MAA FOCUS March 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 3 Editor: Fernando Gouvêa, Colby College; [email protected] Inside Managing Editor: Carol Baxter, MAA [email protected] 4 Halmos Endowment Fund for the Carriage House

Senior Writer: Harry Waldman, MAA By Gerald L. Alexanderson [email protected] Please address advertising inquiries to: 6 An Online Registration System for Section Meetings [email protected] By G. Jay Kerns, Jonathan Duran, Thomas E. Price, and D. P. Story President: Joseph Gallian 8 Teaching Time Savers: Using Preview Problems to Get Ahead of the Syllabus First Vice President: Elizabeth Mayfield, Second Vice President: Daniel J. Teague, By Marion Deutsche Cohen Secretary: Associate Martha J. Siegel, 9 Funding Available for Students Going to MathFest Secretary: James J. Tattersall, Treasurer: John W. Kenelly By Robert W. Vallin Executive Director: Tina H. Straley 10 Project NExT Reaches 1000 Fellows

Director of Publications for Journals and 11 Developing Mathematical Habits of Mind Communications: Ivars Peterson By Annie Selden and Kien Lim MAA FOCUS Editorial Board: Donald J. Albers; Robert Bradley; Joseph Gallian; 12 Robert P. Balles Awards for IMO Team Participants Jacqueline Giles; Colm Mulcahy; Michael By Stephen Dunbar Orrison; Peter Renz; Sharon Cutler Ross; An- nie Selden; Hortensia Soto-Johnson; Peter 13 Undergraduates Win Awards at San Diego Stanek; Ravi Vakil. By Joe Gallian Letters to the editor should be addressed to Fernando Gouvêa, Colby College, Dept. of 14 Mathematicians Recruited for Climate Change Research Mathematics, Waterville, ME 04901, or by By Pat Kenschaft email to [email protected]. 15 Mathematics in Voting Theory at the 2008 Joint Meetings Subscription and membership questions By Eric Gottlieb, Brian Hopkins, and Michael A. Jones should be directed to the MAA Customer Service Center, 800-331-1622; email: 18 Mathematical Experiences in Business, Industry, and Government [email protected]; (301) 617-7800 (outside By Phil Gustafson U.S. and Canada); fax: (301) 206-9789. MAA Headquarters: (202) 387-5200. 20 San Diego Joint Mathematics Meetings in Pictures Copyright © 2008 by the Mathematical 24 The Undergraduate Poster Session at JMM 2008 Association of America (Incorporated). By Diana Thomas Educational institutions may reproduce articles for their own use, but not for sale, 25 My First Year as MAA President: A Report provided that the following citation is used: By Joe Gallian “Reprinted with permission of MAA FOCUS, 27 MAA Business at the 2008 Joint Mathematics Meetings: A Report the newsmagazine of the Mathematical Association of America (Incorporated).” By Martha Siegel, MAA Secretary Periodicals postage paid at Washington, DC On the cover: The MAA Carriage House at dusk. Photograph by Paul Burk and additional mailing offices. Postmaster: Photography. 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 August/September October November of America. Editorial Copy July 8 September 16 Display Ads July 10 August 20 September 24 Employment Ads June 11 August 13 September 10

2 March 2008 MAA FOCUS Mathematics Awareness Month to Focus on Voting By Ryan Miller

hat makes your vote matter?” summer’s MathFest entitled “Math- “W ematics and the Geometry of Voting.” That‘s the question those eye-grab- bing orange posters you’ve probably As mentioned above, mathaware.org noticed around collegiate math depart- also hosts a voting exercise that is ments are asking. This is the theme for meant to show the outcome of an elec- April, which is Mathematics Aware- tion may more accurately reflect the ness Month. It’s a fitting choice. With voting method rather than the voters’ the ongoing primaries and caucuses wishes. The exercise provides three and upcoming presidential election separate ways to cast your ballot for on everyone’s mind, 2008 makes the your presidential candidate of choice, perfect year to display the relevance and then reveals the winners as chosen of mathematics to the many issues by each method. They are usually dif- raised by the processes of voting and ferent! vote-counting. Resources for this year’s Mathematics The Joint Policy Board for Mathemat- Awareness Month are designed to help ics (JPBM, comprised of the MAA, the explain what makes your vote matter, American Mathematical Society, the as well as how the voting system used American Statistical Association, and affects the outcome, regardless of the the Society for Industrial and Applied context of the voting. At mathaware. Mathematics), has chosen “Mathematics which preferences are expressed and org, you can download articles and es- and Voting” as the theme for this year’s some attempt is made to combine those says that deal with the relation between Mathematics Awareness Month. JPBM preferences into a general preference, math and voting, as well as an 8.5" x 11" has put together a web site at http://www. voting has occurred. When you apply for copy of the 2008 poster, titled “What mathaware.org containing a variety of a job, for example, those responsible for Makes My Vote Matter?” resources to help increase public under- hiring are voting for the candidate they standing of and appreciation for math- like most. The idea of Mathematics Awareness ematics. One of the pages asks visitors Month came about in 1986 when Presi- to vote for their favorite presidential This year’s featured activity is a video dent Ronald Reagan issued a proclama- candidates in three different ways, and contest encouraging participants to create tion establishing National Mathematics displays the results of the voting so far as a short piece to be hosted on YouTube Awareness Week. Extended to a month in tabulated by three different methods. that conveys their feeling on the connec- 1999, MAM activities are generally orga- tion between math and voting. The site nized on local, state and regional levels It’s important to note that “voting” is also hosts a number of video lectures on by college and university departments, something that happens in many contexts math and voting given by University of institutional public information offices, not at all related to politics. According California-Irvine professor Don Saari. student groups, and related associations to the MAM web site, any situation in Saari will be teaching a minicourse at this and interest groups.

Alder Award Nomination Deadline Changed MAA Dues Will Not Increase for 2009 ominations for the Henry L. Alder The Awards Committee encourages N t the Joint Meetings, the MAA Awards for Distinguished Teaching by nominations in accordance with the pro- A Board of Governors approved a pro- MAA FOCUS Deadlines a Beginning College/University Faculty cedures outlined on the MAA web site. posal to put off increasing MAA dues Member are due October 1, 2008. The The Alder Awards are announced every August/September October November for 2009, reversing a decision made original deadline for these nominations year at the MAA MathFest, and winners Editorial Copy July 8 September 16 last year. We hope this will encourage was in December. are invited to make a presentation at that Display Ads July 10 August 20 September 24 MAA members to renew early and of- meeting. Employment Ads June 11 August 13 September 10 ten!

 MAA FOCUS March 2008

Halmos Endowment Fund for the Carriage House

By Gerald L. Alexanderson

In 2003, Paul and Virginia In the fall of 2007, a Halmos gave $3 million to the small informal com- MAA to restore the Carriage mittee was formed to House, part of the headquarters discuss the launch of complex of the MAA in Wash- a capital campaign to ington, DC. A historic structure raise money for an en- dating back to the late 19th cen- dowment. When Vir- tury, the building over recent ginia Halmos heard of years had been used for a few this, she immediately offices and storage but it was decided to help out. not in good condition. Execu- After some discussion tive Director Tina Straley came she offered to establish up with the idea that it could, a matching grant that with a good deal of work, be would match gifts made converted to a small meetings by MAA members and center that would complement friends, offering two the two existing townhouses, The conference room on the first floor of the Carriage House hosts dollars for every dollar the Vaughn and Pólya Build- the larger meetings and distinguished lectures. Photograph by Paul contributed by others, ings. After extensive work on Burk Photography. up to a maximum of the outside and the construction $600,000. This would of a whole new interior, the require contributions Carriage House Conference of $300,000 from the Center was dedicated in April members, which would 2007. The donors’ interests fit go a long way to reach exactly with Straley’s ideas the desired total for for the building: to establish an endowment fund of a national center to promote over $1 million. More mathematical exposition, would be required to something that fits perfectly fund the programs ad- with the MAA’s mission. Paul equately, but that would Halmos was long known and be a good start. admired as a promoter of fine exposition, as evidenced by As is usually true of his writings on the subject and capital campaigns, be- his editing of book series and fore a public announce- journals over many years. Art Benjamin, with a group of students, during a Math Fair held last ment of the campaign, spring. Photograph by Ivars Peterson. the officers of the MAA Already the Carriage House were to approach in- is functioning well, with a and various planning subcommittees dividuals and founda- wide range of events being held there: a chaired by Art Benjamin (dedication tions that would be most likely to give funded lecture series (supported by the ceremonies), Bob Megginson (national significant amounts. The public campaign NSA), workshops, committee meetings, activities), and Dan Ullman (local pro- would follow once roughly half the goal short courses, and symposia sponsored gramming), have put an ambitious plan was reached. The campaign was set up by the MAA. It is also, when not needed in place for future use of the facility. But to last for two years. Would the goal of for MAA activities, available to other it became clear that in order to fulfill the $900,000 be reached in that time? No one mathematical organizations for similar projected goals, an endowment would be could predict. This kind of fund raising events. For more lasting effect outside the needed to make the center truly national had not been tried before by the MAA. Washington area, and given the quality of and to ensure its success now and into the lectures presented so far, the MAA the future. Funds would be needed to On November 1, 2007, Executive Direc- could introduce a series of published lec- support honoraria for speakers, travel tor Tina Straley, President Joe Gallian, ture notes based on talks at the Carriage funds for workshops, research seminars, and Treasurer John Kenelly, started House. An advisory board (co-chaired and similar activities that would have a making a few phone calls to past offi- by Jerry Alexanderson and Ken Ross), national impact. cers and editors of the MAA, and others

 March 2008 MAA FOCUS who have a record of giving generously at the level of $9,000 or above become Gerald L. Alexanderson is a former and regularly to the Association. Very Founding Patrons and will be recognized president of the MAA (1998-99) and pre- quickly contributions and pledges came on a plaque in the Carriage House and in viously served as secretary (1990-97). in. By the beginning of December orga- materials for supported programs. nizers began to speculate about whether $200,000 from MAA contributors might be received or pledged by the time of the A Selection of 2008 Carriage House Events San Diego meetings in January. Those involved were not at all certain that this March could be achieved. It turns out that by the end of December, the MAA had pledges 5th: Margaret Wright Distinguished Lecture 8th- 9th: United States American Math- or actual contributions in hand for the (http://www.maa.org/dist%2Dlecture/) ematical Olympiad Awards Ceremony full $300,000! 6th-7th: TENSOR/Strengthening Un- 10th-14th: Professional Enhancement Pro- The MAA must, in the view of its mem- derrepresented Minority Mathematics gram (PREP 986-8J) - Teaching Arithme- bers, be doing something right. The level Achievements tic to College Students (http://www.maa. of commitment to the goals of promoting org/prep/2008/) mathematical exposition was very, very 7th-9th: American Mathematical Invita- gratifying. What the organizers expected tional Examination Committee meeting 19th-22nd: Professional Enhancement would take two years, took only two Program (PREP 986-8I) - Leading the months. April Academic Department (http://www.maa. org/prep/2008/) When Virginia Halmos was told of the 26th: “Mathematical Thinking for the th th success of the campaign, she promptly Rest of Us,” seminar led by Ed Burger, 25 -29 : Professional Enhancement Pro- offered an additional $200,000 to match Williams College, co-sponsored with gram (PREP 986-8L) - Linear Optimiza- further contributions, one to one, so that Smithsonian Institute Resident Associ- tion (http://www.maa.org/prep/2008/) the total amount of the endowment, if the ate Program th rd drive were completed successfully, would 30 -3 of July: Professional Enhance- be $1.3 million. This would be enough 28th- 29th: Workshop on Proactive Re- ment Program (PREP 986-8M) - Exposi- to guarantee the ongoing program at the cruiting in the Lower Division / National tory Writing to Communicate Mathemat- http://www.maa.org/prep/2008/ Carriage House at a base level, without Science Foundation, Division of Math- ics ( ) relying completely on the success of ematical Sciences grant applications. The work on the Car- July riage House did not deplete the funds May th th from the original Halmos gift, so the 17 -19 : Professional Enhancement Pro- remainder of those funds will assure suf- 2nd: Conference Board of the Mathemati- gram (PREP 986-8N) - How to Build and ficient money needed for upkeep of the cal Sciences meeting Run a Successful Emerging Scholars Pro- http://www.maa.org/prep/2008/) Carriage House and provide replacement gram ( rd th equipment as needs arise. For example, 3 -4 : Committee on Committees meet- st th the current meeting rooms are furnished ing 21 -25 : Teacher’s Circle meeting (tenta- http://www.aimath.org/ARCC/ with state-of-the-art audio-visual equip- tive date) ( workshops/tcircle.html/) ment, which in three or four years will 9th-11th: United States American Math- no longer be state-of-the-art and will ematical Olympiad grading October have to be replaced. That is assured by the original grant. 15th-18th: Executive Committee meeting 3rd-4th: American Mathematical Competi- With funds already obtained along with 22nd-24th: New Governor’s Orientation tions Advisory Board meeting anticipated additional contributions, the (or early June) November Carriage House can be expected to be th th another major contributor to the success 28 -30 : Mathematics Education Re- st of the MAA in promoting mathematical search Conference/Knowles Science 21 : Reception for American Mathemati- exposition. To contribute to the Halmos Teaching Foundation cal Association of Two-Year Colleges Endowment Fund go to http://www.maa. national meeting participants org/development/pledgeform.pdf. Your June donation will be doubled thanks to the match to be made by Virginia Halmos. 6th-7th: American Mathematical Competi- Pledges may be made to spread your tions Testing donation over three years. All donors

