The Art of Teaching Mathematics Garikai Campbell Swarthmore College

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1-1-2008 The Art of Teaching Mathematics Garikai Campbell Swarthmore College

Jon T. Jacobsen Harvey Mudd College

Aimee S A Johnson Swarthmore College

Michael E. Orrison Jr. Harvey Mudd College

Recommended Citation Campbell, G. Jacobsen, J., Johnson, A., and Orrison, M. "The Art of Teaching Mathematics." MAA FOCUS 28.1 (2008): 10-11.

This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. MAA FOCUS January 2008

MAA FOCUS is published by the Mathematical Association of America in January, February, March, April, May/June, MAA FOCUS August/September, October, November, and December. Volume 28 Issue 1 Editor: Fernando Gouvêa, Colby College; [email protected] Inside Managing Editor: Carol Baxter, MAA [email protected] 4 “Mathematicians Have a Different Perspective” Senior Writer: Harry Waldman, MAA An Interview with Bernd Sturmfels [email protected] By Joe Gallian and Ivars Peterson Please address advertising inquiries to: [email protected] 8 FOCUS on Students: Networking for the non-Networker President: Joseph Gallian By Robert W. Vallin First Vice President: Carl Pomerance, 10 The Art of Teaching Mathematics Second Vice President: Deanna Haunsperger, Secretary: Martha J. Siegel, Associate By Garikai Campbell, Jon Jacobsen, Aimee Johnson, and Michael Orrison Secretary: James J. Tattersall, Treasurer: John W. Kenelly 12 What Students Say About Their REU Experience By Frank Connolly and Joseph A. Gallian Executive Director: Tina H. Straley Director of Publications for Journals and 13 Costs and Benefits to Faculty From Undergraduate Research Communications: Ivars Peterson and the Institution MAA FOCUS Editorial Board: Donald By James Davis, Suzanne Lenhart, and Margaret Robinson J. Albers; Robert Bradley; Joseph Gallian; Jacqueline Giles; Colm Mulcahy; Michael 14 Top Ten Signs of Retirement Orrison; Peter Renz; Sharon Cutler Ross; An- By Roseanne Hoffman nie Selden; Hortensia Soto-Johnson; Peter Stanek; Ravi Vakil. 15 MAA on the Road: At the AMATYC Meeting in Minneapolis By Michael Pearson and Candace Baumann Letters to the editor should be addressed to Fernando Gouvêa, Colby College, Dept. of 16 What I Learned…From Shutting Up and Listening Mathematics, Waterville, ME 04901, or by email to [email protected]. By David Kung Subscription and membership questions 18 The 2008 IAS/Park City Mathematics Institute: Analytic and should be directed to the MAA Customer Service Center, 800-331-1622; email: Algebraic Geometry [email protected]; (301) 617-7800 (outside By William H. Barker U.S. and Canada); fax: (301) 206-9789. MAA Headquarters: (202) 387-5200. 20 Calculus Placement that Really Works! Copyright © 2008 by the Mathematical By Todd Retchless, Randy Boucher, and Donald Outing Association of America (Incorporated). Educational institutions may reproduce 21 Employment Opportunities articles for their own use, but not for sale, provided that the following citation is used: “Reprinted with permission of MAA FOCUS, the newsmagazine of the Mathematical Association of America (Incorporated).” On the cover: A tube of coaxial circles tunnels through the San Diego Convention Periodicals postage paid at Washington, DC Center. Photo by John Tweed, Old Dominion University. and additional mailing offices. Postmaster: Send address changes to MAA FOCUS, Mathematical Association of America, P.O. Box 90973, Washington, DC 20090-0973. MAA FOCUS Deadlines ISSN: 0731-2040; Printed in the United States April May/June August/September of America. Editorial Copy February 8 March 10 June 16 Display Ads February 22 March 24 June 23 Employment Ads February 8 March 10 June 16

January 2008 MAA FOCUS MAA’s Commemorative River of Bricks By Ryan Miller The brick paths that traverse the court- Each member of the MAA’s Executive ence Center and its activities. For just yard in front of the MAA’s Carriage Committee has a brick. John Kenelly $303 (the price was chosen to honor Paul House Conference Center in Washington, bought seven bricks, one for each mem- Halmos’ birthday, on March 3), anyone D.C. represent the confluence of two bod- ber of his family. Six MAA sections have can have a name or dedication inscribed ies of water that help define the District purchased bricks. (That leaves only 23 on a brick to become a part of MAA of Columbia: the Potomac and Anacostia more!) Five current MAA employees history. Bricks will be laid in sets of 100 Rivers. On Nov. 1, 2007, these pathways who worked for former Executive Direc- until the walk is completely full. took on additional significance with the tor Al Wilcox purchased a brick in his official unveiling of the Paul R. Halmos honor. Individual members, colleges, The response by members thus far has Commemorative Walk. universities, and various organizations been great, according Lisa Kolbe, the with ties to the MAA are all now part of MAA’s Development Manager. “I’ve had To create the Commemorative Walk, the the commemorative walk. nothing but very positive reactions from nearly 1370 bricks that now pave the everybody who has seen the walk so far,” paths will gradually be replaced by bricks A generous donation in 2003 by Paul and she said. The second set of 100 bricks is inscribed with the names and memories Virginia Halmos made the restoration almost ready to be put down. of those close to the MAA. The first 100 and renovation of the Carriage House commemorative bricks have now been possible. The Conference Center is now The MAA encourages members who are put into place, and they already tell all actively being used, both by the MAA visiting Washington, D.C., to stop by the kinds of stories. and by other organizations. The MAA is headquarters, visit the Carriage House now raising funds to support the Confer- Conference Center, and stroll down the Paul R. Halmos Commemorative Walk.

A worker in the process of laying the first 100 bricks. Some of the bricks that appear in the river of bricks.

MAA FOCUS Deadlines April May/June August/September Editorial Copy February 8 March 10 June 16 Display Ads February 22 March 24 June 23 Employment Ads February 8 March 10 June 16 MAA Executive Committee and staff members prepare MAA President Joe Gallian does the honors. for the unveiling of the first 100 bricks. MAA FOCUS January 2008 “Mathematicians Have a Different Perspective” An Interview with Bernd Sturmfels

By Joe Gallian and Ivars Peterson

After receiving his PhD from the for very focused studies; students tend University of Washington in 1987, Ger- to reach a higher level rather quickly. man-born Bernd Sturmfels held positions In the traditional German system, the at Cornell University and New York first degree was called the “Diplom,” University before landing at the Univer- and one obtained this degree after about sity of California at Berkeley, where he five years. It used to be a rather rigorous currently teaches. Sturmfels has received program, comparable to a strong Masters numerous honors, including the MAA’s Lester R. Ford Award for expository writ- program with a written thesis. ing in 1999 and designation as a George Pólya Lecturer. Sturmfels served as the In the last 20 years, there have been Hewlett-Packard Research Professor at many changes in the university systems MSRI Berkeley in 2003/04, and was a in Europe, not just in Germany, and now Clay Institute Senior Scholar in Summer most schools offer Bachelor and Masters 2004. A leading experimentalist among Degrees more similar to the system in the mathematicians, Sturmfels has authored States. Not everyone is happy about these eight books and about 140 articles, in changes, needless to say. the areas of combinatorics, algebraic geometry, symbolic computation and FOCUS: How do you describe your their applications. He currently works research area? on algebraic methods in statistics and of Washington and the University of To- computational biology. ronto, so I was told to go to one of these Sturmfels: I work in combinatorics two schools. He was absolutely right: and algebraic geometry, with a special MAA FOCUS: Did you become inter- Victor Klee, a former MAA president, emphasis on applications outside of ested in mathematics at a young age? turned out to be a most wonderful thesis pure mathematics. Recently, I have been advisor for me. especially interested in applications in Bernd Sturmfels: I was a pretty good statistics, optimization and computational student in school, interested in almost I obtained a fellowship from the Ger- biology. Computational biology offers all subjects. There was never anything man government to study for one year fascinating opportunities to mathemati- special about mathematics, no competi- in Seattle. Three months after arriving, cians, and can ultimately lead to new tions or Olympiads or any such in my I fell madly in love with this wonderful and unexpected developments within high school. After graduating from high undergraduate English major from Korea, mathematics. school, I was in the German army for 15 and I just had to stay awhile longer in months to do my compulsory military Seattle. That’s why I ended up getting FOCUS: Which mathematicians have service. It was during those lonely eve- my doctoral degree at the University of had a large influence on you? nings in the barracks that I tried to figure Washington. My now-wife and I stayed out what I was really interested in. That together all those years, and, after mov- Sturmfels: Victor Klee, my advisor turned out to be mathematics and the ing around for many years, we finally in Seattle, ranks first. Others that have emerging field of computer science. returned to the West Coast in 1995. influenced me greatly include Louis Billera, Richard Stanley, I.M. Gel’fand FOCUS: You came from Germany to at- FOCUS: How does mathematics educa- and Bill Fulton. tend graduate school at the University of tion at the college and graduate school Washington. What drew you there? level in the U.S. differ from that in FOCUS: You have been involved in Germany? “experimental mathematics.” What, to Sturmfels: My German advisor sent me you, is experimental? What are the ex- to Seattle in 1985. I was interested in Sturmfels: The main difference is that periments that you are doing? spending a year abroad, but I was think- in Germany, students declare their major Sturmfels: Well, computer experiments. ing of France or England, something before entering university. There are few I can experiment by hypothesis testing or closer to home. My advisor strongly breadth requirements, so once someone it could be just plain old exploration, just urged me to go to North America. His is a university student most of the classes to see what happens. “Gee I wonder what contacts were primarily at the University taken are in mathematics. This makes

