MAA FOCUS December 2007
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 27 Issue 9 Editor: Fernando Gouvêa, Colby College; [email protected] Managing Editor: Carol Baxter, MAA Inside [email protected] Senior Writer: Harry Waldman, MAA 4 The Dangers of Dual Enrollment [email protected] By David M. Bressoud Please address advertising inquiries to: [email protected] 6 A Case Study of Concurrent Enrollment Partnership President: Joseph Gallian in Mathematics First Vice President: Carl Pomerance, By Philip Cheifetz and Ellen Schmierer Second Vice President: Deanna Haunsperger, Secretary: Martha J. Siegel, Associate 9 Teaching Time Savers: The Exam Practically Wrote Itself! Secretary: James J. Tattersall, Treasurer: By Michael E. Orrison John W. Kenelly Executive Director: Tina H. Straley 10 The MAA Euler Study Tour Reports and Comments from the Participants Director of Publications for Journals and Communications: Ivars Peterson 12 Eulogy of Leonhard Euler (1707-1783) MAA FOCUS Editorial Board: Donald By Siu Man Keung J. Albers; Robert Bradley; Joseph Gallian; Jacqueline Giles; Colm Mulcahy; Michael 14 Asking Questions: Improving Prospective Teachers’ Orrison; Peter Renz; Sharon Cutler Ross; An- Knowledge of Mathematics nie Selden; Hortensia Soto-Johnson; Peter By Wendy A. Weber Stanek; Ravi Vakil. Letters to the editor should be addressed to 16 First Annual TENSOR-SUMMA Grant Program a Success Fernando Gouvêa, Colby College, Dept. of Mathematics, Waterville, ME 04901, or by 17 TENSOR-MAA Grants for Women and Girls in Mathematics email to [email protected]. 19 Remember the Alamonoid: The Art of Factorization in Subscription and membership questions Multiplicative Systems in San Antonio should be directed to the MAA Customer Service Center, 800-331-1622; email: By Scott T. Chapman [email protected]; (301) 617-7800 (outside U.S. and Canada); fax: (301) 206-9789. MAA 21 Remembering Paul Cohen Headquarters: (202) 387-5200. By Peter Sarnak Copyright © 2007 by the Mathematical 23 Statistics: A Key to Student Success in College and Life Association of America (Incorporated). Educational institutions may reproduce By John Loase articles for their own use, but not for sale, provided that the following citation is used: 24 It is Time to Support Extremes “Reprinted with permission of MAA FOCUS, By Reza Noubary the newsmagazine of the Mathematical Association of America (Incorporated).” 26 Employment Opportunities Periodicals postage paid at Washington, DC and additional mailing offices. Postmaster: Send address changes to MAA FOCUS, On the cover: To escape the hurly-burly of the court in Berlin, Frederick the Great Mathematical Association of America, P.O. built himself a palace in Potsdam and named it “Sans Souci,” meaning “without a Box 90973, Washington, DC 20090-0973. care.” There, Euler was asked to help create a fountain that would shoot water high ISSN: 0731-2040; Printed in the United States up in the air. Unfortunately, the engineers did not heed his advice to test the ability of America. of the tubes to withstand the required water pressure, and the fountain never worked successfully. (Photograph of Sans Souci Palace provided by Jim Smith)
2 December 2007 MAA FOCUS
Mathematics at the AAAS, February 2008
By Edward Aboufadel
The 2008 Annual Meeting of the media awareness.) The AAAS Program Symposia Sponsored by Section A American Association for the Advance- Committee is genuinely interested in ment of Science will be February 14–18, offering symposia on pure and applied Mathematics and the Brain in Boston, MA. This year’s program fea- mathematical topics of current interest, (organized by Jack Cowan, Chicago) tures many outstanding expository talks and in previous years there have been by prominent mathematicians, including symposia on subjects such as the chang- Design of Mechanical Puzzles a topical lecture by Curtis T. McMullen ing nature of mathematical proof, math- (organized by Peter Winkler, Dart- on “The Geometry of 3-Manifolds,” fo- ematical ecology, and recent results in mouth) cusing on the Poincaré Conjecture. The prime number research. overall theme of this year’s meeting is Modeling the Dynamics of the Drug- “Science and Technology from a Global The 2009 meeting will be February Resistant Killers of the 21st Century Perspective”; many of the symposia 12-16, 2009 in Chicago. The Steering (organized by Sally Blower, UCLA) sponsored by Section A (Mathematics) Committee for Section A seeks organiz- are interdisciplinary sessions that fit this ers and speakers who can present sub- Quantum Information Theory theme. The Section A symposia are only stantial new material in an accessible (organized by Mary Beth Ruskai, a few of the nearly 200 AAAS program manner to a large scientific audience. Tufts) offerings in the physical, life, social, and All are invited to attend the Section A biological sciences. For further informa- Committee business meeting in Boston New Techniques in the Evaluation and tion, including the schedule of talks, go on Friday, February 15, 2008, at 7:45 Prediction of Baseball Performance to http://www.aaasmeeting.org. PM, where we will brainstorm ideas for (organized by Edward Aboufadel, symposia. In addition, I invite you to Grand Valley State) AAAS annual meetings are the show- send me, and encourage your colleagues cases of American science, and the to send me, proposals for future AAAS Other Symposia of Interest to the participation by mathematicians and annual meetings. Mathematical Community mathematics educators is encouraged. (AAAS acknowledges the generous Edward Aboufadel is the Secretary Biometrics in Border Management: contributions of AMS and MAA for of Section A of the AAAS. He can be Grand Challenges for Security, Iden- travel support and SIAM for support of reached at [email protected]. tity, and Privacy
Atomic Detectives: Nuclear Forensics Steering Committee for Section A Members at Large: and Combating Illicit Trafficking
Chair: Carl Pomerance Walter Craig Ethical Issues in Scientific Publishing (Dartmouth College) (McMaster University) Enhancing Science Globally Through Chair-Elect: William Jaco Mary Beth Ruskai High-Performance Computing and (Oklahoma State University) (Tufts University) Simulation Retiring Chair: Jack Cowan David Isaacson (University of Chicago) (Rensselaer Polytechnic Institute) Information, Computing, and Commu- nications: Keys to Sustainable Global Secretary: Edward Aboufadel Claudia Neuhauser Development (Grand Valley State University) (University of Minnesota) Collaboratively Developing Student Mathematical Thinking Among APEC Member Economies
Looking for Hard Problems? Promoting the Success of Minority Graduate Students
The world premiere of George Csicsery’s film Hard Problems: The Inside the Double Bind: Women of Road to the World’s Toughest Math Contest will happen at the Joint Color in Science, Technology, Engi- Mathematics Meetings in January. See page 31 for more details. neering, and Mathematics
3 MAA FOCUS December 2007
The Dangers of Dual Enrollment By David M. Bressoud
Dual or concurrent enrollment pro- What has been the level of accountabil- colleges and 12% of those affiliated grams are arrangements between a local ity in these programs? How has the mas- with 2-year colleges are taught by col- college and high school whereby stu- tery of this material been assessed? lege faculty. Even the label “college fac- dents enrolled in certain classes receive ulty” can be problematic because there both high school and college credit. For This column looks at data from the are cases where high school teachers are the purposes of this article, the term will CBMS Survey, Fall 2005, references the designated as faculty of the college. be used exclusively for classes that are example of Nassau Community College taken at the high school and receive dual which has shown what it takes to build College departments were asked to credit. a good dual enrollment program (see self-report on whether they maintained also the separate article “A Case Study control over four aspects of these pro- Though they have existed for a long time, of Concurrent Enrollment Patnerships grams: selection of textbook, syllabus, dual enrollment programs have taken off in Mathematics” in this issue), and de- final exam, and choice of instructor. The in the past decade as high schools and scribes the Early College High School table on page 5 shows the results. I find colleges have come to see them as mu- Initiative, a national dual enrollment it very disturbing that the most effective tually beneficial. In fact, they seem to be program with both promise and danger. means of ensuring quality control over a win for everyone: Colleges like them the final examination and the choice of because, even at reduced tuition, they CBMS Survey Data instructor, are so seldom used. are a source of additional revenue and a way of increasing enrollment numbers While a small number of students take This lack of accountability can lead to at very low cost. High schools like them statistics under dual enrollment, most of serious problems for both the students because of the prestige of offering col- the dual enrollment courses in mathe- and the colleges at which they are ac- lege-credit courses. Students like them matics are either college algebra/precal- cepted. While there are many excellent because they eliminate the need to take a culus or calculus. Just offering college high school teachers capable of teaching high stakes national test, such as the AP algebra/precalculus as a college-credit these courses as well as or better than Calculus exam, in order to earn credit, course taught in a high school can be the instruction at college, and while and they look good on a transcript when problematic. Many colleges, Macalester even the term “college-level” is prob- applying to college. Parents like them among them, no longer consider these to lematic because of the great disparity because they can shorten the number of be college-level courses. Even colleges among colleges in how such courses are years their child will be in college, and and universities that do offer college taught, the goal of any dual enrollment thus the number of tuition payments. credit for these courses usually treat course is that students who successfully them as preliminary to the mathematics complete it are at least as well prepared I know that college material can be taught required for mathematically intensive for the next course in the sequence as in high schools at a level that is compa- majors. If these courses really are high those who have taken it at the college rable to what is taught in college. This school mathematics, what does it mean from which credit is given. Without is the premise of the College Board’s to offer college credit for them when close oversight and supervision, there is Advanced Placement Program, and it they are taken in high school? danger that credit might be given even is justified in the high success rates of when the level of preparation is below students who use the college credit they To get an idea of the annual numbers, that required for the next course. have earned to move directly on to more the CBMS survey counted the number advanced courses. But not everyone of students in dual enrollment programs Maintaining Quality who studies college-level material in in both spring and fall terms of 2005 [1]. high school achieves college-level pro- There were 63,986 students studying There are dual enrollment programs that ficiency, as witnessed by the fact that at college algebra/precalculus and 33,436 do maintain the control that is needed. most a third of those who study calculus studying calculus. Both 2-year and 4- Nassau Community College on Long while in high school earn college credit year colleges are involved in awarding Island, New York, is one that is doing for this course, and many of those who dual enrollment credit: 22% of the stu- a good job. Further information on their do not earn college-level credit struggle dents in college algebra/precalculus and program can be found elsewhere in this when they take a real college course on 42% of those in calculus I received their issue [2]. this same material. For this reason, a red credit from a 4-year college. Occasion- flag goes up when colleges see students ally, these courses are taught by college There now is an accreditation agency coming to them with credit for courses faculty, but that is the exception. Only for dual enrollment programs, the Na- taken under dual enrollment programs. 4% of the courses affiliated with 4-year tional Alliance of Concurrent Enroll-
4 December 2007 MAA FOCUS ment Partnerships (NACEP) [3]. For that carry college credit are taught by them a sense of what college is like and those readers who know more about this college faculty (again, this can be an what it will take to succeed while pro- organization, I would be very interested ambiguous designation), but usually to viding an environment that helps them in your assessment of its usefulness. classes made up entirely of high school to succeed. This is a very worthy goal. What impresses me most about NACEP students. Of eight ECHS programs that I do worry about the promise of be- is that its accreditation is accompanied were studied in detail in 2006, five teach ing able to complete two years of col- by a rating system that describes the de- at least some of the college-credit cours- lege-level courses while in high school. gree of control along several different es in classes that enroll both college and Students who successfully complete dimensions. It appears that NACEP ac- ECHS students, but only one of the pro- this program and arrive at college with creditation may be a useful tool to help grams teaches all of its college-credit the intention of graduating in only two colleges decide when dual enrollment courses in this manner. [5, p. 33] more years will find themselves shut out credit should be accepted. of any mathematically intensive major. Few students, Early College College Control 4-year colleges 2-year colleges even among the High School Over most talented (ECHS) and acceler- never sometimes always never sometimes always ated, arrive at In learning Textbook 41% 15% 44% 14% 12% 74% college with the about dual en- three semesters Syllabus 2% 6% 92% 4% 7% 89% rollment pro- of calculus and grams, I have Final Exam 40% 30% 30% 36% 28% 37% the semester also been alert- Instructor 32% 20% 48% 35% 13% 52% of linear alge- ed to a move- Level of control by college mathematics departments over dual-enrollment courses. bra required ment that has before taking the potential junior/senior to greatly impact the educational sys- This makes a difference. In an indepen- level courses in mathematics, physics, tem as we know it. The Early College dent study of this program undertaken or chemistry. High School Initiative is a nation-wide during 2004–05, “interviews revealed collection of programs that award col- that college instructors were more like- The independent report does suggest lege credit for courses taken while in ly to maintain their usual standards if backing off from the promise of two high school. It is targeted specifically ECHS students were enrolled alongside years of college credit earned while in at “low-income youth, first-generation other college students, rather than com- high school as a “core principle” and college goers, English language learn- prising the entire class.” [5, p. 35] The replacing it with a commitment to com- ers, students of color, and other young program is new enough that as yet there press the time needed to complete a people underrepresented in higher edu- are no reliable statistics on its effective- post-secondary degree. [5, p. 93] Even cation.”[4] It is committed to instruction ness at getting at-risk students into and that commitment may make it more dif- that is “rigorous, relevant, and relation- through college. ficult for these students to enter- math ship-based” (its 3R’s). Alarmingly, one ematically intensive fields. of its primary goals is to allow these In an independent evaluation prepared students to earn up to two full years of for The Bill & Melinda Gates Founda- This initiative is already encountering college credit while still in high school. tion, the American Institute for Research some problems in maintaining rigor. I Spearheaded by the Bill & Melinda noted that college faculty make a very am concerned about what will happen Gates Foundation with support from conscious effort to maintain college- to the expectations for achievement lev- Carnegie Corporation of New York, The level expectations in these courses, but els as this initiative expands and attracts Ford Foundation, and The W.K. Kellogg that “rigorous instruction was elusive imitators who sell their program as a Foundation, it is a serious undertaking [in the ECHS high school classes], par- way of cutting college costs. that is already operating in 130 schools, ticularly in mathematics classes.” [5, p. involving over 16,000 students. The 91] Of eight programs offering college Many forces have combined to drive goal by 2012 is 239 schools and 96,400 credit that were studied in some detail in both dual enrollment and the ECHS students. the spring of 2006, five did teach at least promise of a high school diploma that some classes that enrolled both college comes with an associate’s degree at- Unlike the dual enrollment programs de- and high school students, but only one tached. High college tuition is one of scribed above, most of the college-level of the colleges did this exclusively. them. Until now, demand for college classes (58% of all programs) are taught degrees has seemed perfectly inelas- on college campuses, while another As I see it, the primary goal of this ini- tic; people will pay whatever colleges 22% are in special facilities devoted to tiative is to introduce rigorous, college- charge. As everyone knows, that cannot ECHS. For the most part, the courses level material to at-risk students, giving be true. People are not going to stop go-
5 MAA FOCUS December 2007 ing to college, but they may well seek [2] Philip Cheifetz and Ellen Schmierer, SRI International. Evaluation of the perverse and counter-productive routes A Case Study of Concurrent Enrollment Early High School Initiative. Prepared to its completion. For those of us who Partnership in Mathematics, see below. for The Bill & Melinda Gates Founda- worry about the low number of students tion. 2007. http://www.earlycolleges.org/ going into mathematically intensive ma- [3] National Alliance of Concurrent publications.html jors, this should be a matter of concern. Enrollment Partnerships. http://www. nacep.org David M. Bressoud teaches at Macales- References ter College. He will be President-Elect [4] The Early College High School Ini- of the MAA beginning in January 2008. [1] CBMS Survey, Fall 2005, tables tiative. http://www.earlycolleges.org/ A version of this article also appeared SP.16 and SP.17, pages 62–63. http:// as a Launchings column in July, 2007 www.ams.org/cbms/report05/ [5] American Institute for Research and on MAA Online, at http://www.maa.org/ columns/launchings.
