pp. 331-340. authored with Istvan Szalkai.)
Photographs of Lester R. Ford Awardees, S. Bhargava and Istwin Szalkai were unavailable at press time.
Governors Express Concern Over Mathematics Job Market
At the suggestion of incoming MAA President Ken Ross, the AssociatiOb'S Board of Govemors hosted a visit to its meeting in San Francisco by two members oftbe YOUJia Mathematicians' Network. Steve Kennedy of Carleton College and Frank Sottile from the University of Toronto. The two explained the problems and frustrations of young mathematicians unable to find suitable employ• ment. In a separate but related agenda item, the Board unanimously passed the following three recommendations to departments and institutions. • It is incumbent on mathematics departments to make all their potential Ph.D.s aware of the realities of the job market and to encourage them to prepare for a broad range of jobs in the matlJematical sciences. • Colleges and universities should strenuously seek means to offer new Ph.D.s positions of at least two years' duration in order to allow for the crucial career development in teaching, research, and service that our profession expects at this stage. • Colleges and universities should recognize that part-time appointments and the practice of hiring unemployed Ph.D.s by the course, without integrating them into the scholarly life of the department, are detrimental to the individuals and the profession.
12 February 1995 FOCUS
It seems that under• New Undergraduate graduate research can thrive in diverse loca• Research Prize tions and in diverse ar• eas of mathematics. Frank Morgan Good work and acces• The new prize for undergraduate research sible questions are pro• announced in December's FOCUS stands liferating. Students cite to recognize and encourage the excellent theorems of previous mathematical research that undergraduates students and pose con• are already doing. Undergraduates are jectures for their suc• working on problems of current research cessors. The I 994AMS interest, proving theorems, writing up re• What's Happening in k h Wi'II' C II "SM'ALL" nd ad sults for publication, and giving talks on the Mathematical Sci- Undergraduates at wor at tel lams 0 ege No u erg,; uate their work. They are making significant ences features some research project. Photograph by Cheryl LeClaire. Williams Alumni Review. contributions to mathematics. At the same work by undergradu- time, they find out what real mathematics ates in an article titled "Soap Solution." mitted by June 30, 1995. Undergraduates is like. They gain a new perspective and a History In 1990 the Association for in the United States, Canada, or Mexico new appreciation for their courses and for Women in Mathematics established the as of December 1994, are eligible. their teachers. They can make more intel• Alice T. Schafer Prize for excellence in Faculty or students interested in organiz• ligent decisions about whether to go on to mathematics by an undergraduate woman. ing or participating in undergraduate re• graduate school. And when they do not go The Schafer prize inspired Joe Gallian, search might contact Jack Ryffat the NSF on in mathematics, they become a new who for seventeen years has run the (jryff@nsfgov)foradvice and a list ofthe breed of politicians, business persons, granddaddy of modern undergraduate current NSF Research Experiences for magazine editors, or other members of the mathematics research programs at the U ni• Undergraduates. general populace who have an apprecia• versity of Minnesota in Duluth, and Stan tion of mathematics. Wagon to propose an undergraduate re• Frank Morgan is chair of the Mathemat• Undergraduate research in mathematics is search prize to the MAA. Joe took the idea ics Department at Williams College and no longer an oddity. Dozens of colleges to the CUPM Subcommittee on Under• codirector with Colin Adams of "the and universities sponsor summer under• graduate Research in Mathematics, SMALL" NSF undergraduate research graduate mathematics research programs, chaired by John Greever. Stan Wagon took project. He is also a member-at-large of many supported by the National Science the idea to the MAA Coordinating Coun• the AMS Council. cil on Awards, chaired by Henry Alder. In Foundation. The National Security This article also appeared in the January an astonishing tribute to our mathemati• Agency, the Institute for Defense Analy• issue of AMS Notices. sis,AT&T, Bellcore, and NSF Centers and cal organizations and committees, the idea Institutes have expanded or created un• developed rapidly to fruition, with the help Summer dergraduate research programs. Many na• of Aparna Higgins (chair of the MAA tional and regional meetings now include Committee on Student Chapters), Beth Mathematics Camp sessions on undergraduate research. Pi Mu Ruskai (chair of the Joint Committee on Epsilon and the Mathematical Association Women in the Mathematical Sciences), The third annual camp for mathemati• of America award prizes and support to Ron Graham (AMS President), Gil Strang cally talented high school students will undergraduate speakers. The Council on (SIAM Vice-President for Education), and be held this year in Canada: July 3rd- Undergraduate Research, which sponsors many others throughout the mathematics 28th in Vancouver (University of British research conferences for undergraduates community. Columbia) and July 31 st-August 25th in Toronto (University of Toronto). The from all fields, now has a Mathematical The Prize The prize has some interesting four-week camp, known as MathCamp, and Computer Sciences Division. At the features. To proclaim the existence of ex• is a reeducation of mathematically pre• first annual Hudson River regional under• cellent undergraduate research, there is a cocious teenagers by university math• graduate mathematics conference at Siena single prize of $1000. To recognize the ematicians who are also experts in College near Albany last year, faculty and keenness of the competition, there is pro• communicating concepts to these teen• students ranging from introductory calcu• vision for a few honorable mentions. To agers and in helping them do mathemat• lus to senior mathematics majors partici• recognize the important role collaboration ics. The fee is $550 (Can$690). Meals pated as equals in giving talks, chairing often plays, the prize may be shared by a and supervised university dorm rooms sessions, and discussing mathematics (see group of students working together. the article on page 946 of the October 1994 are U .S.$30 extra per day. Enrollment is Notices by Douglas Briggs, one of the Nominations For the first award, research limited to forty. The application dead• undergraduate participants). papers (submitted by the student or a nomi• line is May 15th. For a brochure which nator) and a supporting letter must be sub- also contains the application form, call (519) 672-7990. 13 FOCUS February 1995
school, and produces uniformly high New GRE Mathematical Reasoning Test scores among college seniors in math• ematically oriented disciplines. These high Alan Tucker admission decisions in their disciplines. scores limit the value ofthe test for admis• Thus mathematics departments should This article is being published simulta• sion decisions in mathematically oriented become familiar with the general types of neously in Notices of the American Math• graduate programs. At the same time, non• mathematical problem-solving on this new ematical Society, and an earlier version mathematical graduate programs are also test. In particular, the new test will include appeared in the September 1994 issue of concerned because they feel these high problems of the sort found in calculus re• UMETrends. quantitative scores bias the GRE General form texts such as the Harvard Consor• Test to favor mathematically oriented dis• The General Test portion of the Graduate tium text. ciplines. Such a bias may influence deci• Record Examination will be reorganized Readers might be interested to know that sions about university-wide fellowships in fall 1997 to consist of the following five the GRE testing program is likely to be at the many institutions that use the GRE parts: fully computerized in 1997; that is, exam• General Test in selecting fellowship re• I. A revised version of the current quan- inees will be required to take the test on cipients. titative test computers. Currently the GRE General Following is the proposed content of the Test is available on computer only at se• 2. A revised verbal test new text in terms of both reasoning skills 3. A revised analytical reasoning test lected locations. The current computerized tested and the mathematical background 4. A new mathematical reasoning test General Test is an adaptive test, as each of assumed. Many questions may involve 5. A new writing sample test. the new General Test measures will be more than one category of reasoning skill (with the exception of writing). In an adap• These five General Test measures will be and also possibly more than one content tive test, each question an examinee re• offered in two packages: (1) verbal, ana• category. ceives will depend to some extent on the lytical, writing, and revised quantitative; answers to previous questions. Examin• and (2) verbal, analytical, writing, and Mathematical Reasoning ees recei ve fewer questions, since an adap• mathematical reasoning. Graduate depart• tive test allows the test to estimate the Mathematical Modeling ments will select the package of measures examinee's approximate ability level and • Recognize whether a mathematical most appropriate for graduate study in their focus questions around this level. There is model applies to a given situation (e.g., disciplines. Fall 1994 freshmen will be the one serious drawback, in the minds of linear function) first graduating class to take this new GRE many faculty, to the adaptive testing • Develop a mathematical model (e.g., exam. The new mathematical reasoning scheme--examinees may not skip a ques• graph, equation, schematic, geometric test will include problems of the type as• tion and return to it later (or change an figure) from a description of a system sociated with "reform calculus." answer after it has been confirmed). • Recognize what assumptions underlie a The new GRE mathematical reasoning test The current GRE quantitative test assumes particular mathematical model and how is expected to replace the current quanti• only two years of high school algebra so those assumptions can affect the valid• tative test for students planning graduate as to accommodate humanities students ity of the model (e.g., employing a con• study in engineering, the physical sciences, who have had no mathematics since high tinuous model for a discrete situation, the mathematical sciences, computer sci• ence, economics, and some areas of biol• ogy. Unlike most widely used tests for y post-baccalaureate study, such as the LSAT, MCAT, GMAT, and current GRE exams, the new GRE mathematical rea• soning test will draw on knowledge ac• quired in a first-year course in calculus. This writer was a member of the advisory committee that recently set the specifica• ++++ ...... x tions for this new mathematical reasoning test. It is important for mathematics depart• ments to be aware of this direct link be• tween learning in calculus and performance on the new GRE mathemati• cal reasoning test. The new test has drawn high praise from faculty in the sciences Given the graph of y = r(x). produce the graph of and engineering, and they expect the test y = r(2x) - 3 to carry considerable weight in graduate by selecting one of the 5 curves and positioning it in the coordinate system.
14 February 1995 FOCUS
neglecting aspects of a physical situa• between points, relationship between computation of limits tion in developing a mathematical model rectangular and polar coordinates) • Rates of change (e.g., speed, average rate of it) Functions and Their Graphs of change) • Reason with symbols rather than num• • Relationship between algebraic and • The derivative of a function and its rela• bers (e.g., determine the influence of a graphic representations of functional re• tionship to the behavior of the function parameter on a mathematical system) lations (continuous or discrete) and to the slope of its graph • Express in everyday language relation• • Properties of functions and their graphs • The definite integral and its relationship ships that are given mathematically (e.g., domain, range, continuity, symme• to the area of a region in the plane Logical Reasoning try, composites, inverses, intercepts, as• • Relationship between integration and • Formulate a conjecture or draw conclu• ymptotes) differentiation sions from a given set of results or obser• • Elementary functions, their properties, • Differentiation and integration of el• vations and their graphs (e.g., polynomials, ra• ementary functions (e.g., polynomial, • Construct a valid argument to support or tional functions, trigonometric functions, trigonometric, exponential, and logarith• refute a conjecture or hypothesis exponential functions, logarithmic func• mic functions) tions) • Determine the validity of an argument • Standard applications ofthe differential or identify the flaw in an invalid argu• Calculus and integral calculus (e.g., velocity and ment • The limit concept, properties of limits, acceleration, exponential growth, work, Patterns and Similarities • Recognize patterns, trends, or symme• tries; continue a pattern • Investigate relationships between vari• • x 2 + 3x + 2 > 2 ous mathematical ideas and processes (e.g., relationship between slope and rate Statements: • 1x < 0 of change) • 0 < x < , Problem-Solving Strategies • Reduce a problem to a simpler case • Determine when a certain procedure is appropriate for solving a problem (e.g., If ______' then ______differentiation, integration, averaging) Estimation and Approximation Place two of the three statements in the underlined positions so that the resulting if-then statement is true. • Determine when estimation techniques are appropriate and determine the degree of accuracy needed in an estimate • Recognize the reasonableness of results 33 of 3B #26 test Elapsed TIme: --:-- (e.g., orders of magnitude, signs of physi• Cllek an an -- and."ter or edit cal quantities) I your ...- ... 1"9 the ...--.on k¥<> • Laws of exponents What is c in terms of /1 ? • Ratio, proportion, and percent c = I Geometry and Analytic Geometry • Properties of geometric figures (e.g., CD00~00GJ0000 perimeter, area, volume, Pythagorean 000 0C!J0 8c:Jc:JCD Backspace theorem, similarity) 000 [ Space 1 0000 '---- • Coordinate geometry (e.g., equations of lines, planes, and conic sections, distance 15 FOCUS February 1995 compound interest, optimization prob• lems) x + .by = d Probability and Statistics ex + y = e • Measures of central tendency (mean, median, mode) and dispersion (range, .b, e, d, and e are nonzero constants. standard deviation, variance) If the system of equations has infinitely many solutions, • Interpretation of tables, graphs, histo- what is e in terms of .b ? grams, etc. The new mathematical reasoning test is e = IL... ______--' designed to test mathematical reasoning skills and not mathematical content knowl• edge, and so its calculus prerequisite will not result in questions directed at knowl• edge of calculus techniques. The types of ;(x) q(x) ;(q(x)) q(f"(x» questions to appear on this test were de• mathematical reasoning test, contact D veloped somewhat independently of the Jacqueline Briel, GRE Testing Pro• 2 3 9 BO 16 calculus reform movement but reflect the gram, Educational Testing Service, 4 B1 same concern for an understanding of con• Princeton NJ 08541. 0 0 0 cepts and their interpretation in graphical 2 and applications settings. Alan Tucker is the associate chair• ;(x) = x - 1 man of the Department of Applied q(x) = (x + 1)2 Shown are sample test items generated by Mathematics at the State University ETS staff. of New York at Stony Brook. Complete the table. For further information about the new GRE 1996 NSF-CBMS Regional Research Support Your Conferences in the Mathematical Sciences AAASStamp Preparations are underway to mark a his• toric event in 1998: the 150th anniversary Request for Proposals of the American Association for the Ad• vancement of Science (AAAS). Recently Closing Date: April 3, 1995 launched was a campaign for a 1998 U.S. The National Science Foundation intends to support from five to seven NSF-CBMS postage stamp commemorating not only regional research conferences in 1996. A panel chosen by the Conference Board of the AAAS but the last 150 years of scientific Mathematical Sciences will make the selections from among the submitted proposals. achievement. A proposal will be submit• In the twenty-six-year history of this NSF-CBMS Regional Research Conference ted to the Citizen's Stamp Advisory Com• Series, a total of 247 such conferences have been supported. mittee (which screens new stamp Each five-day conference features a distinguished lecturer who delivers ten lectures on proposals for the U.S. Postal Service) soon a topic of important current research in one sharply focused area of the mathematical after the AAAS February 1995 annual sciences. The lecturer subsequently prepares an expository monograph based upon meeting. It takes about three years to go these lectures, which is normally published as a part of a regional conference series. from a stamp's approval to its design and Depending on the conference topic, the monograph is published by the American production, and in order for the proposal Mathematical Society, the Society for Industrial and Applied Mathematics, or jointly to be successful, we must show signifi• by the American Statistical Association and the Institute of Mathematical Statistics. cant and widespread public support for the new stamp. Letters of support would help Support is provided for about thirty participants at each conference and the conference a great deal. Send them, as well as any organizer invites both established researchers and interested newcomers, including suggestions for the stamp's theme and postdoctoral fellows, graduate students, and underrepresented groups, to attend. design, to: Inquiries concerning this conference series or the preparation of proposals for confer• AAAS Commemorative Stamp Commit• ences should be directed to Conference Board of the Mathematical Sciences, 1529 18th tee; Office of Communications, Rm 801; St. NW, Washington, DC 20036; (202) 293-1170. 