May/June 2004

FOCUS May/June 2004

FOCUS is published by the Mathematical Association of America in January, February, March, April, May/June, FOCUS August/September, October, November, and December. Volume 24 Issue 5 Editor: Fernando Gouvêa, Colby College; [email protected] Managing Editor: Carol Baxter, MAA Inside [email protected] Senior Writer: Harry Waldman, MAA [email protected] 4 Convergence: A New Online Magazine from the MAA Please address advertising inquiries to: By Victor Katz Frank Peterson [email protected] President: Ronald L. Graham 5 SIGMAA QL is Formed By Rick Gillman President-Elect: Carl C. Cowen First Vice-President: Barbara T. Faires, 6 The Archives of Mathematics at the Center for American History Second Vice-President: Jean Bee Chan, By Kristy Sorensen Secretary: Martha J. Siegel, Associate Secretary: James J. Tattersall, Treasurer: John W. Kenelly 8 The Teacher as Artist: A Letter to My Colleagues Executive Director: Tina H. Straley By Peter Taylor Associate Executive Director and Director of Publications: Donald J. Albers 10 What I Learned by Teaching Proof and Logic By Peter Ruane FOCUS Editorial Board: Rob Bradley; J. Kevin Colligan; Sharon Cutler Ross; Joe Gallian; Jackie Giles; Maeve McCarthy; Colm 12 Mathematics Magazine Editor Search Mulcahy; Peter Renz; Annie Selden; Hortensia Soto-Johnson; Ravi Vakil. 13 2004 Summer Short Course—Teaching and Doing Knot Theory Letters to the editor should be addressed to Fernando Gouvêa, Colby College, Dept. of 14 Remembering Bill Chinn Mathematics, Waterville, ME 04901, or by email to [email protected]. By Jean Bee Chan Subscription and membership questions should be directed to the MAA Customer 15 Letters to the Editor Service Center, 800-331-1622; e-mail: [email protected]; (301) 617-7800 (outside 16 MAA Contributed Paper Sessions: U.S. and Canada); fax: (301) 206-9789. MAA Atlanta Joint Mathematics Meeting, January 5-8, 2005 Headquarters: (202) 387-5200. Copyright © 2004 by the Mathematical 21 MAA President Ronald L. Graham Testifies Before House Association of America (Incorporated). Educational institutions may reproduce Appropriations Subcommittee on Funding for articles for their own use, but not for sale, Mathematics Education provided that the following citation is used: By Harry Waldman “Reprinted with permission of FOCUS, the newsletter of the Mathematical Association of America (Incorporated).” On the cover: The MAA has just launched Convergence, an online magazine about the History of Mathematics. See the article on page 4. Periodicals postage paid at Washington, DC and additional mailing offices. Postmaster: Send address changes to FOCUS, Mathematical Association of America, P.O. FOCUS Deadlines Box 90973, Washington, DC 20090-0973. August/September October November ISSN: 0731-2040; Printed in the United States Editorial Copy July 8 September 16 of America. Display Ads July 9 August 20 September 24 Employment Ads June 11 August 13 September 10