A Case Study of Concurrent Enrollment Partnership in Mathematics
By Philip Cheifetz and Ellen Schmierer A concurrent enrollment partnership us to alert the high schools in the county come to NCC once a month to discuss (CEP) is an association between a col- of the dramatic changes their students the progress of the partnership.” Meet- lege and one or more high schools that would face if they enrolled in calculus ings with the NCC administration and allow qualified high school students to courses at NCC. the College union were held at NCC to earn college credit by taking college examine the viability and ramifications courses taught in their high schools. As In 1993, a workshop was offered to high of such an arrangement. After a year of reported by David Bressoud elsewhere school mathematics chairpersons and meetings, NCC began a pilot partner- in this issue [1], enrollment in such pro- their department members. The calculus ship program in precalculus in the fall grams has grown exponentially in the workshop consisted of two eight-hour of 1997 with two high schools and 47 past few years. days in the fall and spring semesters. students. The model we developed had These four workshops were attended the key components that follow. The last This article reports on the Partnership by approximately 25 teachers, many of item was recently added. Program in Mathematics at Nassau whom were department chairs. The at- Community College (NCC) on Long tendees received hands-on instruction • Any high school that wishes to par- Island, NY, as an example of what we on the new “fewer but deeper” ideas in ticipate in our CEP must send the believe the parameters of a success- the NCC curriculum. A second set of teachers who would teach the course ful CEP with integrity should embody. four calculus workshops was given the to a one-week summer workshop NCC, the largest single campus com- following year to a new set of partici- at NCC to prepare them to teach the munity college in New York, serves pants. course. Implicit in this preparation is Nassau County, whose population is the introduction to the NCC course over 1.3 million people. The College With the success of the calculus reform, curriculum and the textbook we enrolls over 21,000 students per year. the Harvard Consortium developed a use, as well as working out many of The Partnership Program in Mathemat- precalculus curriculum and associated the homework exercises. ics [2] began as an outreach program to materials. Again, NCC adopted this cur- • These teachers must agree to local high schools in 1997 as a result of riculum and ran four new workshops in attend a regularly scheduled a bold redesigning of NCC’s calculus precalculus. These workshops were at- users’ group at NCC to discuss courses in 1993. At that time, the NCC tended by many of the same attendees issues, examinations, upcoming mathematics department began teach- who had attended the calculus work- material, etc. ing what was then referred to as “reform shops. • School districts must agree that the calculus.” In particular, the department high school class will be visited adopted the calculus curriculum and During one of the sessions in 1996, one every second week by an NCC materials developed under a National of the high school chairs asked if he faculty member who will teach the Science Foundation grant by the consor- could offer precalculus in his high school next scheduled lesson to the high tium based at Harvard University. Since and have his students receive NCC cred- school class. NCC captured approximately 28% of all it. His reasoning was “you trained us, • School districts must agree that the Nassau County college-bound seniors at we will use the same textbook as NCC, final course grade will be computed that time, we felt in was incumbent upon we will use your exams, and we will using examinations written and graded by faculty at NCC.
6 December 2007 MAA FOCUS
• Students will be taught using The at the College. Three other course ex- dent centered instruction, the multiple Way of Archimedes: formal aminations (each worth 10% of the final representations of concepts, the differ- definitions and procedures evolve grade) are given at the high school. All ence between superficial and deep- un from the investigation of practical these exams are constructed and graded derstanding, and the shift from drill to problems. by the NCC professor. The remaining interpretation and the absence of weeks • The responsibility for learning will 35% of the final course grade is com- of review for standardized exams. This be gently shifted from teacher posed of the average of many quizzes interaction provides an appreciation for centered to student centered. and tests constructed by the high school the issues of each group, and is very • Faculty will continually encour- teacher in collaboration with the NCC useful for establishing a collegial rela- age students to adapt to ever professor. tionship. changing challenges in their academic lives. Furthermore, the As we pioneered this CEP in 1997, the In 2006, 11 NCC instructors and 19 high techniques for learning to meet these above four points seemed reasonable to school teachers were responsible for 504 various challenges may have to all parties. However, today, ten years enrollees in seven public high schools. change. later, there is an overwhelming national This year, we expect over 600 students • Course texts, syllabi and pressure to pass all students and more to enroll in one of our three mathematics materials will reflect the spirit of likely than not, award every high school courses. To date, over 3,500 high school the standards established by the and college student a grade of A or B, students have been through the program. National Council of Teachers of which, in many CEPs, translates into the In 2000, the program was recognized by Mathematics (NCTM), the use of easier texts than on the college the American Mathematics Association American Mathematical campus and a compromise on rigor and of Two-Year Colleges and was awarded Association of Two-Year Colleges curriculum. its Annenberg/CPB Foundation’s IN- (AMATYC), and meet the PUT Award for “innovative curriculum recommendations of the In [1], Bressoud has reported the data and pedagogy in introductory college Mathematical Association of that shows how few colleges maintain courses before calculus.” America (MAA). control over textbook, syllabus, final • Course quality and standards should exam, and instructor, Given this data, it References meet the recommendations appears that today the NCC program is published by The National Alliance rather uncommon, since we completely [1] David Bressoud, The Dangers of of Concurrent Enrollment control textbook selection, course syl- Dual Enrollment. See page 4. Partnerships [3] and endorsed by the labus, the final exam, and the NCC Chancellor of the State University of instructor. Our model does not depu- [2] The NCC Partnership Program in New York (SUNY). tize the high school teacher to assign Mathematics. http://www.matcmp.sun- the NCC final course grade, nor does ynassau.edu/~cheifp/partners.htm The first step in the implementation of the high school teacher have adjunct our CEP is the one-week summer work- status in the NCC mathematics depart- [3] National Alliance of Concurrent shop, which is open to all high school ment. Rather, the high school teacher Enrollment Partnerships. http://www. teachers in the county whether or not has the shared responsibility for impart- nacep.org they wish to participate in our CEP. The ing instruction and therein produces the workshop serves as a brush-up for many partnership. Although the students who Phil Cheifetz has been teaching at Nas- participants, as well as an exposure to have enrolled in finite mathematics, pre- sau Community College (NCC) for 41 the rigor and performance we expect. calculus or calculus are among the more years. He is the co-author of five mathe- mathematically talented students in matics texts and numerous articles. Phil After the completion of the course, each their respective high schools, only 88% was a founder of AMATYC and served new participating high school teacher receive a C or higher. as its president in 1978. He is the co- is introduced to his or her NCC partner director of the Partnership Program in professor. Telephone numbers and email An unexpected benefit of our regularly Mathematics. addresses are exchanged and plans are scheduled users’ group at NCC is the made for the professor’s first visit. It interaction between high school teach- Ellen Schmierer has been teaching at should be noted that the NCC professor ers and NCC faculty on topics that are NCC for 12 years. She is the co-author has this high school partnership course tangentially related to the curriculum. of three mathematics texts and has pre- as part of his or her full-time schedule. The NCC professors learn about the sented many papers at both local and At the first visit, the professor outlines awarding of significant partial credit, national mathematics meetings. Among the course expectations, hands out the parental pressure, administrative pres- her other duties, Ellen is co-director of NCC course outline, discusses policy, sure, and statewide mandates, while the Partnership Program in Mathemat- and informs the class that the NCC the high school teachers learn about the ics and handles the day-to-day opera- course-wide final examination (worth lack of student motivation, resistance to tions of the program. 35% of the final course grade) is given the shift from teacher centered to stu-
7 MAA FOCUS STT06-10_4.8125x9.qxd 11/3/06 2:49 PM Page 1 December 2007
Found Math
NATIONAL SECURITY AGENCY Do you have to pick me up in your NSA Acme uniform? It’s, like, soooooo embarrassing.”
“I certainly do,” I replied, giving her a big smoochy kiss to embarrass her further, something that didn’t really work, as the pupils in her math class were all grown up and too obsessed with number sets and parameterized elliptic curves to be bothered by a daughter’s embarrassment over her mother.
“They’re all a bit slow,” she said as we walked to the van. “Some of them can barely count.”
“Sweetheart, they are the finest minds in mathematics today; you should be happy that they’re coming to you for tutoring. It must have been a bit of Put Your Math a shock to the mathematics fraternity when you revealed that there were Intelligence to Work sixteen more odd numbers than even ones.” When you join NSA, you join a highly talented group of Mathematicians who deduce structure where it is not apparent, “Seventeen,” she corrected me. “I find patterns in seemingly random sets, and create order out of thought of another one on the bus this morning. The odd-even disparity is chaos. They apply Number Theory, Group Theory, Finite Field the easy bit,” she explained. “The Theory, Linear Algebra, Probability Theory, Mathematical hard part is trying to explain that there Statistics, Combinatorics, and more to a world of challenges. actually is a highest number, a fact that tends to throw all work regarding They exchange ideas and work with some of the finest minds infinite sets into a flat spin.” and most powerful computers in the country. And you can too, when you put your math intelligence to work at NSA. Thursday Next in First Among Se- quels, by Jasper Fforde, p. 108. For more information and to apply online, visit our Thanks to Henry Ricardo Web site.
1188 Reviews So Far!
So many books... So little cash... It helps to know what others think. MAA Reviews is the place to find reviews of www.NSA.gov/Careers mathematics books. WHERE INTELLIGENCE GOES TO WORK
http://mathdl.maa.org/mathDL/19/ U.S. citizenship is required. NSA is an equal opportunity employer. All applicants for employment are considered without regard to race, color, religion, sex, national origin, age, marital status, disability, sexual orientation, or status as a parent.
8 December 2007 MAA FOCUS
Teaching Time Savers: The Exam Practically Wrote Itself!