1333 H St. NW; Washington, DC 20005. 16 February 1995 FOCUS Search Committee Diary Edward F. Aboufadel uted to all members of the department for comments. As we bidfarewell again to our diarist, we cannot help but wonder, will he be back to The School of Education here was look• clue us in when he goes up for tenure? As ing to hire for a position starting January for now, read on and find that intrepidity 1994, but the funding for that position was has paid off. The final months of the cut over the holidays. The Search Com• committee ssearch slowly wind down until mittee wondered if that meant that fund• Dr. M arrives on the scene and says, "Yes. " ing for our position was also in jeopardy, but the chair said that the dean has reas• Some books also may be read by deputy, sured him there is no problem. picked. The committee is a little anxious and extracts made of them by others. that we may lose a prime candidate to These are the things that you at home Francis Bacon another school if we don't act quick! y. One need not even try to understand. "Of Studies," Essays tactic we have undertaken is to call some Ernie Pyle of our better candidates, to tell them that January 31, 1994: The pace of the "On Victory in Europe," Ernie s War we are interested in them, and to ask if search has slowed to a crawl these past they are still available. So far none of those two weeks, for a number of reasons. First February 16, 1994: Today the Search that we have called said that they found a has been the dangerous weather of the past Committee met to make our second cut. position elsewhere. few weeks. Southern was closed two dif• After some discussion and voting, we sepa• ferent mornings last week, and the Search rated our seventy quarterfinalists into four The committee is also getting a little anx• Committee meeting we had scheduled has groups. The first group, which we labelled ious about funding for this position. Last been postponed until next week. The sec• as "Double Stars," are the best candidates, week a different department at Southern ond reason has bee.n the beginning of and there are seven of them. Next are the completed a search, decided on a candi• spring semester, which has distracted all "Single Stars," which number eleven. The date, and made the candidate an offer. Soon of us with on-the-spot advising and roster third group is the "Pluses," and there are after, the offer was "put on hold" for fi• juggling. nine of them. The remaining forty-three nancial reasons. The chair asked the dean we have now rejected. The eighteen what the implications were for our search. The third reason is that we now have over "Stars" are our semifinalists (which is a The dean told us to proceed, but we are 450 applications, 250 of which haven't little more than the fifteen that we planned, apprehensive. We are ready to defend our been read yet. In order to speed up our but that's OK), while the "Pluses" are our need for this position vigorously! decision making, we have decided to split alternate candidates. The next step is to the unread applications among the five of The Search Committee also decided on give members of the department the op• us. The chair has asked us to "try to iden• the wording of a rejection letter to send to portunity to comment on the applications tify the clear and evident rejections since those applicants who did not make our first of these eighteen candidates. we have been consistently agreeing on cut. Thinking about my comments in these." We will meet soon to come up with Although it would be both difficult and FOCUS two years ago, I now realize that a manageable number of viable files. inappropriate for me to go into the details writing a good rejection letter is not easy. about how we are making these decisions, On the one hand, you want to treat these February 8, 1994: The Search Com• I do want to point out that our reduction applicants with courtesy, and perhaps even mittee yesterday settled on seventy from nearly five hundred applications to sympathy. The opportunity is there to use quarterfinalists-the thirty from Decem• fifteen semifinalists is based upon the the rejection letter to educate (provided ber plus another forty from our individual application materials only. (In the case of that the applicant reads it), and you don't readings this past two weeks. The other those that we saw in Cincinnati, our ob• want people to go away with a bad attitude four hundred or so applicants we have servations from those interviews were about Southern Connecticut State Univer• rejected completely, and we will start send• factored into today's decisions.) Only af• sity-after all, we may meet again. On the ing them rejection letters soon. Many of ter we receive the comments from the other hand, you don't want to get into an the rejected applicants were simply not department will we start doing telephone argument with anyone about how deci• who we were looking for. interviews. Then, to decide on our three sions were made. With these ideas in mind, Our next task is to reduce the seventy finalists, we will use the combination of we drafted the following letter: quarterfinalists to fifteen or so semifinal• the department's thoughts, telephone in• Dear 19 FOCUS February 1995 bucks for this six-day week! der, but I see that the question involves the LEITERS TO THE symbols < , x , 0 , and the word "sen• Student 1. What class trip would cost forty• tence" in a way that takes some getting five dollars? Some trips might, but not no EDITOR used to. It requires some knowledge of class trip. jargon and vocabulary more than of solv• Student 2. Hey man, remember this is not ing or thinking. I would be interested in A Second Opinion real; this is math. You're not supposed to comparing responses to this question with think about real trips and real pay. So, say responses to From Anneli Lax she earn her fifteen bucks in one week, Find two different numbers, call them a In "Personal Opinion" (June 1994), Pro• and say she be dumb enough to not spend and b, which, when multiplied by 3, give fessor Gillman criticizes some suggested a cent until she has enough for the forty• numbers bigger than 54. model solutions of mathematical problems five-dollar trip. designed to test problem solving ability of Next Gillman discusses an extended re• Student 1. She won't even have enough 4th, 8th, and 12th graders. He is reacting sponse problem which asks 12th graders after two weeks! to a report on the 1992 Mathematics As• why the square of a number ending in 5 sessment published by the U.S. Depart• Student 4. Right. She will have thirty dol• must end in 25. The comments that follow ment of Education in 1993. lars. And now she's got to slave another sound very much as if the only attack ex• week at that weirdjob before she has forty• pected of the students is expanding (IOn + I am sufficiently critical of Gillman's ap• five. 5f. I should have liked to read some of the proach in "Personal Opinion" to write a extended responses to see what other second opinion; his attitude typifies that At this point, the class may have to be thoughts might have been expressed. For of many professional mathematicians so reminded that it has, in fact, answered the example, somebody might have noticed preoccupied with their ways of thinking question as earlier interpreted. that all numbers ending in 5 are odd mul• that they are not open to students' home• In the meantime, a fifth student, who was tiples of 5, and could have developed that spun solutions. My remarks are based on not paying attention to the conversation thought. Amore likely response, especially my personal experiences. between the other four, may say, "Wait a on the part of students who are comfort• "Jill's Class Trip" is the first example cited minute. Even if she made three bucks on able with multiplication but not with alge• or paraphrased: each ofthe six days, she would make only braic methods, would be simply to square eighteen bucks, but since she makes only numbers ending in 5 and to see if the prod• How many weeks must Jill work to earn two bucks on three of these days, that's ucts follow a pattern: $45 for the trip ifevery week she earns $2 one buck less three times, so three bucks I 35 on each of 3 days, $3 on each of 3 other less a week, so only fifteen bucks a week. 5 15 25 95 xl2 days, and nothing on Sunday? And look, fifteen times three is forty-five, u x 6175 Now, when colleagues and I experimented so in three weeks she will have the money 25 6175 with word problems in a class of 9th grad• she needs. ers in a New York inner city school, we 3705 Gillman remarks that "the solution re• found that after the students demystified a quires more than one step." In any of these question (Le., after they interpreted it, criti• and other scenarios, how many steps were cized it, made fun of it, altered it to make used? What is a "one step" problem? Who ----- 25 it more interesting), the youngsters seemed measures or counts the steps? Let us count Some youngsters would observe, even if to have no trouble solving it. Extrapolat• the ways of solving, not the number of they performed the multiplications with a ing from our experience, let me demon• steps. The difficult part of dealing with calculator, that the products certainly end strate what might take place. "Jill's Class Trip" may well be swallow• in 25; and if they looked at the pencil and Student 1. How come she earns only two ing the unbelievable 'given data' rather paper algorithm as in the above example, dollars a day? Don't her boss pay no mini• than having to play the school math game. they might discover that no matter how mum wage? Or does she work less than large the numbers we square, the last two Gillman reports that only 49% of 8th grad• half an hour those days? digits of the factors. So it suffices to try ers gave an acceptable response to the Student 2. Yeah. What does she do any• question 05, 15,25, ... , 85, 95. way for that kind of pay? Find two numbers that make the sentence I am not suggesting that such an attack Student 3. So okay. She works three days would be better or more efficient than 54< 3 xD a week for the miserable sum of two bucks. squaring IOn + 5, but I do suspect that in So if she doesn't spend any, she will have true. asking for "extended solutions," we sub• six bucks for three days. consciously look for extensions of our way Gillman rightly criticizes the suggested of thinking and are not sufficiently open Student 4. And another nine bucks for the answer: 19,20. He concludes, "apparently to our students' thoughts, often poorly three days when she gets the big fat three students are trained to solve first, think expressed. dollars a day. Big deal! She gets fifteen later." I see no evidence that they are trained either to solve or think in any or- 20 February 1995 FOCUS Finally, let me take issue with Gillman's Suppose somebody has absolutely no in• I applaud Lax's emphasis on experiment• treatment of the last illustration he cites, come. Then he pays no tax. His effective ing with examples of squares ending in an extended response problem for 12th tax rate can be anything, say t%, since t% 25; the actual test question takes this very graders: of 0 isO. So whycan'tt= 6?Why doesn't approach, displaying the squares of 15, this dilemma show itself in the equation 25, and 35, conjecturing the general re• Under a proposed income tax, you pay (cluttered or uncluttered)? sult, and requesting a proof. But then it nothing up to $10,000, then 6% on any provides the hint: (IOn + 5)2 X ? On reread• excess over $10,000. The effective tax rate Because the expression for "in excess of ing my article, I find no suggestion that 10,000" was expressed by (x - 10,000), is the percent o/total income that you pay the only way to proceed is via the formula, in tax. Can this be 5%? Can it be 6%? which words fine when x ~ 10,000; but nor any hint that I am not open to students' when x < 1O,OOO~ we say there was no Again Gillman conjectures, on the basis homespun solutions. The report itself is excess, i.e., we think of excess as of "the suggested solution" that the "abys• wedded to the binomial expansion, assert• mal performance is due in part to the plod• ing that the correct formula is "central" in ding way students are taught to handle x-l0000 for x:5; 100000 solving the given problem. Excess = 0 equations." He then gives his solution and { for x < 100000 In the context ofthe tax problem, expressing ends the article with, "The reader can now income in units of $1 000 seems reasonably see something of the spirit and power of and so the graph of .06 Excess looks like straightforward, requiring no clever scaling mathematics, as well as its elegance, this: (and the device may be familiar from charts beauty, and charm." This reader is not and tables in newspapers and magazines). I charmed; I doubt that many are, and here have for many years watched calculus stu• is why. / income dents struggle with dispensable symbols as In showing us his solution, Gillman be• the notion is cemented in their minds that Thus, for somebody making less than gins with "clearing clutter as soon as pos• math is drudgery. The typos in the tables $10,000, the effective tax rate and the tax sible," i.e., adjusting the units in which come from gremlins, not the manuscript. specified by the 6% rule give the same the independent variable is expressed in a Rate is a ratio, the effective tax rate being the result: no taxes due. convenient way. I wonder how many times ratio of tax to total income; when the in• an experienced mathematician has had the Gillman's verbal solution does not help; come, and therefore the tax, are zero, this x-axis of a graph run off the paper or dis• he defines tax rate as tax divided by in• ratio is undefined. My reference to charm covered that his sequence of equations all come, which does not handle the case of 0 was intended only for the geometric solu• contained some large factor k before he or income. tion (otherwise I would have made it a para• she learned to scale or divided both sides graph of its own). It may be useful to analyze extended re• of an equation by k, or multiplied both sponses in order to learn what students are The report states: "One-fifth of the high sides by 10" to clear "decimal clutter." thinking even when they are not thinking school seniors left their papers blank. Nearly Clever scaling and choice of coordinates what we think they ought to be thinking. two-thirds (66 percent) did not demonstrate are subtle skills crucial in many applica• understanding of the initial premise of the tions of mathematics and should be nur• AnneliLax question-that only income in excess of tured. Yet Gillman uses the contemptuous Vermontville, New York $10,000 would be taxed." This may be due expression "clutter" when he sees a rather Gillman replies: to the phrasing that introduced the problem: laudable literal translation of words into "One plan for a state income tax requires The report concludes that the results on symbols: income is x dollars, its excess those persons with income of $10,000 or Jill's Class Trip "provide insight into stu• over 10,000 is x - 10,000, 6% of that is less to pay no tax and those persons with .06(x-l0,OOO) dents' difficulty with numbers and opera• while 5% of the income is income greater than $10,000 to pay a tax of .05x. tions questions calling for more than one So the first question indeed seems 6 percent only on the part of their income step to the solution." A one-step problem equivalent to: Does the equation that exceeds $IO,OOO"-a contorted, 45- would provide the information that Jill .06(x - 10,(00) .05x word, comma-less sentence "requiring" cer• = earns $15 a week. tain citizens to pay no tax. have a solution? Incidentally, in spite of The actual test calls the inequality Gillman's claim that his alternative "in• Leonard Gillman hibits copying errors," it has an unfortu• 54<3xO Austin, Texas nate typo in the first line of both tables. a "number sentence," a friendly term I met, Gillman then accepts the invitation, given along with boxes for variables, at SMSG From Hugh McGuire in the instructions, to answer the second in 1964. I think both are widely used to• question in words: "Can the effective rate day. "Solve first, think later" refers to the I write in response to the article "Can Stu• be 6%? Of course not." The "of course" is common tactic of first solving the corre• dents Solve Math Problems?" by Leonard