January 2008 MAA FOCUS properties the following mathematical who has many of these diploma students; learned over the years the value of com- object has.” Actually, that’s often the first that setting lends itself more easily to munication skills. Of course you have to step. I think you can use computation as a maintaining software. So the software in be good in math, but I think when I was tool to form hypotheses, or to test. my field is quite difficult for an academic younger I wasn’t as keenly aware of how math person to develop and even more important social skills and communica- I don’t trust humans a lot. You know, difficult to maintain. That’s a real chal- tion skills are. One doesn’t think about it people think that a written proof is the lenge. I admire anybody who does it and in the math context, but that’s something gold standard. I think many mathematical hope they get a lot of support. that I also pay attention to. So if there’s papers and arguments contain errors and a student I feel would have a very hard gaps and the only reason we don’t find FOCUS: Now that you’re at Berkeley, time communicating in either writing or them is because they don’t get read. On a prestigious institution that attracts verbally, then I’m more cautious. the whole, the building of mathematics very good students, how do you find the is sound, but if a mathematical state- students and their capabilities? FOCUS: Have you had any significant ment works out in a computer test, then involvement with undergraduates in I believe it a lot more. So I would say Sturmfels: I think the students are won- research? exploration, verification, and falsifica- derful, but of course we have a large tion. And then the last stage is once you program in Berkeley. Compared to some Sturmfels: Some, but not as much as have the conjecture or hypothesis then of the smaller private schools we compete I would like to. Lior Pachter and I are you can sort of go case by case and see: with, we have more of a spectrum, simply teaching an upper division undergraduate Is it true for n=8? And so on. because our program is much larger and course on mathematical biology at Berke- also because we try to be open. We have ley, where the students work in teams on The design of experiments is very im- a tradition of letting in students that might specific projects. This replaces the final portant. I think that’s where I sort of see not otherwise make the cut, so there’s a exams, and some interesting research my strength. Just like a lab scientist, a lot spectrum. But I would say that the stu- projects have sprung out of that. In fact, of work needs to go into creating a well dents I worked with are wonderful. I’ve thought out model system and model learned a lot from, and with, them. I just taught that course in the spring organism, to design the experiment, to of 2007, and I think that three or four pick a range of test problems that are FOCUS: And it sounds like you work a students have continued their research not too easy and not too hard, which lot with them. over the summer and beyond. However, will reveal the right phenomenon. That’s I must say that supervising undergraduate the challenge that distinguishes a good Sturmfels: Yeah, it’s the best part of research in mathematics is challenging, experimentalist from one that’s maybe the job. and essentially impossible for me, if and not so good. when it requires very regular one-on-one FOCUS: I’ve known faculty who take meetings. A laboratory setting, where FOCUS: Is that influenced by changes the approach of identifying the gradu- undergrads learn from graduate students in technology? ate students they want and approaching and post-docs, always works better for them. Is that similar to your approach? the problems of interest to me. Sturmfels: Definitely! Certainly now you can do a lot more than when I Sturmfels: Yeah I think that’s probably FOCUS: What about the sense of doing started. But I would say that experiments similar. It’s a little bit like dating. It helps mathematics in a broader context, for ex- in mathematics are not as technology- to be proactive. But finding a good match ample in biology and so on. Do students driven as in, say, molecular biology. Our is a two-way street. I think it helps the come in with that kind of interest or do technology is not progressing at the rapid student along if this relationship is built they develop it? rate it is in molecular biology, where it’s early on, so I try to play as active a role crucial to have the latest equipment just as possible, both in recruiting when they Sturmfels: It works both ways. I had to compete… It’s a little less crucial in come to Berkeley, and also in the first some students who were very pure and mathematics. You don’t need the biggest year, to identify who might be a good then found it interesting. There’s a good computer! fit. example of a student I work with. He just finished his first year in graduate school. FOCUS: What about on the software FOCUS: Has the kind of student you He was an undergraduate at Harvard, and side? want changed over the years? he had a very strong undergraduate math background: a very pure, typical Harvard Sturmfels: The software side… that’s Sturmfels: That’s a good question. Since math undergraduate background. Then he actually quite hard because developing I have become a bit more of an applied went back to his hometown and worked and maintaining software is extremely mathematician, I would also expect my in the biotech industry for a while. So he difficult in an academic math environ- students to be open to engage with people got a job and worked for a small private ment. It’s a little bit easier in the old Ger- outside of math. Also, I want them to company that was involved in the Grape man system where you have a professor have good communication skills! I’ve Genome project, so they got funding from

5 MAA FOCUS January 2008 the province of Trento in Northern Italy ics are very different subjects, and that FOCUS: I think Dick Karp and others and they sequenced the pinot noir grape. statistics, computer science, theoretical have spoken very strongly about the That was a two and a half year project physics, and mathematics are very very computer side of these things and how and in the course of that work he picked different subjects. To a typical biologist important those questions are. Are there up a lot of statistics, genetics, and com- these are indistinguishable. similar things on the math side? putational biology. Then he wanted to go back and get a math PhD, so he ended up So I think we have a lot of educating to Sturmfels: I think so. I think math- working with me in Berkeley and it’s a do and to explain that there are different ematicians have a different perspective. very good fit, because he has the biology areas of mathematics like there are dif- Mathematicians can contribute a lot by background but he’s now interested in ferent areas of biology, and they have dif- clarifying the notions and helping to pure algebraic geometry studies. He has ferent points of view. Partial differential gain precise definitions of the objects the option to go both ways. equations can do this while combinatorics and concepts involved, and that’s kind of will do that. And so I think that is very where I see my role. In the conversation, FOCUS: How did you become interested important to explain just what it is you do as a mathematician, I think I’m trained to in mathematical biology? and what techniques you use, and it’s a give precision and rigor to the concepts long process, but an interesting one. and to the discussion, while computer sci- Sturmfels: It all started with my ju- entists thinks more about the large scale nior colleague Lior Pachter. Lior is a FOCUS: In terms of say, computational model and the complexity of carrying computational biologist who ended up biology, what do you see as the important out a certain computation. A mathemati- in Berkeley’s mathematics department. questions and the places where progress cian would ask “what’s meaningful?” or Four years ago, we started talking about is possible? “what is it we’re really computing?” A phylogenetics, and we soon discovered mathematician would say: “suppose we Sturmfels: that the combinatorial structures he was I think in evolutionary biol- had a suffiently powerful computer, what ogy there are key questions. How does would you do with the output?” using for problems such as gene predic- evolution really work for biological tion or sequence alignments are very systems? What drives it? What’s the no- FOCUS: What kind of questions can similar to things I knew about. We started tion of fitness landscapes? I think for the biology suggest for mathematicians? a joint seminar in the fall of 2003 and it first time people have serious data, so has been a fascinating journey for me there’s statistical genetics. Evolutionary Sturmfels: I wrote an opinion piece on and my students ever since. Last year, theory has existed for 80 or 100 years, this for the Clay Institute called, “Can bi- we published a book called Algebraic going back to Fisher and Wright and ology lead to new theorems?” and I high- Statistics for Computational Biology. other people in the early 20th century, lighted four examples of such theorems. but I think for the first time we really see One that I was involved in was compu- FOCUS: When there’s a need for com- a significant amount of data and I think tational algebraic geometry and the study munication between mathematicians, there’s some really interesting problems of certain families of algebraic varieties and for example, biologists, what kinds we can now address. Basically, how does that come from phylogenetics. It turns out of barriers are there? What makes it dif- evolution work? that statistical models for evolution can ficult? be described by algebraic varieties (in the FOCUS: Some recent genome com- sense of algebraic geometry). These are Sturmfels: The language and the back- parisons were showing that there isn’t ground are both very difficult and very that much difference in the genomes of different. First of all, people don’t un- related species, and that more and more derstand that mathematicians will speak important is the significance of what parts about biology as a single discipline. But are turned on and off. That whole process the concept of biology being one field is changing. makes no sense to biologists, because obviously there are a thousand different Sturmfels: Exactly, the networks and the branches that are vastly different from regulatory mechanisms are what really each other. And conversely! At this differentiate cells, and differentiate or- stage most serious senior researchers in ganisms and species. I think the amazing molecular biology in particular realize variety we see in living systems cannot that they have massive data sets and that be explained by a static sequence which they need quantitative help, so often is relatively short. The human genome is they will say “we are really interested in less than 3 billion base pairs. If you see working with mathematicians,” but they the complexity of us, that’s not enough don’t quite know what mathematics is. to explain it. As mathematicians, we think that partial differential equations and combinator-

January 2008 918937_MathFin.qxp 11/19/07 2:27 PM Page 1 MAA FOCUS generalizations of classically known varieties, but very interesting new objects. In the last couple of years, we’ve seen four or five really interesting papers. So BOSTON UNIVERSITY I think biology and models for studying biological systems can suggest new and MATHEMATICAL FINANCE PROGRAM interesting mathematical objects that gen- eralize known objects that have already been classically studied in math. Earn a Master of Arts degree in FOCUS: On a side issue, one of the things mathematicians feel compelled Mathematical Finance in less than one year to become involved in are issues about math education, especially at pre-college levels. Do you have any interests or If you have an affinity for logical thinking and analytical reasoning, are concerns in that way? highly motivated, and aspire to a fascinating and rewarding career in the