2 FOCUS Sir Michael Atiyah and Prof. Isadore Singer Win the 2004 Abel Prize The Norwegian Academy of Sciences and Letters has awarded the 2004 Abel Prize jointly to Sir Michael Francis Atiyah, of the University of Edinburgh, and Isadore M. Singer, of the Massachu- setts Institute of Technology. Atiyah and Singer will receive the prize, wrote the Academy, “for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.” The Atiyah-Singer Index Isadore M. Singer Theorem is considered one of the great Sir Michael Francis Atiyah landmarks of twentieth-century mathematics, influencing many of the most as one of their most important contri- Michael Francis Atiyah was born in 1929 important developments in topology, butions. in London and was educated at Trinity differential geometry, and quantum field College, Cambridge. He spent most of his theory. An account of the theorem aimed Isadore M. Singer was born in 1924 in career at Cambridge and Oxford. He oc- at a general mathematical public can be Detroit and was educated at the Univer- cupied the famous Savilian Chair of Ge- found on the Abel Prize web site at http:/ sity of Michigan and the University of ometry at Oxford and later was Master /www.abelprisen.no/nedlastning/2004/ Chicago. He joined the faculty at the of Trinity College in Cambridge. popular_english_2004.pdf. Massachusetts Institute of Technology in 1950, and has been there ever since. He While in Cambridge, he helped create the A few decades ago, physicist Freeman has received several prizes for his work, Isaac Newton Institute for Mathemati- Dyson pointed out that “the marriage including the Steele Prize for Lifetime cal Sciences and became its first direc- between mathematics and physics, which Achievement. Accepting that prize, he tor. Atiyah received the Fields Medal in was so enormously fruitful in past cen- said, “For me the classroom is an impor- 1966. He has been Sir Michael Atiyah turies, has recently ended in divorce.” The tant counterpart to research. I enjoy since 1983, and was made a member of situation has changed dramatically since teaching undergraduates at all levels, and the Order of Merit in 1992. then; Dyson’s “divorce” turned out to be I have a host of graduate students, many only a temporary separation. Atiyah and of whom have ended up teaching me For more information on the Abel Prize Singer, both jointly and individually, have more than I have taught them.” Singer and on this year’s winners, visit http:// been instrumental in repairing the rift, has also written several important text- www.abelprisen.no/en. See also the April and the Norwegian Academy noted this books. 2004 Devlin’s Angle on the MAA web site.

An Update on Hales’ Proof of Kepler’s Conjecture

In 1998, Thomas C. Hales announced it. An attempt to check everything by per will be split in two parts. The more that he had proved Kepler’s Conjecture hand ended badly: “[The referees] have theoretical part will be published by the about the best possible sphere packing not been able to certify the correctness Annals and the computer-intensive part in three-dimensional space. His claim of the proof, and will not be able to cer- will be published in another journal, Dis- generated controversy, mostly because of tify it in the future, because they have run crete and Computational Geometry. the crucial role of computers in the (very out of energy to devote to the problem,” long) proof. As Hales says in the most said Annals editor Robert McPherson. Hales himself has opted for a more radi- recent version, “nearly every aspect of the The Annals then considered publishing cal route. He has started the Flyspeck proof relies on computer verification.” the paper with a disclaimer, to the effect Project, an attempt to create a formal The issue was put into sharp relief by the that the editors could not certify that proof of Kepler’s Conjecture. The idea is editors of the Annals of Mathematics, who every detail was correct. to create (via computer) a version of the were leery of publishing a paper whose proof of the conjecture that can be au- correctness they could not verify. An article in the April 6 issue of The New tomatically verified line-by-line. You can York Times, entitled “In Math, Comput- read more about this at http:// The problem, for the editors, was to what ers Don’t Lie. Or Do They?,”reports that www.math.pitt.edu/~thales/flyspeck/ extent they should trust mathematics the Annals editors have decided to solve index.html. done on the computer without checking the problem in a different way. The pa- 3 FOCUS May/June 2004 Convergence: A New Online Magazine from the MAA By Victor Katz