an uncalled for put-down. A thoughtful sponding equation (54 = 3x) and then pro• Gillman, which appears in the June 1994 student may well struggle with the fol• ceeding from there. issue of FOCUS (pp. 12-l3). lowing dilemma. My conjecture (or personal opinion) is that 21 FOCUS February 1995 perhaps the poorness of students' math• day, and Saturday. Regardless of which • Here's a rephrasing: ematical performances described in that day of the week Jill starts this work sched• A simple scheme for assessing tax on article is due to the lack of clarity and rel• ule, after how many weeks will she have people's income is that the first $10,000 evance in mathematical problems-in the earned $45? earned by any person is not taxed, but then problems tested, as described in that ar• 2. A short-answer problem for Grade 8 6% of any further amount that is earned ticle, and conceivably in the problems used asks for two numbers that make the sen• must be paid as tax. Now, the "effective" in schools. tence 54 < 3 x D true. tax rate for a person is the percent of total Gillman's article comments on a U.S. income that the person pays as tax. Under • "54 < 3 D " is not a sentence! (Even Department of Education report which I x the described scheme for taxation, is it symbolic logic would encounter difficul• have not read, so my remarks are not fully possible for some person to have an effec• ties with the D)And numbers don't 'make' informed; I comment only on the contents tive tax rate of5%? Is it possible for some sentences true! And requesting two num• of Gillman's article. person to have an effctive tax rate of 6%? bers in that manner (i.e., with those words) I discuss some mathematical problems confuses matters: will both somehow si• Gillman mentions the manner in which specified in the article; in each case, I quote multaneously be used in the 'sentence'? I mathematics is taught as a root cause of the problem from the article. believe the problem could be more clearly students' poor peformances on those prob• stated, e.g., as follows: lems. I agree with this assessment of the 1. "Jill's Class Trip" asks how many weeks situation; but I have a somewhat different Jill must work to earn $45 for the trip if The following incomplete mathematical stance on the particulars. I believe students every week she earns$2 on each of three formula is missing a number, in the D: learn to cope with most problems by ap• days, $3 on each of three other days, and 54< 3 x D plying almost mechanical rules. An ex• nothing on Sunday. ample of such a rule which might 'appear Give two numbers such that either one of • While reading this problem, I 'stumbled' in' a student's head is as follows: them can be put in the D to make the com• (in reading) on the words "the trip"; I pleted formula true. When faced with a "percent" problem, look wondered: what trip? Then I realized that for a large amount and a small amount and "the trip" was merely an excuse for the 3. An extended-response problem for divide the former by the latter to find the necessity to earn $45. But by then, my Grade 12 asks why the square of a number percentage; or look for an amount and a concentration was broken. ending in 5 must end in 25. percentage, and either multiply or divide • I wondered on which three days $2 was • What does it mean for a number to "end the former by the latter to find another earned, and on which $3 was earned. As in 5"? Similarly for 25. Must the problem amount; otherwise ... 'flail.' stated, the situation somehow feels unreal: be solved for decimal numbers such as Standard teaching may fail to prevent stu• in real life, a statement of facts begins with 3.5? (Regarding other possibilities, hope• dents' operating so mechanically. particulars-such as amounts earned on fully bases other than 10 are not being the specific days Monday, Tuesday, considered.) The following statement of More pragmatic (and probably more ef• Wednesday, etc.-and then proceeds to the problem may be clearer: fective) mathematics requires understand• abstractions. This issue actually became ing areal situation, formalizing it, applying For any whole number whose last or 'ones' significant when I started solving the prob• formal methods to calculate something, lem, for if particular amounts are not tied digit is "5", the last two digits of the square and then translating the result of the cal• of the number are "25". For example, the to particular days, the answer to the prob• culation into a pragmatic result. (This last square of75 is 5625. Why is this true (for lem can be-instead of "3 weeks"-"it step may comprise merely specifying any number whose last digit is "5")? depends." For suppose work starts with units, e.g., "weeks.") Ifonly calculation is some $3-days at the end of one week and 4. Under a proposed income tax, you pay taught, then only that should be tested (e.g., ends, a little less than three weeks later, nothing on income up to $1 0,000, then 6% "What is 6% of the difference between with some $3-days at the beginning of on any excess over $1 0,000. The effective $60,000 and $10,000?") But if we want to another week. Then an amount of time tax rate is the percent of total income that include the other aspects of complete strictly less than three weeks suffices. It's you pay in tax. Can this be 5 percent? Can mathematics, then we should do so coher• analogously possible that a few days more it be 6 percent? ently. My rephrasings of those problems than three weeks are necessary. probably can be improved; to the extent • What does "income" mean? possible, they should be improved. I be• • I would restate the problem somewhat lieve a guiding principle should be the goal like the following: • Asking "can" here is tremendously vague. For any specific person, with a specific of phrasing exercises in a manner com• Jill wants to earn $45 to pay for a field trip income, the effective tax rate either is or is prehensible to those who must solve them. with her class. She finds regular work at not 5 percent; "can" makes no sense in Sincerely, various odd jobs (raking leaves, such a specific case; the numbers yield a babysitting, helping an elderly neighbor certain value, and that's it. Terminology Hugh McGuire buy weekly groceries, etc.) which, in sum• other than "can" may help the student Graduate student mary, yield $2 each Monday, Wednesday, consider possibilities. Stanford, California and Friday, and $3 each Tuesday, Thurs- 22 February 1995 FOCUS your graduates to provide specific evi• A Career Kit for dence that there is life after graduation. There is a high likelihood that your col• Advisors lege or university placement and alumni offices will help you identify some of your Andrew Sterrett graduates to invite back to the campus. Underclass Student: "I like math, but I Also check with your career advising and don't think that I want to teach. What do placement office to determine whether most math graduates do?" they have a reasonable collection of ca• reer information about the mathematical Professor: "Many math majors become Attention April White; or call (800) 331- sciences-more than to suggest that math actuaries. " 1622. majors find careers in banking, insurance Underclass Student: "Yes, I've heard that. 2. CBMS Career Information Kit Again industries, and as computer programmers. What else do they do?" with support from the Department of En• ergy, CBMS has accumulated kits of ca• Professor: "--" reer information produced by the 5. Math Horizons Every issue of Math And on the same day ... mathematical sciences professional soci• Horizons contains articles about careers eties. In addition to the Resource Guide, and/or finding jobs. To date we have seen Another Upper-class Student: "This has these kits contain information about ca• "Seven Steps to Finding a Job," "Math been a great major, but I think I'll post• reers in applied mathematics, statistics, and Jobs: Look What You CanDo!" "Math pone going to graduate school. How do I actuarial science, operations research, and Majors Making Money?" "Interviewing go about getting a job?" teaching. CBMS will distribute one thou• Tips From the Pros," and "How to Really Professor: "Better talk with people in the sand kits without charge at forthcoming Get a Job." This most recent article pro• placement office." annual meetings of the American Math• vides an excellent model for obtaining ematical Association of Two-Year Col• useful information from your own gradu• Upper-class Student: "But don't you have leges, the American Counseling ates and can be easily replicated on indi• any suggestions specifically for math Association, and the AMS-MAA Joint vidual campuses. Editor Don Albers majors?" Annual Meetings in San Francisco; and promises to continue the emphasis on ca• Professor: "--" five thousand kits at the annual meeting of reer information in future issues of Math NCTM in Boston. If you attend one of Horizons. For information on obtaining these meetings, pick up your kit at the Math Horizons for your students (and for If you are as stumped as this professor CBMS booth in the exhibit area. Resources yourself), write to the MAA, Attention Lisa when your students inquire about what do not permit the distribution of these kits Johnson. they can do with a math major or how they to individuals or to departments, but all of 6. Computer- and Mathematics-Related should go about finding ajob, perhaps you the CBMS materials are listed in the Re• Occupations Bulletin 2450-4, #029-001- and your department need to put together source Guide. 03163-7; 1994-95; $2.00, check payable a career information kit. This "kit" should 3. MathematiciansAt Work These are one• to "Superintendent of Documents"; Su• contain not only printed material, but some page essays written by mathematical sci• perintendent of Documents, U.S. Govern• real live people-graduates, local place• entists and distributed to faculty advisors ment Printing Office, Washington DC ment personnel, and even faculty mem• ofMAA student chapters. Advisors of stu• 20402. This bulletin includes information bers from other departments who will be dent chapters have received fifty-five es• about actuaries, computer programmers, happy to discuss how a mathematics back• says to date. New ones are distributed each systems analysts, economists, marketing ground can be used in their disciplines. fall to chapter advisors. The essays are research analysts, engineering, data pro• Here are a few suggestions for starting your printed on high quality paper so that cop• cessing managers, mathematicians, opera• own kit: ies are easily made for individual faculty tions research analysts, and statisticians. I. Career Information in the Mathemati• members and students, the career guid• For each occupation, information is pro• cal Sciences: A Resource Guide This ance office, bulletin boards, and loose-leaf vided on working conditions; employ• booklet, produced by the Conference folders. For information on starting an ment; training, qualifications, and Board of the Mathematical Sciences MAA student chapter, write to Jane Heck• advancement; job outlook; earnings; and (CBMS) with support from the Depart• ler at the MAA (see address in 1). related occupations. ment of Energy, contains fifty-five refer• 4. Recent Graduates in Your Own De• 7. Combined Membership List The CML ences for career information intended for partment There is no better way to in• lists twenty-eight companies, laboratories, students ranging from junior high school form your majors of potential careers in and agencies in Virginia where mathema• through college. It is available, in any the mathematical sciences than to let them ticians are employed, ninety-four sites in number and without charge, from the hear from your own successful gradu• California, and twenty-eight sites in Texas. Mathematical Association of America, ates-what they are doing and how they These lists, found in the back of the CML, 1529 18th St. NW, Washington, DC 20036, found their jobs. Bring back a panel of provide many suggestions for your job- 23 FOCUS February 1995 hunting seniors regardless of their locales of interest. NSF -CBMS Regional Research There is a good chance that your place• ment office has copies of the following: Conferences in the Mathematical Sciences 8. Job Choices in Science & Engineer• Contingent upon National Science Foun• Approximation Dynamics with ing College Placement Council, Inc.; (800) dation funding, it is anticipated that seven Applications to Numerical Analysis 544-5272; 1995; $16.75. A directory of NSF-CBMS regional research confer• George R. Sell, lecturer employment opportunities for college ences will be held in the summer of 1995. June I st-5th, University of Missouri, graduates in engineering, sciences, the These seven will bring to 254 the total Columbia computer field, and other technical op• number of such conferences held in the Carmen Chicone and Yuri Latushkin, or• tions. twenty-six-year history of this NSF• ganizers; (314) 882-6331; (314) 882-8275; CBMS Regional Research Conference 9. Peterson's Job Opportunities for En• [email protected]; Series. gineering, Science, and Computer [email protected] Graduates Peterson's customer service at These conferences are intended to stimu• Nondestructive Evaluation and (800) 338-3282; 1995; $18.95 + $5.75 late interest and activity in mathematical Inverse Problems shipping. research. Each five-day conference fea• Gunther Uhlmann, lecturer tures a distinguished lecturer who deliv• For those students interested in graduate ers ten lectures on a topic of important June 12th-16th, University of Kentucky school, include these in your kit: current research in one sharply focused Peter D. Hislop and Russell M. Brown, 10. Assistantships and Graduate Fellow• area of the mathematical sciences. The lec• organizers; (606) 257-5637; (606) 257- ships in the Mathematical Sciences Pub• turer subsequently prepares an expository 3951; [email protected]; rbrown@ms. lished by the American Mathematical monograph based upon these lectures, uky.edu Society; (800) 321-4267; $18.00; AMS, which is normally published as a part of a Probabilistic Aspects of Single P.O. Box 6248, Providence, RI 02940. regional conference series. Depending Orbit Dynamics upon the conference topic, the monograph II. What's Happening in the Mathemati• Benjamin Weiss, lecturer is published by the American Mathemati• cal Sciences Also published by the AMS; June 18th-22nd, California State cal Society, the Society for Industrial and free, $4.00 for postage and handling. University, Bakersfield Applied Mathematics, or jointly by the 12. Prentice Hall Guidefor Fellowships American Statistical Association and the Kamel N. Haddad, organizer; (805) 664- for Math and Science Students Published Institute of Mathematical Statistics. 2150; [email protected] by Prentice Hall, Inc.; 1993; $29.95; Or• Tight Closure, Big Cohen-Macaulay Support for about thirty participants is der Processing Center, PO Box 11071, Des provided and the conference organizer Algebras, and Uniform Artin Rees Moines, IA 50336-1071. invites both established researchers and Theorem Finally, there is one creature to add to your interested newcomers, including Craig Huneke, lecturer kit of career information: postdoctoral fellows and graduate stu• June 25th-29th, North Dakota State dents, to attend. University 13. MAA Gopher Many of the Mathema• ticians at Work essays are available from Pending NSF funding, the seven confer• Joseph Brennan, organizer; (701) 238- the MAA Gopher, as are career articles ences anticipated to be held in 1995 are 8195; [email protected] from Math Horizons. As information be• listed below. Information about an indi• Probability,Algorithms, and Combina• comes available about REUs and other vidual conference may be obtained by torial Optimization opportunities for students, that too, will contacting the conference organizer. In• J. Michael Steele, lecturer be available on the MAA Gopher. Instruc• formation about the series and guidelines July 17th-21 st, Michigan Technological tions for reaching the MAA Gopher are for submitting proposals for future con• University found in the August 1994 issue of FO• ferences may be obtained by writing or Anant P. Godbole, organizer; (906) 487- CUS. calling the Conference Board of the Math• 2884; [email protected] ematical Sciences, 1529 18th St NW, Andrew Sterrett is Professor Emeritus of Washington, DC 20036; (202) 293-1170. Index Theory, Coarse Geometry, mathematical sciences at Denison Uni• and Topology of Manifolds versity. andAssistant Director ofPrograms Numerical Linear Algebra on Parallel John Roe, lecturer at the MAA. Processors August 15th-19th, University of James W. Demmel, lecturer Colorado, Boulder June 12th-16th, University of San Fran• Carla E. Farsi, Guoliang Yu, and Jeffrey cisco S. Fox, organizers; (303) 492-7422; (303) Peter S. Pacheco, organizer; (415) 666- 492-2396; (303) 492-6418; farsi@euclid. 