5 MAA FOCUS March 2008 An Online Registration System for Section Meetings By G. Jay Kerns, Jonathan Duran, Thomas E. Price, and D. P. Story Approximately seven years ago, the Perhaps the easiest and most efficient way ments chair should be responsible for Ohio Section began using an online reg- for a section to implement a registration modifying the database after the registra- istration system to pre-register section program is to take advantage of the strate- tion time period has passed. meeting participants. Registrants’ contact gies, software, and design employed by and professional information as well as existing registration administrators. The The initial registration page contains tests presentation submissions and automated purpose of this article is to assist those to ensure that sufficient and, in some fee calculations were stored in a crude da- interested in developing an online reg- cases, proper information is entered. tabase until the registration deadline. The istration system by providing them with For example, the page contains a form data were then converted to a spreadsheet insights resulting from our experience field in which the registrant’s email ad- format for use by the section’s program and making our current code available dress is entered. This field is checked to chairs and local arrangements committee to them. In the next section we describe ensure that some text is entered and that chairs. An automated email confirmation the lessons we have learned through the the entered text represents a valid email was sent to each registrant. years. Then we briefly describe a web site address. However, these checks are not containing links to sample (but active) sufficient to avoid many of the difficulties The original program was written by Tom registration pages and database view that arise during an online registration Price using php code. A few years later pages. Links to downloadable files con- process. The first major revision of the the program was rewritten by Don Story taining the ASP and ASP.NET code can program included code that generated a using ASP and JavaScript. This program be found at this site. Once understood, verification web page that reproduced the added important features including a the program is easy to implement at most entered information. This page allowed “confirmation page” to be reviewed by installations. It is also easily modified for users to review and check their infor- registrants before submitting their in- other uses. For example, a copy of the mation for accuracy and completeness. formation to the database and a strategy ASP program was customized several Once the initial form data were entered for avoiding duplicate registrations. This years ago to handle online applications the user would then click a “continue” modified version also included fields for for the mathematics REU program at The button which would direct the browser special requests such as video or comput- University of Akron. to this check page. This page notifies the er equipment for presenters. Data were user that it is for checking purposes only stored in a centralized database server lo- Lessons learned and that he or she is not registered until cated in the Department of Mathematics the “submit” button is clicked. The im- at the University of Akron. Registration The initial system did not include data- portance of the verification strategy can data were available to section members at base queries designed to avoid duplicate not be overemphasized. Its implementa- anytime from any computer with internet registrations. This caused several prob- tion noticeably decreased the number of access via the freeware online database lems. For example, some people would input errors. viewer GenericDB. register twice: once indicating that they were submitting a presentation and once Automated total cost calculation for Price and Story retired in 2006. At that indicating that they did not wish to give each registrant has been a part of the time Jay Kerns took the responsibility a talk leaving them the “flexibility” to system used in Ohio since its inception. for updating and maintaining the online make a decision at a later date. Of course However, there is currently no vehicle registration process. With Jonathan duplicate registrations of any type cause for online payment. Developing one Duran, he rewrote the code in ASP.NET difficulties. In 2003 a database query would probably not be worth the cost. using Microsoft Visual Studio 2003, was included in the program that was Information on the final web page and Dreamweaver 8 (Adobe), and Flash designed to avoid duplicate registrations. on the confirmation email that result Professional 8 (Adobe). The updated If a person attempted to register more after a successful registration encourages program is currently built on a Windows than once, the browser would be directed attendants to bring a check with them to server equipped with a Microsoft Ac- to a page indicating that duplicate reg- the meeting. This approach has worked cess Database located at Youngstown istrations are not permitted and that the well at the section level and there are no State University, and then uploaded registrant should contact the program plans for a change. It is worth noting that to a commercial web hosting service, chair to make any changes. The success automated email reminders were sent namely, DiscountASP.NET, for section of this strategy prompted the section to to all registrants immediately after the registration. A session is generated for include this feature on all later versions. registration deadline. These automated each user, which both remembers entered The manager should not assume respon- emails contained their submitted infor- information for use in the session, and sibility for making changes in registration mation, a printable parking pass, and which automatically times-out after a information contained in the database. other instructions including the amount reasonable period. The program chair and local arrange- of money due on arrival at the meeting

 March 2008 MAA FOCUS and preferred payment method — “Bring Information Page (RIP) located at http:// a list of current registrants along with your check to the meeting.” Personalized www.math.uakron.edu/~tprice/rip/. One is specific fields such as titles and abstracts email reminders are very easy to imple- a link to a sample, but active, registration of submitted presentations. Permissions ment using Microsoft Office. page. Viewers can enter the required in- can be granted giving, for example, the formation into the form fields. Once this program chair and local arrangements A second option easily accomplished is done a participant may click the “con- chair privileges to update certain fields in using MS Office and often implemented tinue” button which will transfer control the database such as the time and location in our section is name tag generation. A to a verification page. The verification of a presentation. GenericDB is flexible, file designed to print name tags was sent page lists all information as entered for works with most databases, and is easy to to the local arrangements chair for print- a last check. If some fields are incorrect, implement. It can be found at http://www. ing on name tag stock. Often, institutions then the user may return to the first page genericdb.com. generated their own name tags so that a by pressing the “edit” button. Once the particular logo or other information could user is satisfied with the input data, the The ASP.NET version uses a customized be included on each tag. Such things as “confirm” button on the second page is database display page. At this time, only banquet tickets and receipts were gener- clicked which completes the registration viewing capability is implemented. A ated in a similar way. process by storing all input information task for the near future is to add editing into the database. A confirmation email is functionality to the existing view-page to The national MAA organization has spe- sent to the registrant and a copy is sent to allow program chairs and local arrange- cific abbreviations for colleges and uni- the address [email protected]. ments coordinators to modify submitted versities. These can be downloaded from information as needed. the MAA web site and used to create the This sample system is active so all in- Affiliation field for the registration page. formation is literally stored and may be Once the deadline for online submission This simplifies transferring information viewed by others as described below. of contributed talks has passed, the Mi- about section meeting attendees to the (Visitors are invited to register more crosoft Access database is downloaded national organization. The use of these than once if desired since the duplicate and, if desired, exported into an amenable abbreviations is strongly recommended registration variable is set to “False.” file format such as Excel. Next, the ap- and since they can be downloaded they However, as mentioned above, we rec- propriate file is forwarded to the program are easy to include in the code. ommend that sections implement a strat- chair, who uses the information to sched- egy that restricts multiple registrations.) ule contributed sessions. Similarly, when The current program does not include The sample registration page contains the the deadline for online registration has a procedure for registrants to log onto minimum information a system should expired, the database is downloaded, ex- a page that will permit them to make contain. ported, and forwarded to the chair of local changes to previously submitted informa- arrangements, who uses the registrant tion before registration or presentation A second link on the RIP transfers brows- information to print name tags and as- deadlines. This addition to the system ers to a page containing current registrant semble conference folders which include would require considerable additional information. Normally, several such receipts and banquet tickets. It should code to generate a password system and pages with different filters are posted be noted that the ability to view and, to an automated strategy for resetting or since users have various needs. For ex- some extent, edit database information reminding users of their passwords. This ample, program chairs may wish to obtain gives these chairs the flexibility to begin level of sophistication is not suitable for information about submitted presenta- planning before the deadlines. local, smaller meetings. tions while the local arrangements chairs are interested in those who plan to attend Acknowledgements Finally, we mention that the current banquets. The GenericDB program used program does not parse data and, for with the ASP files allows users to delete The authors would like to thank the math- example, replace lower case, first letters and/or edit registration information. Us- ematics departments at the University of of names with the upper case letters. ers are permitted to edit only two fields Akron and Youngstown State University Some people input all information using on the demonstration page. The third link for their support in the development and only lower case letters. Since the data are redirects to gmail.com, where users may deployment of the software packages. used to generate name tags and receipts, view sample confirmation emails that are Story and Price owe a special debt of the registrants’ names and affiliations automatically sent to the successful reg- gratitude to Jianping Zhu for his assis- should begin with upper case letters so a istrants. The last link on the RIP permits tance in making the vision of an online replacement strategy of this nature would one to download a zip file containing the registration process a reality. be valuable. latest code. G. Jay Kerns and Jonathan Duran are Online information For years the freeware package Gener- at Youngstown State University, while icDB was used with the ASP program Thomas E. Price and D. P. Story are at Several instructive links dealing with to allow others to view and potentially the University of Akron. To contact the both ASP.NET and ASP code can be ac- edit the database. For example, Ohio authors, email Price at teprice@uakron. cessed by navigating to the Registration section members were allowed to view edu. 7 MAA FOCUS March 2008

Teaching Time Savers: Using Preview Problems to Get Ahead of the Syllabus By Marion Deutsche Cohen As extra-credit at the end of most many. This kind of extra-credit can entail next on the agenda, and since the students tests, I include a “preview problem,” relaxed “quiet time” with the new mate- have already experienced it, they’re more meaning a problem about the next topic rial, since they’ve already finished the comfortable with it — even if they didn’t to be covered. For example, for the lat- test. (The lecture format doesn’t provide make any progress with it on the test, and est Elementary Statistics quiz, which quite the same atmosphere.) Students even if they got it wrong (although I usu- covered hypothesis testing about two know that it’s extra credit so they have ally give a small amount of extra-credit means, I included a problem about two nothing to lose. Also, they get the chance just for trying). proportions. (I also included the TI83+ to try it out themselves, without being keystrokes). taught or guided, and many find out that During this class I soon see that many they’re up to, or almost up to, the task. students have completely learned the This practice has worked wonders for topic via the “preview extra credit,” and me, but it’s probably not really a time- I’m always on the lookout for ways in are ready for the topic after that. (When saver in the same sense that most of the which testing can be used as an aid in that happens, though, I make sure to go articles for this column provide. Rather the learning process; at any rate, it’s a over — or have them go over — one other than saving preparation or grading time shame when testing is detrimental to the example of the topic.) for the teacher, it saves time with respect learning process. The extra-credit pre- to keeping up with, or getting ahead of, view problems seem to me a good way Moreover, I find that often, when some the syllabus. to make testing become learning, to “kill of the students in the class know a topic two birds with one stone.” that moves things along for the rest of I give an in-class quiz every Friday, as the class (though how and why is subtle; well as a test at least every four weeks, so One thing to be careful about is that the a lot of it is probably psychological, the idea of teaching through the quizzes problem isn’t so difficult, or so far afield and student interaction must have a lot and tests can go a long way. In actuality, from the familiar, that they wind up doing to do with it). I do, however, need to be I usually do finish the syllabus at least it wrong or getting too bogged down on wary that some students don’t get lost in a week ahead of time, and am often (to it, thus practicing and cementing their the drift, or feel pressured to pretend to my knowledge) farther along than many misconceptions (along, perhaps, with understand that topic. In my experience, of my colleagues teaching other sections negative feelings). To this end, I make this rarely happens, and can be easily of the same course. (This semester, this these problems not-hard — meaning, dealt with. happened despite the fact that I had to problems that the students are ready for. miss two classes, and also lectured on I’m very careful to word the problems A variation: Sometimes I do “preview clarifying sub-topics not included in the as strategically as I can, and I realize problems” as in-class “seatwork,” rather syllabus.) that such problems aren’t advisable for than on tests. That is, I give the students all topics. a problem to do at their seats (not to be Some might feel that, as far as the collected) before showing them how, teacher’s time is concerned, this idea is Also, I make sure that the test itself (with- or before actually teaching the topic. In more of a time sink than a “swim.” After out the extra-credit) is short enough so general, students seem to be more likely all, one does have to make up and grade that there’s time left over to do the extra- to do seatwork than they are to do oral the problem. However (and this might credit. (Also, for classes containing more recitation, and sometimes they’re more be because I’m a poet and writer), I find than their usual share of math-anxious likely to do seatwork that they are to do that such “extra credit preview problems” students, I make the extra-credit problem extra-credit problems on a test! It all de- simply occur to me; I don’t need to put a separate handout, so that students who pends, sometimes on the personalities of time into making them up. And grading don’t want to think about them don’t the students in the class, or on how long doesn’t seem to me to be a “time sink,” have to. (Some students have, admittedly, the class period is, or on the particular since it’s only one problem (sometimes remarked, “They make me nervous.” topic involved. In-class seatwork might two), and also since those students who Many of these same students, though, not offer the same relaxed “quiet time” choose to do the problem usually get it have eventually come around.) aspect as test-taking; on the other hand, right. (As we know, it’s a lot easier to working together offers other advantages. grade when the student gets it right.) That this “preview extra credit” con- There are pros and cons to each; it’s not stitutes a “syllabus-moving-along-er” hard to gauge, either is a change of pace Most students in my classes do, or par- becomes evident when we get to the class from the other, and from the straight lec- tially do, or try to do, the extra credit after the test. The “new” material (upon ture style — and the two practices save problems, so what I’m saying applies to which the extra-credit was based) is now time and energy in different ways.