Sturmfels: Many concerns. My kid is in world of finance, we invite you to find out more about Boston University’s school right now so I can see it from that Master of Arts in Mathematical Finance Program at www.bu.edu/mathfn. end, but I have not done much with that. I feel that it is more important to build What distinguishes us? a continuous spectrum of research. We now have K–12 and education research programs, and I feel it is all a little dis- � Intensive one-year M.A. program connected from the mathematics research � Focused and interdisciplinary cutting-edge curriculum community. I think there should be a dedicated to mathematical finance layer in between; maybe the MAA could play a big role. Maybe also high school � Rigorous treatment of the subject based on stochastic calculus math teachers who are very interested in � Integration of advanced, abstract theory with practical applications more advanced issues can help bridge the gap. For instance, there is the Park � Faculty dedicated to the field of mathematical finance City Mathematics Institute, the summer program where they have a high school program and a college program, and that The next program will begin on June 30, 2008. Applications accepted I think overall works pretty well. They until April 30, 2008. have some joint activities that I think are pretty good. I personally don’t see myself diving into the deep sea and the deep waters of K–12 education. Mathematical Finance Program 143 Bay State Road, Fourth Floor FOCUS: You were saying before that Boston, MA 02215 your impression is that too few MAA members participate in NSF and institute programs. An equal opportunity, affirmative action institution.

Sturmfels: And conversely, I think the NSF and the institutes could do a better Sturmfels: Yeah, I don’t necessarily have The institutes, for instance MBI at Ohio job in reaching out to the MAA commu- the ideal answer; I think that should be a State, could help find a way to facilitate nity. And in fact I just discussed this with discussion… which format works best? this. Perhaps they can bring in people, Peter March and he agrees, so I hope that Workshops could work, but maybe other give people a chance to speak about Peter and Joe Gallian will get together formats too? As a Pólya Lecturer, I’ve research results, to learn new things. I and have a chat about this. seen a lot of faculty at four-year col- don’t have the answer, but I’d like to leges interested in doing research with raise it as a question. I feel that MAA FOCUS: Is it just more participation in undergraduates to stay connected to the members are amazing and that they are a the actual programs? That means taking research community, maybe to engage in slightly underrepresented group at these time off to spend time for workshops and interdisciplinary research with faculty at institutes. that kind of thing. their college in say, biology for instance.

7 MAA FOCUS January 2008 FOCUS on Students: Networking for the non-Networker

By Robert W. Vallin What comes to mind when you think php?gid=5110769515. As for ‘profes- of the phrase ‘working the room?’ May- sional’ networking sites, the premiere be you just think, ‘That’s not something one is LinkedIn. Created in 2001, I could do.’ For me, it brings to mind LinkedIn, at http://www.linkedin.com, Eric Stratton (played by Tim Matheson), has over 12 million users. It’s a free a character in the classic movie National service; you just have to fill out an Lampoon’s Animal House. There is a online profile. This is perfect for the pledge party in full swing, but Stratton, nervous networker, you can ‘work the president of Delta House, wants to skip room’ at your own pace and commun- out for a date. Walking through the ing on the Internet does not have the crowded room, he shakes every hand negative stereotype of chit-chat with he finds on the way out each time stat- people you don’t really know. Some ing, ‘Eric Stratton, damned glad to meet other online places to look include the you,’ with absolutely zero sincerity. alumni sites for your schools, professional association web pages, resource This is networking, but it’s really bad sites (e.g. directories of people of the networking. To network is not to meet profession), and bulletin boards where everyone as superficially as possible, you can post questions and comments. nor is it a game of gathering more Remember when emailing or com- names and email addresses than anyone Illustration by Brad Fitzpatrick municating online you need to make else, nor is it to exhaust yourself doing a good impression. Be clear, concise, something which makes you horribly 60% of the graduate students echoed the honest, and courteous. Don’t use slang uncomfortable. So what is networking? feelings of the undergraduates on social or even worse, text language; ‘plsd 2 The Oxford English Dictionary reads, events, the Student Hospitality Center, meet u,’ gets you nothing. ‘network/nétwerk (n. & v.) group of peo- and SIGMAAs (Special Interest Groups ple who exchange information, contacts, within the MAA). In other words, they Suppose you are at the Joint Meetings. and experience for professional or social say networking is important, but feel that You want to go to the First Timer’s Re- purposes.’ Is this helpful? Not really. So these events —where they could be doing ception, but don’t know how to mingle. we’ll just say this: networking is mostly it — are unimportant. Let’s go over some rules. To begin with, doing the things you’d do anyway, but building a network takes time. You will in such a way as to help you know more Why does networking get such a bad not leave the room with a network in people whom you can help and who can rap? Probably because of the dread that place. That is okay. Also, there is no help you. fills the body of the networker as he/she recipe for success, this is not one size (or enters the room. The sweaty palms, ner- one rule) fits all. Networking is a very funny thing. In vous stomach, feeling of having nothing a survey given to students at the Joint to add to the conversation, and idea that Say your name clearly. Smile. Mathematics Meeting in 2007, 65.7% somewhere there is something more of undergraduates rated networking very important to be doing are all symptoms Make sure you have a good handshake important or important, while 80.8% of of this dread. That’s why we want to talk — firm grip, eye contact, right duration. graduate students ranked networking in about how to network so you can real- those categories. However, almost half ize successful communicating is within It’s not just for you, think of what you of the undergraduates rated the social your grasp. can do for others. events as somewhat important or unimportant and over half felt that way about First off, where do you network? Meet- Listen. Listen for real, it’s not just nod the Student Hospitality Center. Almost ings are a great place. That’s pretty your head time. obvious. Another place, quickly gaining This is the fourth in a series of short on first place, is on the Internet. Most Ask people about themselves. Then let articles for students. The overall title students already do some form of social them answer. for the series is FOCUS on Students. networking by having an account on Some of these articles will be for under- Facebook or MySpace. Not only can you Be prepared to talk about yourself. Intro- graduates, others for graduate students, do social things there, but professional verts find this difficult, so have something and many for all students. These articles as well. Check out the MAA’s Facebook already in mind. You can even practice will also be posted in the Student sec- group at http://www.facebook.com/group. what you want to say. (See the discussion tion of the MAA web site.

January 2008 MAA FOCUS of the elevator talk below.)

If you connect well with someone, get an email address. Then, within 24 hours of getting home, email the person reiterating how nice it was to meet. DIRECTOR OF FIRST YEAR MATHEMATICS

Don’t stay too long in one spot, but stay Arizona State University announces a national search to recruit a Director long enough to make a good impression. for its First Year Mathematics (FYM) Program. The Director of FYM will be Don’t be an Eric Stratton. responsible for all aspects of the FYM program, including leading the creation and delivery of innovative curricula, hiring and evaluating teachers, You don’t have to go it alone. If you want, and overseeing a highly effective and qualified staff. The Director may also bring a friend, but just remember, you are engage in an active personal research program. The successful candidate not there to talk with your friend, but to may hold an academic appointment at the rank of associate or full professor meet others. or a senior administrative appointment as appropriate.

Now, on to the elevator talk. A lesson Candidates must have an earned doctorate in mathematics education, from Business School, the premise is statistics, or mathematics and a commitment to and strong interest in simple. You have a project that you fostering student success in mathematics at the first- and second-year think is dynamite. One morning you find levels. Desirable qualifications include documented leadership skills, previous administrative experience, a history and awareness of external yourself riding up in the elevator with the funding for educational projects, experience with program development in company CEO. During that 30-second education, evidence of excellent communication and organizational skills, ride you have the undivided attention of and a commitment to working with and supporting a diverse student and the boss of bosses. What will you say, faculty population. in that time, to convince the CEO to ap- prove your project? That’s exactly what The position is available beginning 1 July 2008. Salary and start-up funding you want to do when you meet someone will be competitive and commensurate with qualifications. Review of except what you’re selling is yourself. applications will begin on 4 January 2008; if not filled, applications will be The elevator talk is not some over the evaluated every two weeks thereafter until the search is closed. Applicants top ‘Look at ME!’ rehearsed speech, but must submit electronically (in MS Word or PDF format) a cover letter, instead something that will tell the people current curriculum vitae, and names and contact information (phone and e- around you that you’re an interesting per- mail) for three references to Mr. Gabriel Escontrias son. Talk about where you are in school ([email protected]). Inquiries and nominations should be and what subjects you really like. If you directed to: Sid P. Bacon, Dean of Natural Sciences, College of Liberal Arts are giving a talk or presenting a poster, and Sciences, Arizona State University, PO Box 876505, Tempe, AZ 85287- tell that. This can be difficult for intro- 6505; or email: [email protected]. A background check is required for verts, who like to talk about ideas rather employment. Arizona State University is an affirmative action, equal than themselves, but it is necessary. opportunity employer committed to excellence through diversity. Arizona State University (see http://www.asu.edu) is a dynamic, Some last tips for the reticent. Join a internationally recognized research university serving more than 60,000 group and participate. Although it can students in the Phoenix metropolitan area, one of the fastest growing urban be tough to be active in professional centers in the nation. The FYM Program (see http://math.asu.edu/fym) organizations there are ways to be suc- currently has more than 50 lecturers and instructors teaching FYM courses cessful. Like learning a foreign language, defined as College Algebra, Finite Mathematics, College Mathematics, the key is immersion. If you can help out Theory of Elementary Mathematics, Elementary Statistics, Pre-Calculus, in some position in the group, then you and several sequences of Calculus. get to learn about the organization from the inside. Then people will be seeking at official events. Talk to people. People you out. Also, if there is an event, go you meet on airplanes, people waiting in Correction early, if possible, even volunteer to help line with you, the guy in the seat next to set up. This helps because if you are you at the game, just talk with them. You At the August meeting of the Board of among the first to arrive the crowd will may be surprised where it leads. Governors only Mary Sheppard was be small and not so intimidating. Set an elected to the Audit Committee. We achievable goal; e.g. by the end of this Robert W. Vallin is the MAA Associate apologize for the incorrect informa- meeting I will know five new people Director for Student Programs. He wel- tion in the October issue of MAA having chatted with each for at least 5 comes questions and comments by email FOCUS. minutes and leaving them with my card. at [email protected]. Lastly, networking does not just happen