By Michael E. Orrison
When I first started teaching, creating I made it clear that the exercise was com- at least inspired) by the classes them- an exam for my upper division courses pletely optional, and that there would be selves. Thanks to my students, crafting was a genuinely exciting process. The no “extra credit points” assigned to it. the perfect exam is again a real joy. material felt fresh and relatively unex- I did, however, tell them that I would plored (at least by me), and I remember carefully read each submission, and that Time spent: About one minute to read often feeling pleasantly overwhelmed there was a good chance that one or two through each submission. with what seemed like a vast supply of of their problems would make their way intriguing and engrossing exam-ready onto the exam, especially if the justifi- Time saved: About 30 to 90 minutes problems. Crafting the perfect exam, cation was solid and compelling. I also of creating or searching for exam ques- one that was noticeably inviting, ex- told them that I thought this was a great tions. ceedingly fair, and unavoidably illumi- way to begin studying for the exam, and nating, was a real joy. that they were free to share their ideas Michael Orrison teaches at Harvey with their classmates. Mudd College. He is the editor of the As the years went by, however, the pro- Teaching Time Savers series. cess of creating an exam began to feel The response was wonderful. First of all, more and more like a chore. What had almost every student submitted a prob- once seemed like a vast supply of great lem together with a solid justification; problems now began to look like a nice, from their responses, I could quickly get tidy, but ultimately small collection of a sense of what they thought were the simple and uninspiring questions. The major themes of the course. Moreover, excitement had worn off, and I was be- their justifications were thoughtful and, Teaching Time Savers are articles ginning to feel like I was spending too more often than not, made their point designed to share easy-to-implement much time crafting or looking for prob- while cleverly connecting together sev- activities for streamlining the day-to- lems. I still wanted to create perfect ex- eral of those major themes. day tasks of faculty members every- ams, though, so I decided that I needed where. If you would like to share your to find a new source of inspiration. From an exam writing perspective, it favorite time savers with the readers was impossible for me not to feel as of FOCUS, then send a separate email So, a few years ago, in anticipation of though the exam was essentially writing description of each activity to Michael what seemed like another uneventful itself as I read through their responses. Orrison at [email protected]. Make exam making session, I decided to get The questions were at the right level, sure to include a comment on “time my students involved in the process. A used the right notation, and were fo- spent” and “time saved” for each ac- few days before I was to write the exam, cused on the right ideas. As you might tivity, and to include pictures and/or I gave each student a sheet of paper that expect, I found a lot of my old stand- figures if at all possible. had, printed at the top, the following: bys in the mix, but they now seemed re- freshed and ready for action. It was like Problem Proposal: Propose a problem being handed a draft of the exam, and I for the upcoming exam and explain why was the editor. It was the inspirational it should be on the exam. spark I needed. Have You Moved? The MAA makes it easy to change I explained to them that (and I really do I now use the Problem Proposal sheet in your address. Please inform the MAA believe this) one characteristic of a good all of my classes, and at least 90% of my Service Center about your change student is his/her ability to anticipate students submit a problem each time. I of address by using the electronic what an upcoming exam will look like. have found that in addition to speeding combined membership list at MAA He/she should have a solid feel for the up the process of creating an exam, it Online http://www.maa.org) or call big ideas in a course, and be able to sug- gives me an easy way to gauge student gest several questions that could act as understanding at key points through- (800) 331-1622, fax (301) 206-9789, [email protected] vehicles to demonstrate a thorough un- out the course. Moreover, the resulting email: , or mail derstanding of those ideas. This exercise exams seem to be ultimately more fair to the MAA, PO Box 90973, Wash- would, therefore, give them an opportu- and connected to my classes because, in ington, DC 20090. nity to go through that process. a very real sense, they are derived (or
9 MAA FOCUS December 2007 The MAA Euler Study Tour
The MAA’s Leonhard On July 9, we went on to Euler Study Tour took place Berlin, the city where Euler from July 1–14, 2007. The spent close to a quarter of a group of 26 intrepid travelers century in the prime of his assembled initially in the An- life in the employ of Fred- dersen Hotel in Saint Peters- erick II (sometimes known, burg, Russia on July 1. Led incorrectly according to one by Victor and Phyllis Katz, of our guides, as Frederick the group then spent six the Great). There we heard days in the former Russian several lectures on Euler’s capital, visiting places where life and work while visiting Euler lived and worked, as the Academy of Sciences well as his tomb. We also and the Humboldt Uni- were privileged to examine versity, formerly the Uni- some of Euler’s notebooks View of Basel versity of Berlin. We were and other original documents treated to another exhibit in the Archives of the Rus- of Euler’s papers, including sian Academy of Sciences. Naturally, we also visited some of the maps he worked on as part of his duties in the Academy. the main tourist sites of St. Petersburg, and also learned about At the Technical University, we learned about German gradu- mathematics and mathematics education in Russia today. ate mathematical education today and had a tour of their vir- tual reality exhibit as well as their three-dimensional printing On July 7, we flew to Basel, Euler’s birthplace. There, besides equipment. We concluded our tour with a visit to Frederick’s seeing the house where Euler grew up and the gymnasium that summer palace in Potsdam, where Euler had tried unsuccess- he attended, we also visited the houses and graves of several fully to persuade Frederick that he needed better materials in of the members of the Bernoulli family. In addition, we had the pumping system if he wanted the fountains to work. Victor a great time exploring the mathematics exhibit that had been was responsible for putting together most of the academic por- originally put together for the ICM in Madrid in 2004. tions of the tour, while Phyllis led several discussions on how to use the photographs and other information gleaned on the tour in teaching classes.
Reports and Comments from the Participants
Thoughts on the eulogy before his tomb in the Alexander Nevsky Monastery Euler Tour Cemetery (see page 12). Siu Man-Keung Thanks to the kind arrangement of a Russian friend of Andrew For me the most Potter, a few of us went to visit the Pulkovo Observatory on worthwhile part of a free afternoon. The Observatory is situated in a large bushy this Euler Study estate on the outskirts of St. Petersburg. Two things about the Tour was the op- visit particularly impressed me. The first is the spirit of enthu- portunity to read the siasm and dedication the staff possess despite the not-so-afflu- many original manu- ent condition (compared with some institutions which howev- scripts from the pen er do not accomplish as much). When I saw the old-fashioned of Euler now kept in Man Keung Siu reading his eulogy to observatory towers hidden in the bush I thought at first they the archives in St. Euler. were only historical remnants, but after explanation by the Petersburg and in staff I realized that those were still working well and had ac- Berlin. The neat and complished good work in the past. In one telescope a part is beautiful handwriting, the meticulous and copious writing and actually a bicycle chain cleverly converted to fit the purpose. the substantial content enhance my admiration and respect of It reminds me of the kind of self-reliance practiced by Chinese this great mathematician and personality. While I walked on scientists from the 1950s to the early 1980s to cope with the the spiral stone staircase and the floor of the place that the financially inadequate situation at the time, but in high spirits great man once worked in, the feeling was both strange and and with full enthusiasm. uplifting. I regarded it an honour to be able to read a short
10 December 2007 MAA FOCUS
The second impressive thing is a copy of an old book that Two Weeks Studying the bears the scar of war in the defense of the motherland dur- Life of Euler ing WWII. The bullet went straight through the pages! At the John Wilkins time there was nothing but books to build a blockade for the Observatory under heavy German fire. I have heard and read Leonhard Euler was a re- about the 29-month siege of Leningrad (former name of St. markable mathematician. Petersburg from 1924 to 1991), but this time I actually saw the Although very few math- relics of the siege for myself. ematicians would disagree with this statement, that is As someone interested in history of mathematics, the other not to say our community (even older) books in the library certainly fascinated me also. appreciates him fully. Par- Fortunately the staff succeeded in removing those precious ticipating in the tour was John Wilkins at the Idiot Café in old books to a safe place at the time of the siege; otherwise a chance to develop my St. Petersburg. we would not be able to still read them. In particular, it was appreciation of his work. fascinating to read a copy of Astronomicum Caesareum (1540) One experience that left a written by Peter Apianus for Charles V, which is full of illus- significant impression on me was the day we visited the St. Pe- trations in the form of movable hands-on devices, like those tersburg Academy. The room that contained Euler’s archived seen in a modern pop-up book for children! notebooks was small, so we had to take turns viewing them. One group toured the surrounding buildings, while the other viewed yet to be published manuscripts. To hold a notebook that Euler wrote over 250 years ago in his own hand was a very The Euler exciting moment. Some members of our talented group were Adventure able to translate a portion of the pages we were viewing from Carole Lacampagne their original Latin and German. Those translations allow us the opportunity to discuss Euler’s thinking about a particular The most exciting problem in mathematics. You can see photos of this experience event of the Euler at: http://www.maa.org/EulerTour/StP_Academy_Euler_Archives/ trip was our visit to index.htm. the Euler Archives at the Russian Academy This experience and many others have supplied me with a of Sciences. We were great wealth of material that I can use to enrich my History of actually allowed to Mathematics course. turn the pages of Euler’s notebooks Carole Lacampagne at the “Farewell and read his math- to St. Petersburg” dinner with a bala- ematical ideas and laika player. A Non-Mathematical formulas written in Moment his own hand or that Madeleine Long of his scribe. Other great experiences included hearing talks by Fred and Joel in the Old Aula in Museum lecture hall in Basil, The sky kept darkening as surrounded by busts of Euler, the Bernoulli brothers, Gauss, the rain interrupted the pace and other famous mathematicians; and going to the Humboldt of the visit through the streets University in Berlin to hear lectures by Helmut Koch and Nor- of Basel. As we stood in front bert Chapter on various aspects of Euler’s work. of one of the many Euler sights, the group of twenty Of course the sightseeing was also terrific. St. Petersburg is or so, uncomfortable yet in- an absolutely beautiful city, filled with palaces and cathedrals, tensely interested in the town rivers and bridges. Basel retains its quaint charm with old and its many historical sights, buildings and the Rhine River running though it. And Berlin, our guide, clad in a bright red Madeleine Long outside of the with all its beautiful museums, still has a flavor of the devasta- jacket that contrasted mark- Hermitage in St. Petersburg. tion of World War II. edly with the gray black of the day, raised a question. And I did this marvelous trip on what I wore on the plane from Would it be ok, she asked, if she pointed out something that Washington, a change of underwear, and a couple of borrowed was totally non-mathematical, even non-scientific? clothes loaned to me by good friends! My luggage never ar- rived. “Yes,” we all said, as if in a chorus. She then spoke of a phi- losopher she felt might be virtually unknown to most of us
11 MAA FOCUS December 2007 but was evidently quite important to her, a man who had lived ery corner. The view from and worked in Basel longer than in any other place, a phi- a 31st floor window was losopher she was extremely eager, yet quite hesitant, to tell us breathtaking. A ground about. Does anyone know who Nietzsche is? The responses level view of Basel re- that came from our group surprised and pleased our guide, vealed its old charm. I and produced a radiance brighter than any we were to see that look forward to returning entire day. Mathematicians are interested in matters that pass to Basel some day. beyond Euler and mathematicians! There is balance in the world after all! Being in Berlin conjured up memories of families Herb Kasube in front of Euler’s separated by “The Wall” tomb. Reflections on the and thoughts about how Euler Tour, 2007 far this city had come. One Herb Kasube day, my morning run took me through the infamous “Check- point Charlie” in both directions. Forty years ago that would The 2007 Euler Tour was inspiring as well as breathtaking. have been a lot more dangerous. Tears welled up in my eyes. To look at Euler’s papers in the archive in St. Petersburg was a once in a lifetime opportunity. His notebooks were journals I end these reflections where they began, with the man of the chronicling his life. What better way to look into the soul of year, Leonhard Euler. Standing next to his tomb In St. Peters- an individual? burg, I felt his presence.
The charm of the city of Basel, Switzerland radiated from ev-
Eulogy of Leonhard Euler (1707–1783)
By Siu Man Keung Respectfully read before the tomb of Euler at the Alexander Nevsky Monastery Cemetery in St. Petersburg on July 3, 2007.
To commemorate the three- Academy in St. Petersburg hundredth anniversary of the and the Prussian Academy great Swiss mathematician Le- in Berlin. The French math- onhard Euler we would have to ematician-physicist François transport ourselves to the Eu- Arago said of him, “Euler rope in the eighteenth century, calculated without apparent which can be justly hailed as effort, as men breathe, or as “the Age of Euler.” As the late eagles sustain themselves Clifford Truesdell, himself a in the wind.” The French renowned mathematician and a philosopher-mathematician life-long admirer of Euler, put Marquis de Condorcet said it, “To study the work of Euler in a eulogy of Euler, “He is to survey all the scientific life, ceased to calculate and to and much of the intellectual life breathe.” That happened on generally, of the central half of the evening of September 18, the eighteenth century.”1 It was 1783 when Euler suffered a time when academic activities Euler Notebook 6. Photograph by Herb Kasube. from a stroke at his home and scientific progress in- Eu in St. Petersburg. Today we rope took place most promi- come in homage to his burial nently in the few royal academies rather at the same academy. Among them was ground. than in the many universities. It was also the “incomparable Leonhard Euler” (in a time when no national distinction nor the words of Johann Bernoulli). Euler is well known for his prolific geographical boundary was recognized output of an order of magnitude unpar- in the pursuit of science, so that eminent Euler left his homeland at the age of alleled in history, both in quality and scholars came from different parts of twenty and spent the next fifty-six -in quantity. There is this amusing story Europe to study and work side by side credibly productive years in the Russian that the accumulation of the stack of his
12 December 2007 MAA FOCUS finished memoirs increased the virtuous deed we emu- more rapidly than the rate late : “The high mountain I these memoirs went to press, look up at it. The great road so that very often they were I travel on it.” Euler was a published in the reversed or- genius, whose height very der of completion with the few can hope to reach. But strange consequence that the we can all learn from his content of a published mem- great zest for life, work oir was more extended and and study, his insatiable improved than that of anoth- curiosity to know and to er that appeared later! probe, his determination to procure deeper and deeper Approximately one-third of understanding, his indus- the research in mathematics, try, his modesty, his gener- mathematical physics and osity, and his toughness in engineering mechanics pub- facing adversity with tran- lished in the last three-quar- quility. These are qualities ters of the eighteenth century Instruments. Photograph by Phil Scalisi. we strive for. was authored by Euler. The modern revision of his col- discoveries.3 No wonder Pierre-Simon The inscription on a plaque lected works (Opera Omnia) began in Laplace said of him, “Read Euler, read put up in his memory in his hometown 1911 and is not yet finished, with - al Euler. He is the master of us all.” His (Riehen near Basel) sums up succinctly ready near to eighty volumes published primary interest was in the wonder of the life of this simple yet great man, “Er by 1994. In 1983 a special issue of discovery and its explication, but cared war ein grosser Gelehrter und ein güt- Mathematics Magazine dedicated to his little whether he or somebody else made iger Mensch (He was a great scholar and memory listed forty-four items of math- the discovery. The late Russain historian a kind man).” ematical terms or theorems that bear his of mathematics, Adolf P. Youschkevitch, name. His versatility as an all-round re- described Euler as a well disposed man searcher is phenomenal, both in the con- not given to envy, and further borrowed 1. C. Truesdell, Leonhard Euler, Su- tinuous and in the discrete, so that he was from the famous eulogy by Bernard le preme Geometer, in C. Truesdell, An Idi- truly an expert in “con-crete” mathemat- ot’s Fugitive Essays on Science, Spring- 2 Bouyer de Fontenelle on Gottfried Wil- ics! The influence of Euler goes far be- helm Leibniz to highlight this noble er-Verlag, New York, 1984, 337-379. yond the eighteenth century and into the trait of Euler, “He was glad to observe era we live in. Just to cite two examples. the flowering in other gardens of plants 2. Concrete Mathematics is the title of His idea on the famous seven bridges of whose seeds he provided.”4 the book written by R.L. Graham, D.E. Königsberg led to the Guan-Edmonds Knuth, O. Patashnik, published by Ad- algorithm on the Chinese Postman Prob- One more thing about the man that earns dison-Wesley in 1989. The authors say lem in the early 1960s. It also led to the our admiration is the way he led a fruit- that they are not bold enough to try “dis- efficient Christofides algorithm in the ful life despite adversity. He was seri- tinuous mathematics!” mid-1970s that yields a solution to the ously ill at twenty-eight that led to loss (Euclidean) Travelling Salesman Prob- of sight in his right eye at thirty-one. He 3. Many more examples, besides the lem no worse than three-halves of the was struck by illness again at fifty-nine computation of ζ (2), can be found in optimal solution. The second example is that led to partial loss of sight in his left M.K. Siu, Euler and heuristic reasoning; in number theory. Euler calculated the eye, and at sixty-four became totally Mathematical thinking and history of famous infinite sum of the reciprocal of mathematics, in Learn From the Mas- ζ blind for the last twelve years of his life. consecutive squares, also known as At sixty-four he was further hit by mis- ters! Proceedings of the Kristiansand (2), to be π squared over six, and treated Conference on History of Mathematics ζ fortune in losing his house and proper- successfully (n) for n even. More than ties in a fire, followed two years later by and its Place in Teaching, August 1988, 240 years later Roger Apéry proved that the death of his first wife. Euler had thir- edited by F. Swetz et al, Mathematical ζ (3) is irrational, but nobody yet knows Association of America, Washington ζ teen children, but eight died in infancy. what (3) is. In recent years both in- He took all the misfortune in stride and D.C., 1995, 256-275; 279-282. terest and progress are rekindled in the was so dedicated to his work that nearly investigation of (n) for odd n. half of his research was produced when 4. A.P. Youschkevitch, Euler, in Bio- he was close to sixty. graphical Dictionary of Mathemati- As an author and teacher Euler is noted cians, Volume 2 (from Dictionary of for his clarity in exposition, but more In the ancient Chinese Book of Odes one Scientific Biography, Volume 4), edited than that he was eager to share with reads the verses that have come to con- by C.C. Gillispie, Scriber’s, New York, his readers his ideas that led to those note the noble character we admire and 1970-80, 736-753.