6630; [email protected] colorado.edu; [email protected]; [email protected] 24 February 1995 FOCUS 1995 Small Grants for Development of Mathematics-Based Intervention Projects The MAA plans to award small grants for agencies, and industry for support of the • Rationale: In what way will your institu• development of mathematics-based inter• project. tion be a welcome host for the project? vention projects in spring 1995. SUMMA • Objectives: What are the objective(s) of (Strengthening Underrepresented Minor• Nature of the Grant your planning project? ity Mathematics Achievement) is solicit• Grants are less than or equal to $5000 and ing college and university mathematicians will be made to the institution of the project • Activities: What tasks do you plan to and their departments and institutions to director to be spent within the year. (Project undertake to design your project? submit planning proposals for the advance directors have an obligation in accepting • Personnel: What are the name, position, work necessary to host mathematics-based the grant to design a project and submit and qualifications of the proposed project intervention projects for middle and high versions of a proposal beginning in fall director? Who else will be involved in school students, targeting underrepre• 1995 until funded, with the assistance of the planning? How? sented minority students. These projects the SUMMA staff.) To provide maximum may replicate already existing successful flexibility, unexpended funds may be car• • Evaluation: How will you judge the suc• projects, adapt components of such ried forward. An institution is expected to cess of this planning? projects, or be innovative. In any case, the supply matching funds or in kind support • Budget: How will your planning funds planned activities should include those as an indication of commitment to the be spent-personnel, travel, materials, characteristics that are known to exist in development of the project. While the telephone, workshop attendance (math• successful projects as documented in Les• MAAISUMMA will fund the planning ematicians receiving grants are expected sons for HBCUs from Precollege Math• activities, it will not fund the project to attend a workshop on August 3-4, ematics and Science Programs. These itself. These grants will not support any 1995), release time, etc? An additional lessons are summarized at the end of this institutional indirect costs. budget page is requested from all announcement. The outcome of the plan• proposers. ning should be a formal proposal for a Who May Apply project to be submitted to public or pri• • Commitment: What is the potential for vate agencies. Grants will be made to faculty at long-term commitment of the host insti• tution? Objectives • minority institutions; or • colleges and universities which have stu• • Future Funding: What is the likelihood Specifically the objectives of the SUMMA dent bodies with a high percentage of of institutionalization through local or Small Grants Program are to: underrepresented minorities (at least state funding (perhaps after start -up fed• • encourage mathematicians to develop 20%) and a track record of success in eral funding from a variety of programs)? projects to increase minority participa• developing minority students' interest in • Support: Can you demonstrate an ability tion in mathematics; mathematics and science; or to mobilize internal and community re• and to provide support to project directors • colleges and universities in which the sources to sustain an intervention including the following institution or department has demon• project? strated that the faculty have the willing• • Adaptation: How might a successful • provide funds for project directors to visit ness and capacity to replicate or adapt established projects existing project be adapted to your local successful projects. conditions, or what is the need for inno• • make it possible for the project director Evaluation of Proposals vation? to work with the host institution to re• • Timeline: When will you carry out the cruit faculty and in other ways develop Proposals will be evaluated by members planning activities? the foundation for the project of the MAA Committee on Minority Par• • carry out a feasibility study ticipation in Mathematics, the director of Submission of Proposals SUMMA, and the executive director of • provide the project director the opportu• theMAA. Proposals should be submitted as soon as nity to participate in a proposal writing possible, but no later than March 15, 1995. workshop to be held in August 1995 in The (maximum) three-page (single• All proposers will be notified by March Burlington, Vermont just before the spaced) proposed plan should include: 31, 1995. The MANSUMMA intends to AMS-MAA Joint Summer Meetings • Concept: What ideas and philosophy make ten to twelve grants. The SUMMA about mathematics and education do you staff is available to discuss your potential • secure technical assistance in proposal proposal at any time. writing and fund raising have for a project? Describe the math• ematics content and activities you ex• For further information, contact: Dr. Wil• • make it possible for the project director pect to provide participants. liam Hawkins, SUMMA Director, or Dr. to contact private foundations, public 25 FOCUS February 1995 Florence Fasanelli, Director of SUMMA Intervention Programs, MAA, 1529 18th 1995 Summer Workshops for St., NW, Washington, DC 20036; (202) 387-5200; fax: (202) 265-2384; e-mail: Mathematicians who Teach Statistics [email protected]. Again this year, the MAA, with support workshop site and reunion. Each work• Characteristics of Effective from the NSF, is sponsoring a series of shop will provide room and board for Projects summer workshops for mathematicians twenty-four participants. In addition, par• who teach statistics but who do not have ticipants will receive course materials What has been learned? Despite the dif• recent or advanced training in the subject. which will include copies of Perspectives ferences among current highly successful (See the November 1993 issue of UME in Contemporary Statistics, Statistics for intervention projects, there are some char• Trends for more details.) In the first two the Twenty-First Century, a personal copy acteristics which all effective projects tar• years of this project, we received 350 ap• of the Minitab statistical analysis package geting minority students seem to have in plications for the 120 places at five work• used at the workshop, and $100 towards common. These have been identified in shops. For 1995, the final year of the the cost of travel, room, and board for the several reports and are synthesized in this project, there will be four workshops with one-day reunion. list. a total of ninety-six places available. The application deadline for the 1995 • Project goals are clearly articulated and According to a recent survey by the Con• workshops is March 3, 1995. Project Di• measurable ference Board of the Mathematical Sci• rectors George Cobb of Mount Holyoke • Strong academic component; focus on ences, introductory statistics courses are College and Mary Parker of Austin Com• enrichment, not remediation much more often taught in departments of munity College will send more detailed mathematics than in departments of sta• information and an application form to • Hands-on learning opportunities and use tistics. It is a worthy but demanding chal• department chairs. To receive a personal of computers lenge for a faculty member who is not copy, please contact Jane Heckler, STATS • Teachers highly competent in the sub• trained as a statistician to shape such a Project Registrar, MAA, 1529 18th St. ject matter who believe that students can course to meet the spirit of the various NW, Washington, DC 20036-1385; (202) learn the material guidelines and recommendations from the 387-5200; fax: (202) 265-2384. MAA and the American Statistical Asso• • Heavy emphasis on everyday applica• ciation. The goal of these workshops is to tions of mathematics and on careers in provide background and resources that will the field help faculty to meet that challenge more • Teaching strategies that take into account effectively. 1995 Statistical Thinking the needs, socioeconomic background, Four days of each week-long workshop and Teaching Statistics and cognitive development of adoles• will be divided between presentations by (STATS) Workshops cents leading applied statisticians and related Colorado State University, Ft. • Multi-year involvement with students computer lab activities. The rest of the Collins, CO May 31st-June 7th time, teams of four participants each, with • Strong directors and a committed and stable staff; guidance and assistance from the work• Thomas Boardman, Coordinator; shop coordinator, will design and carry James zumBrunnen, Site Manager • Involvement of staff members who look out projects involving collection and University of Wisconsin, Oshkosh, like the target population analysis of statistical data or activities WI June 4th-11th • Stable, long-term funding base with suitable for a statistics lab. After the work• shop ends, team members will remain mUltiple funding sources K.L.D. Gunawardena, Coordinator; networked bye-mail with each other, with Norbert Kuenzi, Site Manager • Recruitment of participants from a rel• the statistician presenters, and with the University of Delaware, Newark, DE evant target population in a defined area workshop coordinator, first during the June 18th-25th • Involvement with universities, colleges, remainder of the summer as they develop industries, schools, community-based or• and adapt their projects for use in their John Schuenemeyer, Coordinator; own teaching, and then throughout the fall ganizations Lillian Russell, Site Manager as they try out their projects in the class• University of California, Santa • Parental and community support room. Each regional workshop will hold a Barbara, CA June 25th-July 2nd one-day reunion after the tryout period, in • Development of a peer support system connection with a regional or national Carolyn Dobler, Coordinator; Ann • Evaluation, long-term follow-up, care• meeting of the MAA. Watkins, Site Manager ful data collection Participants or their home institutions are • Networking through a consortium expected to cover the cost of travel to the 26 February 1995 FOCUS ematics and Statistics, Washburn Univer• MAA Contributed Papers in Burlington sity, Topeka, KS 66621; (913) 231-1010 The Mathematical Association ofAmerica emphasis on student understanding of X1491; e-mail: [email protected] and the American Mathematical Society mathematics. As a result of these projects, Howard L. Penn, U.S. Naval Academy will hold the annual MathFest from Sun• students will arrive at post-secondary in• day, August 6, 1995 through Tuesday, stitutions with new expectations and ex- ' This session will present talks describing August 8, 1995 in Burlington, Vermont. periences. This session will address project innovative techniques in the teaching of mathematics courses typically taught in The complete meetings program will ap• approaches, teacher development issues, the first year of college. Innovative tech• pear in the April 1995 issue of FOCUS evaluation, and student learning. niques include (but are not limited to) the and the May 1995 issue of the AMS No• Symbolic Computation in the Under• use of technology, writing projects, and tices. This preliminary announcement is graduate Mathematics Classroom designed to alert participants about the cooperative learning. Sunday and Monday afternoons MAA's contributed papers sessions and Achieving K-12 and Higher Education their deadlines. Ronald Sklar* Department of Computer Collaboration in Systemic Reform Science, St. John's University, St. Programs Please note that the days scheduled for Vincent's College, Grand Central & Uto• these sessions, as well as other details, pia Parkways, Jamaica, NY 11439; (718) Sunday and Monday afternoons remain tentative. The organizers listed 990-6161 X7212; fax: (718) 990-1882; e• R. D. Anderson* Louisiana Systemic Ini• below solicit contributed papers pertinent mail: [email protected] tiative Program, 1885 Wooddale Boule• to their sessions; proposals should be di• vard, II th floor, Baton Rouge, LA 70806; rected to the organizer whose name is fol• Richard O'Lander, St. John's University (504) 922-0690; fax: (504) 922-0688 lowed by an asterisk (*). For additional The creative use of computer algebra sys• More information on this session is avail• instructions, see the Submission Proce• tems is resulting in a revolution in the way able on the MAA Gopher or from the or• dures box below. we teach mathematics. These systems have ganizer. Sessions generally must limit presenta• been effectively used in teaching calcu• Use of Interactive Video Games in tions to ten minutes, but selected partici• lus, linear algebra, differential equations, pants may extend their contributions up to combinatorics, probability, and statistics. TeachinglLearning Mathematics twenty minutes. Each session room con• This session invites papers on experiences Sunday and Monday afternoons tains an overhead projector and screen; with the use of a symbolic computation Richard Stephens* Department of Math• blackboards will not be available. You may system in teaching mathematics. Subject ematics and Computer Science, Western request one additional overhead projec• areas not mentioned above are especially Carolina University, Cullowhee, NC tor, a 35mm slide projector, or a 112-inch welcome. 28723-9049; (704) 227-7245 or 3/4-inch VHS VCR with one color Innovative Teaching in First-year More information on this session is avail• monitor. Persons needing additional equip• College Mathematics Courses able on the MAA Gopher or from the or• ment should contact, as soon as possible, Sunday and Tuesday afternoons ganizer. but prior to May 5, 1995: Donovan H. Van I. Osdol, Department of Mathematics, Uni• Aaron Stucker* Department of Math- See Papers on page 28 versity of New Hampshire, Durham, NH 03824; e-mail: [email protected]. Submission Procedures for Contributed Paper Proposals Implications of the NCTM Standards After you have selected a session to which you wish to contribute a paper, forward the for College Level Teaching following directly to the organizer (indicated above with an asterisk (*»: Sunday and Monday afternoons • the name(s) and address(es) of the author(s); and Darien Lauten* Department of Mathernat• • a one-page summary of your paper. ics, University of New Hampshire, Durham, NH 03824-3591; (603) 868- The summary should enable the organizer(s) to evaluate the appropriateness of your 7133; fax: (603) 868-4096; e-mail: paper for the selected session. Consequently, you should include as much detailed dlauten@ christa.unh.edu information as possible within the one-page limitation. New secondary mathematics curriculum Your summary must reach the designated organizer by Monday, April 14, 1995. projects funded by the National Science The organizer will acknowledge receipt of all paper summaries. If the organizer Foundation engage students in learning accepts your paper, you will receive a standardized abstract form. Use this form to situations aligned with the NCTM Cur• prepare a brief abstract. Please return the completed form to the organizer by Monday, riculum and Evaluation Standards for May 8, 1995. Abstracts received after the deadline will not be published in the booklet School Mathematics. Characteristics of of abstracts which will be available in the meetings registration area during the con• these projects include changes in math• ference. ematics content, use of technology, new pedagogical approaches, mathematics in DO NOT FORWARD COMPLETED ABSTRACTS TO THE MAA OR TO context, inclusion of all students, and THE AMS. THEY ARE TO BE SENT TO THE SESSION ORGANIZER. 27 FOCUS February 1995 265-2384; email: rickey@ MAA Professional Development Activities bgsuvax.bgsu.edu. SPRING-SUMMER 1995 June 15-17 Short Course: Symmetry and Group Theory. Doris Schattschneider; The following is a listing of all MAA or 18th St. NW, Washington, DC 20036- University of Dayton, Dayton, Ohio. Con• MAA section sponsored professional de• 1385; (202) 387-5200; fax: (202) 265- tact Tom Gantner, Department of Math• velopment activities for spring and sum• 2384; e-mail: [email protected]. ematics, University of Dayton, Dayton, mer 1995 about which information was OH 45469-2316; (513) 229-2511; fax: June 1-11 Workshop: Constructivist received by December 5, 1994. Each list• (513) 229-2566; e-mail: gantner@udavxb. Methods in Undergraduate Mathematics ing contains the date, title, principal oca.udayton.edu. Teaching: AbstractAlgebra. I Ed Dubinsky, speaker(s), location, and contact person. Keith Schwingendorf, and David June 18-23 Short Course: The Teaching More information on some of these ac• Mathews; Purdue University, West of Calculus. Arnold Ostebee and Paul tivities will be found elsewhere in this issue Lafayette, Indiana. Contact Terry Loro, Zorn; University of Maine, Orono, Maine. of FOCUS. A more up-to-date listing of Department of Mathematics, Purdue U ni• Contact Clayton W. Dodge, 5752 Neville/ professional development activities is versity, West Lafayette, IN 47907; (317) Math, University of Maine, Orono, ME posted on the MAA Gopher. 