 March 2008 MAA FOCUS

This “preview extra-credit” is not only an extra-credit challenge problem, nor credit preview problems, plus five to ten a time-saver (in whatever sense…); it’s is it the same thing as learning by doing. minutes grading them. also an energy-saver, and a stress-saver. Besides, when I think of the term “time- Students simply don’t have to work as saver,” I’m also thinking of saving the Time Saved: 15 minutes to an entire hard at it, and I myself feel more re- students’ time. Moreover, my extra-credit class – or more than one class. It’s not laxed, knowing that there’s already been previews often convey to students the always possible to measure. progress made in this new topic. Also, (true) information that a particular topic I believe that it’s possible for a topic to is not “as big of a deal” (meaning as diffi- Aggravation Saved: in my experience, be “over-taught” — that is, it sometimes cult) as they might think. A teacher could, a lot! happens that all a student might need, in of course, do both — ask the students to order to learn something, is a problem read ahead, with the incentive that there Marion Cohen is the author of Crossing or two clearly worked out, rather than will be an extra-credit problem on the the Equal Sign (Plain View Press, TX), a explanation or even motivation (or with new material. (In fact, that might be an collection of poetry about the experience minimal explanation, perhaps simply idea for me to consider for the future in of mathematics. She teaches part time at a formula). The extra-credit preview- select situations.) In general, there are Arcadia University. problems often prevent “over-teaching,” many ways to do things, and the ways while simultaneously allowing for as can be combined. much “over-teaching” as necessary. Teaching Time Savers are articles Some students choose not to touch extra- designed to share easy-to-implement Some might suggest that I simply have credit problems (although they probably activities for streamlining the day-to- the students read something in advance do look at them, which can be beneficial). day tasks of faculty members every- before every class (or before some At the other extreme, some students find where. If you would like to share your classes). My first-approximation answer them fun. (Students have asked me, “How favorite time savers with the readers would be: that’s just not my style. If come there weren’t any extra-credit prob- of FOCUS, then send a separate email pressed further, I’d say that the textbook’s lems on that test? I was looking forward description of each activity to Michael handling of the new topic is often not to to them!”) In between are those who are Orrison at [email protected]. Make my liking, or to the students’ liking. (Of motivated by the “extra credit” aspect of sure to include a comment on “time course, that depends.) Also, the spirit is the extra credit. All in all, students who spent” and “time saved” for each ac- different; my “extra-credit previews” choose to can feel very positive when tivity, and to include pictures and/or seem less invasive, and they involve the they see what they’re capable of doing figures if at all possible. element of choice, which sits well with on their own, at the same time knowing me and probably with students. that they have the option not to.

Certainly reading portions of the book Time Spent: Zero to five to fifteen min- beforehand isn’t the same thing as doing utes (depending) making up the extra-

Funding Available for Students Going to MathFest

By Robert W. Vallin Maybe it sounds too good to be true, “Students who wish to present at the available at http://www.maa.org/students/ but this is real. The MAA has a grant MAA Student Paper Sessions at MathFest undergrad/.” from the National Science Foundation to 2008 in Madison, Wisconsin, must be fund undergraduate travel to MathFest, sponsored by a faculty advisor familiar Graduate Students Get in on held this year in Madison, Wisconsin with the work to be presented. Some This Too from July 30th to August 2nd. MathFest funding to cover costs (up to $600) for features talks by undergraduate students, student presenters is available. At most Graduate students also have opportuni- graduate students, and faculty, along one student from each institution or REU ties. In addition to the annual Graduate with special student activities sessions, can receive full funding; additional such Student Poster Session there will be a Math Jeopardy, and the MAA Lecture students may be funded at a lower rate. Graduate Student Paper Session. Speak- for Students (the topic will be “Sudoku: All presenters are expected to take full ers in the paper session will be able to Questions, Variations, and Research”). part in the meeting and attend indicated practice their expository job search talk These travel grants are for students who activities sponsored for students on all and then get individual help improv- are giving talks at the meeting and could three days of the conference. Nomination ing the presentation. Participants in use some assistance. The official call forms and more detailed information for both events are also eligible for travel reads as follows: the MAA Student Paper Sessions are funds and can apply when registering to speak. 9 MAA FOCUS March 2008 Project NExT Reaches 1000 Fellows

Project NExT (New Experiences in Teaching) is the MAA’s professional development program for new or recent Ph.D.s in the mathematical sciences. It addresses all aspects of an academic ca- reer: improving the teaching and learning of mathematics, maintaining research and scholarship, and participating in service and professional activities. In 2007–08, Project NExT passed a significant mile- stone when it welcomed its 1000th new faculty member into the profession. The 1002 Project NExT Fellows who have participated in the program in the last fourteen years include:

Seven Fellows who received the MAA’s Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member.

One Fellow who received the MAA’s Haimo Award for Distinguished College or University Teaching of Mathematics.

Ten Fellows who won prizes for out- standing papers in MAA journals. T. Christine Stevens, director of Project NExT, stands next to a poster recording One Fellow who won the MAA’s Selden the location of the 1000+ NExT Fellows. Prize for Research in Undergraduate Mathematics Education.

Five Fellows who were elected to the MAA Board of Governors.

One Fellow who gave the James R.C. Leitzel Lecture at MathFest.

Faculty for whom the 2008–2009 aca- demic year will be the first or second year of full-time teaching at the college/ university level are invited to apply to become Project NExT Fellows. Further information about Project NExT, as well as application instructions and forms, can be found on the Project NExT web site http://archives.math.utk.edu/projnext/. In 2008, applicants may choose whichever application format they prefer: hard copy or online. Applications must be received on or before April 18, 2008. Project NExTers staff the booth in San Diego.

10 March 2008 MAA FOCUS Developing Mathematical Habits of Mind

By Annie Selden and Kien Lim A Project NeXT panel on “Helping highlighting the deep structural similarity ¾ requires an act of interpretation that Students Develop Mathematical Habits between Z and Q[x]. Cuoco has a joint could include many possible ways of of Mind without Compromising Key paper with Goldenberg and Mark on this understanding: three objects out of four Concepts from the Syllabus” was held topic. [Habits of Mind: An Organizing objects, the sum ¼+¼ +¼, the measure at the San Diego Joint Mathematics Principle for a Mathematics Curriculum, of the quantity resulting from dividing 3 Meetings. It was organized by Kristin Journal of Mathematical Behavior, 15(4), units into 4 equal parts, the solution of Camenga of Houghton College and Kien 375-401.] the equation 4x = 3, or the equivalence Lim of the University of Texas at El Paso. class {3n/4n | n ≠ 0}. In contrast, a way The panel addressed the tension between Hyman Bass, University of Michigan, of thinking is a characteristic of a mental teaching all the mathematical concepts continued by considering things math- act. For example one’s interpretation listed in a syllabus and incorporating ematicians do, which have variously been of symbols might be characteristically opportunities for students to develop called “mathematical habits of mind,” inflexible or flexible, non-referential or mathematical habits of mind. “ways of thinking,” including disposi- referential. tions and sensibilities, and “practices.” Al Cuoco, Education Development Cen- These practices include: (1) asking In answer to his initial question, Harel ter, opened with some ideas for helping “natural” questions, (2) seeking patterns proposed that mathematics consist of two students develop mathematical habits of or structure, (3) consulting the literature complementary subsets. The first subset mind. These included: (1) being explicit and experts, (4) making connections, (5) is a collection, or a structure of structures, about one’s own thinking, (2) working using mathematical language with care consisting of particular axioms, defini- on problems with students, (3) making and precision, (6) seeking and analyzing tions, theorems, proofs, problems, and thought experiments an integral part of proofs, (7) generalizing, and (8) exercis- solutions — the institutionalized ways of one’s courses, (4) providing concrete ing aesthetic sensibility and taste. understanding in mathematics through- experiences before introducing formal- out history. The second subset consists of ity, and (5) looking for habits of mind Bass claimed that instructors can culti- all ways of thinking that characterize the characteristic of various branches of vate these practices while treating the ba- mental acts whose products comprise the mathematics. He presented various ana- sic curriculum responsibly and that such first set. Harel has a more detailed paper, lytic and algebraic habits, with one of the instruction can, and must, start very early. “What is Mathematics?” It can be down- latter being the habit of seeking structural The knowledge and skills demanded of loaded from his website: http://math.ucsd. similarities. a teacher are considerable, but they can edu/~harel/Research/1Publications.htm. It be learned, with proper support. Bass will also appear in Current Issues in the For example, the problem of finding a concluded with a video of a third-grade Philosophy of Mathematics from the Per- polynomial f that goes through a list of class that was discovering, exploring, and spective of Mathematicians, edited by R. points speculating about even and odd numbers, B. Gold and R. Simons, to be published illustrating how a teacher might encour- soon by the MAA. age such habits. (a1,b1), (a2, b2), … (an,bn) Finally, Annie Selden focused on two spe- is, by the Remainder Theorem, the same Next Guershon Harel, of the University cific habits of mind and how one might as finding a polynomialf that satisfies of California at San Diego, asked the encourage them without compromising question: “What is mathematics?” He the syllabus. The first was persistence, emphasized that mathematics teaching a habit that is widely applicable, even f(x) = (x–a1)q1(x) + b1 should not appeal to gimmicks, entertain- outside of mathematics. The second was f(x) = (x–a2)q2(x) + b2 … ment, rewards, or punishment, but rather writing “Let x be a (fixed, but arbitrary) focus on the learner’s intellectual need by number” into the proof of a universally f(x) = (x–an)qn(x)+bn fully utilizing humans’ remarkable capac- quantified statement, a habit that is nar- This, in turn, is the same as looking for ity to be puzzled. Nor should mathemat- rowly applicable. a solution to the simultaneous set of ics curricula compromise mathematical congruences integrity, which is determined by ways How might persistence be cultivated in of understanding and ways of thinking one’s class? Selden gave an example that have evolved over many centuries from her sophomore-level transition-to- f(x) ≡ b1(mod (x–a1)) of mathematical practice. proof courses. At every assessment, she f(x) ≡ b2(mod (x–a2)) …, gives both a take-home and an in-class For Harel, a way of understanding is a exam. On the final take-home exam, she f(x) ≡ bn(mod (x–an)) particular product of a mental act car- asks students to tell her what they have This “sameness” can be made precise, ried out by an individual. For example, gotten out of the course, but not to give

11 MAA FOCUS March 2008 back a list of topics. One response she something that students are reluctant to students to adopt this habit of mind. In- particularly liked was, “I’ve learned that do, preferring instead to consider all x in stilling good habits of mind takes time. I can wake up in the middle of the night the proof. Dr. K requires students to hand She proposed that in some cases students thinking about a math problem.” She in only three proofs per week and me- must “just do it” and understanding often conjectures that this student’s persistence ticulously grades just one. Students are follows somewhat later. resulted from having multiple opportuni- allowed one additional week to rewrite ties and motivation to work on problems and resubmit the proof for more points. The slides from all four presentations are for a substantial period of time such as The rewritten proof is invariably better, available at http://www2.edc.org/cme/ take-home tests doable over one week. often incorporating Dr. K’s suggestions, showcase.html. such as ending the proof with, “Since x For the second habit of mind, she present- was arbitrary, we have now shown the Annie Selden is Professor Emerita at ed a vignette of how Dr. K helps students theorem to be true for all x.” Selden again Tennessee Technological University develop the habit of taking a fixed, but conjectured that multiple opportunities and Adjunct Professor at New Mexico arbitrary x in a proof where the definition and motivation to consider a fixed, but State University. Kien Lim is a Project reads for all ε > 0 , there is a δ > 0 arbitrary object, as well as writing a NeXT Fellow and Assistant Professor such that for all real numbers x…This is rationale, were instrumental in getting of Mathematics at University of Texas at El Paso.