13 MAA FOCUS December 2007
Asking Questions: Improving Prospective Teachers’ Knowledge of Mathematics
By Wendy A. Weber In 1987 Richard J. Crouse and Clif- age them to discuss their answers with A student asks you the following ques- ford W. Sloyer published the (now out me before their presentation. tion, “When multiplying decimals, why of print) two-volume book Mathemati- do we count off decimal places from cal Questions from the Classroom (Jan- Most of the class time is spent discuss- right to left in each factor and then add son Publications, Inc.). Using this book ing the questions. Students are asked to them in order to find out where to put the as a guide, I designed a one semester read through the questions before com- decimal point in the answer?” How do hour course focused on the mathemat- ing to class and to think about how to you respond? ics needed for teaching middle and high answer them. I intentionally choose school. This course helps prospective questions that are rich and offer extend- [What is a factor? What are other repre- teachers deepen their own understand- ed conversation. We typically spend sentations of decimals? When might one ing of fundamental mathematics, make only 2–3 minutes per question, though representation of a decimal (or a frac- connections between the courses they some questions (like those included be- tion) be more useful than another? Why have taken as mathematics majors, and low) are so rich they take up to 10 min- do we line up decimal points when add- learn how to handle questions that arise utes to cover. As students become more ing and subtracting decimals? Why do in the classroom. We explore questions proficient at unpacking their knowledge, we move decimal places when perform- from the middle school level, introduc- many questions that would have taken ing long division of decimals?] tory through advanced algebra, geom- several minutes at the beginning of the etry, trigonometry, functions, probabil- course only take 30 seconds to discuss. In response to your question of how to ity, and calculus (statistics and discrete In a 50 minute class period we can usu- define yk, a student says that y is mul- mathematics are undeveloped areas in ally cover 20–25 questions. tiplied by itself k times. How do you re- the original text). ply? After each class students are expected Course Design to write solutions to all the questions [What does the student mean versus discussed from the day. I periodically what s/he says? How could you deter- The questions we explore could certain- collect the solutions and examine them mine if the student has the correct idea ly fill more than one semester hour. We for accuracy. I make comments and cor- and is only explaining it incorrectly? nevertheless chose to make the course rections on a separate piece of paper so Does the context of the question make only one semester hour, in part because that students may fix their errors before a difference as to whether you pick up our students already take a full load of they turn in all solutions at the end of this thread with the student and correct courses from our department as well as the semester. him/her?] the education department. We did not want to overburden them with more The Questions Tony can’t understand why division by hours. In addition, because all math- zero is undefined. He wants 0 ÷ 0 = 0. ematics faculty already have a full load We begin with questions from the mid- His explanation is: “If you have nothing and we did not want to reduce offerings dle school level; understanding these in one hand and you divide it by noth- that all majors could take, we restricted questions provides a foundation for sub- ing, why don’t you still end up with noth- the number of hours faculty would be sequent questions. Mathematics majors ing?” teaching. know and can perform fundamental al- gorithms, but they no longer question [What is the operation of division? Is The structure of the course is simple. why they do what they do. These ques- “zero” the same as “nothing?” What is We meet once a week for fifty minutes. tions help them unpack their knowledge 0 ÷ 0? How is this problem (or your ex- We either start class with a short quiz and rebuild their understanding into a planation) the same or different from 1 containing questions similar to ones we usable form for teaching mathematics. ÷ 0?] discussed in the previous class, or we begin with student presentations of the Examples of questions we explore fol- “If it is true that the square root of 25 is questions. In these presentations, stu- low. To illustrate the depth of questions ± 5, then why doesn’t √25 = ± 5?” dents are expected to respond as if they and the type of conversation these ques- were actually asked such a question by a tions can generate, I have included in [What does the square root symbol student in one of their classes. Students brackets after the original question the mean? What is a function? How do we know ahead of time which questions warm-up or follow-up questions I might solve x2 = 25? What are we really do- they will need to prepare, and I encour- ask. ing when we take the square root of both sides of this equation? What is √(x2)?]
14 December 2007 MAA FOCUS
The students in your class are asked to you assume an irrational number even- [What does a probability of 1/3 (or 1/4) solve the following problem. “Find the tually repeats?] mean in this situation? How can you il- roots of x3 = 2x2 + 8x.” One student does lustrate the solution?] the following work: x3 = 2x2 + 8x → x2 What is the difference between the terms – 2x – 8 = 0 → (x – 4)(x + 2) = 0 → x polynomial, polynomial equation and A student asks you why sin2(x) + cos2(x) = 4 or x = -2. Is the student correct? polynomial function? = 1.
[How many solutions should the equa- [How important is the language we use [What is the Cartesian equation of the tion have? Why can’t s/he divide by x?] as teachers?] unit circle? What are the functions sine and cosine? This question is not found A student hands in the following work: Does lim x → 0 (1/x) = ∞ or does this in Crouse and Sloyer’s text, but we be- (6t)√3 = (6√3)(t√3). How would you limit not exist? lieve it is needed.] help this student? [What does the symbol ∞ mean? Is The class has been very successful. It [What property is s/he attempting to there a difference between saying a limit helps prospective teachers learn to draw use? How would you help this student equals infinity and it does not exist? Is on their considerable mathematical see that his/her work is incorrect? How there always a difference?] knowledge to accurately answer student would you help this student see that questions at the correct level. Students multiplication distributes over addition Why is an inverse function a reflection enter their student teaching experience and subtraction but not over multiplica- about the line y = x? armed with 200 questions that cover tion?] nearly all the topics they will teach. [Geometrically, what is a reflection? More importantly, they enter the teach- How do you know for sure that some ir- What is an inverse function? If the origi- ing profession able to answer their stu- rational number does not have a repeat- nal point is (x,y) and the reflected point dents’ questions with much more than a ing decimal eventually? is (y,x), how do we find the line of re- simple “because.” flection and what is it?] [What is an irrational number? A ratio- Wendy A. Weber is Associate Professor nal number? What is a series that con- You ask your students to find the prob- of Mathematics at Central College in verges to a rational number? How can ability of both children being boys in a Pella, Iowa. She can be reached at we- we write any repeating decimal as a two child family. One student says the [email protected]. rational number? Why are we allowed probability is 1/3; another says it is 1/4. to do these operations? What happens if Is either student correct?
Calculators at the Smithsonian
On September 25, 2007, in commemoration of the 40th anniversary of its invention of the handheld calculator, Texas Instru- ments donated one of the original handhelds to the Smithsonian. Speakers at the donation ceremony included Dr. Brent D. Glass, Director of the National Museum of American History, Melendy Lovett, President of Texas Instruments Education Technology Division, and Jerry Merryman, one of the three inventors. The original, along with an array of the calculators that followed, was on display. Photographs and caption courtesy of Robert Vallin.
15 MAA FOCUS December 2007 First Annual TENSOR–SUMMA Grant Program a Success By Efraim Armendariz and Carole Lacampagne
The first TENSOR-SUMMA grants zona, and North Carolina A&T State urday morning program designed to ex- were awarded in April 2007, thanks University (an HBCU) are concerned pose high school sophomores to math- to the generous support of the Tensor with the transition from high school to ematics in fun and innovative ways. As Foundation. The awards program was college. The University of Texas at San part of the SLAM Math Career Series, designed to provide grants to support Antonio’s Number Olympics was held three actuaries from the International college or university faculty conduct- in September. Project director Eduardo Society of Black Actuaries, a Boston ing programs to encourage the pursuit Duenez reports that they attracted a university bioinformatics professor, and enjoyment of mathematics among diverse pool of almost 50 participants. and an information technology special- middle, high school, and beginning col- Duenez recruited UTSA freshmen at ist spoke to the students who will later lege students from groups traditionally orientations days in his “Tensor Hut” present a poster session on these topics. under-represented in mathematics. The with currency in TENSOR bills which quality of the proposals submitted was came in denominations of 101 and 1010 Bemidji State University’s Colleen Liv- very high and had strong relevance to tensors. Students earned tensors by solv- ingston and Fort Lewis College’s Carl the goal of the grant program: engaging ing problems and used them to purchase Lienert are both holding week-long more minority students in activities that prizes. summer mathematics camps for middle potentially open doors into mathemati- or high school students, many of whom cally based professions. Twelve propos- The University of Arizona’s David Sav- come from near-by Indian reservations. als were recommended for funding. itt and William Yslas Velez held a one- week, four hour-a-day workshop for Fayetteville State University, an HBCU, Two awards were given for Math Cir- entering calculus students. Twenty-one the University of Minnesota, and Mount cles to serve inner city school students: students participated. An added bonus Holyoke College received grants to sup- the DC and the Oakland Math Circles. was that many of the participants en- plement existing successful programs. Project Director Jamylle Carter, of San joyed the interaction with the math ma- Fayetteville’s Project Director Kim- Francisco State University, reported that jors so much that they chose to major in berly Smith Burton is adding a techni- the first five weeks of the Oakland Math mathematics! North Carolina A&T State cal literacy component to her Saturday Circle were spent on plane symmetries University’s project director Alexandra Academy for middle school students. and tessellations. A seventh grade boy Kurepa is also looking to the transition In September their teachers received in- from the Circle said, “I love everything from high school and college. Kurepa struction in using the National Science you’re doing because it’s stuff I’ve never has designed a program to provide en- Digital Library which they will pass seen before. It rocked!” Project Direc- richment activities for high school stu- on to their students. The University of tor Dan Ullman of George Washington dents enrolled in AP or Honors Calculus Minnesota’s Project Director Harvey University is conducting his Math Cir- and precalculus as well as for students Keynes and Mount Holyoke College’s cle at the Carriage House of the MAA. in the Early College program. She says Project Director D. James Morrow are DC middle school students have a lively that students from the Early College making a concerted effort to attract more time experimenting, conjecturing, find- program find the experience especially young women of color to their summer ing counterexamples, debating, proving, helpful with their transition from high program programs. and generalizing. A grant for another school to college. Math Circle went to Joseph Meyinsse The deadline for receipt of the second from Southern University, an HBCU. Wentworth Institute of Technology’s annual TENSOR-SUMMA Grant ap- Amanda Hattaway directs Students plications is February 15, 2008. All in- Projects from the University of Texas Loving Adventures in Mathematics novative projects are welcomed. See the at San Antonio, the University of Ari- (SLAM), a yearlong, twice a month Sat- MAA website for details. Found Ego-Boost
Today I discovered the Public Library of Science journals, free, open-access academic journals in Biology, Medicine, Computational Biology (Sweet!), Pathogens, Neglected Tropical Diseases, and Clinical Trials. For once Google was not my informer, but FOCUS (the MAA newsmagazine).