The MAA announced the launching of magazine will report on past meetings a new online magazine on the history of and give notice of future meetings. mathematics and its use in teaching, en- Where abstracts are available for a par- titled Convergence: Where Mathematics, ticular meeting, these will be included. History and Teaching Interact, with the We may also include copies of handouts financial support of the National Science for easy access, as well as links to the Foundation. The target audience is author’s webpage, if available. teachers of grades 9-14 mathematics, be they secondary teachers, two- or four- Initially the magazine will be free to all. year college teachers, or college teachers However, a subscription fee will be nec- preparing secondary teachers. (“Grade essary after the initial period. We hope 9–14 mathematics” encompasses algebra, Frank Swetz and Victor Katz to secure institutional as well as personal synthetic and analytic geometry, trigo- Reviews of current and past books, ar- subscriptions. Information as to cost and nometry, probability and statistics, el- ticles, and teaching aids on the history access will be provided on the Conver- ementary functions, calculus, linear al- of mathematics of use to teachers, as well gence site shortly. gebra, and differential equations.) The as reviews of websites providing infor- initial editors of the magazine are Victor mation on the history of mathematics. Currently, we have a limited supply of J. Katz, from the University of the Dis- articles in our pipeline. Because our goal trict of Columbia, and Frank Swetz, from Classroom suggestions. These may be self- is to bring out new material on a regular Penn State University, Harrisburg. contained articles showing how to use basis, we need a continual flow of articles history in the teaching of a particular and classroom suggestions. We therefore Among the types of material that will topic or they may be materials closely welcome your ideas for articles as well as appear in the magazine are the follow- related to a main article, showing in some your completed manuscripts. In particu- ing: detail how to use the article in a class- lar, we welcome short classroom sugges- room setting. tions that can immediately be imple- Expository articles dealing with the his- mented by teachers. Materials should be tory of various topics in mathematics Historical problems. These problems will sent both in hardcopy and electronically. curriculum. They will usually contain appear in a section entitled “Problems Hardcopy should be sent to Victor Katz, interactive components and color graph- from another time,” with new problems Convergence, Mathematical Association th ics, to take advantage of the capabilities appearing frequently. After publication, of America, 1529 18 St. N.W., Washing- of the Web. Frequently, articles will be the problems will be archived in sections ton, DC 20036. We can take articles in designed, through use of hyperlinks, to based on the main topic of the problem, Word or TeX, but please include illustra- appeal to multiple audiences. In addition, such as algebra, geometry, trigonometry, tions (in jpg format), applets, etc. as sepa- each article will have a discussion group or calculus. Answers will appear sepa- rate files, and give explicit instructions attached, where readers can share sug- rately. for both internal and external hyperlinks. gestions as to how the material can be If there are many illustrations or applets, used in the classroom and point out What Happened Today in History? Each it is best to send the electronic version strong points and possible pitfalls; au- day, there will be a listing of 2-3 “math- on a CD to the same address. Short ar- thors would also have a chance to re- ematical events” which happened on that ticles, as well as ideas for articles, can be spond. date in history. Many of the items in this sent electronically to Victor Katz at section will have links to other websites, [email protected]. If you have an idea for Translations of original sources. These will so teachers can find out more about the an article, but do not know how to pro- generally be accompanied by commen- particular person or event. duce applets for it, we suggest that you tary from experts showing the context of contact an expert on your own campus the works. If possible, interactive com- Quotation of the day. A new and interest- for help. If necessary, however, we can ponents will be used to help with the ing quotation about mathematics from provide help in the editorial office, pro- understanding of these materials. But the a historical figure will appear in this sec- vided you give us very explicit instruc- goal of the translation is always to show tion each day. The reader will also be able tions as to what you need. The editors teachers how ideas were developed in to search our database of quotations to look forward to producing this magazine various cultures and how knowledge of find additional ones. for the mathematics community. this development is useful to teaching the same ideas to today’s students. An up-to-date guide to what is happen- Convergence can be accessed through the ing around the world in the history of MAA home page, http://www.maa.org, or mathematics and its use in teaching. The directly at http://convergence.mathdl.org.