Robert P. Balles Awards for IMO Team Participants

By Steven Dunbar In 2005, Robert P. Balles, a Brian Lawrence, a gradu- mathematics instructor and ate of Montgomery Blair businessman from Califor- High School, Silver Spring, nia, created an award for MD, received a Silver members of the USA team medal. Lawrence is now at the International Math- majoring in mathematics ematical Olympiad. Balles at CalTech. admires the brilliance and dedication of these out- Alex Zhai, who attends standing “mathletes,” so he University Laboratory made a fund at the MAA High School, Champaign, to honor and reward their IL, won a Gold medal. achievement in mathemat- ics with a $1000 “Einstein” Tedrick Leung, a gradu- Series I savings bond. ate of North Hollywood High School, Hollywood, In the accompanying photo, The 2007 team was: CA, won a Bronze medal. you can see the 2007 IMO team from the Leung is now majoring in mathematics USA team on the stage at the Awards Eric Larson who attends South Eugene at MIT. ceremony in Hanoi, Vietnam on July 30, High School in Eugene, OR, received a 2007. They are holding the certificate Silver medal. In July 2008, the MAA will sponsor announcing that they are winners of the the USA team to the 49th Annual In- 2007 Robert P. Balles IMO Award. Arnav Tripathy, a graduate of East ternational Mathematical Olympiad in Chapel Hill High School, Chapel Hill, Madrid, Spain. The USA team members In addition, each received a $1000 I-bond NC, won a Siliver medal. Tripathy is now will be named in June 2008, following bearing the picture of Einstein. Also vis- majoring in mathematics at Harvard. the results of the USA Mathematical ible in the picture, each student is wearing Olympiad in May 2008 and preliminary the ribbon holding the medal won at the Sherry Gong, a graduate of Philips Ex- training and testing at the Mathematical IMO for their superior solutions of the eter Academy, Exeter, NH, is from Puerto Olympiad Summer Program. Once again six challenging problems. Rico. Gong received a Gold medal at the this year, Robert Balles has donated so IMO. She now attends Harvard, majoring that the MAA can recognize the team in mathematics. with an award.

12 March 2008 MAA FOCUS Undergraduates Win Awards at San Diego AMS Prizes at the By Joe Gallian Joint Meetings: A Correction Alice T. Schafer Prize AMS/MAA/SIAM Morgan Prize

Galyna Dobrovolska Alison Miller Nathan Kaplan Avi Wigderson

At the Joint Meetings in San Diego, that happened to our field since Novem- In last issue’s listing of prizes and Galyna Dobrovolska of MIT and Alison ber 2003.” awards given at the Joint Meetings we Miller of Harvard were co-winners of the inadvertently omitted the second Levi L. Alice T. Schafer Prize given by the As- Both Dobrovolska and Miller were Conant Prize. The Conant Prize, which sociation for Women in Mathematics for gold medal winners in the International recognizes the best expository paper Excellence in Mathematics. Dobrovolska Mathematical Olympiad and plan to enter published in either the Notices or the was cited for her exceptional perfor- graduate school in mathematics in the Bulletin of the AMS, was split between mance in undergraduate and graduate fall of 2008. Brian Conrey’s article we listed, and the courses and her research in algebra. She following: is a coauthor of a paper published in the The AMS/MAA/SIAM Morgan Prize for Journal of Algebra that solves an impor- Outstanding Research in Mathematics by Shlomo Hoory, Nathan Linial, and Avi tant conjecture. She has solved another a an Undergraduate Student was awarded Wigderson “Expander Graphs and their notable conjecture in a work in progress. to Nathan Kaplan for four professional- Applications” Bulletin of the AMS, 43 Alison was cited for her research in num- level papers in algebraic number theory (2006), 439–561. ber theory and combinatorics, and for her done at REUs at Trinity University, performance in Putnam Competitions. Williams College and Duluth. Kaplan’s Only Wigderson was present at the meet- papers, two of which are coauthored, will ing. We apologize for the omission. As a participant in the REU program at appear in the Journal of Number Theory, the University of Wisconsin in 2006, the Journal of Algebra and its Applica- Miller coauthored two papers on infinite tions, and Acta Arithmetica. A referee for product expansions of modular forms. the Journal of Number Theory wrote that Found Math In a paper that has been published in his paper “contains… definitive results the Proceedings of the American Math- substantially advancing our understand- ematical Society, Miller and her coauthor ing of cyclotomic polynomials of order Elementary Math Grows answered a deep and difficult question three.” Exponentially Tougher: originating in the Fields Medal work of Borcherds. Kaplan graduated from Princeton in Students, Teachers Tackle 2007 and is currently doing Part III of Algebra Miller’s work in combinatorics was done the Mathematical Tripos at Cambridge at the University of Minnesota Duluth University. He will enter the graduate REU between 2005 and 2007. One of program in mathematics at Harvard in — headline, The Washington her letters of support called her results the fall of 2008. Post, December 26, 2007 on permutation patterns “the best thing

13 MAA FOCUS March 2008

Mathematicians Recruited for Climate Change Research

By Pat Kenschaft “The influence of climate change on nine PDEs and two accompanying alge- GDP by reducing greenhouse gas emis- mathematical research in the 21st century braic equations that roughly describe the sions. Such equilibria (some of which could be comparable to physics’ a cen- “convective overturning,” which is the have been identified by the analysis) tury ago,” claimed Gerald North of the process whereby surface water warmed could be templates for self-enforcing Department of Atmospheric Sciences at in the tropics flows poleward, where it is treaties. Texas A&M in his introductory talk on cooled and consequently becomes more the “Status of Climate Change Research” dense because of thermodynamics, so it Swaters’s work incorporates a wide at the Joint Mathematics Meetings in sinks to the deep ocean and flows back to- range of applied mathematical themes San Diego. He was the first of dozens ward the equator. The Gulf Stream is the including asymptotic reduction, physi- of speakers who appealed to mathemati- best known example of such a poleward cal modeling, variational principles for cians to use their professional abilities to flowing current. Swaters concentrated his Hamiltonian PDEs, and hydrodynamic analyze climate change. efforts on the deep equatorward flowing stability theory for non-parallel shear currents. flows. He believes there are opportunities Inez Fung, award-winning professor of to extend classical Hamiltonian dynam- atmospheric science at the University of He described his governing equations as ics in finite dimensions to the infinite California at Berkeley, gave the SIAM “hopelessly simple” and added, “I’m not dimensional (Hilbert) function (phase) invited plenary address. She said that truly modeling climate change, but these spaces where the solutions to the PDEs mathematicians are needed to formulate equations do get a large chunk of the large live. He believes that there are many better ways of modeling both planet-wide scale physics for these flows correct.” opportunities for both pure and applied and local interactions as well as to devise Swaters said that being an applied math- mathematicians to become involved in better approaches for understanding ematician working in climate change understanding climate change. uncertainty and risk. We also need faster was “between a rock and a hard place. computers (another mathematical oppor- The numerical ocean modeler is working Mathematicians interested in learning tunity) and more and better observations. with large sets of complicated PDEs that more can see videos of the talks at the Unfortunately, as the “grid” of observa- appear all but hopelessly intractable to first joint workshop mathematicians and tions becomes more dense, the current classical mathematical analysis. On the climate scientists, to be held at the Math- mathematical models diverge more in other hand, mathematical reductions that ematical Sciences Research Institute in their local predictions, in contrast to what result in tractable equations often seem Berkeley in April 2007. There will also be we would hope for. We urgently need bet- to be completely missing the point to the a summer graduate workshop on climate ter ways to predict flooding, drought, and computational oceanographer.” change at MSRI July 14–25. Visit the intense rains, all of which are increasing MSRI web site at http://www.msri.org/ for and are statistically predicted to acceler- Similar problems arise in related fields. more information on both events. ate. We need to predict where and when Some models that try to predict the im- individual extreme weather events will pact of climate change on the economy occur so people can plan for them. include from 300 to 20,000 parameters, Found Math none of them precise. Maximilan Auff- Understanding climate dynamics requires hammer, an environmental economist at Greg’s textbooks had queer titles understanding the movements within and the University of California at Berkeley, among the atmosphere, the oceans, the pointed out a cycle: economics affects like First Course in Modular solid earth (especially volcanoes), and public and private decisions, which affect Forms, Calculus of Variation and the biosphere (which consists of plants, the climate, which affects economics. Homogenization, Methods for animals, and other living organisms). There are many opportunities for math- Structural Optimization, Applied Individual scientists and mathematicians ematical research here. analyze various parts of Earth’s systems, Complex Variables, and Asymp- and some are considering the whole. Roy Radner, who has spent most of his totic Methods. I remember the Coupled atmosphere-ocean climate career as an economist after earning titles because I remember leafing models numerically solve the governing a doctorate in mathematical statistics, through the books, page after page equations of fluid motion that model this presented a model of a dynamic “climate- process as realistically as possible. change game” in which the countries are filled with an alien typography. the players, and there are infinitely many Gordon Swaters is an applied mathema- Nash equilibria, one of which is “business — from the short story “Land- tician and theoretical physical oceanog- as usual.” The goal of the analysis is the slide,” by David McGlynn rapher who is trying to understand the characterization of equilibria that yield dynamics of ocean currents. He presented each country a higher present value of 14 March 2008 MAA FOCUS Mathematics in Voting Theory at the 2008 Joint Meetings

By Eric Gottlieb, Brian Hopkins, and Michael A. Jones Voting has been getting of voting theory attractive to a lot of attention lately. In MathSciNet publications with “voting” in the title mathematicians, but many are addition to the presidential 90 surprised by and drawn to the primaries now underway, 80 breadth of mathematical areas it is the subject of Math that can be applied to questions Awareness Month this 70 in voting theory. year. Besides being topi- 60 cal, voting theory has been Many election procedures attracting more interest 50 require voters to rank order among mathematicians, as all candidates from most- to evidenced by the growing 40 least-preferred and return a