—Eric “SigmaX” Scott Thinking Aloud, 11/07/2007. http://sigmax.no-ip.org/blog/?p=966
16 December 2007 MAA FOCUS
TENSOR-MAA Grants for Women and Girls in Mathematics
In April 2007, the TENSOR-MAA pro- Paul, Minnesota, held its annual summer with plans for another CaMeW (Career gram made grants of up to $6,000 to 17 residential camp for sixteen high school Mentoring Workshop) for women fin- institutions for faculty members to con- girls June 24-29, 2007. Activities under ishing their PhD’s in mathematics. duct programs to encourage women and the direction of Lisa Rezac, included girls to continue the study of mathemat- math mini-courses, talks on connections An entirely different program was held ics. Among these, seven were renewals in biology and mathematics, symmetry, for a group of 50 high school women at of previously funded programs. topology and mathematical sculpture as Bloomsburg University of Pennsylva- well as a career panel and a trip to the nia, each of whom was given a scholar- Renewal Grants Science Museum of Minnesota to see ship for the First Annual Computer Fo- the calculus exhibit. rensics Summer Experience for Young The Joy of Thinking girls’ math club pro- Women (CFSEYW). In this week-long gram at Texas Tech University is now in CSU Stanislaus, under the direction of program directed by Elizabeth Mauch, its fifth year and can boast a record num- Viji Sundar, conducted a week-long in- young women were introduced to the ber of seven schools participating in fall tensive Summer Academy called Prepar- mathematics and computer science 2007. Five of these schools have student ing Women for Mathematical Modeling- necessary for the field of computer fo- populations almost entirely members of Robotics (PWMMR) which included a rensics by studying numerical analysis, underrepresented groups. Project Direc- one day field trip to The Tech Museum statistics, mathematical modeling, and tor, Jerry Dwyer, wrote of his work in of Innovation in San Jose. There will finite mathematics. FOCUS, January 2006. be a one-day follow-up in March 2008 when the participants will return to cam- For many years Hampshire College has The MPower program at Metropolitan pus to participate in a math conference had a highly successful summer pro- State University (Saint Paul, MN) of- on “Mathematical Moments” based on gram for pre-college students who love fered its second annual, five-day summer the AMS series. mathematics. In an effort to enroll more camp to 21 middle school girls living in women, HCSSiM recruited its female Saint Paul. Director Rikki Wagstrom is The Girls Math & Technology Camp alumnae to mail customized postcards preparing to reunite these girls for a day was held July 22-28, 2007, on the Uni- to recruit new girls to apply to HCSSiM of mathematics in November 2007. versity of Nevada, Reno campus for 60 2007. Under the direction of sarah-ma- middle school girls of diverse back- rie belcastro, they held mathematical The TENSOR-MAA program usually grounds from across Northern Nevada. and discussion activities with alumnae makes one grant for a Math Day. In 2006 A web site will provide participants and during the program to enhance the pro- and again in 2007, Hood College hosted their parents year-round support and gram for girls. their Sonya Kovalevsky Day for local encouragement in math and technology high school girls. Betty Mayfield will from director, Lynda Wiest. The first day of the USD Math Days for direct the next day March 10th, 2008. Women program was in October 2007 Members of the AWM Student Chapter New Grants at the University of South Dakota, Ver- help with the day of workshops, panels million SD; a second day will be in De- and fun. The ten grants for new programs were cember and the third in February. The made. These programs range in scope as young women participants from the sur- This is the second year of the Piney their directors seek new ways to reach rounding area high schools worked on Woods Lecture Series in Mathematics, candidates who would have benefit from origami constructions and Geometer’s a program which invites well-known this work. Sketchpad exercises in addition to learn- female mathematicians to campus to ing about the usefulness of mathematics present talks which are accessible to Rachelle De Coste (now at Wheaton in various careers. Communication be- undergraduates. The women then meet College), recognized a need for assis- tween the director, Violetta Vasilevska, informally with students during a recep- tance in the transition year from com- program leaders, students and their tion, and have dinner with nominated pleting the dissertation to gaining em- teachers is by email where weekly prob- female students. In the past 18 months ployment. With the help of the Military lems are posed. there were five speakers and four are Academy where she was employed she scheduled for spring 2008. More infor- conducted a highly successful program At the University of California Santa mation is available at http://www.shsu. in August 2007 which brought together Barbara, graduate students are making edu/~mth_jaj/pwls/ or by contacting at West Point ten women participants, plans to attend the Joint Meetings in San project director, Jacqueline Jensen. five junior faculty “mentors” in- addi Diego utilizing some of the grant funds. tion to senior faculty mentors from West The group, let by Maria Isabel Bueno Another summer program is in its sixth Point for a two day conference. They Cachadina and Birge Huisgen-Zimmer- year: The University of St. Thomas in St. will reconvene at the Joint Meetings mann, organized activities in the fall of
17 MAA FOCUS December 2007
2007 which included a documentary in their algebra preparedness under the sented minority women in the math- film showing of The Gender Chip, five guidance of project director, Alejandra ematical sciences. Conference activi- professors talking about their research Sorto. ties included plenary talks by Iris Mack and, of course, accompanying social ac- and Freda Porter, research talks, student tivities one of which was with women At the time this report was written the poster presentations, roundtable discus- undergraduates. second three-day residential workshop sions, and panels on graduate studies planned by Janet Kaahwa to refresh 20 and professional development along Catherine O’Neil directed a new pro- teachers of mathematics from ten schools with workshops for high school stu- gram at Barnard College which funded in three districts (Masindi, Kibaale, and dents and the development of mentoring a day of mathematics and physics in Hoima) in Uganda was completed. The circles at every stage of the educational partnership with the nearby Urban As- facilitators were six women mathema- and professional pipeline. sembly Institute for Math & Science for ticians who acted as role models af- Young Women. The entire school con- ter participating in the first workshop In Fall 2007, under the leadership of sisting of 78 sixth grade girls came to held at Makerere University, School of Jacqueline Dewar, Women and Math- Barnard campus for a day and listened Education, Department of Science, and ematics for Future Teachers collabo- to lectures, did puzzles, won prizes, and Technical Education. rated with various programs at Loyola competed in a group math competition Marymount University (the future K-8 for tickets to Six Flags. The Infinite Possibilities Conference teacher preparation program, Women’s (IPC) sponsored by the non-profit Build- Studies, Honors program) to cross- Fifteen Hispanic and economically dis- ing Diversity in Science, North Carolina list a team-taught course on women advantaged young girls, Grades five and State University, and the Statistical and and mathematics in Spring 2008. This six, were granted scholarships to partici- Applied Mathematical Sciences Insti- course will examine historical and cur- pate in a two-week long summer Math tute (SAMSI) directed by Kimberly rent equity issues in mathematics educa- Camp at Texas State University San Weems was held at the North Carolina tion and math-related careers through a Marcos. The girls were challenged to State University, November 2–3, 2007. study of the biographies and mathemati- think and do mathematics on their own, This conference was designed to foster cal contributions of nine women mathe- and demonstrated a significant increase increased participation of underrepre- maticians from the 4th through the 20th centuries.
18 December 2007 MAA FOCUS Remember the Alamonoid: The Art of Factorization in Multiplicative Systems in San Antonio
By Scott T. Chapman In a typical beginning abstract algebra class, students are in- ters level work, summer REUs, or even capstone undergradu- troduced to non-unique factorizations into products of irreduc- ate projects. ibles by the example In keeping with the theme of the workshop, the participants 6= 2 ⋅3=() 1 + ––5() 1 –5 represented a broad mathematical spectrum: about 60% col- lege or university faculty and 40% undergraduate or graduate D = Z– 5 in the ring . Here the elements 2, 3, 1,+ students. Trinity University’s Holt Conference Center offered –,5 1––5 are all irreducible and are pairwise nonassociate, an intimate setting for the meeting. i.e., the two different factorizations are not generated simply by multiplying factors by invertible elements. A careful analy- Why might this topic be worthy of such a general gathering of sis of this example shows that ()1+ –– 5 ()1– 5 and 2⋅ 3 interested mathematicians? Since 1988 several hundred papers are, up to associates, the only factorizations of into products of dedicated to non-unique factorizations in commutative rings irreducibles in D. Moreover, D has the following nice proper- or monoid have appeared in refereed mathematical journals ty: if x is a nonzero nonunit of D and a1,, a 2 KK,,an b1,,b 2 , bm or conference proceedings (see the references in [4] for a irreducible elements of D with x= a1 LLan = b1 bm , then n complete listing). Two major conference proceedings ([1] and = m . [3] ) dedicated primarily to factorization problems have been recently published by Marcel Dekker and Chapman & Hall. It is observations such as these which led mathematicians to Additionally, 2006 has seen the publication of a major mono- become interested in the behavior of non-unique factorizations graph by Geroldinger and Halter-Koch [4] which details the in other mathematical structures. Problems which involve current state of research in this field. non-unique factorizations of elements in integral domains and monoids have become frequent topics in undergraduate and The workshop began with keynote addresses by the workshop graduate mathematics over the past 35 years. Solving such leaders, Scott Chapman of Trinity University and Jim Coyk- problems requires techniques from multiple areas of algebra endall of North Dakota State University. The morning sessions and discrete mathematics. consisted of a series of lectures by three leading researchers in this area: David Anderson from the University of Tennessee A group of 33 participants gathered for five days from May at Knoxville, Alfred Geroldinger from Karl-Franzens-Univer- 20th to May 25th 2007 at Trinity University in San Antonio, sität in Graz, Austria, and Ulrich Krause from Universität Bre- Texas for an MAA-PREP Workshop entitled The Art of Fac- men in Bremen, Germany. In combination, the main speakers torization in Multiplicative Structures. The workshop, unlike a offered over 80 years of classroom experience, in addition to regular research conference, had multiple goals: over 200 refereed publications. The afternoon sessions includ- ed seminars by two graduate assistants, Paul Baginski from • To present in accessible fashion the fundamental results of the University of California at Berkeley and Amanda Matson the theory of non-unique factorizations. from North Dakota State University, who provided extensive experience in working with factorization problems at both the • To demonstrate that such results involve not only commu- undergraduate and graduate level. The afternoons concluded tative algebra, but also rational and additive number theory, with a series of problem/discussion sessions which not only combinatorics, discrete structures and the geometry of Nk . addressed problems posed by the main speakers, but also how factorization material can best be used in the classroom and in • To provide to college and university level faculty and gradu- undergraduate research projects. ate students a rich source of material which not only can sup- plement existing courses in algebra, number theory and com- The three main speakers focused on four different approaches binatorics, but also can be taught as a stand-alone advanced to factorization theory prevalent in the literature today. undergraduate course or as an introductory graduate level course. Algebraic: How do ring properties influence factorizations? Do polynomial rings, power series rings and other such struc- • To demonstrate that while advanced results in this area in- tures inherit the factorization properties of their base rings? volve deep mathematics, many problems are suitable for mas-
19 MAA FOCUS December 2007
Combinatorial: An in-depth analysis of the factorization and L(m) can be computed with elementary number theory as properties of arithmetically motivated monoids (such as sub- seen in the table below. It turns out (and isn’t hard to check) monoids of the natural numbers). A study of the connection of that ρ()m ≤ 3/2 for all m and ρ()m = 3 2 if m ≡ 0()mod 6 . factorization problems to problems involving zero-sum theory Thus ρ()S = 3 2 and the elasticity is accepted. in finite abelian groups. This includes a study of the calcula- tion of the elusive Davenport Constant and cross number. The workshop’s website, located at http://www.trinity.edu/ schapman/ArtofFactorization.htm, is a multipurpose electronic Geometric A study of factorization properties in submonoids resource for not only participants, but all those interested in of Nk . An exploration of the connection between factorization the subject. This site contains almost all of the materials used properties of integral domains and the structure of cones in during the meeting including background material, slides, ex- Nk. ercises, and a list of principal open problems. A list of projects appropriate for undergraduate students is under development. Number Theoretic: How do number theoretic properties in- fluence factorizations? What role do factorization properties Despite the workshop’s full schedule, participants were able play in some classical problems from number theory (e.g. rep- to enjoy an afternoon off to explore San Antonio’s celebrated resentation of integers by quadratic forms)? How completely Riverwalk and the Alamo. A group of participants even braved are orders in rings of integers determined by only considering the 60 mile trek to Luckenbach, Texas, but (alas) did not catch factorization properties? a glimpse of Willie Nelson.
We offer the reader one example of the combinatorial/number References theoretic aspects described above. This approach to factoriza- tion theory studies combinatorial sets and constants associ- [1] D. D. Anderson (ed.), Factorization in Integral Domains, ated to particular multiplicative structures. Suppose M is an Lect. Notes Pure. Appl. Math. 189(1997), Marcel Dekker, monoid in which every element has an irreducible factoriza- New York. tion. If x∈ M is not a unit, then the set of lengths of x is the set of integers n such that x can be factored as a product of n irre- [2] S. T. Chapman, On the Davenport constant, the cross num- ducibles. Write these lengths in increasing order: {n1,n2,…,nk}. ber, and their application to factorization theory, Lecture Notes We can look at several ways of describing this set. We define in Pure and Applied Mathematics, Marcel Dekker, 171(1995), l ()x= n1 to be the length of the shortest factorization and let 167-190. x L(x) = nk to be the longest. We define ∆ () to be the set of M differences ni – ni–1, and ∆ () to be the union of all the sets [3] S. T. Chapman (ed.), Arithmetical Properties of Com- of differences as x ranges through the non-units in M. mutative Rings and Monoids, Lect. Notes Pure. Appl. Math. 241(2005), Chapman & Hall/CRC, Boca Raton. The elasticity of an element x∈ M , denoted ρ()x , is the quotient L() x / l ()x . The elasticity of M is then defined as [4] A. Geroldinger and F. Halter-Koch, Non-Unique Fac- x ρ(M)= sup{ ρ ( x) x∈ M\} M We say that M has accepted torizations: Algebraic, Combinatorial and Analytic Theory, elasticity if there exists x∈ M such that ρ()x = ρ()M . Chapman & Hall, Boca Raton, 2006.