9 MAA FOCUS January 2008 The Art of Teaching Mathematics By Garikai Campbell, Jon Jacobsen, Aimee Johnson, and Michael Orrison On June 10–12, 2007, Harvey Mudd ics among the students. It was, she said, in which one learns to accommodate for College hosted A Conference on the Art of an experience that not only significantly all these variations. This parallels the Teaching Mathematics. The conference broadened her understanding of both subtle variations we see in our students, brought together approximately thirty teaching and learning mathematics and classrooms, and college environments. mathematicians from the Claremont improved her own teaching, but also To this end, George argues we should Colleges, Denison, DePauw, Furman, one that confirmed her sense that teach- be developing our teaching through the Middlebury, Penn State, Swarthmore, and ing mathematics is an incredibly artistic relationship of master and apprentice. Vassar to explore the topic of teaching as endeavor. an art. Assuming there is an element of With this in mind, the conference turned artistic creativity in teaching mathemat- After Maria set the tone for the confer- to theater to discuss ideas from that realm ics, in what ways does it surface and ence, George Andrews (Penn State) that carry over to the classroom. Ann what should we be doing to develop this framed the discussion by challenging the Woodworth, a theater professor from creativity? notion of teaching as a science and argu- Northwestern University introduced these ing why we must consider teaching as an ideas in “The Classroom as a Stage,” a The first day of the conference focused art. Although there are many lessons we workshop in which we both talked about on the art of teaching and the second day can learn from scientific studies of teach- and were required to perform the physical

Garikai Campbell Ann Woodworth Tom Garrity focused on building community within ing, George maintained that because of act of teaching. this craft. The conference was organized the personal nature of teaching, there will by Jon Jacobsen and Michael Orrison at always be aspects of these studies which Ann pointed out that according to some, Harvey Mudd College with assistance fail to be repeatable or transferable. much of what we communicate is non- from John Harris and Mark Woodard verbal and of the remainder, up to half is (Furman), Ben Lotto (Vassar), and Lew The classroom environment is filled with tone. So, in the end, a very small percent- Ludwig (Denison). It was funded by empathy, self-awareness, subtle adjust- age of what we actually communicate is a grant from the Andrew W. Mellon ments, and other tacit knowledge that is content. Ann demonstrated this, for ex- Foundation. communicated but not stated. Alluding ample, by saying the same simple phrase to the work of the philosopher Michael in three different tones with three distinct Maria Klawe (Harvey Mudd) opened Polanyi, George said, “We know more sets of body language. The mathemati- the conference with “Lessons Learned than we can tell.” George prompted us to cians in the audience recognized, perhaps from a Third Grade Classroom.” Maria consider, for example, creating a violin. surprisingly, the multitude of similarities recounted her four years collaborating One needs to work with wood, cat gut and connections between acting in the- with a third grade teacher to explore the and a number of other materials with ater and teaching in the classroom and effectiveness of mathematical computer high degrees of variations. Teaching one the value of paying attention to one’s games and other activities in enhancing to create a violin from these raw materi- physical presence and voice. This is not the motivation and learning of mathemat- als requires a hands-on sort of training to suggest that we don’t do this already,

10 January 2008 MAA FOCUS but there are tricks one a good question one could can employ to do even ask is: what are the ways a better. department can facilitate the continued development From there, the con- of this mathematical matu- ference shifted focus rity? Tom shared examples and on the second day, from Williams such as Fac- concentrated on build- ulty Jam Sessions, in which ing community. Garikai a few people give talks, in Campbell (Swarthmore) one sitting, around a single reflected on community topic, and Faculty Dada- building through the Pro- ism, where over lunch, fessional Development the department picks three Program (PDP) Sum- areas out of a hat and tries mer Math Institute at to make connections. Berkeley. The program was developed by Leon An interesting theme that Henkin and Uri Treis- appeared throughout the man to, in part, encour- Participants at a breakout session. conference was the impor- age underrepresented tance of tacit knowledge minorities to consider in learning — whether it pursuing a PhD in math- be from the teacher, the ematics. Garikai talked classroom environment, about his sense of the key the social environment communal components on campus, or the college of the program that made environment. There was a the year he spent there as common thread through- a student, one of the most out the conference that rewarding, exciting, and suggests we need to do invigorating mathemati- more thinking about what cal experiences he has is happening outside our ever had. classes. Focusing only on the lectures and homework In particular, he recalled is a limiting factor. that the intellectual engagement instigated by In our mathematical cul- the organizers, encour- ture teaching is generally aged by the faculty and not viewed as an art or Group photo. actualized by the students craft and master teachers extended well beyond are somewhat reluctant the courses being taken ics’ has the potential to be one’s primary to take on the title and/or — students talked about mathematics as identity. to articulate what makes them a master. a whole; mathematical ideas that were However programs based on this concept related to things touched on in other Finally, Tom Garrity (Williams) talked on such as Project NExT clearly demonstrate students’ classes; and even meta-math- “Using Mathematical Maturity to Shape the progress that can be made by embrac- ematical and philosophical ideas. Our Departments.” Other disciplines ing this model. don’t seem to have a corresponding no- Students talked walking back from tion of mathematical maturity, he argued, Garikai Campbell and Aimee Johnson classes, over meals, on the way to con- so what is this thing we seem to have so are Associate Professors of Mathematics certs, and out at the weekend picnics. uniquely in mathematics? One can trace at Swarthmore College, where Garikai Moreover, the program seemed to be built through the time a student is in elemen- is also Associate Dean for Academic around a triumvirate of high intensity, tary school through undergraduate years, Affairs. Jon Jacobsen is an Assistant high expectations and high accountability through graduate school to beginning Professor of Mathematics, and Michael and on the principle that a truly successful professor to full professor to emeritus and Orrison is an Associate Professor of community should provide students the outline the specific skills and reasoning Mathematics, at Harvey Mudd College. opportunity to transform how they see ability one should have minimally at- themselves so that ‘student of mathemat- tained at each level. Once one does this,

11 MAA FOCUS January 2008 What Students Say About Their REU Experience By Frank Connolly and Joseph A. Gallian In 2006, the American Mathematical It is sometimes said that REUs are only admissions committees saw their REU Society sent a 30-question survey to getting the students who have already experience as valuable. Three out of ev- 444 individuals who had been student decided to go to graduate school in ery four say that it helped them get into participants in a Research Experience for mathematics. But the survey results a better graduate program. Moreover two Undergraduates program (REU) between do not support that viewpoint. Out of out of every three respondents who had 1997 and 2001 asking about the value of 262 respondents, a total of 231 entered won a fellowship thought that the REU the programs. Responses were received graduate school (82%), and of these, was a factor in the award. from 262 students (59%). Of these, 55% 78% said that the REU was definitely were male, 9% were Latino, and 2% were a factor, or somewhat a factor, in their This article is adapted from one with African Americans. Almost without ex- decision to do so. Only 32% of those the same title written by the authors ception, these respondents were reporting who did not go to graduate school say that appeared in the Proceedings of the on a summer research program, and for the REU was definitely or somewhat a Conference on Promoting Undergradu- a large majority (75%) this program was factor in their decision. This might be ate Research in Mathematics, edited by held at a college or university different the most significant finding of the whole Joseph A. Gallian and published by the from their own. Because the respondents survey. It suggests that REU’s are valu- American Mathematical Society. The were commenting on an event at least able precisely in the way they nurture volume can be downloaded free at http:// five years past, they had had time to the commitment of a student to pursue a www.ams.org/employment/REUproceed- gain perspective on their undergraduate career in mathematics. ings.html. Free bound copies are available research experience, and were unlikely by writing to Dana Chyung, Assistant to to be swayed in their responses by an The respondents seem quite clear-eyed, the Associate Executive Director, at the emotional afterglow of a summer’s ca- however, about what an REU can and American Mathematical Society (201 maraderie. cannot accomplish. An overwhelming Charles St., Providence, RI 02904). majority (82%) of those going to graduate Copies will also be available at the Joint The first striking result emerges when school say the REU did not shorten their Mathematics Meetings. these students are asked whether they time there. Another clear majority (53%) think their REU was valuable to them. of these say the REU did not influence Frank Connolly has been director of an They were offered three choices: Not their choice of a thesis area (although NSF funded REU at Notre Dame since valuable, somewhat valuable, or valu- 36% report their REU had at least some 2000 and has been director of Notre able. The answers exhibit a strong influence on their choice). But two out of Dame’s Seminar for Undergraduate consensus. Fully 84% answered that it every three of these say that the REU had Mathematical Research for many years. was valuable; only one respondent said some effect in accelerating their develop- Joe Gallian has been been director of an it was not valuable; the rest answered ment as research mathematicians. REU program at the University of Min- somewhat valuable. (Interestingly, 100% nesota Duluth since 1977 and is currently of Hispanic students declared their REU Most respondents believe that graduate President of the MAA. experience to be valuable.) This is a clear endorsement of REU programs.