Publications number of articles contain- 30 total ordering or permutation ing the word “voting” in of the candidates as the elec- the title, as measured by a 20 tion result. Therefore permuta- search of MathSciNet (see 10 tion groups are often useful in the graph on this page). analyzing election procedures. The 2008 Joint Meetings 0 Karl-Dieter Crisman (Gordon featured a number of events 1993-1995 1996-1998 1999-2001 2002-2004 2005-2007 College) focused on pairwise related to voting theory. Year changes in preferences and Our goal in this article is to corresponding hyperplane in the U.S. House of Representatives so give the reader some sense of issues in arrangements. Symmetry is one of the that each person’s vote will have equal voting theory, and the surprising breadth axioms considered in the work of both weight. Edelman gave a historical per- of mathematics being used to study it, by Vincent Merlin (University of Caen) and spective which included the history of reflecting on two invited addresses and a Alan Taylor (Union College). Michael apportionment methods in the United special session. Orrison (Harvey Mudd College) and his States, including those developed by students have gone farther, moving into Thomas Jefferson, Daniel Webster, Al- Donald Saari (University of California at the group ring of the symmetry group, exander Hamilton, and John Q. Adams. Irvine) gave an AMS Invited Address on which adds a second operation and brings Edelman explained how each of the the mathematics of the social and behav- the power of representation theory to bear above methods, as well as two others, ioral sciences that prominently featured on certain problems in voting theory. including the Hill-Huntington method voting theory. Saari motivated his talk that is currently used to apportion the with an apparently simple example that Geometry is often used to analyze elec- House, solves an optimization problem demonstrates how many different elec- tion procedures, especially when there that tries to mathematically satisfy the tion outcomes may be possible even with are three candidates. In these situations, goal of “one man, one vote.” Edelman few voters (see the sidebar on page 17). the election outcome can be visualized also described some of the paradoxical In his address, Saari described how alge- as a point in an equilateral triangle or behavior of apportionment methods and braic varieties and symmetry are used to 2-simplex, described by barycentric focused on how the methods may be bi- analyze election procedures and to make coordinates. Bill Zwicker (Union Col- ased toward smaller or larger states (see sense of such paradoxical behavior. Saari lege) prefers hexagons to triangles, since the sidebar on page 16). championed the Borda Count, an election the hexagon’s vertices represent all six procedure published by Jean Charles possible preference orderings of three These invited addresses were not the only Borda in 1781 (although recent historical candidates. By bending the hexagons in voting theory at the Joint Meetings. The work suggests the system was developed 3-space, he changes notions of closeness. authors of this article organized a well- in the 13th century by Ramon Llull). In Higher dimensional permutahedra are attended special session on voting theory fact, Saari argued that the Borda Count used to model situations with more than whose speakers included economists, is the only procedure to respect some three candidates. His talk also included political scientists, and mathematicians compelling symmetry requirements. multivariable generalizations of the from Austria, Canada, France, Spain, and median used to devise voting systems the United States, representing liberal Paul Edelman (Vanderbilt University) that are less manipulative and more de- arts colleges, comprehensive universi- gave an MAA Invited Address on ap- cisive. In the work of Jason Kronewetter ties, and research universities. Not only portionment. One example is determin- (University of California at Irvine), the is the inherently interdisciplinary nature ing the number of seats a state receives geometric setting moves to 2-manifolds

15 MAA FOCUS March 2008

Hamilton Apportionment and the Alabama Paradox How many Representatives does each U. S. state get in Congress? There is a rich history of how this question has been answered as new states were brought into the country and census data recorded change in populations. In this example, we explain one method and one of its problems.

Alexander Hamilton, the first Secretary of the Treasury, devised the apportionment system described below. Congress voted to use it for the 1790 apportionment, but George Washington blocked it with the first presidential veto. Hamilton’s method was adopted in 1852 to apportion representatives in light of the 1850 census, and was the law of the land for some fifty years.

We will explain Hamilton’s by examining a few states and their 1880 population data:

Alabama Illinois New York Texas 1880 pop. 1,262,505 3,077,871 5,082,871 1,591,749 % US pop. 2.5571 6.2341 10.2952 3.2240

The House of Representatives has long been fixed at 435 members, but through the 19th century, the size increased with each reapportionment. The 1870 apportionment had 292 representatives. If the number were increased to 299 for the 1880 apportion- ment, how many representatives would each of these four states have? Multiplying 299 by each state’s percentage of the total population gives a non-integer quota. Hamilton’s method is to initially drop the decimal part, i.e., use the integer floor function. Adding up these numbers for all 38 states gives 277 representatives. What do you do about the other 22? Give one to each of the 22 states with the highest decimal part in their quota (and any state assigned 0 representatives). For 299 representatives, each state with quota decimal part .646 or greater would receive an additional delegate, as follows:

Alabama Illinois New York Texas 299 quota 7.646 18.640 30.783 9.640 299 floor 7 18 30 9 299 adjusted 8 18 31 9

300 quota 7.671 18.702 30.886 9.672 300 floor 7 18 30 9 300 adjusted 7 19 31 10

This list also shows what would happen if the number of representatives were increased to 300. The quotas all increase slightly, relative to the percentage of total population. The sum of the integer floors of all the quotas is 280, so 20 states would receive an extra representative. For 300 delegates, that works out to be the states with a quota decimal part of .672 or greater. Illinois and Texas would each receive an additional representative in the move from 299 to 300 members, and Alabama would lose one. (New York and the other 34 states would see no change.) An “Alabama paradox” is a situation where increasing the numbers of seats results in a state having fewer seats.

Charles Seaton, chief clerk of the U.S. Census Office in the early 1880s, recognized this paradox and notified Congress; the reapportionment of 1883 increased membership to 325 representatives, avoiding the problem. Two other paradoxes were dis- covered for Hamilton’s apportionment method, both from using the technique with real data. The “population paradox,” where a state increases population but loses seats, was a threat to Maine in the 1900 apportionment, and the “new state paradox,” where adding seats for a new state causes other states to lose seats, arose when Oklahoma became a state in 1907. By the 1910 apportionment, a different method was in effect. such as a torus, in which geodesics come systems that combine approval (in which to analyze how committees form under to play. voters separate the candidates into those approval voting. Both applied measures who are acceptable and those who are of distance on discrete structures (such as Not all election procedures have voters’ not) and preference orders for candidates. hypercubes) and viewed an outcome as preferences as inputs and a linear order Marc Kilgour (Wilfrid Laurier Universi- the solution of an optimization problem. of candidates as an output. Steve Brams ty) used notions of error-correcting codes Francis Su’s (Harvey Mudd College) (New York University) examined voting such as weights and Hamming distance study of agreeable societies also related

16 March 2008 MAA FOCUS

And the winner is… well, it depends Suppose your department of eleven Suppose instead that your top two choices saw above that this leads to the overall faculty is choosing a new hire among count; equivalently, you vote against your ranking ACB. Suppose that, in the three candidates, Anneliese, Binh, and last choice. Notice that no one ranked midst of the process, Binh calls to say Carlos. Each of you is asked to rank the Anneliese second, so she still has 5 votes that he has accepted another offer and candidates in preference order. Three in this system. Binh has a total of 9 first should be removed from consideration of you prefer Anneliese best, followed or second place rankings, Carlos 8. So for the job at your institution. He was by Binh, with Carlos your least favored this system gives a composite ranking last in your composite list, anyway. You choice. We’ll abbreviate this choice as the reverse of before, BCA, and the offer still call Anneliese, right? ABC, and do the same for the other goes to Binh. orderings. Consider what has happened to the In a compromise between the two sys- votes when Binh is removed. Keeping Here are all eleven votes: tems discussed so far, suppose that your the same relative rankings for An- top choice gets 2 points and your second neliese and Carlos leaves 3 ABC 2 ACB choice gets 1 point (this is an example 2 BCA 4 CBA of what is known as the Borda count). 5 AC 6 CA Anneliese, with 5 first place rankings What is the department’s composite and 6 third place rankings, gets 10 points. so that Carlos now has the most first ranking? Which candidate gets the job Binh, with 2 first place, 7 second place, place rankings. Counter-intuitively, offer? It depends on how you tabulate and 2 third place rankings, gets 11 points. Binh dropping out moves Carlos the votes. Verify for yourself that Carlos gets 12 above Anneliese in this voting scheme. points. This system gives yet another Would you have thought to retabulate Suppose that only your top choice composite ranking, CBA, suggesting the votes? counts, known as the plurality rule. that Carlos is the faculty’s choice for the Anneliese was ranked first by 5 of you, new hire. This demonstrates a formal result which Binh by 2, and Carlos by 4. This gives an says that over half of reasonably close overall ranking of ACB, and Anneliese There can be problems even if you agree three-person races have outcomes that gets the call. on a tabulation method in advance, say change with the tabulation procedure. only using the voters’ top choices. We

to discrete mathematics. In particular, Su voting theory to address questions in Eric Gottlieb is Associate Professor and applies Helly’s theorem from convex- judgment aggregation, in which indi- Chair of the Department of Mathematics ity theory and ideas from graph theory, viduals’ judgments on interconnected and Computer Science at Rhodes Col- including cliques, chromatic number, propositions are aggregated to form a lege in Memphis, TN. He enjoys playing interval graphs and perfect graphs. collective judgment. Anna Bargagliotti poker, volunteering at the local farmer’s (University of Memphis) applied Saari’s market, doing projects with his twin sons, Another type of voting is when a col- techniques to determine data sets on and commuting to work without the use lection of voters must arrive at a “yes” which certain nonparametric statistics of gasoline. or “no” decision — often modeled as a disagree. Her work also utilized magic voting game. Josep Freixas (Technical squares when examining the structure of Brian Hopkins is an Associate Professor University of Catalonia) and Jennifer nonparametric data. in the Department of Mathematics at Wilson (Eugene Lang College of the Saint Peter’s College in Jersey City, NJ. New School) used linear algebra and Talks in our session featured connections He works hard at optimizing his cultural combinatorics to examine situations in to topology, nonparametric statistics, experience in New York City. which voters have various ranges of op- representation theory, convexity, discrete tions from “yes” to “no.” mathematics, linear algebra, and other Michael A. Jones is an Associate Profes- fields. A quick review of the articles in sor in the Department of Mathematical Finally, there were two talks that applied MathSciNet on voting shows that there Sciences at Montclair State University voting theory to answer questions in other are many other areas of mathematics in Montclair, NJ. He is the proud, but fields. Christian Klamler (University of have been applied to voting theory, and tired parent of two young children. His Graz) applied geometric approaches to we believe there are many more connec- daughter’s name had been slipped into tions to be made. one of the examples. Can you find it?