Determining these sets and constants, even for simple monoids, Scott Chapman teaches at Trinity University in San Antonio, can be tricky, as we demonstrate with the set of integers S = TX. He can be reached at [email protected]. {0,2,3,4,5,6,7,…} under regular addition. Notice that 2 and 3 are the only irreducible elements of S. Hence, an irreducible m()mod 6 L ()ml () m factorization of n∈ S is of the form n= x ⋅ 2+n ⋅ 3. Factor- m m 1 2 0 izations in S are far from unique; for example, 2 3 m− 3 m − 4 1 +1 + 2 17 =7 ⋅ 2 +1 ⋅ 3 =4 ⋅ 2 +3 ⋅ 3 =1 ⋅ 2 +5 ⋅ 3 . 2 3 m m − 2 2 +1 Thus, 2 3 m − 3 m l()17 = 6 , L(17) =8. ∆ 17 = 1 and ρ 17 =8 6 = 4 3. 3 +1 () {} () 2 3 m m − 4 Notice that the longest factorization of an element n∈ S con- 4 + 2 tains the most possible copies of 2 and the shortest the most 2 3 l m m− 2 m − 2 possible copies of 3. Hence, modulo 6 the exact values of () 5 +1 +1 2 3 20 December 2007 MAA FOCUS Remembering Paul Cohen
By Peter Sarnak Paul Joseph Cohen, one of the stars of As an instuctor at MIT, Paul was in- 20th century mathematics, passed away troduced to the question of uniqueness in March, 2007 at the age of 72. Blessed for the Cauchy problem in linear partial with a unique mathematical gift for differential equations. Alberto Calderon solving difficult and central problems, and others had obtained uniqueness re- he made fundamental breakthroughs in sults under some hypotheses, but it was a number of fields, the most spectacu- unclear whether the various assump- lar being his resolution of Hilbert’s first tions were essential. Paul clarified this problem — the continuum hypothesis. much studied problem by constructing examples where uniqueness failed in the Like many of the mathematical giants of context of smooth functions, showing in the past, Paul did not restrict his atten- particular that the various assumptions tion to any one specialty. To him math- that were being made were in fact nec- ematics was a unified subject which one essary. He never published this work could master broadly. He had a deep un- Paul Cohen. Photograph (other than putting out an ONR Techni- derstanding of most areas, and he taught courtesy of Stanford News cal report) but Lars Hörmander incor- advanced courses in logic, analysis, dif- Service. porated it into his 1963 book on linear ferential equations, algebra, topology, In Chicago he formed many long last- partial differential equations, so that it Lie theory, and number theory on a reg- ing friendships with some of his fellow became well known. This was one of ular basis. He felt that good mathemat- students (John Thompson, for example, many instances in which Paul’s impact ics should be easy to understand and that who remained a lifelong close friend). on mathematics went far beyond his it is always based on simple ideas once published papers. He continued to have you got to the bottom of the issue. This The period after he graduated with a a keen interest in linear PDEs and taught attitude extended to a strong belief that PhD was very productive: he enjoyed a graduate courses and seminars on Fou- the well recognized unsolved problems series of successes in his research. He rier Integral Operators during the 70s, in mathematics are, firstly, the heart of solved a problem of Walter Rudin’s in 80s, and 90s. the subject and, secondly, that they have Group Algebras and soon after that he clear and transparent solutions once obtained his first breakthrough on what After spending some time in Princeton the right new ideas and viewpoints are was considered to be a very difficult at the Institute for Advanced Study, Paul found. This belief gave him courage to problem — the Littlewood Conjecture. moved to Stanford in 1960; there he re- work on notoriously difficult problems He gave the first nontrivial lower bound mained for the rest of his life. He said throughout his career. for the L1 norm of trigonometric poly- that getting away from the lively but nomials on the circle whose Fourier co- hectic mathematical atmosphere on the Paul’s mathematical life began early. efficients are either 0 or 1. The British East Coast allowed him to sit back and As a child and a teenager in New York number theorist-analyst Harold Daven- think freely about other fundamental City he was recognized as a mathemati- port wrote to Paul saying that if Paul’s problems. He has described in a num- cal prodigy. He excelled in mathemat- proof held up, he would have bettered a ber of places (see, for example [Yan- ics competitions, impressing everyone generation of British analysts who had dell]) his turning to work on problems around him with his rare talent. worked hard on this problem. Paul’s in the foundations of mathematics. By proof did hold up; in fact, Davenport 1963 he had produced his proof of the After finishing high school at a young was the first to improve on Paul’s result. independence of the continuum hypoth- age and spending two years at Brooklyn This was followed by work of a num- esis, as well as of the axiom of choice, College he went to graduate school in ber of people, with the complete solu- from the axioms of set theory. His ba- mathematics at the University of Chica- tion of the Littlewood Conjecture being sic technique to do so, that of “forc- go. He arrived there with a keen interest achieved separately by Konyagin and ing,” revolutionized set theory as well in number theory, which he had learned Mcgehee-Pigno-Smith in 1981. as related areas. To quote Hugh Woodin by reading some classic texts. There he in a recent lecture “it will remain with got his first exposure to modern math- In the same paper, Paul also resolved us for as long as humans continue to ematics and it molded him as a math- completely the idempotent problem for think about mathematics and truth.” A ematician. He tried working with André measures on a locally compact abelian few years later Paul combined his inter- Weil in number theory but that didn’t group. Both of the topics from this paper est in logic and number theory giving a pan out. Instead, he studied with Antoni continue to be very actively researched new decision procedure for polynomial Zygmund, writing a thesis in Fourier today especially in connection with ad- equations over the p-adics and the reals. Series on the topic of sets of uniqueness. ditive combinatorics. His very direct and effective solution of
21 MAA FOCUS December 2007 the decision problem has proved to be Atle Selberg and especially the “trace central in many recent developments in formula”. This became my thesis topic. the subject [Macintyre]. Paul and I spent one or two years go- ing through this paper of Selberg and After 1970 Paul published little, but he providing detailed proofs of the many continued to tackle the hardest prob- theorems that were announced there. We lems, to learn and to teach mathemat- wrote these up as lecture notes, which ics and to inspire many generations of both Paul and I used repeatedly over mathematicians. I was a member of the the years as lecture notes for classes new generation who was lucky enough that we taught. Sections of these notes to be around Paul. have found their way into print (in some cases with incorrect attribution) but un- I first heard of Paul when I was still fortunately we never polished them for an undergraduate in Johannesburg in publication. the 1970s. I was taken by the works of Gauss, Dirichlet and Riemann, but Paul continued to work on the Riemann studying with them was not an option. Hypothesis till the end, not for the glory, However Paul, whose work on the con- but because he believed in the beauty of tinuum problem is recorded in any good the problem and expected that a solution introductory text to mathematics, was would bring a deep new understanding apparently alive and well and living of the integers. As mentioned above, it in California. Moreover his reputation The Stuyvesant Math Team, Fall 1948. was his strong belief that such problems and stories of his genius had reached Top row, left to right: Unknown, Un- have simple solutions once properly all corners of the mathematical world. known, Elias Stein, Harold Widom, Paul understood. This gave him the courage Kathy Driver (then Kathy Owen and to- Cohen. Bottom row: Unknown, Martin to continue this life long pursuit. When day Dean of Science at the University Brilliant, Unknown. Photograph cour- working on such problems one is out of Cape Town) had just returned from tesy of Martin Brilliant, from Indicator, there alone, with nothing to fall back on. Stanford with the following sort of de- January 1949. Most professional mathematicians sim- scription of Paul, which she e mailed me so in an instant my interests changed ply don’t take this kind of risk. recently. and moved in that direction too. At Stanford Paul and his wife Christi- “Paul was an astonishing man. Impa- Given his stature and challenging style na hosted many dinners and parties for tient, restless, competitive, provocative he was naturally intimidating to students students (graduate and undergraduate), and brilliant. He was a regular at cof- (and faculty!). This bothered some; it is faculty and visitors to the department. I fee hour for the graduate students and probably the reason he had few graduate remember many occasions where Paul the faculty. He loved the cut-and -thrust students over the years. I have always would treat a visitor to a personal guid- of debate and argument on any topic felt that this was a pity, because one ed tour of San Francisco and the bay and was relentless if he found a logical could learn so much from him (as I did) area. This opening of their house and weakness in an opposing point of view. and he was eager to pass on the wealth their hospitality is remembered fondly There was simply nowhere to hide! He of understanding that he had acquired. by many mathematicians around the stood out with his razor-sharp intellect, Once one got talking to him he was al- world. his fascination for the big questions, his ways very open and welcomed with en- strange interest in “perfect pitch” (he thusiasm and appreciation your insights Paul’s passing marks the end of an era at brought a tuning fork to coffee hour and when they were keener than his (which Stanford. The world has lost one of its tested everyone) and his mild irritation I must confess wasn’t that frequent). As finest mathematicians and for the many with the few who do have perfect pitch. a student I could learn from him results of us who learned so much from him He was a remarkable man, a dear friend in any mathematical area. Even if he and spent quality time with him, it is dif- who had a big impact on my life, a light didn’t know a particular result he would ficult to come to terms with this loss. with the full spectrum of colours.” eagerly go read the original paper (or, I should say, he would skim the paper, of- References I set my goal to study with Paul in the ten creating his own improvised proofs foundations of mathematics and was of key lemmas and theorems) and then B. H. Yandell, The Honors Class, A.K. lucky enough to get this opportunity. rush back to explain it. Peters, Ltd. 2002, pages 59--84. Paul lived up to all that I expected. Soon after I met him he told me that his inter- His ideas on the Riemann Hypothesis A. Macintyre, in records of proceedings ests had moved to number theory and in (about which I will write elsewhere) at meetings of the London Mathematical particular the Riemann Hypothesis, and led him to study much of the work of Society meeting, June 22, 2007.
22 December 2007 MAA FOCUS Statistics: A Key to Student Success in College and Life
By John Loase
A few years ago, Ben Fusaro asked multiple regression, and nonparametric and require students to write essays and me what I thought was the most impor- statistics for a second, optional course. attend book discussions on the readings. tant college course for an undergraduate. If they fail to attend the discussion (with I replied in an instant: Statistics. Ben Some might feel that this amounts to the essay as ticket for admission), they agreed. His innovations in mathematical lowering standards. But consider the lose a letter grade. My assigned books modeling and environmental mathemat- collateral damage of finishing the book. are Man’s Search for Meaning (Frankl), ics appeal to our advanced students. Sta- Do you really reach the students by cov- Our Neglect, Denial, and Fear (Loase), tistics, on the other hand, could help all ering a different topic every day with and The Millionaire Next Door (Stanley undergraduates in both their future per- little time to apply the concept to the and Danko). The three essays, which go sonal and professional lives. student’s future personal and profes- through a revise-rewrite cycle, coupled sional life? Are we denying students an with a required short research paper, In 2005, the Department of Education opportunity to become an English teach- constitute 1/3 of the students’ grade. released a report that revealed mathe- er or social worker because he/she can- Each student is required to write a two matics to be the most serious obstacle to not understand Chebyshev’s Theorem? page (400-500) word essay critiquing college graduation in the United States. each book and discussing the statistical The report described mathematics as an I confess to ignorance as to where Iraq issues that I raise from each book. “insurmountable barrier” for economi- is on the modern globe. My high school cally disadvantaged college students. and college French were not adequate Man’s Search for Meaning is one of the We should be looking to fix this. for me to obtain routine directions while twentieth century’s great works. I relate in Paris. We believe that students need statistics to the book by having each We can make a difference in student arithmetic and algebra as life skills. As a student complete a survey to quantify retention, effect long-term influence to- consequence, we frequently end up de- students’ perceptions of their positive ward our students, and share an enjoy- nying them a college degree, which is influence toward others. We discuss able, almost fun, experience. Students essential for economic survival. Are we this concept at the book seminars. As a in my statistics courses are successful. abusing our power by maintaining the mathematician-psychologist, the books Concordia College–NY has wonder- status quo without scrutinizing the role I assign may be different from the ones ful students with mean SAT scores in mathematics plays in winnowing the you assign. Statistics has the potential to mathematics at approximately the na- preferred jobs? prepare students for their future person- tional average, but this approach could al and professional life with relevance be replicated at any college. I came up Rely on the TI-83 Calculator that few courses can rival, and using it with many of these strategies during to think about books such as these high- my eighteen years teaching statistics at Many (if not most) of our students can- lights that potential. SUNY–Westchester Community Col- not read their textbook. Some students lege. The resulting course was enjoy- will never learn elementary algebra, If you find the readings interesting, your able for the students and for me. despite our best efforts. The calcula- students may as well. At worst, they will tor compensates for their mathematical be amazed by our idiosyncratic tastes. Here is how. I rely on four moves that handicaps. I wrote a 92 page book, Es- Experiment. Have fun. If you are not are the results of two decades of experi- sentials of Elementary Statistics (TI-83 having fun, your students are not. ence. Based), that I give each student for free. For every topic I teach in statistics, my I highly recommend The Millionaire Shorten the List of Topics book shows the students the step-by- Next Door, an outgrowth of extensive step key strokes to perform the statistical data mining and statistical analysis, as If I did not use a topic from statistics in analysis. This becomes my main text for one of your required readings. It fo- my first five published interdisciplin- statistics. Write me and I will send you a cuses on money and how to become ary books or my consulting work (both copy of my book, that you can duplicate rich in America — a high interest topic college and legal), then I did not teach and distribute to your students, until that for everyone. I suggest to the students that topic. We are still left with seven or blessed day when (or if) it is published. that they advise their “Baby Boomer” eight key topics: probability, small and I assign Triola’s masterful Elementary parents about retirement, using material large sample confidence intervals, sam- Statistics for additional homework and from the book. I also caution them that ple size, one and two sample hypothesis as an invaluable lifelong reference text. their parents do not expect us to teach testing of the difference of means and their children anything that could help proportions, correlation, and linear re- Have Fun them. Be gentle with parents with weak gression. That is plenty. Postpone anova, hearts: the shock of being counseled by I assign three interdisciplinary books their children may be too much to bear. 23 MAA FOCUS December 2007