The most important question about REUs is whether they are successful in achiev- CURM Center for Undergraduate Research in Mathematics ing their goals. Of course goals vary. But the goal of the National Science Founda- tion and the National Security Agency, Promoting Research in Mathematics by: the two principal financial sponsors of the REU programs, is to enhance the research •Training professors as mentors for undergraduate research projects infrastructure of mathematics within the •Advising professors on how to organize and operate undergraduate United States. It is therefore impressive research at their own institutions that 78% of all of these respondents entered graduate school intending to obtain •Awarding mini-grants from $12,000 - $20,000 to all a PhD, and 14% more entered graduate participating professors school with the intention of obtaining a Masters degree. Nearly all of those Current Application Deadline: March 15, 2008 entering graduate school (93%) were For more information please visit our website at curm.byu.edu seeking degrees in one of the mathematical sciences. These are remarkably high CURM Director: Michael Dorff numbers. Brigham Young University•292 TMCB•Provo, UT•84602 *Funding for CURM provided by the NSF and BYU 12 January 2008 MAA FOCUS Costs and Benefits of Undergraduate Research to Faculty and the Institution

By James Davis, Suzanne Lenhart, and Margaret Robinson

Opportunities for undergraduates to a framework for students, faculty, and supported and rewarded for their supervi- do mathematics research have increased administrators to follow as they seek to sion of student work. Possibilities of such dramatically over the past twenty years. participate in undergraduate research support would be one course teaching The National Science Foundation and activities in mathematics. reduction or a light load of service duties. the National Security Agency sponsor Rewards might include summer salary programs to encourage involvement in Here is an example of a difference or travel funding to visit a collaborator. summer and regular semester programs, between undergraduate research in This work should contribute positively to and individual schools have supported mathematics and in other disciplines. tenure and promotion decisions. Thus in- local efforts. The Subcommittee on “While undergraduate researchers in stitutional support is a critical component Research by Undergraduates of the Com- other disciplines often assist the project for a sustainable undergraduate research mittee on the Undergraduate Program mentor in his/her own research projects, program in mathematics. in Mathematics (CUPM) feels that the undergraduate research projects in math- community and administrators have ac- ematics are often created specifically for The committee’s document which has cepted the value of this research to the undergraduates. Thus, time spent on lead- been approved by the CUPM can be students, but many administrators are ing an undergraduate research project is found at http://www.maa.org/cupm/ not aware of the difference between this often time taken away from the mentor’s CUPM-UG-research.pdf. The CUPM work and research by undergraduates in own research endeavors. While under- Subcommittee on Research by Under- other fields. graduate research is extremely beneficial graduates would be glad to have input and to the student and to the department, continuing dialogue about these issues. Administrators are encouraging depart- it does not always advance the faculty Please contact the authors of this article ments to increase the involvement of member’s own research program.” with your comments. faculty and students in research activities, but are not always aware of the likely Of course, we note that in some projects, Suzanne Lenhart is the chair of the resulting costs and benefits to the faculty. publications do result with significant CUPM subcommittee on research by This subcommittee wrote a document, contributions from the undergraduates undergraduates; she can be reached Mathematics Research by Undergradu- involved. The benefits to the faculty at [email protected]. James Davis ates: Costs and Benefits to Faculty and have a wide range in the mathematics ([email protected]) and Margaret the Institution, with the goal to provide community, and many faculty need to be Robinson ([email protected]) are committee members.

Recommendation on Interviews at the Joint Meetings

At its meeting in San Jose in August 2007, the MAA Board of Governors approved a resolution dealing with procedures for job interviews at the Joint Mathematics Meetings and at MathFest. The resolution reads as follows:

The MAA strongly discourages the use of personal hotel sleeping rooms as the site for professional interviews of prospective employees. This practice is intimidating for some job-seekers, particularly those who find the situation uncomfortable and possibly unsafe.

This recommendation will appear in MAA FOCUS before each national meeting of the Association. The Board has also requested that it be printed in the official materials provided for employers and job seekers using the Employment Register at the Joint Mathematics Meetings.

13 MAA FOCUS January 2008 Top Ten Signs of Retirement By Roseanne Hoffman I attended a very small grade school: 5th and 6th grades in one classroom, one teacher. I was in 5th grade and my teacher handed me the chalk and said to go to the 6th grade side of the classroom and teach decimals, while she worked with the 5th graders. I think that determined my future.

I have been teaching since 1963, the last 29 years at a community college, at four year universities before that. I still love teaching, but realized it was time. I have been asked how I knew it was time to lay down my chalk. You should certainly not wait until you don’t enjoy teaching! This is the list I created.

Signs that one needs to lay down their chalk (or electronic pen)

1. When a former student sees you and asks: “Are you still teaching?” 2. When a student asks, “Do we need to know this?” and you reply no. 3. When all the students get the correct answer when you use the polling devices in the classroom. 4. When you create a mathematics lesson so that you can incorporate the Wii in the classroom. 5. When you need two carts to get your course materials into the classroom. 6. When you need space on the H drive and you can’t decide which files to delete, so you delete them all. 7. When you realize you will never get that grant of 25 tablet PCs for students to use for a semester. 8. When you finally get a classroom with individual tables and chairs (not CSMC Hosting International Conference tablet chairs) for students to use. 9. When you get the biggest SMART- The Center for the Study of Mathematics Curriculum (CSMC) is hosting an in- Board on campus. ternational conference on the subject “Future Curricular Trends in School Algebra 10. When you ask for a “rotating” laptop and Geometry” on May 2-4, 2008, at The Field Museum in downtown Chicago and computer loaded with mathematics on The University of Chicago campus. The conference will feature international software for mathematics faculty experts from Australia, Europe, and South America, and national expertise from the to borrow over weekends and the United States. answer is “yes.” The conference is open to all, but limited in size. Registrations will be accepted on Roseanne Hoffman laid down her chalk a first come/first-served basis. If you are interested in attending, please check the at Montgomery County Community CSMC web site, http://www.mathcurriculumcenter.org/conferences.php,for further College. details and a registration form. 14 January 2008 MAA FOCUS MAA on the Road: At the AMATYC Annual Meeting in Minneapolis

By Michael Pearson and Candace Baumann

The American Mathemati- development program cal Association of Two-Year modeled on Project Colleges (AMATYC) held NExT. The first three its 33rd Annual Meeting in years of this program Minneapolis on Nov. 1–4, were funded by Exx- 2007. The meeting had origi- onMobil in response nally been scheduled for New to a proposal submit- Orleans, but the facility that ted jointly by MAA was to host the meeting was and AMATYC. Project permanently closed as a result ACCCESS is currently of Hurricane Katrina. AMA- led by Karen Gaines of TYC decided to move to the St. Louis Community other end of the Mississippi College and Brad Chin River and hold “New Orleans of West Valley College. in Minneapolis, 2007.” Fellows participate in workshops and ses- About 1,000 people attended sions at two consecu- this year’s conference. High- tive AMATYC meet- lights of the meeting included Michael Pearson talks with a visitor to the MAA’s booth at the 33rd ings. Alhough the initial Bernie Madison’s invited lec- AMATYC Annual Meeting. Photo by Candace Baumann. funding period is over, ture on the increasing impor- AMATYC plans to con- tance of quantitative literacy tinue the program, and for effective participation in mathematical ideas from that show. He the MAA will continue the modern world, both in the workplace also noted other examples from recent to support Project ACCCESS Fellows by and as a citizen. Wayne Roberts’ talk, popular culture, including A Beautiful providing a one-year membership and “The Derivative as a Linear Transfor- Mind and Proof, in which mathematics limited funds for travel to MAA Section mation” was well-received. Wayne also plays a significant role. As Dan observed, meetings. spoke about the importance of his experi- the mathematical community should not ence as a student (and later as a teacher) at miss the “teachable moments” afforded The MAA booth buzzed with activity Morton Junior College to his outlook as a by these events. throughout the entire meeting, and sev- faculty member at Macalester College. eral MAA members volunteered their The 2007 conference marked the end of time at the booth to talk to AMATYC Dan Kennedy gave a delightful early- the MAA’s formal involvement with Proj- members about the benefits of MAA morning lecture, “My Year with NUM- ect ACCCESS (Advancing Community membership. Thanks in particular to to B3RS: Mathematics Goes to Hollywood” College Careers: Education, Scholarship, Steve Blasberg, Joanne Peeples, Abe describing some of the interesting Service), an early-career professional Mantell, Ray Collings, Dennis Ebersole, and Jay Malstrom. Keep Up with ICMI News

ICMI News is the new electronic news- ICMI News will also report on major education around the world. The editor letter of the International Commission activities by the ICMI Affiliated Study will be Jaime Carvalho e Silva of the on Mathematical Instruction (ICMI). It Groups (HPM, PME, IOWME, WFNMC Department of Mathematics of the Fac- aims to improve communication between and ICTMA), on major international ulty of Sciences and Technology at the ICMI and people interested in mathemat- events related to mathematics education University of Coimbra (FCTUC), in Por- ics education worldwide. It will include and on other topics of general interest to tugal. He is a member of the ICMI Ex- information about actions and recom- the community of educational research- ecutive Committee. He can be contacted mendations of ICMI, highlighting issues ers, curriculum designers, educational at [email protected]. that are under discussion and reporting policy makers, teachers of mathematics, about ongoing activities. mathematicians, mathematic educators To subscribe, visit http://www.mathunion. and others interested in mathematical org/ICMI/Mailinglist.