494-1982; fax: (317) 494-6318; e-mail: 04469-5752; (207) 581-3908; e-mail: April 22 Minicourse: An Introduction to [email protected]. [email protected]. Research in the Teaching and Learning of June 4-11 Workshop: Summer Workshop June 18-25 Workshop: Summer Work• Undergraduate Mathematics: Examples in for Mathematicians Who Teach Statistics. shop for Mathematicians Who Teach Sta• Calculus. Joan Ferrini-Mundy and Karen K. L. D. Gunawardena and Norbert tistics. John Schuenemeyer and Lillian Graham; Simmons College, Boston, Mas• Kuenzi; University of Wisconsin, Russell; University of Delaware, Newark, sachusetts. Contact Donna Beers, Oshkosh, Wisconsin. Contact Jane Heck• Delaware. Contact Jane Heckler, STATS Simmons College, 300 The Fenway, Bos• ler, STATS Project Registrar, Mathemati• Project Registrar, MAA, 1529 18th St., ton, MA02115-5898; (617) 521-2389; e• cal Association of America, 1529 18th St. NW, Washington, DC 20036-1385; (202) mail: [email protected] NW, Washington, DC 20036-1385; (202) 387-5200; fax: (202) 265-2384; e-mail: May 25-June 11 Workshop: Con• 387-5200; fax: (202) 265-2384; e-mail: [email protected]. structivist Methods in Undergraduate [email protected]. June 25-July 2 Workshop: Summer Mathematics Teaching: Calculus. I Ed June 5-9 Workshop: Mathematics Labo• Workshop for Mathematicians Who Teach Dubinsky, Keith Schwingendorf, and ratories with DERIVE. Marvin Brubaker Statistics. Carolyn Dobler and Ann David Mathews; Purdue University, West and Carl Leinbach; Frostburg State Uni• Watkins; University of California, Santa Lafayette, Indiana. Contact Terry Loro, versity, Frostburg, MD. Contact Jack Barbara, California. Contact Jane Heck• Department of Mathematics, Purdue Uni• Biggs, Department of Mathematics, ler, STATS Project Registrar, MAA, 1529 versity, West Lafayette, IN 47907; (317) Frostburg State University, Frostburg, MD 18th St NW, Washington, DC 20036-1385; 494-1982; fax: (317) 494-6318; e-mail: 21801; (410) 543-6740, e-mail: (202) 387-5200; fax: (202) 265-2384; e• [email protected]. bafusaro@sae,towson.edu. mail: [email protected]. May 31-June 7 Workshop: Summer June 5-23 Summer Institute: Institute in June 26-30 Short Course: Modular Workshop for Mathematicians Who Teach the History of Mathematics and its Use in Forms, Elliptic Curves, and Fermat's Last Statistics. Thomas Boardman and James Teaching. V. Frederick Rickey and Victor Theorem. Fernando Gouvea; Allegheny zumBrunnen; Colorado State University, J. Katz; The American University, Wash• College, Meadville, Pennsylvania. Con• Ft. Collins, Colorado. Contact Jane Heck• ington, DC. Contact V. Frederick Rickey, tact George Bradley, Mathematics and ler, STATS Project Registrar, The Math• MAA, 1529 18th St. NW, Washington, DC Computer Science Department, Duquesne ematical Association of America, 1529 20036-1385; (202) 387-5200; fax: (202) University, Pittsburgh, PA 15282-0001; (412) 396-5115; e-mail: bradley@ Papers from page 27 duq3.cc.duq.edu. Popularizing Mathematics ing mathematics both outside the class• July 10-21 Workshop: Cooperative Sunday and Tuesday afternoons room, among families, seniors, policy Learning in Undergraduate Mathematics Eric Muller* Department of Mathemat• makers, etc., and in the university class• Education. Ed Dubinsky and Janet Ray; ics, Brock University, St. Catharines, ONT room for students enrolled in mathemat• Purdue University, West Lafayette, Indi• L2S 3A 1 Canada; (905) 688-5550 X3297; ics "service" courses, including ana. Contact Terry Loro, Department of fax: (905) 682-9020; e-mail: emuller@ prospective elementary school teachers. Mathematics, Purdue University, West spartan.ac.brocku.ca Submissions should include some docu• Lafayette, IN 47907; (317) 494-1982; fax: mentation of the activities, some informa• Bernard R. Hodgson, Universite Laval (317) 494-6318; e-mail: loro@math. tion on who developed and ran them, the purdue.edu. Papers on successful activities to popular• segment of the population at whom they ize mathematics are solicited. Of special were aimed, how often the programs have I Not sponsored by the MAA or a section, but interest are activities aimed at populariz- run, and some indication of their degree of will be of special interest to MAA members. success. 28 February 1995 FOCUS It is through the study of groups that dif• Mathematics Mathematics and ferent types of patterns are distinguished. Using group theory, mathematicians can Awareness Week, Symmetry prove that there are exactly seventeen ways to construct repeating wallpaper patterns April 23-29, 1995 Paul Davis and Martin Golubitsky (or periodic planar tilings). Indeed, group theory has been one of the most exciting Symmetry is all around us. We see sym• Mathematics Awareness Week is held each branches of mathematics during the past metry in everyday objects-in buildings, year to increase public understanding of century, beginning with Lie's discovery in floor and wall tiles, in gears, and even and appreciation for mathematics. This of continuous groups through to the re• in automobile hubcaps. We see symmetry year's theme is "Mathematics and Sym• cent classification of finite simple groups. in many natural forms-in the bilateral metry." More information can be obtained Investigating the vast connections between symmetry of the human form, in the rota• from the Joint Policy Board for Mathemat• group theory and topology, geometry, and tional and kaleidoscopic symmetry of blos• ics «202) 234-9570) and on the MAA analysis continues to be a central theme in soms, in the sinuous spiral symmetry of Gopher (gopher.maa.org). MAW visuals mathematics research. Beginning with vines and shells, and in the translation are available via the WorldWide Web; in• Galois theory and continuing to current symmetry of honeycombs and fish scales. formation can be found on the MAA Go• research, symmetry enables us to find Symmetry also reveals itself in the deco• pher. (See page 7.) solutions to equations-first to algebraic rative arts of many cultures, including, for equations and now to differential equa• Activities for Mathematics Awareness example, in the decorative designs of the tions. Week are generally organized by college Moors at the Alhambra, in the weavings and university departments, institutional of the Indians of the American Southwest, Symmetry is also central to the mathemati• public information offices, student groups, and in the curious interlocked creatures cal description of many natural phenom• and related associations and interest that are found in the work of the math• ena. The catalog of three-dimensional groups. They include a wide variety of ematically perceptive modem graphic art• repeating patterns is identical to the cata• workshops, competitions, exhibits, festi• ist M. C. Escher. Indeed, the symmetry log of ways that atoms can arrange them• vals, lectures, and symposia. evident in the decorati ve art of a particular selves on crystal lattices. Chemists and The number and breadth of activities in• culture may serve as a cultural marker. mathematicians have classified the 230 crystallographic groups-the 230 forms creases annually. For example, one col• Symmetry appears on a grand scale in the of crystal structure-by analyzing in de• lege regularly sponsors a high school formulation of distance in special relativ• tail the possible combinations of rotations, mathematics day to encourage women to ity and even in the shapes of galaxies, and reflections, and translations that leave a continue their studies in mathematics. At on a microscopic scale in the classifica• crystal lattice unchanged. Symmetry is one university, two departments-math• tion of crystal structure. Symmetry also important in material science and elastic• ematics and architecture--cooperated to plays a pivotal role in mathematics-from ity where it is incorporated into the consti• plan and produce an interactive traveling the verification that a general polynomial tutive relations that govern the structure exhibit that provides hands-on experience of degree five or higher cannot be solved and behavior of solids and liquids. for such topics as codes, tilings, chaos, by formula, to the classification of types geometry, graphs, and computer science. of geometry, to the existence of conserva• Symmetry is basic to our understanding Coordination and support for Mathemat• tion laws. of the hydrogen atom and molecular spec• troscopy, to elementary particles and the ics Awareness Week is provided by the To mathematicians, symmetries are de• theory of quarks, as well as to the two Joint Policy Board for Mathematics. Ad• fined as transformations that leave an crowning achievements of twentieth cen• ditional financial support comes from the object or a picture or an equation un• tury physics-the theory of relativity and United States Army Research Office. changed. These transformations are called quantum mechanics. Indeed, one might the symmetries of the object; together they The accompanying article (see right) on even characterize the current search for a form a mathematical structure known as a the theme for Mathematics Awareness 'unified field theory' -a single theory to group-the symmetry group of the object. Week 1995 was written by Paul Davis describe all forces of nature-as a search (Worcester Polytechnic Institute) and Objects such as the human form or a per• for the fundamental symmetry group of Martin Golubitsky (University of Hous• fectly symmetric butterfly have bilateral the physical universe, from which all the ton). Review and ideas were provided by symmetry because they cannot be distin• basic laws of physics will follow. Jeffrey Adams (University of Maryland), guished from their reflections across a Symmetry has appeared in technology in Martin Isaacs (University of Wisconsin), mirror plane. Similarly, repeating patterns, surprising ways. In computer vision the Peter Olver (University of Minnesota), or more vividly wallpaper patterns, are symmetries of human perception (projec• Marjorie Senechal (Smith College), and pictures that can be picked up, shifted, and ti ve transformations) are incorporated into Doris Schattschneider (Moravian Col• put down again so that the picture is un• the design of mathematicall y based image lege). disturbed. Objects with helical symmetry processing systems which may have im• are those that are invariant under screw portant applications to medical imaging. motions about a central axis. 29 FOCUS February 1995 In applications to control theory, rotation and translation symmetries must be taken MSEB Launches Study of Mathematics into account when designing feedback controllers for both aircraft and satellites. for the World of Work Just as the absence of symmetry has strik• On November 18th the Mathematical Sciences Education Board of the National Acad• ing effects in art and music, the absence or emy of Sciences held a workshop at which mathematics and vocational educators, math• loss of symmetry is of great interest in ematicians, business and industry representatives, and policy makers discussed the models of natural phenomena, often with mathematical preparation of the technical workforce, including questions of core cur• dramatic consequences. Symmetry break• riculum, tracking, articulation, and student concerns. ing occurs when structures buckle, when water boils, and (possibly even) when spots The workshop, supported by the Alfred P. Sloan Foundation, is the first step in a long• form on leopards and stripes form on ti• term MSEB project to build the dialogue between the communities of mathematics and gers. Twenty years ago, mathematicians vocational educators in order to develop mathematics programs that address student and physicists demonstrated a route to needs for both academic and vocational preparation. turbulence that involves the development Workshop participants agreed on the need to: of more complicated fluid flow patterns • Integrate vocational and academic programs; signaled by a succession of losses of sym• metry. These kinds of exploration have • Identify mathematics that is latent in the workplace; given symmetry an apparently paradoxi• • Disseminate concrete examples of mathematics in work situations; cal role-the role of charting the onset of complicated or chaotic behavior. The im• • Align disciplinary and occupational skills standards; ages shown on this year's Mathematics • Promote examples of successful programs; and Awareness Week poster are formed using a combination of symmetry and chaotic • Improve communication between vocational and mathematics educators. dynamics. Their detailed complex struc• They also identified many issues that need further exploration, including: ture is due to chaotic dynamics while their apparent regularity and familiarity is due • Is vocational education a path to a different goal or a different path to the same goal? to symmetry. • Can a single mathematics curriculum serve both vocational and academic needs? Arrangements that show a high degree of • Is a "separate and equal" track for vocational education appropriate or possible? order may fail to have any global symme• try, yet symmetry may have a role in de• • Who will develop mathematics curricula for vocational education? scribing and classifying these patterns. • Which are more important as preparation for work-basic or higher order skills? Tiles that can only fit together in ways that have no translation symmetry and newly • Is mastery of basic skills a prerequisite to learning higher order skills? found 'quasicrystals' that display symme• The MSEB will publish a workshop report early in 1995 which will be for sale from the try forbidden by the conventional model National Academy Press. For further information about the workshop and its follow-up, of crystals are two exciting areas of cur• contact Susan Forman bye-mail at [email protected], or by mail at the Mathematical rent research. Sciences Education Board, 210 1 Constitution Ave. NW, HA476, Washington, DC 20418. Institute in the History of Mathematics and its Use in Teaching The MAA will organize the Insti• ena Pycior. Activities at the insti• institutions that prepare secondary tute in the History of Mathematics tute will include reading of original teachers. Facilities at American and its Use in Teaching at American sources, survey lectures, small University are fully accessible. University, Washington, DC, June group projects, field trips to three Dormitory space for families of par• 5-23, 1995, and for three additional great libraries, and discussions of ticipants is available. Completed ap• weeks in June 1996, with work con• methods of conducting a history of plications will be due by March 15th tinuing through an electronic net• mathematics course. Participants and applicants will be notified of work during the academic year will be prepared to make presenta• their acceptance by early April. For 1995-96. The teaching staff will tions on their work at the Joint more information and application consist of well known historians of Mathematics Meetings. forms, write to V. Frederick Rickey, mathematics, including, in the first MAA, 1529 18th St. NW, Washing• year, V. Frederick Rickey, Victor J. Applications are welcome from fac• ton, DC 20036-1385; e-mail: Katz, Steven H. Schot, Ronald ulty teaching at small institutions, [email protected]. Calinger, Judith Grabiner, and Hel- at minority-serving institutions, and 30 February 1995 FOCUS NASA's Flight Opportunities for Science Teacher EnRichment: The FOSTER Program for Science Teachers What is FOSTER? The National Aero• prised of a high school and junior high (or vided with Internet accounts and, if nautics and Space Administration is fly• middle) school teacher team, or a junior needed, a modem for use at their school ing science teachers on board NASA's high (or middle) school and elementary site. Kuiper Airborne Observatory (KAO). teacher team. Teachers from the eleven School Site Visits by NASA During the Through the unique capabilities of this western states served by the NASA Ames school year, FOSTER teachers host a visit airborne observatory, teachers experience Research Center Educational Programs by a NASAAerospace Education Special• cutting edge scientific research and the Office are eligible to apply for 1995. These ist for teacher workshops, assemblies, and discovery process on ovemight astronomi• states are Alaska, Arizona, California, classroom visits at their school site. NASA cal research missions on board the KAO. Hawaii, Idaho, Montana, Nevada, Oregon, These teachers are part of the Flight Op• Utah, Washington, and Wyoming. Appli• scientists, engineers, and others are also portunities for Science Teacher cations are available January I, 1995, and available to share their work and experi• EnRichment (FOSTER) project, a pre-col• due in March 1995. ences with students via Internet or for vis• lege teacher enrichment project at NASA its to schools. Summer Workshop at NASA Ames Ames Research Center. Science on board Research Center To prepare for their How Is FOSTER Funded? FOSTER is a the KAO involves many disciplines: as• flights, teachers participate in an inten• joint project of NASA Ames Research trophysics and astronomy, planetary sci• Center and the SETI Institute, and is funded sive workshop at NASA Ames Research ences, atmospheric sciences, aeronautics, by NASA'sAstrophysics Division. For the Center during the summer. The workshop physics, chemistry, engineering, computer includes lectures, lessons, tours, and ac• summer workshop, housing at Stanford sciences, and mathematics. As FOSTER University is provided, plus transportation tivities related to modem astrophysics and participants, teachers attend a summer airborne astronomy. During the workshop, including airfare up to $500 per round trip, workshop at NASA Ames Research Cen• and meals. Teacher flights on board the teachers are provided with curriculum ter, and return to fly with a team of pilots, NASA's KAO are also funded for trans• materials and activities available through scientists, engineers, mission specialists, portation, housing, and meals. NASA and others. They try out new les• and technical staff on board the KAO to sons to bring their FOSTER experiences For further information or to request an observe the infrared universe. to the classroom and their fellow teachers. application, contact FOSTER Project, Who can participate? Teachers in grades College credit is available for participa• SETI Institute, 2035 Landings Dr., Moun• 3 through 12 are eligible to apply for the tion in FOSTER. tain View, CA 94043; voice mail: (415) 1995 program which begins with the sum• 960-4539; fax: (415) 961-7099; e-mail: Internet Training and Access In addi• merworkshop,July5-14,1995,atNASA wendy_horton @ se ti - ins t. ed u; tion, the workshop offers hands-on tele• Ames Research Center. FOSTER teacher [email protected]. The NASA communications training. Teachers leam flights on the KAO will be scheduled July contact is Garth A. Hull, Chief, Educa• to access the Internet to communicate through September. Teachers are required tional Programs Office, NASA Ames between schools, to contact NASA scien• to apply in teams from two separate Research Center, MS 204-12, Moffett tists and engineers, and to access on-line schools in the same district or geographic Field, CA 94035-1000; voice mail: (415) resources such as FOSTER On-Line and region. Generally these teams are com- 604-5543; fax: (415) 604-3445. Spacelink. Participating teachers are pro- Annual Ohio Section Summer Short Course Symmetry and Group Theory Presented by Doris Schattschneider (Moravian CoUege) June 15-17, 1995, University of Dayton in Dayton, Ohio This short course will emphasize a visual, hands-on approach to understanding the symmetry groups of two- and three-dimensional objects through the use of computer software, patterns and tilings, polyhedral models, and videotapes. Many abstract concepts encountered in a first course in group theory can be illustrated in a graphic manner using this approach. The registration fee is $100. Air-conditioned dormitory suites consisting of a living room, bath, and two bedrooms with two single beds in each bedroom are available with a price per night of $20 (half a double), $40 (single), $80 (suite). A Thursday evening banquet will be held; cost of attendance is $16. Other meals may be purchased in the student union. Registration is open until forty $50 deposits are received or May 15th, whichever is earlier. For more information or to register, contact Tom Gantner, Department of Math, University of Dayton, Dayton, OH 45469-2316; (513) 229-2511; fax: (513) 229-2566; e-mail: [email protected]. 