17 MAA FOCUS March 2008 Mathematical Experiences in Business, Industry and Government

By Phil Gustafson

Applications of mathematics to proj- language processing techniques to assign Mark Mills ([email protected]) of Cen- ects in business, industry and govern- a probability to each model sentence that tral College (Pella, Iowa) spoke about ment (BIG) offer a wealth of exciting the model sentence is paraphrased in the his experiences during his 2006-2007 problems for mathematicians. This article response. Based on these probabilities sabbatical. In an effort to gain more real summarizes presentations given at the and the scoring rubric, the scoring model world experience, Mills was able to get MAA Contributed Paper Session entitled assigns a score to the response. a part-time position working at Pella “Mathematical Experiences in Business, Corporation (aka Pella Windows). His Industry and Government,” which took While the scoring rubric specifies how projects included: (1) doing data analysis place at the Joint Mathematics Meetings the item would be scored on an integer on a number of large data sets from vari- in San Diego this past January. The paper scale (0–2, for example), the probabilities ous test sites and trying to draw conclu- session was sponsored by the Business, returned by c-rater can be used to calcu- sions from each individual data set, as Industry and Government Special Interest late a real-number score that incorporates well as trying to find “correlations” (i.e., Group of the MAA (BIG SIGMAA), and the uncertainty associated with the auto- patterns or connections) between the data was organized by Phil Gustafson of Mesa mated score and that can be thought of sets; (2) participating in the planning and State College and Michael Monticino of as a kind of expected value for the score. implementation of a long-term window the University of North Texas. New techniques have been developed to test within the Test Lab; and (3) work- determine the reliability of the c-rater ing with a sales director to help re-align Problems of sensor fusion led Pete De- scores by comparing the real-number sales regions. Long ([email protected]) of scores from c-rater with integer scores Math C2 to discover an elegant semialge- from a human reader. During the year, Mills was able to get bra that allows one to count the numbers both students and faculty on tours of the of set partitions of a union of sets where Greg Coxson ([email protected]) of company’s facilities. He plans to use his disjointness rules apply. Using the exam- Technology Service Corporation dis- experiences to make his courses more ple of five sets of 40 elements each, Pete cussed “Code Imbalance and DC Bias relevant. Mills found that companies are showed that the number of admissible Sensitivity in a Hard-Limited Radar eager to have help from someone with a partitions of the union exhibited a sharp Signal Processor.” He showed how in- mathematical or statistical background— peak at about 70 parts, and noted that tuition based on linear systems can be and they’re even willing to pay. (Pella if a sensor fusion system combined the misleading in considering a nonlinear Corporation paid the 25% of his salary output of five sensors providing 40 tracks system. He discussed a chain of three that he forfeited by taking a year-long each, and didn’t come up with seventy radar signal processor modules, begin- sabbatical.) However, he would encour- tracks in the result, there had better be ning with a rho-theta limiter (a form of age any faculty looking to do something a good reason why. He closed by noting hard limiter that maps complex inputs to similar to start early (at least a year) and that he had just purchased a copy of the the unit circle), followed by linear binary involve college/university payroll people Petkovsek-Wilf-Zeilberger book A = B, pulse compression and divide-by-N, and from the start (if the company will be and hoped to use its techniques to create then a squarer module. paying you). better formulas for use in working sensor fusion algorithms. The DC gain of the linear modules is pro- Peter M. Jarvis (p.jarvis.ee.math@gmail. portional to the difference in the number com) discussed how longer Hamming James Fife ([email protected]) of the Education of ones and minus ones, or imbalance, codes are better over BPSK channels. Testing Service spoke on using math- of the binary code employed in pulse Peter showed through simulation that, ematics to improve the automated scoring compression. Hence, if the nonlinear except beyond 9 dB, the first three Ham- of short constructed-response answers. modules are not accounted for, DC bias at ming codes, of lengths [7, 4], [15, 11], The ETS has developed c-raterTM, which the input means a DC bias in the output, and [31, 26] give increasingly better is an automated scoring engine for scor- implying higher false alarm rate. When performance, with the codes of lengths ing short (one- or two-sentence) answers the nonlinear modules are included, the [63, 57] and above either no better or to constructed-response test questions; relationship of imbalance and DC bias is worse, except at very high dB. Intui- the scoring is based on a scoring rubric reversed — there exists a pair of nonzero tively, as Hamming codes only correct provided by a content expert. code imbalance values for which, given single transmission errors, the shorter any DC bias of any size and additive codes “should” perform better. Intuitive A c-rater scoring model specifies con- noise at the input, the mean and variance counter arguments are that shorter codes cepts that should be present and model of the output is as if there were no DC perform worse since their lower rates sentences for each concept. To score a bias present. make a higher probability of transmit particular response, c-rater uses natural error, and even 1 or 2 message bit er-

18 March 2008 MAA FOCUS rors out of a large number of bits (e.g. functions of layered media in complex found analytically with the use of the 26) yields better performance than 1 acoustic propagation environments. Pontryagin Maximum Principle. A simu- out of 4. Theoretically derived predic- Continued fraction coefficients suffi- lation software package was developed tions match the simulations. The author cient for near double precision accuracy to demonstrate the performance of their feels this shows that plausible, heuristic Airy functions of a complex variable are proposed optimal algorithms. arguments in engineering can be wrong, provided, as well as recommendations and that precise analysis is necessary for for a minor extension of the presented In this article we have seen many ap- true insight. methods to achieve full double precision plications of mathematics to projects in accuracy. business, industry and government. In a Problems and solutions in efficient, accu- variety of settings, mathematics is a key rate computation of the Airy functions in Ellina Grigorieva ([email protected]. component to many important projects in the complex plane was the topic that Jim edu) of Texas Woman’s University spoke the world around us. Who uses math? The Beard ([email protected]) discussed. on the theory of hierarchical differential answer includes many of the mathemati- Airy functions are critical to a problem games and its application to the interac- cians, scientists and engineers whose encountered in sonar propagation in the tions between a manufacturer, retailer, projects and products help improve the ocean, an application important in BIG. and bank. She and Evgenii Khailov of quality of our everyday lives. Moscow State University sought to Airy functions are proportional to Bessel determine the conditions of interaction Acknowledgement: The author grate- functions of order 1/3. Computation to that might provide stable and maximal fully appreciates the input provided by high accuracy in the complex plane is effectiveness over a given planning the speakers for the content appearing in difficult, but a method from the analytic period. Their particular interest was on this article, and for their participation in theory of continued fractions, a field differential games with restriction on the paper session. pioneered by Stieltjes and Hamburger controls (bounded controls). They cre- with major milestones by H.S. Wall and ated and investigated a microeconomic Phil Gustafson is Professor of Mathemat- others, provides a method for solving the model consisting of three differential ics at Mesa State College in Grand Junc- problem with efficiencies sufficient for equations with five controls. The best tion, CO, and is Vice Chair for Programs applications such as broadband transfer optimal strategy for each player was for BIG SIGMAA.

College Algebra and Precalculus: A Call for Papers

Articles are being sought for a planned much clearer picture of who the students (either in mathematics or in other quan- MAA Notes volume to be titled College actually are who take these courses and titative disciplines)? What percentages Algebra and Precalculus: Today’s Stu- how well they are served by the courses. go on to successor courses and how well dents, Tomorrow’s Courses. The volume We hope that this information will be do they perform? will be co-edited by Sheldon Gordon useful to departments that wish to change ([email protected]), Florence the focus of these courses to better meet How well prepared are these students Gordon ([email protected]), Barbara Ed- the needs of their students. for the mathematical needs of today’s wards ([email protected]), quantitative workplace? and Sharon Cutler Ross ([email protected]) We seek papers that report on both quan- and is being developed on behalf of the titative and qualitative studies examining For departments that offer different MAA committee on Curriculum Renewal various aspects of the student populations tracks/emphases for different student Across the First Two Years (CRAFTY). in these courses, including: populations, are there any differences in enrollment, retention, performance, or at- Most college algebra and precalculus How did they come to theses courses? titudes based on the nature of the college courses were originally developed to Did they take the prerequisite classes algebra or precalculus experience? prepare students for mainstream calculus in high school or in college? Are there and many are still offered in that spirit. any differences in performance based on If you are interested in being considered The few available studies suggest that where they took the prerequisites? Why as a contributor to this volume, please the reality may be very different in that are they taking these courses? What are send a brief description of your ideas only an extremely small number of these their majors? How well do they perform to one of the co-editors. Additional de- students ever start Calculus I. Our goal is in these courses? tails can be found at http://farmingdale. to collect a series of articles that will pro- edu/~gordonsp/CurrentProjects.htm. The vide the mathematics community with a What subsequent courses do they take deadline for this initial contact is April 30, 2008. 19 MAA FOCUS March 2008 San Diego Joint Mathematics Meetings

The MAA Board of Governors in action. Three historians from three countries: Ivor Grattan-Guinness, Ubiratan D’Ambrosio, and Alejandro Gar- MAA president Joe Gallian and AMS ciadiego. president James Glimm cut the tape to open the Exhibit Hall.

The first talk was still half an hour in the future, but the big auditorium al- ready had a few people waiting in it. Laptop heaven: outlets and a wi-fi hotspot! Perhaps we should ask Starbucks to be one of the sponsors…

A factor ot ten: Richard Guy, at 91, was probably the oldest participant; Kaavya Jayram, at 9, was certainly Dan Velleman, editor of The mathartfun.com ex- the youngest. the American Mathemati- hibit had posters, games, AWM Noether Lecturer cal Monthly. t-shirts, even ties. Audrey Terras.

Winners and honorable mentions for the AWM Scha- About half of the MAA’s Council on Publications: fer prize: Reagin McNeill, Naomi Brownstein, AWM Lowell Beineke, Steve Maurer, Elaine Pedreira, Don The “Pi Across President Cathy Kessel, Alison Miller, Galyna Dobro- Albers, Art Benjamin, Allen Schwenk, Carol Baxter, America” folks volska. and Ivars Peterson. had pi-swatters to give out. 20 March 2008 MAA FOCUS

The mathematical art exhibit.

The Key College Publishing booth. Penny Pina and the Acme Klein Bottle Man (Cliff Stoll).

Don Albers and Zaven Carl Pomerance spoke about “The Covering Congru- Karian plot new MAA Text- ences of Paul Erdös.” books in the shadow of the Birkhauser booth.

Tal Skloot and George Csicsery pose next to the poster for Julia Robinson and Hilbert’s Tenth Problem.

Mohammad K. Azarian gives a talk in the AMS Session on History of Mathematics. David Zitarelli

Good books and free chocolate at- tracted people to the MAA booth.

Karen Parshall If you got tired of walking around the exhibit hall…

Terence Tao prepares to give his talk. Waiting in line for badge and program was, for many, the first experience of the Joint Meetings.

Nathan Kaplan shows off his Morgan Prize It was standing room only for Terence Tao’s talk. The “dry walk” to the convention certificate. center. 21 MAA FOCUS March 2008

Students compete in “Who Wants to be a Mathematician?”

Scott Williams, the NAM Clayton-Woodward Lec- turer. Brian Conrey gave the MAA Student Lecture. Photograph by Colm Enthusiastic supporters in “Who Mulcahy. Wants to be a Mathematician?”

MAA Director of Programs and Services Michael Pearson says our meeting photos always include a sil- ly picture of him; we aim to please. Getting a massage at the Sophie Germain’s plan for proving Elsevier-sponsored “relaxation Fermat’s Last Theorem, from the talk station.” by David Pengelley.

Rachel Finley, Kim Weems, Sylvia Bozeman, Tuwanna Lamar, and Emille Davie. Photo- The carpet looks fancy, but graph by Colm Mulcahy. the symmetry group is a Jackie Smith and Murli Gupta. fairly standard one. Photograph by Joe Gallian.

Cheryl Adams and Candace Bau- The “Wood Möbius” exhibit. mann at the MAA Membership Noriko Yui Thomas W. Hunger- booth. ford 22 March 2008 MAA FOCUS

Marcia Sward, David Bressoud, and Lida Barrett. T-shirts, mathematical and mun- That’s child’s play! dane.

Joe Gallian (left), George Csicsery (right), and three of the stars of Hard Problems: Brian Lawrence, Yakov Berchenko Kogan, and Zeb Brady.

“Euler” (Colm Mulcahy) and Rob- Victor Katz ert Schneider.

Robert Bozeman, Sylvia Bozeman, Jackie Smith, and Gisela Fränken. Photograph by Joe Gallian.

Fernando Gouvêa was the speaker at the ACMS dinner. The GlassGeometry.com booth also sold greeting cards.

Informal meetings and conversations are a big part of the JMM.

On the message board, proof that mathematicians don’t just do math.

Mary Shepherd has just been elected to Who knew there were JMM Another long line, this time to par- Harold M. Edwards the MAA Audit Com- VIPs? ticipate in the Employment Regis- mittee. 23 ter. MAA FOCUS March 2008

The Undergraduate Poster Session at JMM 2008

By Diana Thomas

The undergraduate research poster ses- she admired Marlow Anderson, her ad- sion at the Joint Mathematics Meetings, visor, for his ability to convey complex sponsored by the MAA–CUPM Subcom- mathematical concepts. She says “I hope mittee on Undergraduate Research and to be like him some day.” the MAA Committee on Undergraduate Student Activities and Chapters, was Many thanks are owed to all the math- again a wonderful and exciting event. ematicians who responded to the call for More than 260 students presented more judges. There were almost 200! Students than 170 posters. For undergraduates really appreciate the in-depth attention the poster session is a huge draw: more given by the judges. Sara Reynolds, a than half of all undergraduates at the student from Nazareth College in Roch- Joint Meetings presented a poster. Nicole ester, NY, said that “having the judges Scholtz of Denison University consid- come around and speak with us about our ers the poster session as an important poster was very beneficial. This gave us reason to attend the JMM, but not the practice speaking and answering ques- only reason. “It’s definitely not my only tions about our research to professors motivation,” she says. “As an undergrad, that had a good understanding of our I’m also interested in networking.” particular mathematical area.” Diana Thomas gets things going. In fact, many students felt that the most The poster session continues to be a suc- important aspect of the poster session was also an opportunity to reunite and cess with the support of many individuals was networking. For students like Dawn catch up with their REU cohorts. and organizations. This year 37 posters Curtis from Arizona State University, won prizes which were collected by 62 this was the first experience at the poster The remaining poster presenters were students. A list of prize winners and se- session. Curtis enjoyed hearing other mentored by advisors at their home lected images of the session are posted students talk about their research and has institutions. The impact our mathemat- at http://www.maa.org/students/under- plans to return next year. ics community has on undergraduates grad/poster08.html. Thanks to the MAA, involved in a research experience is AMS, SIAM, Educational Advancement A little over 50% of the poster present- tremendous. In fact, when students Foundation, and AWM for funding the ers conducted their research at summer were asked who they admired most as a prizes. A larger selection of pictures and Research Experiences for Undergraduate mathematician, all students came back comments can be found on Facebook, in programs (REUs). For REU students like with their own research advisor. Jette the group “MAA Undergraduate Poster Doris Dobi from MIT the poster session Peterson of Colorado College said that Session.” We look forward to seeing you at the poster session next year!