15 MAA FOCUS January 2008

What I Learned from… Shutting Up and Listening

By David Kung

“Can you always inscribe a square in a triangle?” tions, but when it came to the discussions, I mostly shut up and listened from the back row. It seemed like such a simple problem to start out my proofs course. I had adapted the problem from Schoenfeld’s work on Chantelle’s group decided to solve the problem on a special problem solving and planned to spend about 45 minutes letting class of triangles, namely right triangles. Their picture looked them muddle through the poor wording and implicit quantifier something like this: to come up with the two usual solutions. Two groups dutifully produced those solutions, both based on the Intermediate Value Theorem.

Alberto’s group looked at small squares that have three vertices on the triangle and argued that you could expand the squares until the fourth corner met the triangle:

It made for a great discussion between Beryl and Chantelle’s groups when I pointed out the similarities between this picture and Beryl’s graph of the height vs. width of the family of rectangles.

The most interesting part of the discussion came from Debby’s group. In the initial exploration of the problem, she steered her (Their picture brought up an important point: do the “free” group in a unique direction. “Let’s start with the square!” My corners of the squares all lie on a straight line?) reaction came in two parts:

Beryl’s group looked at a family of inscribed rectangles and ar- Internal: “That’s absurd! You can’t start with the square!” gued (again using the IVT) that the family included a square: External: “That’s interesting. If you start with the square, what happens?”

After receiving some feedback from the rest of the class (who thankfully avoided thoughts similar to my internal ones) they refined their idea to the following. You can get “any” triangle by placing a vertex C between the extensions of the two vertical sides of the square, connecting C with the two upper corners of the square, and extending the lines until they meet the extension of the bottom of the square.

When pushed to be more precise, Beryl’s crew produced a nice graph of the height vs. width of the rectangles, which necessarily crossed the y = x line representing the squares.

What I heard from the other students as I wandered among their groups was so interesting that I ditched my previous plans and ended up spending three hours (seriously!) listening to their ideas and having them discuss, debate, and refine them. While they were working in groups I would listen in and ask ques-

16 January 2008 MAA FOCUS

This led to places I never dreamed this problem would touch in the end, they even agreed that the argument does have to start on. The class, with me hiding in the back row to avoid getting with the triangle (even if their solution didn’t). in their way, quickly realized you couldn’t really get “any” triangle, but rather one similar to any triangle (equivalence What did I learn from this day? Two things that I carry with classes anyone?) They also argued back and forth about the me into every class I teach. wisdom of starting with the square: how do you begin a proof of ∀x ∃ y R(x,y)? This last conversation was punctuated by First, I need to spend more time listening and less time talking. Ernesto’s brilliant observation: Of course I know more mathematics than my students do, but that doesn’t mean that talking at them all the time is the best Start with the triangle (1), draw a square below it (2), use way for them to learn. At the end of this episode, those students Debby’s group’s idea, and then rescale the whole picture to get all understood the problem and the solution — and they had the square inside the triangle (3) instead of below it! started grappling with some of the harder material we would face later in the semester. Perhaps most importantly, they felt a sense of ownership over this solution that would not have developed if I had presented a solution to them. Their open ended, meandering conversation wouldn’t have happened if I had been at the front of the room directing things. I needed to step aside and let them think things through.

Second, I need to provide students more opportunities to explore rich problems like this — problems that have multiple solution strategies and subtly contain the content I’m aiming for. This gives them the freedom to argue out their ideas (both good and bad) and experience first-hand the excitement and intrigue of learning new mathematics.

This solution is so much better than what I had in mind! It’s Dave Kung was the 2006 sectional teaching award winner from constructive in the sense that it doesn’t rely on the IVT. It makes the MA/VA/DC section of the MAA. He currently resides at it very clear that you need to start on the longer side (picture Fudan University in bustling Shanghai, enjoying a sabbatical what happens when you put the square adjacent to an obtuse from St. Mary’s College of Maryland. He’s spending the year angle in the triangle). Students who were skeptical of the other improving his Mandarin and co-writing an upcoming book en- methods seemed to understand this method very quickly. And titled What Could They Possibly Be Thinking: Understanding Your College Math Students. He can be reached at dtkung@ smcm.edu.

Do You Want to Use Mathematics in a World of Industry?

A Professional Science Master’s (PSM) degree from the University of Northern Iowa will help you succeed.

Developed in consultation with industry leaders, University of Northern Iowa’s Professional Science Master’s will help graduates excel in today’s competitive climate.

The degree features: x A mathematical focus on either continuous quality improvement or computing and modeling. x Integrated business and experiential components. x An internship or project on a problem of interest to a host industry.

For further information, including specific requirements and course information, visit www.uni.edu/math or contact Dr. Syed Kirmani at [email protected]

17 MAA FOCUS January 2008 The 2008 IAS/Park City Mathematics Institute: Analytic and Algebraic Geometry

By William H. Barker Each summer since 1994, the Institute for Advanced Study in Princeton, NJ has sponsored a unique mathematics event in Park City, Utah: the IAS/Park City Mathematics Institute. For three weeks mathematics researchers, scholars, and students at the post-secondary level, as well as mathematics educators at the secondary level, gather for research and study in Park City, a setting of breath- taking mountain scenery and outdoor opportunities.

This year’s PCMI Summer Session will be held from July 6 to July 26, 2008, with a research theme of Analytic and In 2007, PCMI participants marched in the Park City July 4 parade. Algebraic Geometry: Common Problems – Different Methods and an education theme of Knowledge for Teaching Math- ematics.

The fundamental, driving goal of the IAS/ Park City Mathematics Institute (PCMI) is to provide a consistent and persuasive model for career-long development for all mathematics professionals. Through its summer institute, PCMI offers paral- lel but interconnecting programs for a variety of components of the mathematics community. Cross-program interaction — a central goal of the summer institute — is accomplished through both struc- tured and unstructured activities. The premise on which PCMI was founded The PCMI group gets ready to march. — and which continues as its central cornerstone — is that interaction between research and education strengthens all At the Summer Session these groups aged and nurtured to further enhance the aspects of the mathematical enterprise. meet simultaneously, each following sense of community among participants. an individual program of research and The rich mathematical experience com- The themes of PCMI are developed in study. However, considerable interaction bined with interaction among groups with five separate but connected programs: takes place between the groups due to the different backgrounds and professional fundamental structure of PCMI: lectures concerns results in greatly increased un- • Research Program, for mathematics and courses in each Summer Session pro- derstanding and awareness of the issues researchers, gram are open to all participants and, in confronting contemporary mathematics • Undergraduate Faculty Program, addition to the activities specific to each and mathematics education. for undergraduate college faculty, group, there are daily events of general • Secondary School Teachers interest designed for the full Institute. Each of these programs will be attractive Program, for secondary school Further opportunities for informal and so- and appropriate for many members of teachers, cial interaction are also available, ranging the MAA. However, the Undergraduate • Graduate Summer School, from organized Cross Program activities Faculty Program (UFP), with its focus on for graduate students, to informal conversations over meals. collegiate mathematicians with a strong • Undergraduate Program, for Cross-program mentoring is also encour- interest in undergraduate education, is undergraduate students.

18 January 2008 MAA FOCUS of relevance to a particularly large portion of the MAA. Co- sponsored by NSF’s Chautauqua Program, the UFP offers opportunities for broad professional growth and engagement with the excitement of mathematics by working with peers on new approaches to teaching, tackling research questions, and interact- ing with the broader mathematical community.