31 FOCUS February 1995 THE BEST IN MATHEMATICAL Lion Hunting and Other Algebra and Tiling A Radical Approach to Mathematical Pursuits Sherman Stein and Sandor Szabo Real Analysis A Collection ofMathematics, Verse, Often questions about tiling space or a David Bressoud and Stories by Ralph P. Boas, Jr. polygon lead to other questions. For This book is an undergraduate introduc• Gerald L. Alexanderson instance, tiling by cubes raises questions tion to real analysis. Use this book as a and Dale H. Mugler, Editors about finite abelian groups. Tiling by tri• textbook for an innovative course, or as a As a young man at the Institute for Ad• pods or crosses raises questions about resource for a traditional course. If you vanced Study in Princeton, Ralph Philip cyclic groups. From tiling a polygon with are a student and have been through a tra• Boas, Jr., together with a group of other similar triangles, it is a short step to inves• ditional course, yet still do not understand mathematicians, published a light-hearted tigating automorphisms of real or com• what real analysis is about and why it was article on the "mathematics of lion hunt• plex fields. Tiling by triangles of equal created, read this book. ing" under a pseudonym (1938). This areas soon involves Sperner's lemma from sparked a sequence of articles on the topic, topology and valuations from algebra. This course of analysis is radical; it returns to the roots of the subject, but it is not a several of which are drawn together in this The first six chapters of Algebra and Til• book. history of analysis. It is rather an attempt ing form a self-contained treatment of to follow the injunction of Henri Poincare: Lion Hunting includes an assortment of these topics, beginning with Minkowski 's let history inform pedagogy. The author articles that show the many facets of this conjecture about lattice tiling of Euclid• wrote the book as a first encounter with remarkable mathematician, editor, writer, ean space by unit cubes, and concluding real analysis, laying out its context and and teacher. Along with a variety of his with Laczkowicz's recent work on tiling motivation in terms of the transition from lighter mathematical papers, the collec• by similar triangles. The concluding chap• power series to those that are less tion includes Boas' verse and short sto• terpresents a simplified version ofRedei 's predictable, especially Fourier series. ries, many of which are appearing for the theorem on finite abelian groups: if such Bressoud marks some of the traps into first time. Anecdotes and recollections of a group is factored as a direct product of which even great mathematicians have his numerous experiences and of his work subsets, each containing the identity ele• fallen in exploring this area of and meetings with many distinguished ment, and each prime order, then at least mathematics. mathematicians and scientists of his day one of them is a subgroup. A remarkable are also included. geometric implication of this result is The book begins with Fourier's introduc• developed in Chapter 2. tion of trigonometric series and the prob• The mathematical articles in this collec• lems they created for the mathematicians tion cover a range of topics. They include Algebra and Tiling is accessible to un• of the early nineteenth century. Cauchy's articles on infinite series, the mean value dergraduate mathematics majors, as most attempts to establish a firm foundation for theorem, indeterminate forms, complex of the tools necessary to read the book are calculus follow, and the author considers variables, inverse functions, extremal found in standard upper division algebra his failures and his successes. The book problems for polynomials, and more. courses, but teachers, researchers and pro• culminates with Dirichlet's proof of the fessional mathematicians will find the Boas' wit and playful humor are reflected validity of the Fourier series expansion book equally appealing. Beginners will in the verses included in this collection. and explores some of the counterintuitive find the exercises and the material found The verses reflect the phases of his career results Riemann and Weierstrass were led in the appendices especially useful. The as author, editor, teacher, department chair, to as a result of Dirichlet's proof. "Problems" section will appeal to both and lover of literature. A section of the beginners and experts in the field. The To facilitate graphical and numerical in• book describes the feud that Boas suppos• book could serve as the basis of an under• vestigations, Mathematica commands and edly had with Bourbaki. Also included are graduate or graduate seminar, or a source programs are included in the exercises. many amusing anecdotes about famous of applications to enrich an algebra or However, you may use any mathematical mathematicians. geometry course. tool that has graphing capabilities, includ• We profit from Boas' labor, and treasure it ing the graphing calculator. 224 pp., Hardcover, 1994 as an inheritance. We should allow our• ISBN 0-88385-028-1 336 pp., Paperbound, 1994 selves in some measure to walk along his List: $34.00 MAA Member: $26.00 ISBN 0-88385-701-4 path. This collection will allow the reader Catalog Code: CAM-25/FOC List: $29.00 MAA Member: $22.00 a glimpse of that path. Catalog Code: RAN/FOC 240 pp., Paperbound, 1994, ISBN 0-88385-323-X List: $35.00 MAA Member: $25.00 Catalog Code: DOL-I5/FOC 32 February 1995 FOCUS EXPOSITION • • • JuST PUBLISHED! All the Math Linear Algebra Problem Assessing Calculus That's Fit to Print Book Reform Efforts Articles from the Manchester Paul Halmos A Report to the Community Guardian This is a book for mathematicians at all James R. C. Leitzel and Alan C. Keith Devlin levels. Paul Halmos tells us, "Even if I Tucker, Editors know some answers, I don't think I under• Assessing Calculus Reform Efforts: A Between 1983 and 1989 Keith Devlin, re• stand a subject until I know the questions. Report to the Community provides a re• search mathematician, author, and educa• The questions in mathematics are called view of various aspects of the calculus tor, wrote a semi-monthly column on problems-and although I learned some reform movement. This study gives an mathematics and computing in the English linear algebra a long time ago, until now assessment of the current attitudes and in• national daily newspaper, The Manches• I have made no serious effort to examine volvement of mathematical sciences de• ter Guardian. This book is a compilation the problems that the solutions are based partments-their faculty and students-in of many of those articles. It is witty, enter• on. I wrote this book to organize those efforts to revise calculus instruction. taining, and easy to read. questions-problems-in my own mind." A key finding of the assessment study is The mathematical topics range from This book can be either the main course or that how calculus is taught has changed simple puzzles to deep results including the dessert for someone who needs linear more than what is taught. The changes in open problems such as Faltings Theorem algebra-and nowadays that means every instructional practice, more frequent use and the Riemann Conjecture. You will find user of mathematics. It can be used as the of technology, and increased focus on articles on prime numbers, how to work basis of either an official course or a pro• building students' conceptual understand• out claims for traveling expenses, calcu• gram of private study. ing are finding their way into both pre• lating pi, computer simulation, patterns calculus and post-calculus mathematics and palindromes, cryptology, and much If used as a course, the book can stand by courses. This has encouraged increased more. itself, or if so desired, it can be stirred in with a standard linear algebra course as interest in research about how undergradu• This book is meant for browsing by any• the seasoning that provides the interest, ate students learn mathematics. Many of one who regularly reads a serious news• the challenge, the moti vation that is needed the institutions that currently report using paper and has some interest in matters by experienced scholars as much as by reform materials in experimental sections scientific or mathematical. Keith Devlin beginning students. have plans to move their efforts to course• tells us "from the mail I received I know wide adoption in the near future. The fact that the readers of the column were a var• The best way to learn is to do, and the that in the past two years over 95% of in• ied bunch. They ranged from students at purpose of this book is to get the reader to stitutions using a reform text continue schools in their early teens (occasionally DO linear algebra. The approach is using a reform text the next year (for at even younger!), to retired people in their Socratic: first ask a question, then give a least some sections) indicates that calcu• nineties (often the ones who best succeed hint (if necessary), then, finally, for secu• lus reform is likely to be around for the in cracking the brain teasers I occasion• rity and completeness, provide the detailed foreseeable future. ally included in my articles); from prison answer. Data is provided on the number of institu• inmates to executives in the computer in• 340 pp., 1994, Paperbound tions at each ofthe levels of postsecondary dustry; from truckers to schoolteachers; ISBN 0-88385-322-1 education that are engaged in reform ef• both men and women." List: $35.00 MAA Member: $25.00 forts and also gives information on fac• It you think that nothing of interest has Catalog Code: DOL-16/FOC ulty and student involvement, and reviews happened in mathematics since the time of texts produced by some of the major of Pythagoras, this book will change your curriculum development projects. The ap• mind. Keith Devlin presents mathematics pendices to the report include brief de• as a living human enterprise, both a sci• scriptions of selected calculus reform texts ence and an art. Use order form on and a complete listing of the NSF awards made during the seven years ofthe Calcu• 345 pp., Paperbound, 1994 lus Initiative. ISBN 0-88385-515-1 List: $32.50 MAA Member: $25.00 page 34, or call 100 pp., Paperbound 1994 Catalog Code: ATMAIFOC ISBN 0-88385-093-1 List: $15.00 1-800-331-1622 Catalog Code: NTE-37/FOC 33 FOCUS February 1995 Resources for Calculus Collection Wayne Roberts, Maca/ester College, Project Director Third Printing This five-volume set provides new materials with which instructors can refresh the teaching of college-level calculus. The independent volumes are composed ofcontrilmtions from a wide assortment ofcollege mathematics instructors. The set, produced through a National Science Foundation grant to the Associated Colleges of the Midwest and the Great Lakes Colleges Association, join in spirit other efforts to improve mathematics instruction. All are most useful as curriculum resource materials in the hands of those teaching calculus. -CHOICE Learning By Discovery Calculus Problems Applications of Calculus A Lab Manual For Calculus For a New Century Philip Straffin, Editor Anita Solow, Editor Robert Fraga, Editor Students see how calculus can explain the structure of a rainbow, guide a robot Contains 26 laboratory modules, that can Emphasizes conceptual understanding arm, or analyze the spread ofAIDS. Each be used as lab components in your course, over rote drill. Graphs and tables, rather module starts with a concrete problem or assigned as independent projects. The than rules, are used to define functions, and moves on to provide a solution. The labs are written without specific computer in the belief that "real world" data discussions are detailed, realistic and pay commands, so students read mathemat• generally come that way. The problems careful attention to the process of math• ics, not text. Suggestions are provided for are organized in groups that parallel ematical modeling. Exercises, solutions, implementing these labs on Derive, Maple traditional grouping of ideas, making it and references are provided. and Mathematica. Many can be done on possible to use them as supplements to graphics calculators. most texts. Most of the problems can be 280 pp., Paperbound, 1993 ISBN 0-88385-085-0 184 pp., Paperbound, 1993 done without the use of a calculator or computer. List: $26.00 ISBN 0-88385-083-4 Catalog Code: NTE-29/FOC 448 pp., Paperbound, 1993 List: $26.00 Catalog Code: NTE-27/FOC ISBN 0-88385-084-2 List: $26.00 Problems For Student Catalog Code: NTE-28/FOC Investigation Readings For Calculus Michael B. Jackson and John R. Underwood Dudley, Editor Ramsay, Editors Presents readings on the history of calcu• lus and of mathematics, on the nature of Students will learn how to use calculus to mathematics and its applications, on the solve real problems, how to use the library learning of calculus, and on the place of to find mathematical sources, how to read calculus and mathematics in society. Can and write mathematical material, and how be used as a supplement to any calculus to cooperate with their peers in solving dif• text, showing students that there is more ficult problems. Learning thatthey can solve to calculus than getting the right answer. what at first seems an inscrutable math• Exercises and problems included. ematical problem can only increase students' mathematical confidence. 224 pp., Paperbound, 1993 224 pp., Paperbound, 1993 ISBN 0-88385-087-7 ISBN 0-88385-086-9 List: $26.00 Catalog Code:NTE-3I1FOC List: $26.00 Catalog Code: NTE-30/FOC !Send this order form to: The MAA, 1529 Eighteenth Str;t, NW, Washington, DC 20036,1-800-331-1622; FAX: (202) 265-2384------, I Name I I Membership Code Address (Please wfrY the first six digit. that appear on~h~;;;i;;;e -;;jy-::"r mailing whel) I II City State Zip Code II Quantity Catalog Number Price IPayment 0 Check 0 VISA 0 MASTERCARD I ICard Number I :Signature Expiration Date I Foreign Orders Please add $3.00 per item ordered to cover postage and I I handling fees. The order will be sent via surface mail. If you want your order TOTAL I l!!:.nt by air, ~ will be h~y~s~ ~u .!