Meghan Flink and Jette Petersen pre- Raiza Cortes and Emil Morales wait Judges had to stand in line to register. pare their poster. for the sessions to begin.

24 March 2008 MAA FOCUS

My First Year as MAA President: A Report

By Joe Gallian The close of the business Early in my term I set a goal meeting at the January 2008 of having 1000 undergrad- MAA/AMS Joint Math- uate students participating ematics Meetings in San in the MAA–AMS Joint Diego marked the end of Mathematics Meetings in my first year as President. 2015 and 500 undergradu- It has been an exciting, ate students at MathFest rewarding and challenging in 2015. I chose the year year for me. 2015 because it is the 100th anniversary of the MAA. In April I participated in To work towards reaching the three-day MAA Car- this goal, MAA Associate riage House Conference Director for Student Ac- Center Grand Opening Cer- tivities Robert Vallin and emonies, which featured I submitted a $468,000 talks by MAA First Vice Joe Gallian, Manjul Bhargava, Alison Miller and Nathan Kaplan. five-year grant proposal -President Carl Pomerance, to the NSF to support un- former President and Vice- provements in practices and procedures dergraduates to travel to President Ron Graham, Manjul Bhargava, in the MAA office. The staff will be pilot- the two major meetings. I am thrilled to Art Benjamin, Melanie Matchett Wood, ing a telecommuting program and more report that our grant has been funded. Laura Taalman, Richard Tapia, Brent variable work-week scheduling starting With over 250 undergraduate students Morris, and myself. The event included in February. at MathFest in 2007 and over 500 at the a reception for DC area representatives JMM in 2008, we are well on our way to of science, engineering, and mathemat- In November the MAA undertook an am- reaching my goal. ics societies and government agencies bitious two-year fund raising campaign and MAA members, a day of talks for in response to a 2-1 matching challenge One of the important duties of the mathematicians in the area, and a day of gift of $600,000 from Virginia Halmos. President is to attend section meetings programs for students. A splendid time A gift from Paul and Virginia Halmos, to strengthen the bonds between the was had by all. the largest ever received by the MAA, sections and the national office. I was provided the funds for the renovation honored to speak at the Iowa, Michigan, Even though the grand opening of this of the Carriage House into a Meetings Southeastern, and Northeastern Sections great new MAA facility was in April, Center. The Halmos Endowment Fund this year. many programs were held in it before- will be used to support mathematical hand. In February I was on hand to programming in the Carriage House for At the 2008 MAA–AMS Joint Mathemat- introduce Larry Schumaker as the first now and long into the future. To kick- ics Meetings in San Diego, I had the plea- speaker in the Halmos Distinguished off the fund-raising campaign before sure of inducting former MAA President Lecture Series, funded by a grant from going to our membership at-large, we and Secretary Jerry Alexanderson and the National Security Agency. Other contacted a small number of MAA sup- former MAA Treasurer Jerry Porter and lectures in the series have been given porters whom we asked to be Founding Judith Porter into the MAA Icosahedron by Doron Zeilberger, Trachette Jackson, Patrons of the Halmos Endowment Fund. Society. It was Jerry Alexanderson’s Bernd Strumfels, David Bressoud, and I am delighted to report that the team of friendship with Paul and Virginia Halmos Ravil Vakil. These talks are intended Tina Straley, Jerry Alexanderson, John that made both the renovation of the Car- for the general, mathematically literate, Kenelly and I were able to raise even riage House and the Carriage House Hal- public. I was fortunate to attend all but more than the total goal of $300,000 in mos Endowment Fund possible. Jerry and one of these wonderful programs. just two months. Virginia Halmos was so Judith Porter have given the MAA funds pleased with this success that she wants to endow a public lecture series that will In May, I had the privilege of presid- to provide another $200,000 that we must be inaugurated at the JMM beginning in ing over the awards ceremony for the match on a 1-1 basis. With Virginia’s 2009. Jerry Alexanderson and Jerry and 12 winners of the MAA United States additional pledge, everyone still has an Judy Porter have left a permanent legacy Mathematical Olympiad at the State opportunity to contribute to this effort and for the MAA. Department in Washington DC. In the be Founding Patrons. (See the article on fall I conducted the first ever MAA staff page 5 for more information.) Another highlight for me at the San Di- satisfaction survey, which has led to im- ego meeting was the world premiere of 25 RubiksAd_BW_4.8125x9bhT.qxd 1/25/08 2:21 PM Page 1 MAA FOCUS March 2008 the film Hard Problems, documenting the United States Mathematical Olym- piad team’s participation in the 2006 NATIONAL SECURITY AGENCY International Mathematical Olympiad in Slovenia. It’s a compelling look at the NSA participation of the outstanding students who make it to the top levels of competi- tion. Three of the team members featured in the film and director George Csicsery were on hand to entertain questions from the audience. The MAA was able to produce the film with the support of the Penn Oberlander Family Foundation and the Ellington Management Group, If you want to make a L.L.C. Larry Penn was a member of the 1979 US IMO team and a Putnam Fellow career out of solving in 1980. I am proud to say that Larry was complex mathematical also a participant in my REU in 1981. I challenges, join NSA as am seeking sponsors to make possible the showing of Hard Problems on public a Mathematician. television and to widely distribute the DVD to high school teachers for use in At NSA you can bring the power classes and math clubs. of Mathematics to bear on today's most distinctive challenges and When running for President of the MAA, problems. We identify structure my two stated goals were to increase within the chaotic, and discover public understanding and appreciation patterns among the arbitrary. for mathematics and to secure funding for You will work with the finest Project NExT. The Halmos Distinguished minds and the most powerful Lecture Series at the Carriage House and technology. the Hard Problems film were two efforts towards fulfilling the first goal. With Make the move that puts the sponsorship of Sun Microsystems, your math intelligence to we expect to have a poster on Women work. Apply to NSA. and Mathematics ready for MathFest in July of 2008. Towards the second goal, I secured funding for ten Project NExT Fellows and now that Project NExT has more than 1000 alumni and nearly 500 consultants, I hope to initiate a Project ® NExT endowment fund in the second DISCIPLINES WHERE INTELLIGENCE GOES TO WORK year of my term. > Number Theory > Mathematical Statistics > Linear Algebra > Probability Theory > Finite Field Theory > and More At the meeting in San Diego, I was often > Group Theory > Combinatorics asked how my first year as President went. I responded by citing the titles of Visit our Web site for a complete list of current career opportunities. three Beatle songs: “Eight Days a Week,” “Hard Day’s Night,” and “Help!” The MAA staff has been a great help to me, and I am fortunate to work with my own Your perspective. Your culture. Your intelligence. version of the Fab Four: Tina Straley, Michael Pearson, Martha Siegel and

John Kenelly. I thank the MAA mem- U.S. citizenship is required. NSA is an Equal Opportunity bers for the opportunity to serve as your Employer and abides by applicable employment laws and regulations. Rubik’s Cube® is used by permission President. of Seven Towns Ltd. www.rubiks.com www.NSA.gov/Careers

26 March 2008 MAA FOCUS MAA Business at the 2008 Joint Mathematics Meetings: A Report

By Martha Siegel, MAA Secretary I come home from the Joint Director of Student Ac- Meetings energized by the tivities), and Michael vitality of the mathematical Pearson (MAA Associ- community. The variety of ate Executive Director talks, sessions, activities, and Director of Pro- and social events resulting grams and Services). from the effort of the MAA, Over their 21 months AMS, AWM, SIAM, and of work, the group had others are coordinated and a three-day conference organized by our able com- in Washington, DC, met mittees and staff. We owe at the national meetings our Associate Secretary, twice a year, conducted Jim Tattersall, our speakers, informal surveys of organizers, volunteers, and students at MathFest staff a big thanks for making 2006, conducted break- this the largest-ever Joint out sessions with the Meeting. The attendance Joe Gallian and Martha Siegel prepare for the MAA Business Meeting. Board at the Section (more than 4300 mathemati- Officers meeting and cians and over 500 students) at the Student Chapter was spectacular. Although a lot more than A substantial amount of time at the Board Advisor Breakfast at the Joint Meetings MAA committee meetings took place in meeting was spent on strategic planning. 2007, and mailed questionnaires to five San Diego, I limit my report to the activi- As many of you are aware, the Board of different groups interested in students: ties of the Board of Governors. Governors has helped to direct a serious Undergraduate Student Members (214 effort at strategic planning for the MAA. responses), Graduate Student Members President Joseph A. Gallian announced An important part of the Board meetings (141 responses), Graduate Student Non- that the MAA raised more than $300,000 is devoted to interim and final reports Members (220 responses), Student Chap- in about six weeks in answer to a chal- from the various Strategic Planning ter Advisors (68 responses), and Project lenge grant from Virginia Halmos (widow Working Groups. With this meeting on NExT Fellows (93 responses). th of Paul Halmos). Virginia Halmos offered January 5 , we have almost completed to match our funds 2-to-1 for program- Cycle II in the process. In their roles as The Working Group adopted the follow- ming at the Carriage House. She was so First Vice-President, Barbara Faires (Cy- ing Vision Statement in order to set pri- impressed with the speed with which cle I) and then Carl Pomerance (Cycle II) orities and establish a coherent structure we were able to raise the funds that she have led the strategic planning process. for the MAA student program. has offered to continue the effort with a Two Cycle II working groups presented 1-to-1 challenge. We will continue the their final reports at the San Diego meet- Vision Statement for Students campaign. Donations are welcome! At ing. Both are extremely important to the the Board meeting, members contrib- Association, hence I would like to take Students are an important part uted over $500 to honor Jim Bruening, this opportunity to briefly summarize of the present Mathematical As- Governor of the Missouri Section, who the reports. sociation of America (MAA) and died this fall. We will be buying a brick are the future of the organization. in his memory. Deanna Haunsperger (Carleton College), Thus, the MAA is committed to who served as Second Vice-President being the preeminent resource for The MAA’s assets no doubt looked in 2006 and 2007, chaired the Working all students of mathematics and the better on January 5th than they do now, Group on Students. Other members of faculty who teach and mentor them. but recent appraisal of our buildings in the group were Betty Mayfield (Hood The MAA is further committed Washington, DC revealed that they are College), Richard Marchand (Slippery worth more than $17 million. The Board Rock University), Donna Beers (Sim- • to encouraging students to voted to keep membership dues for 2009 mons College), Yan Wu (Georgia South- reach their full potential in math- at the same level as in 2008 in an attempt ern University), David Manderscheid ematics, to shore up retention rates. We continue (University of Iowa, now at University • to enriching their education to explore electronic memberships, espe- of Nebraska), Robert Vallin (Slippery through programs and publica- cially for international members. Rock University, now MAA Associate tions,