This year’s UFP instructor/coor- dinator will be Thomas Garrity, the William R. Kenan Jr. Profes- sor of Mathematics at Williams College. His program description is as follows:

Algebraic Geometry in the Undergraduate Curriculum One of the popular “pizza and problem solving” sessions run by Andy Bernoff of Harvey Mudd College. This event happens several times at each PCMI institute, and regularly at- Algebraic geometry is one of the tracts close to 200 participants. richest areas in all of mathematics and also has the somewhat legitimate reputation for being one of the most difficult. Still, undergraduates mathematics. Used in class, algebraic director of the Williams College Project can learn and do research in algebraic geometry can highlight connections be- for Effective Teaching. With research geometry. For faculty, knowing even a tween traditionally segregated areas of interests in algebraic geometry, differ- little algebraic geometry can lead to undergraduate mathematics. The UFP ential geometry and number theory, Tom insights in teaching from calculus to the sessions will give participants the needed is the author of All the Mathematics You highest level courses. The 2008 Under- tools to make these connections. Missed [But Need to Know for Graduate graduate Faculty Program intends for its School] and appears with Colin Adams in participants to: UFP participants have traditionally taken the DVD The Great π/e Debate: Which is part in many of the other PCMI programs. the better number? Among his honors is 1. Develop a sufficient understanding For example, some UFP participants like the 2004 Haimo Award for Distinguished to be able to teach an undergraduate to attend the Graduate Summer School Teaching by the Mathematical Associa- course in algebraic geometry. courses or lectures in the Research Pro- tion of America. gram. Some also work actively with the 2. Work on research type problems in Secondary School Teachers program, For more information and application algebraic geometry. particularly concerning pedagogical, instructions for the Undergraduate curricular, or articulation issues. A large Faculty Program, consult the UFP web 3. See how ideas from algebraic geo- number are attracted to the Undergradu- page at http://pcmi.ias.edu/current/pro- metric are used in other courses and ate Program, both for the interesting gram_undergradfaculty.php. Reviewing fields. mathematics in the courses and for the of applications begins January 20, 2008, kind of research experiences for under- and financial support is available for 4. Survey current work in algebraic graduates that are often available. participants in all the programs. For more and analytic geometry, thus allowing general information on all aspects of the meaningful interaction with the other This summer’s UFP instructor/coordina- 2008 Summer Session consult the PCMI PCMI programs. tor, Tom Garrity was an undergraduate web page: http://pcmi.ias.edu/. at the University of Texas at Austin, a In essence, algebraic geometry studies graduate student at Brown University, Bill Barker is Isaac Henry Wing Profes- the zero loci of polynomials, linking al- and a post-doctoral instructor at Rice sor of Mathematics at Bowdoin College gebra and geometry. Without much effort, University. He joined the faculty of in Brunswick, ME, and a member of the topology and analysis also arise natu- Williams College in 1989, serving as PCMI Steering Committee. rally. In fact, algebraic geometry likely the William R. Kenan Jr. Professor of lurks in the background of most areas of Mathematics, department chair, and the

19 MAA FOCUS January 2008

Calculus Placement that Really Works!

By Todd Retchless, Randy Boucher, and Donald Outing

Bored students, overwhelmed stu- instructor, resulting in a final dents, and instructors struggling to enrollment of approximately make their classroom work: these are 175 students (roughly 14% of some of the difficulties faced by col- each entering USMA class). It leges and universities attempting to is important to note that near- develop curriculum for students who ly one-third of the students in have taken calculus in high school. As the advanced program do not the number of students entering college have any AP Calculus scores with calculus experience has increased, on file at the USMA. so has the difficulty of identifying which students should receive credit Selecting the most qualified for single variable calculus and be students — those who dem- enrolled directly into more advanced onstrate strong fundamentals courses. As a result, many instructors and the ability to learn at an are faced with the difficult task of de- increased rate — allows the signing lesson plans that will reach an development of a curriculum audience with vastly different calculus rich in advanced mathematical backgrounds. While the Advanced concepts. As a result, we have Placement (AP) Calculus Exam score been able to create a two se- gives us some indication of a student’s Illustration by Brad Fitzpatrick mester advanced mathematics preparation, by itself it is an inadequate program consisting of courses indicator of future success. So how can in college Calculus I and II) without in multi-variable calculus and we best assess a student’s high school the use of technology (i.e. graphing differential equations. This allows us to calculus foundation and in turn leverage calculator). Upon completion of valida- expose students in our program to topics this at the university level? tion exam grading, approximately 180 not found in the core mathematics cur- to 190 students receive single-variable riculum, such as advanced vector calcu- Over the past three years, the Math- calculus validation credit based first on lus, infinite series, and a full semester of ematics Department at the United States their validation exam score and second differential equations. Additionally, our Military Academy (USMA) has devel- on their AP Calculus Exam score (when students participate in an interdisciplin- oped a course placement process for available). These students also receive ary guest lecture program and a rigorous an advanced math program that cuts advanced placement and are enrolled into technical writing program designed to through this uncertainty. By considering a fall semester offering of multi-variable pique their interest in and prepare them students’ performance on two USMA calculus. for future study in math, science, or en- administered single-variable calculus gineering. We take great pride in the fact validation exams, as well as any AP Student performance is closely monitored that nearly 70% of participating students Calculus Exam Scores on file, we are by instructors during the first two weeks over the last three years chose to major in able to enroll our best qualified first year of the semester. This allows students to a math, science, or an engineering (MSE) students into a mathematics curriculum acclimate to the pace of the course and discipline. rich in multi-variable calculus and dif- permits instructors to assess students’ ferential equations. skills through classroom interaction Although many hours are required to and graded assignments. The culminat- implement a placement process such as Each year, the USMA enrolls approxi- ing event of the placement process is this, we believe the benefits are worth mately 400 students who have taken the administration of a second single the cost. Through the administration of calculus in high school, with about 170 variable calculus exam (again without in-house validation exams, we are able having AP Calculus Exam scores on file the use of technology) at the end of the to best identify students deserving of at USMA. During the summer prior to second week. While similar in nature to advanced placement who, for whatever the first academic semester, students with the summer validation exam, there is a reason, do not have AP Exam scores on calculus experience are administered an much higher performance expectation on file. Each year, students in this category open-invitation single-variable calculus this exam as students have had two weeks account for nearly one-third of our total validation exam. The validation exam to prepare. At this point in the selection enrollment in the course and have histori- is 110 minutes long and tests student process 10 to 20 students will normally cally performed at or above the course comprehension of single-variable cal- transfer out of the program, either by self averages. We also believe that a thorough culus concepts (typically topics covered nomination or by the suggestion of their placement process allows us to include

20 January 2008 MAA FOCUS more rigor into mathematics instruction classroom. Students complete the course from the University of Maryland and is in the classroom. As a result, we have more confident in their advanced skills currently an Assistant Professor. Randy developed a process to more accurately and better prepared for further study at Boucher received his M.S. in Applied assess a student’s high school calculus the university level. Mathematics from the University of education. The assessment permits us Washington and is currently an Instruc- to correctly place entering freshmen The authors teach at the Department tor. Donald Outing received his M.S into the appropriate academic level of of Mathematical Sciences at the United and Ph.D. in Applied Mathematics from instruction. Students are less bored, more States Military Academy. Todd Retchless Rensselaer Polytechnic Institute and is involved, and properly challenged in the received his M.S in Applied Mathematics currently an Academy Professor.

Employment Opportunities KANSAS courses in mathematics or a Ph.D. in the undergraduate mathematics, potential for mathematical sciences with a willingness to professional activity including publication, Tabor College, Hillsboro teach courses in computer science is required. speaking and grant writing. Applicants must Professor of Mathematics A strong commitment to teaching is essential. be willing to work with regional K-12 teachers Tabor College invites applications for the po- Responsibilities center on mathematics and and teacher educators. See http://www.sou. sition of Professor of Mathematics to start fall computer science teaching ranging from edu/hrs for complete position description 2008. Ph.D. in mathematics preferred. Suitable introductory to advanced undergraduate. and salary. Send ALL of the following: vita, teaching experience required. Responsibilities Above all we are seeking candidates who statement of teaching philosophy, description include teaching undergraduate courses in can achieve excellence in teaching and who, of professional goals, summary of teaching the mathematics major, and in the general in an undergraduate environment, will find evaluations, all post-secondary transcripts, education courses offered by the department, ways to grow professionally. All faculty in the completed SOU Faculty Application, and supervising senior projects, advising students Department will be expected to interact with three letters of recommendation to: Dr. Dusty majoring in mathematics, serving on appropri- students on a one-on-one basis in the excite- E. Sabo, Department of Mathematics, SOU, ate committees, and other duties, as assigned. ment and vitality of their growth through such Ashland, OR 97520. SOU is an Affirmative Candidates must understand and support the activities as summer faculty/student research, Action/Equal Opportunity Employer. mission of the department of mathematics and independent studies, or senior capstone computer science and the mission and goals of projects. The Department is supportive of all RHODE ISLAND an evangelical, Anabaptist Christian college forms of scholarship. Applicants should send (refer to www.tabor.edu/about-tabor/em- a cover letter, resume, transcripts of graduate Roger Williams University ployment/faculty#job2), and must be able to and undergraduate work, a statement of per- The Department of Mathematics in the Fein- articulate a personal Christian commitment. sonal teaching philosophy, and three letters of stein College of Arts and Sciences at Roger Salary commensurate with experience and recommendation by January 21st to Professor Williams University invites applications from rank. The mathematics and computer science James Blackburn-Lynch, Chair, Mathematics candidates specializing in the area of Statistics department offers majors in Mathematics, and Computer Science Department, CPO for the position of Visiting Assistant Professor Computer Science and Computer Systems 2146, Berea College, Berea, KY 40404. of Mathematics/Statistics. Administration and includes programs for pre- More information about Berea College and engineering and secondary teaching. To apply, the Mathematics and Computer Science De- Review of applications will begin January send letter of interest, vita, and names of three partment is available at http://www.berea. 15, 2008 and will continue until the position professional references to lawrencer@tabor. edu/mcs/ Women and minority candidates is filled. edu. Review of applications will begin Janu- are especially encouraged to apply. ary 1, 2008; applications will be accepted until For a full description of this position and position is filled. Tabor College is an Equal Berea College, in light of its mission in the how to apply, please go to: http://www.rwu. Opportunity Employer operating under the tradition of impartial love and social equal- edu/depository/hr/jobs/faculty/fcas_visit- auspices of the Mennonite Brethren churches ity, welcomes all people of the earth to learn ing_asst_professor_math-stats.pdf. of the United States. It complies with all ap- and work here. plicable non-discrimination laws. SOUTH CAROLINA OREGON KENTUCKY Central Carolina Technical College Southern Oregon University Mathematics Instructor (2 positions): Berea College Tenure-track Assistant Professor, Southern #020901MA & #139217MA. Mathematics Oregon University, starting Fall 2008. One Responsible for teaching college credit and Computer Science Department tenure-track position in Mathematics Educa- algebra, statistics, trigonometry, geometry Berea College announces a full time, ten- tion. Application review begins January 15, and calculus classes at main and off campus ure-track position in the Mathematics and 2008 and continues until position is filled. locations. Requires Master’s Degree in math- Computer Science Department, beginning Requirements: Doctorate in Mathematics ematics or a Master’s Degree with 18 graduate September, 2008. Appointment will be at Education or Ed. D. in Mathematics Edu- semester hours in mathematics. Post-second- the assistant professor level. A Ph.D. in cation or Mathematics, excellent teaching ary teaching experience with ability to teach Computer Science and willingness to teach record, strong commitment to teaching all levels of college math and Master’s Degree