£r~r.!!!! ~oice for y~ orde.!::.,. ______:...J 34 Buy all five volumes and save: List: $120.00 Catalog Code: CRPC/CAT February 1995 FOCUS What Will Be Next? , • You won't know unless you renew your subscription to Math Horizons. It's easy! Just fill out the form below and send it in. Don't miss any of the exciting articles in the next 4 issues of Math Horizons. ,------, Yes! Renew my subscription to Math Horizons. Individual Subscription (Individual subscriptions must be pre-paid) Please send __ copies. MAAMember Non-Member $20 per year $35 per year Bulk Subscription Minimum order is 20 copies with additional copies available in multiples of 10. Copies 20 30 40 50 60 ... IOn Price $100 $150 $200 $250 $300 ... $50n Please send copies of all four issues at $5 per student to the address below: Bill me __ Payment enclosed Or order by phone at 1-800-331-1622. _Visa MC Card Number Expiration date Name Address Address City State Zip Signature Date Phone( L ___ ~ail tO~Mat~HoriZO~~529~ighteent~treet, N.W., w~hington, DC_20036 ___ .J ------35 FOCUS February 1995 ~\II'LOY\lE"T Ol'l'OIHl "ITIES Bemidji State University sideration. The University is committed to ex• Assistant Professor of Mathematics. Anticipated cellence in teaching, scholarship and professional Rates for FOCUS EmploymentAdvertise• probationary position to begin August 30, 1995, activity, and service to the University and the ments are $65.00 per column inch (one inch subject to availability offunds. Earned doctorate community. Review of applications will begin February 10, 1995, and continue until positions minimum), billed to the nearest 112 inch. in Mathematics at time of appointment with emphasis in any area of pure or applied math• are filled. Submit letter of application indicating All advertisments are set with a one-line ematics or statistics. Expertise in implementing position of interest (I or 11), curriculum vitae, and headline, centered, boldface, consisting of graphic calculators and computers in the class• the names and addresses of three professional the institution name, unless additional room is desirable. All applicants must be able to references to: Dr. David Kinsey, Chair, Math• headline infonnation is provided. Ads will lawfully accept long-term employment in the ematics Department, University of Southern appear in the same language in which they United States at the time of an offer of employ• Indiana, 8600 University Boulevard, Evansville, are submitted. ment. Send letter of application, curriculum vita, IN 47712. AAlEOE. FOCUS offers a 15% discount for the same copies of transcripts (officials required at the time advertisement in three or more consecu• of employment), and three (3) letters of refer• Mathematics/CIS Department-Kutztown Uni• tive issues. All invoices include a tear sheet. ence sent directly by referrer to: Dr. Kenneth versity is inviting applicants for a tenure-track Advertising Copy Deadlines: The first of Lundberg, Interim Dean, College of Social and Assistant Professor in Mathematics for Fall, 1995. the month, two months prior to publica• Natural Sciences, Bemidji State University, Position involves teaching elementary and/or tion. FOCUS is published in February, Bemidji, MN 56601-2699. Inquiries about the secondary mathematics education courses at the April, June, August, October, and Decem• position may be directed to Dr. Tom Richard, graduate and undergraduate levels, teaching mathematics at the undergraduate level, and su• ber. Department Chair. Telephone: (218) 755-2832. Review of applications will begin on February pervising student teachers. Requirements include: Advertisers should contact: Amy Fabbri, 15, 1995, and continue until position is filled. Ph.D. in mathematics or mathematics education FOCUS Advertising Coordinator, The Equal opportunity educator and employer. with a master's degree in mathematics; exten• Mathematical Association of America, sive experience in the use of calculators and 1529 Eighteenth Street, NW, Washington, California State Polytechnic Univ. computers in the teaching of mathematics in el• DC 20036-1385; (202) 387-5200; fax: Tenure-track teaching position in Mathematics ementary/secondary school; evidence of (202) 265-2384; e-mail: [email protected] Education at the Assistant Professor level, sub• continuing scholarly growth and the potential to ject to funding. Doctorate in Mathematics or make contributions in teaching, research, publi• Mathematics Education, experience in K-12 cation and service to the university and Dartmouth College mathematics teaching or teacher education and community at large. Experience supervising stu• John Wesley Young Research Instruc- potential for related research required. Applica• dent teachers in elementary/secondary torship in Mathematics tion, resume, copy of transcripts, and three current mathematics is desirable. Send letter of applica• tion, a current resume, official graduate and The John Wesley Young Research Instructorship reference letters to be postmarked by 3113/95; in Mathematics is a two-year post-doctoral ap• position starting 9/95. Cal Poly Pomona is ac• undergraduate transcripts, three current letters pointment for promising new or recent Ph.D.s tively seeking to maintain its heritage and identity of recommendation and copies of available pub• lications to: Search Committee Chair, Department whose research interests overlap a department as a comprehensive center ofeducation that serves member's. Current departmental interests include a dynamic, culturally diverse region. For infor• of Mathematics and Computer Science, Kutztown areas in algebra, analysis, combinatorics, com• mation or to apply, contact: Math Ed Search University, Kutztown, PA 19530-0730. Applica• puter science, differential geometry, logic and Committee, Mathematics Department. Califor• tions must be received by March 10, 1995. set theory, number theory, probability and topol• nia State Polytechnic University, 380 I W. Temple Kutztown University is an Affirmative Action/ ogy. Teaching duties of four ten-week courses Ave., Pomona,CA91768-4033. (909)869-3467. Equal Opportunity Employer and actively solic• spread over two or three quarters typically in• EOE/AA. its applications from qualified women and clude at least one course in the instructor's minority candidates. specialty and include elementary, advanced and (at instructor's option) graduate courses. Nine• University of Southern Indiana Trenton State College month salary of $35,000 supplemented by Mathematics Department Department of Mathematics and Statistics summer (resident) research stipend of $7,778 Mathematics Faculty Anticipated Faculty Vacancy for Fall 1995 (two-ninths). Send letter of application, resume, Tenure-track position at the Assistant Professor The University of Southern Indiana invites ap• graduate transcripts, thesis abstract, description level (salary $33,732-$50,609): Req'd: Ph.D. in plications fortwo full-time tenure-track positions of other research activities and interests if appro• Statistics or Probability; demonstrated commit• as Assistant! Associate Professor in the Mathemat• priate, and 3 or preferably 4 letters of ment to quality teaching; strong research ics Department beginning Fall 1995. Position I: recommendation (at least one should discuss potential. Preference will be given to candidates Responsibilities include instructing courses in teaching) to Betty Harrington, Department of with post-doctoral experience in teaching and algebra, precalculus, calculus, and abstract alge• Mathematics, 6188 Bradley Hall, Hanover, NH research. Send vita and three letters of recom• bra or analysis. Position requires a Ph.D. in 03755-3551. Applications received by Jan. 15 mendation including at least one letter regarding mathematics or substantial completion of degree. receive first consideration; applications will be candidate's teaching ability to: Chair, Search Candidates with a specialization in the area of accepted until position is filled. Dartmouth Col• Committee, Dept. of Mathematics & Statistics, algebra or analysis will be given favorable con• lege is committed to affirmative action and TRENTON STATE COLLEGE, CN4700- sideration. Position II: Responsibilities include strongly encourages applications from minori• Hillwood Lakes, Trenton, NJ 08650-4700. instructing courses in algebra, precalculus, cal• ties and women. Deadline for application: March I, 1995, review culus, and mathematics for elementary teachers. process will begin February I, 1995. Non-US Position requires a terminal degree in Mathemat• citizens must include a statement of current visa ics or Mathematics Education or substantial status. The department currently enrolls over 250 completion of the degree. Candidates with a back• majors in Mathematics, Mathematics Education, ground in teaching a variety of courses at the and Statistics. To enrich education through di• undergraduate level will be given favorable con- versity, TSC is an AAlEOE. 36 February 1995 FOCUS Marymount University Division of Mathematics and Wheaton College Chair Computer Science Wheaton,IL Department of Mathematics Lander University Department of Mathematics/Computer Science Arlington, VA Greenwood, SC 29649 announces full-time, tenure-track faculty posi• tion beginning August 1995 as Assistant or Marymount University is seeking a chair of the Two positions in mathematics, (1) Asst. Prof. ten• Associate Professor. Doctorate in computer sci• Department of Mathematics at the associate or ure-track. Required: Ph.D., specialization is open. ence desired as well as interest in continued full professor level beginning in the fall of 1995. (2) Instructor, one year appointment with possi• professional growth. Department has seven full• Ph.D. in mathematics, strong teaching skill and bility of annual renewal for up to five years. time faculty; offers major in both mathematics research accomplishments required. Administra• Required: MA or MS in math. Preferred for both: and computer science. Contact Dr. Robert tive experience preferred. The Department eighteen graduate hours in Computer Science, Brabenec, Department ofMathematics/Computer includes five full-time faculty and twenty-three teaching and grant writing experience. Teach Science, Wheaton College, Wheaton,IL60187, majors. undergraduate math. Positions are for nine 7081752-5869, [email protected]. Marymount University is an independent, com• months. Teaching load is 12 cr.h. Position start prehensive university related to the Catholic August 1995. Screening continued until the po• Wheaton College is an evangelical Christian lib• Church. The University enrolls 4,000 students in sition is filled. Supply letter speaking to your eral arts college whose faculty and staff affirm a 22 undergraduate and 20 master's programs. The qualifications, a current curriculum vitae, and Statement of Faith and adhere to lifestyle expec• main campus is located on a hillside in residen• names, addresses, and phone numbers of at least tations. The college complies with federal and tialArlington, VA, ten minutes from Washington, three references to Bruce F. White, Division Chair. state guidelines for nondiscrimination in employ• ment. Women and minority applicants are DC. Lander University is an EO and AA Employer. encouraged to apply. Review will begin on February I, 1995, and con• We are proud of the diversity among our stu• dents, faculty, and staff, and most strongly tinue until the position is filled. Applicants should Department of Mathematics submit a letter of application, a curriculum vitae, encourage women and members of minorities to and five references with telephone numbers to apply. College of Natural Sciences and Chair, Search Committee for Mathematics, c/o Mathematics Office of the Dean of the School of Arts and Central Michigan University Indiana University of Pennsylvania Science, Marymount University, 2807 N. Glebe The Department of Mathematics invites applica• Indiana University of Pennsylvania invites ap• Road, Arlington, Virginia 22207-4299. tions for a tenure-track position in the area of plications for a tenure-track position in Statistics to begin in the Fall of 1995. The appointment is Marymount University is an Equal Opportunity/ collegiate mathematics education. The position anticipated at the Assistant Professor level. As• Affirmative Action Employer. Women and mi• is at the assistant professor level although candi• sociate Professor is possible in the case of norities are strongly encouraged to apply. dates with a strong record of teaching and research in the area may be considered for an appointment exceptional qualifications. Assistant Professor of Mathematics at the associate professor level. Candidates should Responsibilities: have a Ph.D. in Mathematics, Statistics, or Math• Full-time tenure-track Assistant Professor, be• 1. To teach undergraduate and graduate courses ematics Education and show evidence of having ginning Fall 1995, pending budget approval, to with emphasis on courses in statistics. conducted research in the teaching and learning teach 12 credits of undergraduate mathematics of collegiate mathematics. The successful candi• 2. To provide leadership in the development and per semester & share departmental duties. Salary date will be expected to help in the development implementation of statistics courses. range: $28,000 - $32,000, depending upon cre• of the Department's new Ph.D. Program in Math• dentials & experience. 3. To direct students in a graduate internship pro• ematics with Concentration in the Teaching of gram and to direct graduate student projects in MINIMUM REQUIREMENTS: Masters in College Mathematics. Duties include teaching statistics. Mathematics or related field, plus some doctoral and research with a usual teaching load of nine work; strong commitment to undergraduate teach• semester hours, Salary is competitive and ben• 4. To help establish an academic and professional ing. Teaching experience & quality of teaching is efits include University-paid retirement, medical, relationship between the undergraduate and of prime concern. dental, disability, and group life insurance. Cen• graduate programs at IUP and local and regional business/industrial organizations. PREFERRED QUALIFICATIONS: Doctorate in tral Michigan University has an enrollment of Mathematics or Mathematics Education; back• 16,500 of which 1,700 are graduate students, and Required Qualifications: offers Bachelor's, Master's, and Ph.D. degrees. ground in applications of mathematics welcome. 1. Ph.D. in Statistics or Ph.D. in mathematical The Department of Mathematics, which includes area with emphasis in statistics. Direct position inquiries to Dr. Richard C. pure and applied mathematics, statistics, and Weimer, (301) 689-4384, and employment in• mathematics education, has 32 tenure-track fac• 2. Teaching and/or field experience preferred but quiries to Human Resources, (301) 689-4105 Ulty. Please send resume, transcripts and names not required. (VoiceITDD). To apply, send letter of interest, of three references to Professor Richard J. 3. Experience in using statistical packages such resume, transcripts, and three letters of recom• Fleming, Chair, Department of Mathematics, as SAS and SPSS. mendation, not later than February 21,1995, to: Central Michigan University, Mt. Pleasant, MI Frostburg State University, Office of Human 48859. Consideration of applications will begin Review of applications will begin January 2, 1995 Resources,ATIN: Assistant Professor Mathemat• February 27, 1995, but applications will be re• and will continue until the position is filled. ics (Position #95-427-FOCUS), Frostburg, MD ceived until the position is filled. CMU (AAlEO Send letter of application, resume, undergradu• 21532. Institution) encourages diversity and resolves to ate and graduate transcripts, and three current FSU is an Affirmative ActionlEqual Opportunity provide equal opportunity regardless of race, sex, letters of reference (one of which should address Employer. Minority & women candidates are disability, sexual orientation, or other irrelevant teaching potential) to: Search Committee, Math• strongly encouraged to apply. criteria. ematics Department, Indiana University of Pennsylvania, Indiana, PA 15705, telephone 412- 357-2608, internet: ADDAVIS@GROVE. IUP.EDU, bitnet: ADDAVIS@IUP. IUP is an affirmative action/equal opportunity employer. 