27 MAA FOCUS March 2008

• to presenting them with ideas are charged with working with the com- The Board endorsed many of the recom- and information about opportuni- mittees to set priorities in this process. We mendations, including a proposal for ties that exist for them with degrees understand that we cannot do everything major changes in the composition of in mathematics, and at once, but we are pleased that we have the Board of Governors. These changes • to welcoming them into the com- a strategic plan to guide us in this very require bylaws changes and therefore will munity of mathematicians. important area. be discussed at length and for some time before being voted upon by the member- The recommendations were sweeping; The Working Group on Governance also ship at an MAA Business Meeting. The For example, the group recommended reported to the Board at San Diego. The basic committee and council structure creating and distributing a flowchart working group considered the entire will be retained. However, the role of of student-related ac- the Coordinating Council tivities, especially with Chairs will be expanded. respect to Association The Council Chairs will be committees and staff responsibilities for the elected by the Board and various aspects of ser- will serve on the Board of vice and programs for Governors. Furthermore, students. The recom- the Committee on Com- mendations included mittees will be expanded (it suggestions on stu- is currently the President, dent membership, re- the President-elect or Past structuring the student President, the Secretary, chapters, redefining the and the Executive Director) definition and benefits to include all of the Council to students of Institu- Chairs. tional Membership, maintaining an email The Committee on Com- list of students who mittees oversees the MAA would be interested in committee structure that receiving interesting Dan Teague, MAA Second Vice- Betty Mayfield, MAA First Vice- currently consists of Coun- mathematical tidbits or President President cils, Standing Committees, news, sponsoring more Standing Sub-committees, activities and contacts ad hoc committees, and for student chapter advisors, starting a scope of governance of the Association task forces. The Board of Governors student newsletter or having pages in except Sections (a new Working Group creates standing committees and stand- MAA FOCUS devoted to student con- on Sections is now investigating that) ing subcommittees and the President cerns, news, and information, helping the for nearly two years. This group studied appoints the members. The Committee sections to enhance student activities at the structure of the Board of Governors, meetings and in the institutional chapters the Executive Committee, all the stand- on Committees is charged with advising by having specific programs for chapter ing committees and subcommittees, the the President on all MAA Council and advisors, increasing availability of travel abundance of ad hoc committees, the committee appointments; making recom- funds for students to attend MathFest, council structure, and the interaction of mendations to the Board of Governors enhancing the experience of graduate stu- officers, staff, and others. As mentioned regarding the formation and dissolution dents both at the section level and at the above in the case of the student group, of editorial boards, standing committees Association level, creating online com- various committees are charged with and standing subcommittees, and form- munities for students, and much more! investigating the feasibility of implemen- ing task forces and ad hoc committees tation of recommendations. In the case as needed. The report will be sent to all those com- of Governance, the primary committees mittees whose charges cover any part of that are charged with implementation are In order to keep the Board from getting the recommended actions. The Member- the Executive Committee, the Board, and too large, the three journal Editors would ship Committee, the Council on Publica- their subcommittees. Recommendations no longer serve on the Board, former tions, the Committee on Undergraduate fell into three categories: those that could Presidents would have a shorter post- Student Activities and Chapters, and be acted on by the Board and put in place presidency term, and some members of the Committee on Graduate Students immediately, those that could be acted on the Audit Committee would no longer are among those committees who are without a vote of the Board since they serve on the Board. The Audit Commit- called upon to determine the feasibility required a change in practice but not in tee will be an independent subcommittee of implementing the recommendations. policy, and those that required a change of the Board, reporting directly to the The Executive Committee and the staff in the Association bylaws. Board. Reporting procedures, charges

28 March 2008 MAA FOCUS to committees, formation of task forces lege MA), and Tina Straley (Executive VP), Deanna Haunsperger (Second VP), and ad hoc committees, and creation of Director of the MAA). Jim Daniel (Audit and Budget Commit- standing committees and subcommittees tees), Efrain Armendariz (Minorities), will be more regulated and controlled by The final reports of all the Strategic Plan- Susan Schwartz Wildstron (Secondary the Board or the Committee on Commit- ning Working Groups will be posted on Teachers), and Section Governors: Jane tees. Terms of MAA committee members the MAA web site. Members are encour- Arledge (Rocky Mountain Section), will generally run for three years and aged to read them. Many thanks are owed John Bukowski (Allegheny Mountain one month, so that new appointments to those who contributed to the focus Section), Dan Curtin (Kentucky Sec- to committees will serve starting with groups, questionnaires, and discussions tion), Rick Gillman (Indiana Section), the January meetings, and will overlap that informed the work of these groups. Kathleen Hann (N. California, Nevada, for that meeting with those whose terms Hawaii Section), John Koker (Wisconsin are ending. The terms of the Secretary Lots of other things are going on at Section), Chris Masters (Nebraska- SE S. and the Treasurer are to be more clearly the MAA! Continuing its work is the Dakota Section), Jack Winn (Metro NY defined. Strategic Planning Working Group on Section), and Fred Worth (Oklahoma- Membership (soon to deliver its report) Arkansas Section). The three years of The President will appoint a Task Force and four new groups that are just getting their terms have flown by! Most of these on the Revision of the Bylaws, which started. We now have Strategic Planning members are already actively enrolled will be expected to complete its task Working Groups on Sections, Meetings, as committee members and we hope entirely by the end of the Joint Meetings STEM Issues, and on Periodicals and they continue to serve for a long time. in 2011. Starting immediately, a complete Communications. The Board agreed that Many additional personal thanks go to review of all the units in every council after this third cycle is completed, the Carl Cowen (Past President), who steps will take place and while the Working Board should consider the entire mosaic down from the Executive Committee, but Group recommended new councils and of recommendations, priorities, feasibili- continues to serve on the Board. a realignment of committees within ties, and implementations in light of the these Councils, the Board has not yet mission and vision of the MAA. It is We all depend a lot on our staff. Though approved the recommendation. The Com- likely that a subcommittee of the Board small in number, they are big in output mittee on Committees will consider the will be charged with this task. and show tremendous care for the MAA. realignment further and make its recom- We are lucky to be able to work with mendation to the Board. Members of this There was a lot of routine business, but them. I especially want to thank my as- Working Group were: Connie Campbell non-routine, as far as I am concerned, is sistant, Denise Raspa. She has been an (Millsaps College), Carl Cowen (IUPUI, the thanks to the members of the Board indispensable help to me. Denise, Susan former President and Chair of the Group), for whom this is the last Board meeting. Kennedy, and Calluna Euving at the William Haver (Virginia Commonwealth Note that outgoing Section Governors MAA have been working diligently to University), Martha Siegel (Towson Uni- remain on the Board until June 30, 2008, simplify and streamline our operations. versity, Secretary of the MAA), David but officers and at-large governors leave I thank them all for their good cheer and Stone (Georgia Southern University), the Board at the end of the JMM. Heart- for making my job so much easier. Phil Mahler (Middlesex Community Col- felt thanks go to: Carl Pomerance (First

Looking for a great way to join with your peers this summer to explore new ideas? The MAA Professional Enhancement Program (PREP) offers workshops on a variety of topics. Whether you are interested in looking for more effective approaches for your calculus courses or learning about new topics such as mathematical biology, the PREP program has something for you. Visit www.maa.org/prep/2008 for details and to register for this year’s workshops.

29 MAA FOCUS March 2008

30 March 2008 MAA FOCUS

Madison, Wisconsin July 31 – August 2 Save The Dates! Learn more online and in the April issue of MAA FOCUS

Mathematical Association of America

® www.maa.org New MAA Web Pages for Students The MAA has expanded and re- New Pages Include worked it collection of web pages xHigh school students for students. Our goal is no less pages, than to have the best collection xUndergraduate students of pages and links for anyone in- terested in mathematics accessi- pages, ble to undergraduates. xGraduate students pages, xCareer pages, Take a look at the pages and give xConferences links, and us some feedback on what you xSummer opportunities like and what you’d like to see.

www.maa.org/students On facebook now at Student Page for the MAA

31 MAA FOCUS March 2008 The Mathematical Association of America ®

A Radical Approach to Lebesgue’s Theory of Integration David Bressoud Published jointly with Cambridge University Press A Radical Approach to Legesgue's Theory of Integration is a sequel to A Radical Approach to Real Analysis. This book is an introduction to measure theory and Legesgue integration rooted in and motivated by the historical questions that led to its development. It stresses The Mathematicalthe original purpose Association of the definitions and oftheorems, America and highlights some of the difficulties ® that were encountered as these ideas were refined. The text begins with Riemann's defini- A Radicaltion Approach of the integral, to Lebesgue’s a definition Theory created of so Integration that he could understand how broadly one David Bressoudcould define a function and yet have it be integrable. The story then follows the efforts of many different mathematicians who wrestled with the difficulties inherent in the Riemann Published jointly with Cambridge University Press integral, leading to the work of the late 19th and early 20th centuries of Jordan, Borel and Legesgue who final- ly broke with Riemann'sA Radical Approach definition. to Legesgue's In ushering Theory in of a Integrationnew way isof a sequelunderstanding to A Radical integration,Approach they opened the door to to Real Analysis. This book is an introduction to measure theory and Legesgue integration fresh and productiverooted ways in and ofmotivated viewing by themany historical of the questions previously that led intractable to its development. problems It stresses of analysis. the original purpose of the definitions and theorems, and highlights some of the difficulties This is not intendedthat towere be encountered read as a ashistory these ideas of werethe develoment refined. The text of begins anlysis. with Rather,Riemann's it defini- is a textbook informed by histo- tion of the integral, a definition created so that he could understand how broadly one ry, attempting to communicatecould define a function the motivations, and yet have it uncertainties, be integrable. The and story difficulties then follows thesurrounding efforts of the key concepts. many different mathematicians who wrestled with the difficulties inherent in the Riemann integral, leadingMAA to Textbooksthe work of the • late Catalog 19th and Code: early RADL 20th centuries • 320 ofpp., Jordan, Paperbound, Borel and Legesgue 2008 • who ISBN: final- 978-0-52171-183-8 ly broke with Riemann's definition. In ushering in a newList: way $39.99of understanding • MAA integration, Member: $32.00they opened the door to fresh and productive ways of viewing many of the previously intractable problems of analysis.

This is not intended to be read as a history of the develoment of anlysis. Rather, it is a textbook informed by histo- ry, Isattempting Mathematics to communicate the Inevitable? motivations, uncertainties, A Miscellany and difficulties surrounding the key concepts. UnderwoodMAA Textbooks Dudley, • Catalog Editor Code: RADL • 320 pp., Paperbound, 2008 • ISBN: 978-0-52171-183-8 List: $39.99 • MAA Member: $32.00 This is a collection of gems from the literature of mathematics that shine as brightly today as Is whenMathematics they first appearedInevitable? in print. A Miscellany They deserve to be seen and admired. UnderwoodThe selections Dudley, include Editor two opposing views on the purpose of mathematics, The Strong Law of ThisSmall is a collection Numbers of, gems the treatment from the literature of calculus of mathematics in the that1771 shine Encyclopaedia as brightly today Britannica as , several proofs that whenthe they number first appeared of legs in on print. a horseThey deserve is infinite, to be seen a deserved and admired. refutation of the ridiculous Euler-Diderot Theanecdote, selections include the real two story opposing of /viewsand on the the Indiana purpose of Legislature, mathematics, Thethe Strong reason Law why of Theodorus stopped Smallproving Numbers that, the treatmentsquare roots of calculus were in irrationalthe 1771 Encyclopaedia when he Britannica to the ,square several proofsroot ofthat 17, an excerpt from theMathematics number of legs Made on a horse Difficult is infinite,, a glimpse a deserved into refutation the mind of the of ridiculous a calculating Euler-Diderot prodigy….There will be something of inter- anecdote, the real story of / and the Indiana Legislature, the reason why Theodorus stopped provingest here that squarefor almost roots were anyone irrational interested when he toin the mathematics. square root of 17, an excerpt from Mathematics Made Difficult Spectrum, a glimpse • into Catalog the mind Code: of a calculating IMI • 340 prodigy….There pp., Hardbound, will be something2008 • ISBN: of inter- 978-0-88385-566-9 est here for almost anyone interested in mathematics. List: $56.95 • MAA Member: $45.50 Spectrum • Catalog Code: IMI • 340 pp., Hardbound, 2008 • ISBN: 978-0-88385-566-9 List: $56.95 • MAA Member: $45.50

The Mathematical Association of America Periodicals Postage paid at ToTo order order visit visit us us1529 Eighteenth St., NW Washington, DC and Washington, DC 20036 additional mailing offices onlineonline at: at: www.maa.orgwww.maa.org or call us at: 1.800.331.1622or call us at: 1.800.331.1622

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