21 MAA FOCUS January 2008 in mathematics or Master’s Degree with 18 and Real Analysis. Student advising plus graduate transcripts, a statement of teaching graduate semester hours in mathematics pre- department and university service is expected. philosophy, and three letters of recommenda- ferred. Apply online at http://jobs.sc.com or Qualifications include a Ph.D. in mathematics tion to: Dr. Mary Hassinger, Dean, School visit www.cctech.edu. and teaching experience. Viterbo University of Letters and Science, Viterbo University, is a Catholic, Franciscan, ecumenical liberal 900 Viterbo Drive, La Crosse, WI 54601. Questions? Contact the Personnel Office by arts institution with an undergraduate enroll- FAX: 608-796-3391 Email: mchassinger@ email: [email protected] ment of 1,980 located in southwest Wisconsin viterbo.edu. Review begins January 21, 2008 or by mail: in the scenic upper Mississippi River Valley. and continues until position is filled. Viterbo Central Carolina Technical College Relocation assistance is available. Send let- University is an equal opportunity employer 506 N. Guignard Drive ter of application, application form www. and encourages nominations and candidacy of Sumter, SC 29150 viterbo.edu/EmploymentApplication. historically underrepresented groups. www. aspx, curriculum vita, undergraduate and Viterbo.edu. WISCONSIN

Milwaukee School of Engineering Mathematics Faculty Position The Milwaukee School of Engineering invites applications for a full-time mathematics faculty position starting in the fall of 2008. The person selected should be able to teach any DIRECTOR, SCHOOL OF MATHEMATICS AND STATISTICS of the standard courses in an undergraduate mathematics curriculum and possible a course Arizona State University announces an international search to recruit a in graduate level engineering mathematics. MSOE offers degrees in engineering, engi- Founding Director for the proposed School of Mathematics and Statistics. neering technology, technical communication, Arizona State University (see http://www.asu.edu) is a dynamic, business, construction management and nurs- internationally recognized research university serving more than 60,000 ing. Candidates should possess an appropri- students in the Phoenix metropolitan area, one of the fastest growing urban ate doctoral degree and related experience. centers in the nation. The Department of Mathematics and Statistics at Salary and rank will be commensurate with Arizona State University (see http://math.asu.edu) currently has about 50 experience. tenured or tenure-track faculty members and offers a range of undergraduate and graduate programs including pure mathematics, MSOE is located in the heart of downtown computational and applied mathematics, mathematical biology, statistics, Milwaukee and has been recognized in several and mathematics education. national publications for its “applications-ori- ented” approach. Faculty are judged primarily The Department has established strong interdisciplinary ties with University on excellence in teaching. MSOE graduates and community-based research and education initiatives. The Director of the are in high demand as evidenced by our strong School is expected to maintain and expand these by providing visionary job placement rate. leadership, overseeing growth in the faculty and research programs, and supervising the further development of the graduate and undergraduate The review of candidates will begin immedi- programs. ately and continue until the position is filled. To apply, please submit a detailed resume, Candidates must have an earned doctorate in mathematics, statistics, or a evidence of teaching excellence and three closely related field and a distinguished scholarly record appropriate for professional references to: Dr. Karl David, appointment as a full Professor. Desirable qualifications include documented Chair, Mathematics Department, Milwaukee leadership skills, previous administrative experience in a doctoral-granting School of Engineering, 1025 N. Broadway, department, experience with program development in research and Milwaukee, WI 53202-3109. The information education, evidence of excellent communication and organizational skills, can also be faxed to (414) 277-7462. Please visit us at www.msoe.edu. and evidence of commitment to working with and supporting a diverse student and faculty population. Viterbo University Mathematics The position is available beginning 1 July 2008. Salary and start-up funding Assistant Professor, Tenure Track will be competitive and commensurate with qualifications. Review of The Mathematics Department at Viterbo applications will begin on 4 January 2008; if not filled, applications will be University invites applications for a full-time evaluated every two weeks thereafter until the search is closed. Applicants tenure-track position, effective August 2008. must submit electronically (in MS Word or PDF format) a cover letter and Responsibilities include teaching courses such current curriculum vitae to Mr. Gabriel Escontrias as the pre-Calculus sequence and the Calculus ([email protected]). Inquiries and nominations should be sequence, Statistics, “liberal arts” mathemat- directed to: Sid P. Bacon, Dean of Natural Sciences, College of Liberal Arts ics, mathematics for education majors, busi- and Sciences, Arizona State University, PO Box 876505, Tempe, AZ 85287- ness mathematics, and upper-division courses 6505; or email: [email protected]. A background check is required for in the major such as Linear Algebra, Differ- employment. Arizona State University is an affirmative action, equal ential Equations, Abstract Algebra, Geometry opportunity employer committed to excellence through diversity.

22 January 2008 MAA FOCUS New from the ® Mathematical Association of America Number Theory Through Inquiry David C. Marshall, Edward Odell & Michael Starbird

Number Theory Through Inquiry is an innovative textbook that leads students on a guided discovery of introductory number theory. The book has two equally significant goals. One goal is to help students develop mathematical thinking skills, particularly, theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for an independent study experience, or for a course designed as an introduction to abstract mathematics. This text is suitable for mathematics or related majors or anyone interested in exploring mathematical ideas on their own will enjoy Number Theory Through Inquiry.

Number Theory Through Inquiry contains a carefully arranged sequence of challenges that lead students to discover ideas about numbers and to discover methods of proof on their own. It is designed to be used with an instructional tech- nique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). The result of this approach will be that students: o Learn to think independently o Learn to depend on their own reasoning to determine right from wrong o Develop the central, important ideas of introductory number theory on their own.

From that experience, they learn that they can personally create important ideas. They develop an attitude of personal reliance and a sense that they can think effectively about difficult problems. These goals are fundamental to the educational enterprise within and beyond mathematics.

MAA Textbooks • Catalog Code: NTI • 150 pp., Hardbound, 2007 • ISBN 978-0-88385-751-9 List: $51.00 • MAA Member: $41.00

Is Mathematics Inevitable? A Miscellany Underwood Dudley, Editor

This is a collection of gems from the literature of mathematics that shine as brightly today as when they first appeared in print. They deserve to be seen and admired.

The selections include two opposing views on the purpose of mathematics, The Strong Law of Small Numbers, the treatment of calculus in the 1771 Encyclopaedia Britannica, several proofs that the number of legs on a horse is infinite, a deserved refutation of the ridiculous Euler-Diderot anecdote, the real story of ¿ and the Indiana Legislature, the reason why Theodorus stopped proving that square roots were irrational when he got the square root of 17, an excerpt from Mathematics Made Difficult, a glimpse into the mind of a calculating prodigy….There will be something of interest here for almost anyone interested in mathematics.

Underwood Dudley is the bestselling author of several MAA books: Mathematical Cranks, Numerology, and the Trisectors. He has an Erdos number of 1.

Spectrum • Catalog Code: IMI • 160 pp., Hardbound, 2007 • 978-0-88385-566-9 • List: $56.95 • MAA Member: $45.50

Order your copy today! 1(800) 331-1622 • www.maa.org

23 MAA FOCUSDo you know January 2008 someone who loves pi as much as pie?

Would they also love a full-tuition scholarship for a master’s degree in mathematics education, a New York State Teaching Certificate, and a $90,000 stipend in addition to a competitive salary as a New York City secondary school math teacher? Math for America offers fellowships with these benefits and more to individuals who know and love math, enjoy working with young people, have excellent communications skills, www.mathforamerica.org and a strong interest in teaching. Log on to learn more.

The Mathematical Association of America Periodicals Postage paid at 1529 Eighteenth St., NW Washington, DC and Washington, DC 20036 additional mailing ofﬁces