4 FOCUS SIGMAA QL Is Formed The purpose of this SIGMAA is to pro- and Matt DeLong (Taylor University ) as vide a structure within the mathematics webmaster. The SIGMAA will hold its By Rick Gillman community to achieve these goals. It will first electronic election for new officers achieve this by hosting receptions at in November, 2004, with the new offic- At its January, 2004, meeting, the MAA meetings, organizing talks, panels ers assuming their duties in January of Board of Governors approved the forma- and paper presentations at MAA meet- 2005. tion of SIGMAA QL — the special in- ings, providing e-mail bulletins of timely terest group for Quantitative Literacy. events and issues via a listserv restricted SIGMAA QL will hold its first business to members of the SIGMAA, and pro- meeting and sponsor an invited speaker Quantitative literacy (QL) can be de- moting submissions to publications that at the Joint Mathematics Meetings being scribed as the ability to adequately use further the QL movement. We will also held in Atlanta in January 2005. In addi- elementary mathematical tools to inter- maintain a web page of QL information tion, the SIGMAA will host an informal pret and manipulate quantitative data and distribute a semi-annual newsletter gathering of interested individuals at and ideas that arise in an individual’s to the SIGMAA membership. MathFest being held in Providence in private, civic, and work life. Like reading August, 2004. Further information on and writing literacy, quantitative literacy In addition, the leadership of SIGMAA this gathering will be published in the is a habit of mind that is best formed by QL will be coordinating its activities with program for that meeting. exposure in many contexts. the many other groups, both internal and external to the MAA. Internally, these Finally, members of the MAA are encour- While developing a quantitatively liter- groups include the CUPM and its vari- aged to be part of a team from their in- ate citizenry is the responsibility of a ous subcommittees and other SIGMAAs stitution attending the PREP workshop much larger community, it is the obliga- which have an interest in this area. Ex- on Quantitative Literacy being held this tion of the collegiate level mathematics ternally, the SIGMAA will be interacting summer from August 17-20, 2004, at the community to take leadership in identi- with the NCTM, AMATYC, and the Na- Sleeping Lady Mountain Resort in fying the prerequisite mathematical skills tional Numeracy Network. Leavenworth, Washington. for QL, finding innovative ways of developing and implementing QL cur- The initial leadership for SIGMAA QL ricula, assisting colleagues in other dis- includes Judy Moran (Trinity College) as ciplines to infuse appropriate QL expe- Chair, Caren Diefenderfer (Hollins Uni- riences into their courses, and stimulat- versity) as Chair-elect, John Bukowski ing the national dialogue concerning QL. (Juniata College) as secretary-treasurer,

Congress Asks for Teacher Education Study Figure This!

Tucked into a major appropriations advance the field of education research, If you started counting your heartbeats bill this March was a request by Congress making it more rigorous in support of at midnight on January 1, 2000, when that the Institute for Education Sciences evidence-based education.” Grover J. would you count the millionth beat? How assess teacher education programs na- (Russ) Whitehurst is the current Direc- about the billionth beat? This is one of the tionwide. According to the March 3 is- tor of the Institute. For more informa- “math challenges for families” to be sue of Education Week, “Congress in- tion about IES, see http://www.ed.gov/ found at the Figure This! web site at http:/ tends for existing data to be synthesized about/offices/list/ies/index.html. /figurethis.org. The rationale for the site, on the consistency of required which was created by the National Coun- coursework, how reading and math are It will probably be several years until the cil of Teachers of Mathematics, is ex- taught, and the degree to which pro- study is complete. Meanwhile, reactions plained in the family corner: “Family grams are aligned with scientific evi- are mixed. Some education experts are members — as children’s first teachers dence on the subjects.” The study will confident that the study will confirm the — are crucial to student success. And the likely be conducted by the National Re- effectiveness of teacher education pro- more adults become engaged in their search Council. grams. Others fear that the study will be children’s education, the greater the used against Schools of Education, and chances that children will succeed.” The The Institute for Education Sciences is doubt that the concept of “scientific evi- site provides mathematical challenges a branch of the Department of Educa- dence” is applicable to the assessment of both in a format appropriate for the web tion created in 2002. Its purpose is “to teacher education. and in printable format.

5 FOCUS May/June 2004 The Archives of American Mathematics at the Center for American History By Kristy Sorensen

The Archives of American Math- Together they have ematics (AAM), a unit of The Uni- taken major steps versity of Texas at Austin’s Center to enhance the for American History, has been an value of AAM col- important resource for mathema- lections to re- ticians, historians, and sociologists searchers, includ- for nearly thirty years. Thanks to ing the placement the generous financial support re- of over 50 collec- cently provided by the Legacy of R. tion inventories on L. Moore Project, through the the Internet, where Mathematical Association of they can be ac- America this valuable collection cessed by research- will be made more accessible for ers worldwide. In teaching and research. February, 2004, the AAM was awarded In February, 2003, the Center for an additional two American History utilized grant years of funding funds to hire Kristy Sorensen, a from the Legacy of full-time archivist, and Traci R.L. Moore Project. Drummond, a part-time assistant, R.H. Bing, ca. 1960s, from the R.L. Bing Papers, Archives of American In March, Amelia to manage the growing collection. Mathematics, Center for American History, The University of Texas at Abreu replaced Austin.