Allow Students In its educationally disadvantaged stu- students’ executioners, by whom they dents. At another college, the Director are screened out of Engineering, Medi- If a student places in college arithmetic, of Counseling regularly informed me cine, and MBAs. Mathematicians have which is in fact a euphemism for fourth that the high rate of remedial mathemat- the tools and the intellect to scrutinize grade arithmetic, the conditional prob- ics failure was cutting the graduation our largely unchallenged assumptions ability of that student graduating from rate in half. Effective remedial courses about what constitutes essential math- college within five years is in the same are costly, resource intensive, and only ematical knowledge for college gradu- ballpark as one of us becoming a major available to a small fraction of the un- ates. We could lead other professions in league athlete. We are no better than the der-prepared mathematics students. a spirit of societal transformation. elementary, junior high, and senior high Those same students could succeed in a mathematics teachers, who failed to im- statistics course. We can prepare all our students for their part the fundamentals of arithmetic, and future professional challenges by tap- they had 180 days a year. We have 45– Few professions scrutinize the fairness ping the vitality of statistics. We need 60 class sessions. Yes, some students of their credentialing system, which cre- to prune the curriculum, add enrichment may earn a C in statistics without having ates winners and losers. In fact, several activities, empower students with the mastered addition of fractions. But that professions increase their mathematics TI-83 Calculator, and (if remediation student now has an improved chance of requirements whenever supply exceeds does not work) give students a chance to graduating from college, earning a de- demand. Several years ago, I attended succeed in statistics. cent living, and hiring an accountant to an MAA session on “Why is Math so do his/her taxes. Unpopular?” After an hour of missing John Loase is Chair of the Mathematics the painful truth, I raised my hand and Department at Concordia College-NY. I consulted at one college that had a asked whether math was so unpopular failure rate far in excess of 50% among because we mathematicians serve as the
It is Time to Support Extremes
By Reza Noubary Traditionally, statistics has focused on part of their analysis. Specifically, some average the neighboring values and to the study of values with high frequen- methods treat the data as a message and move it along the time axes. cies and on averages. This becomes clear seek to decompose it into a systematic when we examine textbooks and course (deterministic, trend, signal) part and a Why Extreme Values? descriptions for introductory statistics random (stochastic, noise) part. courses. In today’s world, however, it In many applications, it is not appropri- is not sufficient to focus on averages When the form of the systematic part ate to focus on averages alone. In fact, alone. It has become important and even is known, it is easy to separate it from there are many instances where extreme necessary to study extremes and rare the random part. In the absence of in- values and values with low frequen- events, since they are usually critical, formation pertaining to either system- cies are of more concern. Examples in- newsworthy, and are often accompanied atic or random parts, smoothing is used clude: large natural disasters compared by severe consequences. for separation. This is usually carried to moderate ones; weak components or out assuming that the systematic part is links compared to their average counter- The celebrated central limit theorem smooth and the random part is rough. parts; large insurance claims compared has given statistics its focus on averages Smoothing is an exploratory operation, to average claims. — we do what we know how to do. The a means of gaining insight into the na- statistical theories of extremes are less ture of data without precisely-formu- Extreme values are usually analyzed simple, less unified, and more recent. lated models or hypotheses. Smooth- using one of the three major theories: However they are not less important and ing is often achieved by some sort of The “Extreme Value Theory” that deals there is a need for their inclusion in in- averaging (low-pass filtering). Once with maxima or minima of the subsam- troductory statistics courses. the smooth part is determined, the dif- ples, the “Threshold Theory” that deals ference between the original message with values above or below a specified Why Averages? and the smooth part is used to present threshold, and the “Theory of Records” the rough part. The rough part is usu- that deals with values larger (smaller) Most of the elementary mathematics ally utilized to make reliability state- than all the previous values. we teach deals with the “smooth” part ments regarding the systematic part. For of the world. In the same way most data with a time index (time series), one These theories deal with the actual classical statistical methods are based popular smoothing technique is the so- values of the extremes. The frequen- on the idea of smoothing the data as a called moving average. The idea is to cies of extremes are analyzed using the 24 December 2007 MAA FOCUS
“Theory of Exceedances.” This theory attraction of one of the three types are tent, a matter of judgment depending on deals exclusively with number of times known. Only distributions unbounded what is considered large or small. a chosen threshold is exceeded. In most to the right can have a Fréchet distribu- applications, the number of exceedances tion as a limit. Only distributions with The theory is very useful when model- and values of excesses over a threshold finite right end point can have a Weibull ing large values based on observed large are combined to yield a more detailed as a limit. The Gumbel distribution can values is of main concern. Clearly the analysis. For example, insurance com- be the limit of bounded or unbounded modeling and prediction of large earth- panies analyze both the number of times distributions. quakes should be based on past large a “large” claim is made and the amount earthquakes not on the medium or small by which these claims exceed a specific How do we decide which of the three earthquakes. large threshold. Reliability engineers limiting distributions to fit to the data? study both the number of extreme loads, Theoretically, we can use the fact that Theory of Records such as large earthquakes, and their each of the classical distributions falls in magnitudes. the “domain of attraction” of one of the The theory of records deals with values limiting distributions above. This works that are strictly greater than or less than Extreme Value Theory if distribution of the original data is all previous values. Usually the first val- known. Unfortunately, in practice such ue is counted a record. Then a value is a Extreme value theory deals with annual information is not usually available and record (upper record or record high) if it (or any other periodic) maxima or min- decisions should be based on the area of is bigger than all previous values. The ima. Specifically, the theory is based on application or on expert opinion. When study of record values, their frequen- dividing the sample into a subsamples information about the appropriate lim- cies, times of their occurrences, their and fitting a distribution for maxima or iting distribution is absent, statistical distances from each other, etc. consti- minima of the subsamples. For example, goodness-of-fit may be used. tutes the theory of records. data may consist of largest earthquakes in California for each of the last 100 Since extreme value distributions are Formally the theory deals with four main years. fitted to, for example, annual maxima random variables: the number of records or minima, some relevant data related in a sequence of n observations, the re- As in most statistical theories, first the to the years with several large observed cord times, the waiting time between distribution of the largest or the smallest values could be discarded and some less the records, and the record values. It is values was derived for a finite sample. informative data related to the years interesting to note that the first three can Then, by letting sample size tend to with no real large values could be re- be investigated using non-parametric or infinity, the limiting distribution of ex- tained. The threshold theory discussed distribution-free methods whereas the treme values was obtained. For this a next avoids this problem. last one requires parametric methods. typical maximum Yn is reduced with a location parameter βn and a scale pa- Threshold Theory Records in general and sports records in rameter α n (assumed to be positive) particular are of great interest, and their such that the distribution of standard- Threshold theory allows one to make occurrence usually results in a great deal ized extremes (Yn – βn ) /α n is non-de- inferences about the values above or be- of excitement and the media attention. generate. The forms of the limiting dis- low a threshold, that is, the upper or the tributions are specified by the extreme lower tails of a distribution. It considers Traditionally, data values with high fre- value theorem. This theorem states that the excesses, the differences between quencies and averages have been the there are three possible types of limit- the observations over the threshold and focus of statistical analysis and model- ing distributions for maxima. They are the threshold itself. Like extreme value ing. This has rested on the celebrated known as the Gumbel distribution (Type distributions, there are three models for central limit theorem. Extremes, rare I), the Fréchet distribution (Type II), tails. They are long tail Pareto, medium events, and records values have been and the Weibull distribution (Type III). tail exponential, and short tail distribu- mostly avoided and often treated as out- These three forms can be combined to tion with an end point. liers rather than important information. yield the so-called generalized extreme But low-frequency high-consequence value distribution. Again, most classical distributions fall events are very important. We think that in domain of attraction of one these tail there is a need to introduce students to Most classical distributions fall in the models. It has been shown that the natu- these vital concepts. domain of attraction of one of these ral parametric family of distributions to three types. For example, the distribu- consider for excesses (tail) is the Gen- Reza Noubary teaches at Bloomsburg tion of maxima of samples from a nor- eralized Pareto Distribution (GPD). In University, in Bloomsburg, PA. His re- mal distribution tends to the Gumbel practice the proposed method is to treat search interests include time series distribution. More generally, the neces- the excesses as independent random analysis, geostatistics, reliability, risk sary and sufficient conditions for a par- variables and to fit the GPD to them. analysis, and applications of mathemat- ticular distribution to fall in domain of The choice of threshold is, to a large ex- ics in sports.
25 MAA FOCUS December 2007
EMPLOYMENT OPPORTUNITIES
ALASKA contributions to the educational pro- sity invites applications for three tenure grams of the Department at the graduate track assistant professorships, to begin University of Alaska and undergraduate levels. in September 2008. A doctorate in math- Southeast (Juneau) ematics is required. A solid commitment Assistant or Associate Professor of Applications with a curriculum vitae, a to teaching, and strong evidence of re- Mathematics at the University of Alaska list of publications, a research outline, an search potential, are essential. We are Southeast (Juneau): This is a bipartite account of teaching experience, a state- looking for (1) one person who will be (teaching and service), tenure-track ment on teaching, and the names, phone expected to conduct research with un- position beginning August 2008. The numbers and e-mail addresses of at least dergraduate students and (2) two people position will remain open until filled; four references (with one addressing the who will be expected to teach some however, first consideration will be giv- teaching record) should be sent to: courses in our graduate program. Grad- en to applications received by January uate courses include, but are not limited 11, 2008. Members of the search com- Professor Bruce Shepherd to, year-long sequences in Abstract and mittee will participate in the Employ- Chair, Discrete Mathematics or Contin- Linear Algebra, Applied Mathemat- ment Center at the 2008 Joint Meeting. uous Optimization Search Committee ics, Financial Mathematics, Real and Go to www.uakjobs.com/applicants/ Department of Mathematics Complex Analysis, and Probability and Central?quickFind=60450 for position and Statistics Statistics. In addition, the successful announcement and for information McGill University candidates will share a willingness to about the UAS Mathematics Program 805 Sherbrooke Street West participate in the university’s core cur- go to http://www.uas.alaska.edu/math/. Montreal QC H3A 2K6 Canada riculum, which includes two semesters UAS is an AA/EO Employer and Edu- of mathematics for all undergraduates. cational Institution. Candidates must arrange to have the let- ters of recommendation sent directly to Fairfield University, the Jesuit Univer- ARKANSAS the above address. Candidates are en- sity of Southern New England, is a com- couraged to include copies of up to three prehensive university with about 3,200 Henderson State University selected reprints or preprints with their undergraduates and a strong emphasis Mathematics, Assistant Professor or applications. To ensure full consider- on liberal arts education. The department Instructor: Doctorate in mathematics, ation, applications must be received by offers a BS and an MS in mathematics. mathematics education or related field January 15, 2008. The MS program is an evening program preferred; master’s degree in same re- and attracts students from various walks quired; teach full range of undergraduate To facilitate notification of the outcome of life – secondary school teachers, courses. For more information and ap- of the search, candidates should send an eventual Ph.D. candidates, and people plication procedures, please visit www. email to the address DMCOSearch08@ working in industry, among others. The hsu.edu/Affirmative-Action. Henderson math.mcgill.ca at the time of applica- teaching load is 3 courses/9 credit hours State University, Arkadelphia AR is an tion. per semester and consists predominantly affirmative action/ADA, EOE. Women of courses at the undergraduate level. and minorities are especially encour- McGill University is committed to eq- aged to apply. uity in employment and diversity. It Fairfield offers competitive salaries and welcomes applications from indige- compensation benefits. The picturesque CANADA nous peoples, visible minorities, ethnic campus is located on Long Island Sound minorities, persons with disabilities, in southwestern Connecticut, about 50 McGill University women, persons of minority sexual miles from New York City. Fairfield is The Department of Mathematics and orientations and gender identities and an Affirmative Action/Equal Opportu- Statistics invites applications for a ten- others who may contribute to further nity Employer. For further details see ure-track position in discrete mathemat- diversification. All qualified applicants http://cs.fairfield.edu/mathhire. Appli- ics or continuous optimization. While are encouraged to apply; however, in cants should send a letter of applica- the appointment is expected to be made accordance with Canadian immigration tion, a curriculum vitae, teaching and at the level of an Assistant Professor, the requirements, priority will be given to research statements, and three letters Department would consider applicants Canadian citizens and permanent resi- of recommendation commenting on the for a senior position. The candidate dents of Canada. applicant’s experience and promise as a must have a doctoral degree at the date teacher and scholar, to Matt Coleman, of appointment. They are also expected CONNECTICUT Chair of the Department of Mathemat- to have demonstrated the capacity for ics and Computer Science, Fairfield independent research of excellent qual- Fairfield University University, Fairfield CT 06824-5195. ity. Selection criteria include research The Department of Mathematics and Please indicate in your cover letter the accomplishments, as well as potential Computer Science at Fairfield Univer- position for which you are applying.