37 FOCUS February 1995 Georgia Southern University Hendrix College is a private, selective, United Technology, Farmingdale, NY 11736. The Col• Department of Mathematics and Methodist related, liberal arts college of about lege is an Equal Opportunity/Affirmative Action Computer Science 1000 students in Arkansas about 30 miles north Employer. of Little Rock in the foothills ofthe Ozark moun• Several positions starting September I, 1995. tains. The Department of Mathematics, with four Salary dependent upon qualifications. Indicated East Baptist University full-time faculty, has a vigorous undergraduate degrees are required by starting date. All dead• Marshall, Texas research program and currently offers a major in lines are postmark deadlines. Send letter of Applications are invited for a senior-track fac• mathematics with emphasis in computer science. application indicating position desired, curricu• ulty position in mathematics beginning fall 1995. lum vitae, unofficial transcripts of all college Send a letter of application, curriculum vitae, The successful candidate should have the Ph.D. work, evidence of dedication to outstanding statement of teaching interests and scholarship or Ed.D. in mathematics or mathematics educa• teaching, and name, address, telephone number, goals in a liberal arts environment, graduate and tion with an interest in teaching in both areas. and e-mail address of three references by the undergraduate transcripts, and letters from three Candidates near completion of the doctorate will indicated deadline to [search chair], [search num• references (at least one of whom is qualified to be considered. An appreCiation for teaching in an ber], Department of Mathematics and Computer address teaching effectiveness) in hard copy to: institution with a commitment to education in Science, Landrum Box 8093, Georgia Southern Robert C. Eslinger, Chair, Department of Math• the Christian tradition and the ability to excel in University, Statesboro, GA 30460-8093. ematics, Hendrix College, Conway, AR 72032 undergraduate teaching are required. Applications (eslinger@ alpha.hendrix.edu). The names of applicants and nominees, resumes, taken through March 15, 1995, or until position and other general non-evaluative information are Evaluation of applications will begin February is filled. Marshall is located in the rolling hills subject to public inspection under the Georgia 15, 1995 and continue until the position is filled. and piney woods of East Texas, 35 miles west of Open Records Act. Georgia Southern is an Equal Hendrix College is an Equal Opportunity/Affir• Shreveport, Louisiana, and 150 miles west of Opportunity/AffirmativeAction Institution. Per• mative Action employer. Women and minority Dallas. Send resume to Rutledge McClaren, sons who need accommodation(s) in the candidates are especially encouraged to apply. Chairperson, Department of Mathematics and application process under the Americans with Computer Science, 1209 North Grove, Marshall, Texas 75670-1498. Disabilities Act should notify the search chair. Hunter College of The City University of New York Mathematics. Three tenure-track positions. In• Montana State University structor or assistant professor. M.A. or M.S. in Department of Mathematics and Statistics Mathematics. Assistant Professor of Mathemat• mathematics required. Three years teaching ex• Two tenure-track Assistant Professor faculty ics, Computer Science, and Statistics at Montana perience is preferred. Duties include positions available September, 1995. Salary State University-Billings. Full-time, tenure-track freshman-level mathematics courses. Search Range: $29,931-$52,213 depending on expertise. position (contingent on funding) to begin Fall Chair: Dr. John A. Rafter. Search # 29851. Dead• Doctoral degree required. We are interested in Semester 1995 (August 1995). The person for line: March \, 1995. mathematicians (all fields) and statisticians (all this position should hold a Ph.D. in Mathematics fields). Send CV and three letters of reference by Statistics. One tenure-track position. Instructor and provide the department with expertise in 17 March 1995 to: or assistant professor. M.A. or M.S. in statistics computing systems. Responsibilities include, but or mathematics with an emphasis in statistics is Professor Richard Churchill, Chair are not limited to: I) demonstrated ability or required. Three years teaching experience is pre• Department of Mathematics and Statistics potential to teach a variety of courses in math• ferred. Duties include teaching introductory Hunter College ematics, statistics, and computer science - statistics courses and freshman-level mathemat• 695 Park Avenue Montana State University-Billings is primarily a ics courses. Search Chair: Dr. John A. Rafter. New York, NY 10021 teaching institution and encourages applicants Search # 29850. Deadline: March I, 1995. An Equal Opportunity/Affirmative Action Em• who are committed to excellence in teaching; 2) Temporary mathematics positions. Pending a ployer. revising and making curricular and pedagogical funding decision due in April, several temporary improvements to existing mathematics, statistics, positions may be added at the Instructor level to SUNY-Farmingdale and computer science courses to reflect changes and increased use of technology in these fields teach freshman level mathematics courses. Three Assistant Professor, Mathematics - fall 1995 and in their real world application - the Depart• years teaching experience is preferred. Fifteen anticipated tenure-track vacancy -to teach courses ment uses computers in a variety of ways to teach credit hours teaching load per quarter. M.A. or ranging from remedial through all levels of un• mathematics, computer science, and statistics, M.S. in mathematics required. Search Chair: Dr. dergraduate mathematics. Teaching emphasized, and the successful candidate should have experi• John A. Rafter. Search #29852. Deadline: April with scholarship and service expected. Minimum ence with computer mathematical systems such 17,1995. qualifications: Doctorate in mathematics, statis• as Mathematica; 3) adding to the intellectual vi• tics, or operations research. The Department is tality of the Department with original scholarship Hendrix College particularly interested in candidates with exper• and participating in departmental ,and other semi• Computer Science tise in applied mathematics, however, other areas nars; 4) advising students who are seeking a The Department of Mathematics invites applica• will be considered. Preference will be given to major/minor in mathematics or a minor in com• tions for a new tenure-track assistant professor candidates possessing expertise in the use of tech• puter science; 5) participating in departmental position in computer science beginning August nology to teach mathematics. Consideration will and campus committees. For a detailed vacancy 1995. A Ph.D. in computer science, promise of be given to candidates who expect to complete announcement contact the Human Resources excellence in teaching, and a commitment to all degree requirements by August 15,1995 (es• Office at the address/phone number listed below. continued scholarship are essential. Applicants sential for appointment). Correspondence must Screening of applications to begin February 28, must be qualified to teach both computer science identify the title of the position. Letter of appli• 1995, and will continue until position is filled. courses and mathematics courses that support the cation and resume, including the names and Submit cover letter with statement of teaching computer science program. Responsibilities of telephone numbers of three references (no letters philosophy and research interests, curriculum the position include teaching two courses per term ofreference, please) must be received by March vita, copies of transcripts and three (3) letters of (fall, winter, spring), directing undergraduate 31, 1995. Address correspondence to: Dr. Robert recommendation to Chair, Search #95-7, Math research, and providing leadership for program V. Mark, Dean, School ofArts and Sciences, BOX Screening Committee, Human ResourceslEEO• advancement in computer science. A, State University of New York, College of AA Office, Montana State University-Billings, 38 February 1995 FOCUS 1500 North 30th Street, Billings, MT 5910 1-0298, Occidental College remedial mathematics specialist. One leave re• (406) 657-2278. ADAlAAlEEO. Los Angeles, California placement should be able to teach probabilityl Applications are invited for a tenure-track posi• statistics. You should hold an appropriate doctor• Mathematics: Assistant Professor. The Math• tion in the Department of Mathematics at the ate. Please send letter of intent, resume, and ematical Sciences Department of Muhlenberg Assistant Professor level. Occidental College is evidence of creative, innovative teaching to Larry College is accepting applications for a tenure• a selective college of the liberal arts and sciences Copes, Chair, Department of Mathematics, track faculty position at the Assistant Professor which serves a diverse student body of 1600 in an Augsburg College, Minneapolis, MN 55454 rank, starting August, 1995. A Ph.D. in Math• undergraduate teaching environment which en• ([email protected]).Applications will be ac• ematical Sciences in required. Three courses per courages curricular innovation, interactive cepted at least until 3/1195. Augsburg is an Equal semester teaching load. Individuals with train• learning techniques, and methods and programs Opportunity Employer. ing or experience in the use of technology in the for providing students access to a range of pro• classroom will be given priority. Teaching excel• fessional and intellectual pursuits. The Gustavus Adolphus College Depart• lence is required, and all faculty are expected to Mathematics Department consists of nine full• ment of Mathematics and Computer continue their professional activities. Applicants time faculty members. The normal teaching should s}lbmit a resume, a letter detailing their schedule is five semester courses per year, and Science teaching experience and research, and three let• faculty members receive a sabbatical semester Applications are invited for two visiting posi• ters of reference. Send application materials to: every four years. The college is located in north• tions in mathematics. These are one-year positions Dr. John Meyers, Head, Mathematical Sciences east Los Angeles, easily accessible to a number with the strong possibility of renewal for a sec• Department, Muhlenberg College, Allentown, PA of research institutions. Salary is competitive. ond year. A Ph.D. in mathematics is expected for 18104-5586. EOE. An excellent benefits package includes a choice these positions. Preference for one of the posi• of health care plans, tuition grants for children of tions will be given to those candidates with special faculty, and a mortgage subsidy program. Only interests in applied mathematics and/or math• Mathematics ematical physics. Teaching load is three courses Pacific University, Forest Grove, OR, is seeking applications which contain the following infor• mation will be considered: a current resume, three per semester. Candidates should have a record of an Assistant Professor of Mathematics in a ten• effective undergraduate teaching. Review of ure-track position to teach a variety of courses. letters of reference (at least one evaluating teach• ing experience, performance, and potential), a applications will begin Feb. 15 and continue until The department serves a student body half of the positions are filled. Send letter of applica• whom intend to major in a science and all are statement of commitment to teaching in a liberal arts college environment, and a statement of pro• tion, vita, graduate transcripts, and three letters required to take a math course. Applicants with of recommendation to Mike Hvidsten, Chair, an applied analysis emphasis are preferred. Ph.D. fessional goals including a description of current research plans. All materials must be received by Department of Mathematics and Computer Sci• and a strong commitment to and demonstrated ence, Gustavus Adolphus College, St. Peter, MN excellence in undergraduate teaching required. January 23, 1995. Address all materials to Fac• ulty Search Committee, Department of 56082. (e-mail: [email protected])Atleastone Send vitae and three letters of recommendation letter should specifically address teaching. to Professor Mike Clock, Chair, Department of Mathematics, Occidental College, 1600 Campus Mathematical Sciences, Pacific University, For• Road, Los Angeles, CA 90041. Occidental Col• Gustavus Adolphus College is a liberal arts col• est Grove, OR by February I, 1995. lege is committed to equity and excellence in lege affiliated with the Evangelical Lutheran education and strongly encourages applications Church of America and is located about 65 miles Pacific University is a residential liberal arts from women and minorities. southwest of MinneapoliS/St. Paul. The math• college of 1000 undergraduates and expects to ematics and computer science department has 13 have 1200 students by 1999. Half of the under• Valparaiso University members. The department is located in the re• graduates intend to major in science. The faculty Valparaiso University, Dept. of Math and Compo cently constructed Olin Hall of Science and has of75 is dedicated to quality teaching and service Sci., Valparaiso, IN 46383. Located I hour from state-of-the art high-tech classrooms and com• to students. The University also has five profes• Chicago loop. Pending budget approval, appli• puter facilities. It is the policy of Gustavus sional graduate programs with a total of 700 cations are invited for a tenure-track position in Adolphus College to provide equal educational students and is located 25 miles west of Portland. math, at the asst. prof. level beginning August and employment opportunities for all. We spe• OElAAE. 1995. Ph.D. in math desired, required for tenure. cifically encourage applications from women, Teaching experience, experience in preparing minorities and persons with disabilities. Faculty Position secondary teachers, familiarity with calculus Associatelassistant professor, tenure-track, be• reform helpful. App. from women and minorities ginning August 1995. Teach average 12 hoursl esp. encouraged. Candidates should be willing to semester. Requires Ph.D. in statistics or math• work in a scholarly community committed to ematics, evidence of excellent teaching and of Christian higher education and the Lutheran tra• ability for continued research or scholarly activ• dition. Send letter of app. and vita by March 31 ity. Preference given to applicants in applied to: Patrick Sullivan, Chair. statistics or applied mathematics (with back• ground in mathematical modeling). Send ARE YOU DEDICATED TO QUALI• application letter, resume, transcripts, three refer• ence letters to: Dr. Jack Scheidt, Chair, TY MATHEMATICS TEACHING Mathematics Department, University ofWiscon• IN A PRIVATE, LIBERAL ARTS sin-La Crosse, La Crosse, WI 5460 1. All materials COLLEGE NEAR THE HEART OF must be received by March I, 1995. All positions A DYNAMIC CITY? contingent on funding. Names of applicants who We are. If you are, too, then tenure-track and one• haven't requested confidentiality in writing, and year positions are available starting 9/95. You all finalists, must be released on request. Women, should provide evidence that you are an exciting, minorities encouraged to apply. AAlEOE. creative teacher willing to work with an innova• tive mathematics curriculum. Ideally, one tenure-track position is for an analyst, one for a 39 Calendar National MAA Meetings MISSOURI April 7-8, 1995, Central Other Meetings Missouri State University, October 21-24, 1995 Seventeenth an• August 6-8,1995 SeventiethAnnualJoint Warrensburg, MO nual meeting of the North American Summer Meeting. University of Ver• NEBRASKA April 1995, Creighton mont-Burlington, Burlington, VT Chapter of the International Group for University, Omaha, NE the Psychology of Mathematics Edu• NEW JERSEY April 29, 1995, Will• cation (PME-NA XVII), Columbus, Sectional MAA Meetings iam Paterson College, Wayne, NJ Ohio. Deadline for proposals is Febru• ALLEGHENY MOUNTAIN April 7- NORTH CENTRAL April 21-22, ary 20th. For a copy of the first an• 8, 1995, Duquesne University, 1995, Carleton College, Northfield, nouncement and call for papers, contact: Pittsburgh, PA MN Doug Owens, 253 Arps Hall, Ohio State University, 1945 N. High St., Colum• EASTERN PENNSYLVANIA & NORTHEASTERN June 9-10,1995, bus, OH 43210-1172; (614) 292-8021; DELEWARE April 8, 1995, King's Bates College, Lewiston, ME fax: (614) 292-7695; e-mail: owens.93 College, Wilkes-Barre, PA NORTHERN CALIFORNIA October @osu.edu. FLORIDA March 3-4, 1995, Valencia 21-22,1995, Cal Poly tech State Uni• Community College-East Campus, versity, San Luis Obispo, CA (joint July 30-August 4, 1995 Ninth Orlando, FL meeting with S. California Section) Iberamerican Conference on Mathemat• ILLINOIS March 31-April 1, 1995, OKLAHOMA-ARKANSAS March ics Education (IACME IX), Santiago, Monmouth College, Monmouth, IL 31-Aprill, 1995, Southwestern Okla• Chile. Contact: Organizing Committee, INDIANA March 31-Aprill, 1995, Tri• homa State Univ., Weatherford, OK IACME IX, Casilla 33081, Correo 33, State University, Angola, IN PACIFIC NORTHWEST June 15-17, Santiago, Chile; fax: 56-02-6811739; e• mail: [email protected]. INTERMOUNTAIN April 7-8,1995, 1995, Whitman College, Walla Walla, WA Idaho State University, Pocatello, ID August 14-20, 1995 Symmetry: Natural IOWA April 21-22, 1995, University of ROCKY MOUNTAIN April 21-22, and Artificial-Third Interdisciplinary Northern Iowa, Cedar Falls, IA 1995, University of Southern Colo• Symmetry Congress and Exhibition, KANSAS April 14-15, 1995, Wichita rado, Pueblo, CO Washington, DC (Old Town Alexan• State University, Wichita, KS SEAWAY April 21-22, 1995, Hobart & dria). For information, contact Gyorgy KENTUCKY March 31-April1, 1995, William Smith Colleges, Geneva, NY Darvas, Symmetrion-The Institute for Transylvania University, Lexington, SOUTHEASTERN March 31-Aprill, Advanced Symmetry Studies, PO Box KY 1995, University of North Carolina• 4, Budapest H-1361, Hungary; phone: LOUISIANA-MISSISSIPPI March 3- Asheville, Asheville, NC 36-1-131-8326; fax: 36-1-131-3161; e• 4, 1995, Mississippi State University, SOUTHWESTERN April 7-8, 1995, mail: [email protected]; alternate con• Biloxi, MS University of Texas at EI Paso, TX tact: Denes Nagy, Institute of Applied MD-DC-VA April 7-8, 1995, Thomas SOUTHERN CALIFORNIA October Physics, University ofTsukuba, Tsukuba Nelson Community College, Hamp• 21-22, 1995, Cal Poly tech State Uni• Science City, Japan 305; 81-298-53- ton, VA versity, San Luis Obispo, CA (joint 6786; fax: 81-298-53-5205; e-mail: [email protected]. METROPOLITAN NEW YORK May meeting with N. California Section) 6, 1995, Manhattan College, Bronx, TEXAS March 30, 31, April 1, 1995, NY Baylor University, Waco, TX MICHIGAN May 5-6, 1995, Grand WISCONSIN April 7-8, 1995, Univer• Valley State University, Allendale, MI sity of Wisconsin-Green Bay, Green Bay, WI FOCUS FEBRUARY 1995 The Mathematical Association of America Second class postage paid at Washington, DC and 1529 Eighteenth Street, NW additional mailing offices Washington, DC 20036-1385 032325 0 1994 R G 1966 F015 11111111111111111111111111111111111111111111111111111111111111