George Bruce Halsted, date unknown. From the R.L. Moore Pa- pers, Archives of American Mathematics, Center for American R.L. Moore, January, 1969. Photo taken by Homer G. Ellis, History, The University of Texas at Austin. from the R.L. Bing Papers, Archives of American Mathemat- ics, Center for American History, The University of Texas at Austin. 6 FOCUS

Traci Drummond as the archival assistant. She and Kristy Sorensen look foward to continued work with the mathematics collections in the coming years.

Colleagues of R. L. Moore, a prominent mathematician and long-time professor at The University of Texas (1920-1969), established the Archives of American Mathematics in 1975 with the donation of his papers. This core collection at- tracted other donors, and led to the addition of the papers of many of Moore’s students and colleagues, including R. L. Wilder, R. H. Bing, and G. B. Halsted. In 1978 the AAM became the official reposi- tory for the records of the Mathematical Association of America, expanding the collection to include the administrative records of this important professional or- ganization. Adding to the breadth of collections in the AAM are the records of the School Mathematics Study Group, creator of the influential “New Math” pri- Joint AMS/MAA Mathematics Meeting, Cleveland, Ohio, December, 1930. From left to mary and secondary curriculum of the right: Wilfrid Wilson, J.W. Alexander, W.L. Ayres, G.T. Whyburn, R.L. Wilder, P.M. 1960s. Other prominent collections in- Swingle, C.N. Reynolds, W.W. Flexner, R.L. Moore, T.C. Benton, K. Menger, S. Lefschetz. clude the papers of Max Dehn, Emil From the R.L. Moore Papers, Archives of American Mathematics, Center for American Grosswald, and William T. Reid. Major History, The University of Texas at Austin. strengths of the archives are in topology, mathematics education, analysis, number theory, logic, and the mathematical foundations of physics. The AAM currently consists of 70 collections measur- ing more than one-thousand linear feet.

The Center for American History is a special collections library, archive, and museum that facilitates research and sponsors programs on the history of the United States. The Center supports research and education by acquiring, pre- serving, and making available research collections and by sponsoring exhibi- tions, conferences, symposia, oral history projects, publications, and grant-funded initiatives.

Persons interested in conducting research or donating materials or who have general questions about the Archives of American Mathematics should contact the Archivist Kristy Sorensen, [email protected], (512) 495- 4539. For more information on the Ar- chives, visit the archives’ web page at C. Zarankiewicz, W.L. Ayres, and B. Knaster (with unidentified dog), undated. From the http://www.cah.utexas.edu/ R.L. Moore Papers, Archives of American Mathematics, Center for American History, The collectioncomponents/math.html. University of Texas at Austin.