26 December 2007 MAA FOCUS
Full consideration will be given to com- lakeland.edu. Review of applications Colby is a highly selective liberal arts plete applications received by January will begin January 28, 2005 and will college located in central Maine. The 15, 2008. We will be interviewing at the continue until the position is filled. For college is a three-hour drive north of Joint Mathematics Meetings in San Di- additional information on Lakeland Col- Boston and has easy access to lakes, ego, January 6-9. Please let us know if lege, please access our web site at www. skiing, the ocean, and other recreational you will be attending. Lakeland.edu. and cultural activities. For more infor- mation about the position and the de- DISTRICT OF COLUMBIA An affirmative action / equal opportu- partment, visit our web site at www. nity employer colby.edu/math. American University Tenure-track Assistant Professor in the IOWA Colby is an Equal Opportunity/Affirma- Mathematics and Statistics Department tive Action employer, committed to ex- at American University, beginning Fall University of Iowa cellence through diversity, and strongly 2008. Qualified candidates will have Actuarial science tenure-track assistant encourages applications and nomina- a strong background in Mathematics professor starting 8/08. Ph.D. required tions of persons of color, women, and or Statistics. Ph.D. and teaching expe- 8/20/08. Actuarial Fellowship or Asso- members of other under-represented rience required. American University ciateship highly preferred. Industrial ex- groups. For more information about the is an EEO/AA employer. Minority and perience helpful. Duties include teaching College, please visit the Colby Web site women candidates are encouraged to and research in actuarial science and/or at www.colby.edu. apply. See math.american.edu/positions, financial mathematics, involvement in or contact the Department of Mathemat- Ph.D. program, and supervision of Ph.D. MARYLAND ics and Statistics at (202) 885-3120 for students. Selection begins 12/03/07. CV, details. three reference letters, and transcript for Montgomery College recent Ph.D.s to Actuarial Search, Sta- A Learning College FLORIDA tistics & Actuarial Science, Univ. of Office of Human Reaources Iowa, Iowa City, IA 52242-1409.actu- Your Resource to Excellence Lakeland College [email protected] Consider Montgomery College for your Assistant Professor of Mathematics next career move! Lakeland College, an independent lib- Women, minorities encouraged to apply. eral arts institution located in northeast- AA/EOE. Montgomery College, one of Mary- ern Wisconsin, invites applications for a land’s oldest community colleges, is full-time, tenure-track position of Assis- MAINE a multi-campus institution located in tant Professor – Mathematics, beginning Montgomery County, Maryland. It has in August, 2008. The candidates should Colby College earned a reputation as one of the leading have the ability to teach a wide range of Mathematics Department community colleges in the nation by ap- undergraduate courses in Mathematics The Department of Mathematics at Col- plying a winning formula: excellence in and the ability to teach some computer by College invites applications for a one- teaching; innovative partnerships; and science courses will be an advantage. year sabbatical replacement position in unrivaled leadership. The candidates should be committed mathematics at the assistant professor or to quality teaching and advising and be instructor level, beginning September 1, Montgomery College is seeking a full- willing to participate in the College’s 2008. Ph.D. in mathematics preferred; time Mathematics faculty member for interdisciplinary general education pro- A.B.D. considered. Five course teaching the Rockville Campus. Assignment be- gram. load. Evidence of exceptional teaching gins August, 2008. For more details ability is required. The ability to teach a and to apply online, please visit: https:// A PhD in mathematics is required. Some course in the history of mathematics is jobs.montgomerycollege.edu. college teaching experience is preferred. desirable but not required. Compensation will be commensurate Starting salary range is $44,096 to with background and experience and a Send curriculum vitae, a statement on $63,596 per year. Online applications comprehensive benefit program is- of teaching and research, and three letters must be received by 12 noon, Tuesday fered. of recommendation (all in hard copy) January 2, 2007. to: Mathematics Search Chair, Depart- Interested candidates should submit a ment of Mathematics, Colby College, Mathematics Faculty – Position #2688 letter of interest, current resume, a state- 5830 Mayflower Hill, Waterville, ME – Rockville Campus ment of teaching philosophy and three 04901. We cannot accept applications in current letters of recommendation to: electronic form. Review of applications Required Qualifications: Director of Human Resources, Lakeland will begin on January 15, 2008 and will • A Master’s Degree or Ph.D. in Math- College, P.O. Box 359, Sheboygan, WI. continue until the position is filled. ematics, Mathematics Education, or a 53082-0359; or email to: schoemerjr@
27 MAA FOCUS December 2007 strongly related field. stown, MA 01267. Consideration of The John Wesley Young Instructorship • At least 18 semester hours of gradu- applications will begin on November is a postdoctoral, two- to three-year ap- ate level courses in mathematics. 15th and continue until the position is pointment intended for promising Ph.D. • Coursework that supports the can- filled. Williams College is dedicated to graduates with strong interests in both didate’s capacity to teach the spectrum providing a welcoming intellectual en- research and teaching and whose re- of courses offered by the Mathematics vironment for all of its faculty, staff and search interests overlap a department Department. students; as an AA/EOE employer, Wil- member’s. Current research areas in- • At least one year of recent full-time liams especially welcomes applications clude applied mathematics, combinator- experience (or the equivalent) teaching from women and minority candidates. ics, geometry, logic, non-commutative mathematics courses typical of commu- geometry, number theory, operator al- nity college offering. Williams College gebras, probability, set theory and to- Desirable Qualifications : The Williams College Department of pology. Instructors teach four ten-week • Involvement in programs for improv- Mathematics and Statistics invites ap- courses distributed over three terms, ing student success. plications for one tenure track position though one of these terms in residence • Experience in using graphing calcu- in mathematics, beginning fall 2008, may be free of teaching. The assign- lators. at the rank of assistant professor (in an ments normally include introductory, • Experience using computer algebra exceptional case, a more advanced ap- advanced undergraduate, and graduate systems such as Matlab, Maple or Math- pointment may be considered). We are courses. Instructors usually teach at least ematica in mathematics courses. seeking a highly qualified candidate who one course in their own specialty. This Application Procedures: has demonstrated excellence in teaching appointment is for 26 months with a In order to be considered for the posi- and research, and who will have a Ph.D. monthly salary of $4667, and a possible tion, you must complete the following by the time of appointment. 12 month renewal. Salary includes two- steps: month research stipend for Instructors in • Create the Montgomery College on- Williams College is a private, residen- residence during two of the three sum- line employment application. tial, highly selective liberal arts college mer months. To be eligible for a 2008- To create an application and to apply on- with an undergraduate enrollment of ap- 2011 Instructorship, candidate must be line, please visit: https://jobs.montgom- proximately 2,000 students. The teach- able to complete all requirements for the erycollege.edu ing load is two courses per 12-week se- Ph.D. degree before September, 2008. • After you complete your application, mester and a winter term course every Applications may be obtained at http:// please be sure to search jobs and click other January. In addition to excellence www.math.dartmouth.edu/recruiting/ or on the Rockville Campus Mathematics in teaching, an active and successful re- http://www.mathjobs.org Position ID: position and apply to the job. search program is expected. 237-JWY. General inquiries can be di- rected to Annette Luce, Department of PLEASE NOTE: To apply, please send a vita and have Mathematics, Dartmouth College, 6188 • The online application deadline is 12 three letters of recommendation on Kemeny Hall, Hanover, New Hamp- noon, Wednesday, January 2, 2008. teaching and research sent to the Hiring shire 03755-3551. At least one referee • Incomplete applications will not be Committee, Department of Mathemat- should comment on applicant’s teaching considered. ics and Statistics, Williams College, ability; at least two referees should write Office of Human Resources – 240-567- Williamstown, MA 01267. Teach- about applicant’s research ability. Ap- 5353. ing and research statements are also plications received by January 5, 2008 Montgomery College is an equal oppor- welcome. Evaluation of applications receive first consideration; applications tunity employer committed to fostering will begin on or after November 15 will be accepted until position is filled. a diverse academic community among and will continue until the position is Dartmouth College is committed to di- its student body, faculty, and staff. filled. Williams College is dedicated versity and strongly encourages applica- to providing a welcoming intellectual tions from women and minorities. MASSACHUSETTS environment for all of its faculty, staff and students; as an EEO/AA employer, NEW YORK Williams College Williams especially encourages applica- Williams College Department of Math- tions from women and minorities. For Niagara University ematics and Statistics invites applica- more information on the Department of www.niagara.edu The Mathematics De- tions for a newly authorized visiting po- Mathematics and Statistics, visit http:// partment at Niagara University, a pri- sition in mathematics for the 2008-2009 www.williams.edu/Mathematics. vate Catholic institution sponsored by year, at the rank of assistant professor. A the Vincentian Community seeks As- Ph.D. is required. Send a vita and three NEW HAMPSHIRE sistant Professor, tenure track, August letters of recommendation on teaching 2008 start. and research to: Visitor Hiring Com- Dartmouth College mittee, Department of Mathematics and John Wesley Young Research Instruc- Requirements: Strong commitment to Statistics, Williams College, William- torship undergraduate teaching, ability to do
28 December 2007 MAA FOCUS scholarly research, Ph.D. in Mathemat- gust 2008 in a department that has perience with ability to teach all levels ics, Statistics or related field. graduated the western region outstand- of college math and Master’s Degree in ing mathematics education student of mathematics or Master’s Degree with 18 Candidate to teach a broad range of the year for NC for each of the past 17 graduate semester hours in mathematics classes from introductory statistics to years. Doctorate in mathematics educa- preferred. Apply online at http://jobs. senior seminar; should possess the abil- tion or a related field with the equiva- sc.com or visit www.cctech.edu. ity to work with students both inside and lent of a master’s in mathematics is re- outside the classroom, including student quired from an appropriately accredited Questions? Contact the Personnel Of- research; should be interested in teach- institution. Excellence in teaching, a fice by email: [email protected] ing mathematics to pre-service teach- commitment to ongoing research and or by mail: Central Carolina Technical ers. service are required. Screening be- College gins December 10, 2007 and continues Located near scenic Niagara Falls, Ni- until the position is filled. To apply 506 N. Guignard Drive agara University is a predominantly un- visit: https://jobs.wcu.edu/applicants/ Sumter, SC 29150 dergraduate liberal arts university. Central?quickFind=51041. Western Car- EOE/AA olina University, a campus of the Uni- Application letter, vitae and three rec- versity of North Carolina, is an AA/EOE TEXAS ommendation letters: Dr. Richard Cra- employer that conducts background mer-Benjamin, Chairperson, Mathemat- checks before employment. Proper The University of Texas at Tyler ics Department, Niagara University, NY documentation of identity and employ- The University of Texas at Tyler invites 14109-2044. Applications reviewed un- ability are required at the time of em- applications for the position of Chair til position filled. AA/EOE ployment. of the Department of Mathematics to begin fall 2008. The university seeks NORTH CAROLINA PENNSYLVANIA candidates who will energetically lead the department in continuing to build Western Carolina University California University of PA excellent undergraduate and graduate Department Head, Mathematics Tenure-track position at California Uni- programs and will mentor faculty in and Computer Science versity of PA in MATHEMATICS: Ef- teaching, research, and service. Western Carolina University invites ap- fective August 2008. A PhD in Mathe- plications for the position of Head of matics is required. Teaching experience The successful candidate will have a the Mathematics and Computer Science at the college/university level, and PhD in mathematics, an outstanding Department. This is a twelve-month practical experience, such as consult- record of teaching and research com- position, with a requested start date of ing, is preferred. The successful candi- mensurate with a tenured faculty ap- July 1, 2008. A doctorate in the Math- date will have a strong desire to teach, pointment, effective leadership, admin- ematical Sciences from an appropriately actively recruit students, and partici- istrative, and interpersonal skills, and accredited institution, scholarly creden- pate in the development of web-based the ability to lead the faculty in obtain- tials and experience commensurate with courses and/or programs. Candidates ing external funding. appointment at the rank of Full Professor should be experienced in the use of in- or Associate Professor, and evidence of novative approaches that are student- Located 90 miles east of Dallas in the excellence in teaching, research, and ac- centered, inquiry-based, and hands-on beautiful piney woods of East Texas, ademic leadership are required. Screen- oriented. For more information about The University of Texas at Tyler has an ing begins January 10, 2008 and contin- the position, and instructions about how enrollment of about 6000 students. The ues until the position is filled. To apply to apply (deadline is January 11, 2008): Department of Mathematics offers de- visit: https://jobs.wcu.edu/applicants/ http://www.cup.edu/employment/index. grees at the undergraduate and graduate Central?quickFind=51042. Western Caro- jsp . Cal U is M/F/D/V/AA/EEO. levels. For general information about lina University, a campus of the Univer- The University of Texas at Tyler, visit sity of North Carolina, is a vibrant and SOUTH CAROLINA www.uttyler.edu. The Department of growing university, and is an AA/EOE Mathematics has a web site at http:// employer that conducts background Central Carolina Technical College math.uttyler.edu. checks before employment. Proper Mathematics Instructor (2 positions): documentation of identity and employ- #020901MA & #139217MA. Respon- Please submit (electronically as attach- ability are required at the time of em- sible for teaching college credit algebra, ments, if possible) a letter of applica- ployment. statistics, trigonometry, geometry and tion, curriculum vitae, unofficial tran- calculus classes at main and off campus scripts, a brief description of research Western Carolina University locations. Requires Master’s Degree in plans, statement of teaching philosophy, Mathematics Education mathematics or a Master’s Degree with statement of leadership philosophy, Tenure-track Assistant Professor in 18 graduate semester hours in math- and names and email addresses of at mathematics education beginning Au- ematics. Post-secondary teaching ex- least four references to Dr. Don Kille-
29 MAA FOCUS December 2007 brew, Chair, Department of Mathemat- ics Chair Search Committee, dkille@ mail.uttyl.edu. Paper submissions can be sent to Department of Mathematics, The University of Texas at Tyler, 3900 University Blvd., Tyler, Texas 75799. Review of applications will begin imme- diately and continue until the position is filled. Applicants must be prepared to furnish the university with proof of eli- gibility to work in the United States. UT Tyler is an Equal Employment Opportu- nity/Affirmative Action Employer. UTAH Brigham Young University Applications are invited for visiting professorships at any level (assistant, associate, or full professor) in the De- partment of Mathematics at Brigham Young University. The department has a strong commitment to undergraduate education and has a small, but strong, doctoral program.
Qualifications: Ph.D. in mathematics or related field; demonstrated excellence in teaching; desire and ability to teach undergraduate and graduate level math- ematics, including courses at the fresh- man level; desire and ability to conduct high quality mathematical research. Responsibilities: Teaching mathematics at the undergraduate and graduate level, including courses at the freshman level; conducting and directing independent research; collaborating with other fac- ulty at the university; supporting uni- versity, college, and department goals and missions.
Review of applications will begin on December 1, 2007 and will continue until the position is filled. To apply, go to http://yjobs.byu.edu. For further in- formation about the department and the university go to http://math.byu.edu.
Brigham Young University is an equal opportunity employer. Preference is given to qualified candidates who are members in good standing of the affiliated church, The Church of Jesus Christ of Latter-day Saints.
30 December 2007 MAA FOCUS The Mathematical Association of America ® Invites you to The Exclusive World Premiere Screening of Hard Problems The Road to the World’s Toughest Math Contest a film by George Csicsery
Follow the journey of the six high school students who represented the United States in 2006 at the world’s toughest math competition – the International Mathematical Olympiad (IMO).
Hard Problems gives you behind-the-scenes access to an extraordinary group of individuals. Enter a world of dedication, perseverance, and rigorous preparation. Experience the joys and frustrations related to trying to solve incredibly challenging math problems against competitors from 90 countries and against the clock.
Join the MAA for a reception and the world premiere Tuesday, January 8 6:00-8:00 p.m. please refer to the JMM program for room location
Stop by and meet George Csicsery at the American Mathematics Competitions Booth Wednesday, January 9 from 9:00-11:00 a.m.
Pick up your copy of the DVD at the MAA Publications Booth for the discounted price of $17.50* at anytime during the meeting!
All prices final, sale only good during 2008 Joint Mathematics Meetings in San Diego Price includes 7.75% sales tax
31 MAA FOCUS December 2007 Do you know someone who loves π as much as pie?
Would they also love a full-tuition scholarship for a master’s degree in mathematics education, a New York State Teaching Certificate, and a $90,000 stipend in addition to a competitive salary as a New York City secondary school math teacher? Math for America offers fellowships with these benefits and more to individuals who know and love math, enjoy working with young people, have excellent communications skills, www.mathforamerica.org and a strong interest in teaching. Log on to learn more.
The Mathematical Association of America Periodicals Postage paid at 1529 Eighteenth St., NW Washington, DC and Washington, DC 20036 additional mailing offices
32