7 FOCUS May/June 2004 The Teacher as Artist: A Letter to My Colleagues

By Peter Taylor

In a time of diminishing resources, must Well maybe it was –– certainly we give up the attempt to offer a high I am serious about this type of quality mathematics education to our change: less breadth and more students? Peter Taylor, recently chosen to depth, a few critical examples, head the Department of Mathematics and the best we can find, powerful, Statistics at Queen’s University in beautiful, engaging, explor- Kingston, Ontario, addresses this question atory, artistic, as many of those in a letter to his colleagues. fine words as possible, done thoroughly and well, hopefully Department of Mathematics enough of these to capture the and Statistics essentials of the subject, be that Queen’s University calculus or linear algebra. Such February, 2004 examples are not easy to find; but once you have them you Dear Colleagues: have pure gold. In the conversations I had afterwards, many Thanks to all those who have congratu- of you picked up on my phrase lated me and wished me well. We have that we should dare to be “less at Queen’s a versatile and dedicated De- comprehensive and systematic.” partment and I look forward to guiding You pointed out that while it through the next few years of its most of these students will growth. I say “growth” in the faint hope never encounter us again, some that this means more than spiritual and of them do stay to take MATH Peter Taylor intellectual growth, though the Dean 211 or MATH 232 and there are seems unable to say more than that some many things they will need to know. It’s But now I owe you a confession. The but not all of our upcoming retirements true that some will go on to other math changes I have in mind apply, in my view, will be replaced. These are challenging courses and in fact I hope that the equally to the honours courses. In these times. changes I have in mind will encourage courses, I think we can do much less, have even more to stay with us longer and more fun and engagement, and come Indeed, we are called upon now to do learn some more mathematics. In part, away with a much higher retention rate. much more with many fewer resources my purpose in writing this is to address In saying this I am aware that our than were available when I was a student this “prerequisite” worry that many of honours courses are generally very well here. Our response to each new cut over you seem to have. taught and our graduating students on the years has been to squeeze harder and the whole look back on them with posi- fit more in and I am now convinced that On the other hand, perhaps I protest too tive memories. But less than half of the this response can no longer serve us or much. Perhaps it’s not such a worry. Af- students (in fact much less) who enroll our students. It’s time to do old things ter all, these students are majors in an- in those courses wind up with any kind in a new way. I’m not sure just what a other discipline and this is an applied of math concentration. I want more of new system might look like, but my un- course, and I am an applied mathemati- them to stay. I believe that more math- derstanding of how significant advances cian and apparently a good teacher so on ematics would serve them well, and that were made in our evolutionary trajectory the whole you trust me with these stu- they in turn would serve our subject well as a species is that it was at times of tight dents, and my schemes will probably turn in whatever they chose to do in life. constraint that significant “saltations” out fine. And of course I did make it clear were made. at the very beginning that I was not talk- I have already touched on this subject ing about our honours courses –– those, with one or two of you, and in fact, over I was glad that so many of you attended we have a duty to teach “properly,” a duty the years, with other mathematicians at the colloquium last week — my thoughts not only to our students, but to the sub- professional meetings. It is a difficult on some possible changes in our large ject itself. No, the colloquium was very debate, partly because so much of it is first-year service courses. It wasn’t re- much about our service courses. waged abstractly away from concrete ex- ally meant as a directive, though some amples and explicit curriculum models. of you might have wondered about that. But the bottom line –– what my conver-

8 FOCUS sations always seem to come down to –– example I cite my way of treating eigen- challenge to find such ways, but it is the is your worry that my “schemes” will fail values which (I feel) is much more fun fundamental challenge of the teacher as to properly prepare our students for up- than the standard approach but which I artist. per level mathematics courses. Think- am sure gives them a much clearer idea ing about this, I wonder exactly what it of what an eigenvalue really is.] In closing, these remarks of mine focus is that you are worried about. Is it the on what I believe are fundamental prob- students themselves and their future? Is For most of our math majors (from C lems of the area of study known as math- it the reputation of the university? Or is on), it is enough that they learn a little ematics education. For 30 years I have it your own future courses, the readiness math, enough math to see the power and attended professional meetings and read with which the students will be able to beauty, and learn it well and positively journal articles in this field and followed grapple with the many neat things you enough that they can feel it belonging to the big debates and you know what? –– want to teach them? them, feel it part of their domain. After except for an oscillation from right to left that, we simply have to trust the subject and back again, little seems to change. Well, let’s think for a moment about itself to impel them forward as appro- One problem is that the system is com- these students. They come in different priate, and trust them to respond. pletely interconnected –– a powerful cur- flavours. First there’s A and B who are rent of experience and ideas runs from natural problem solvers and might well I know that many of you have important elementary school through high school wind up in research careers in mathemat- stories that you want to tell your senior and university, to teacher’s college and ics or science. These students will thrive students, stories that they would not be right back to the beginning again. Per- with your curriculum as well as with able to appreciate or even understand haps the only hope is a decisive interven- mine –– all they need is good problems without a “proper” grounding in the first tion at some critical point in the system and someone to talk to about them. So and second year, and so you are under- and my target these days is that crucial can we agree that they are not the prob- standably worried by my suggestion that first year in which we play host to a sig- lem? Then there are C, D, E, F, G, H, I we might lighten (that word in all its nificant fraction of the students at this and J. Some of them will go on to law or meanings!) our introductory courses. I university. business, some to government; some are find that the metaphor of “teacher as art- in our teacher program. And then of ist” is helpful here. You’ve noticed a strong thread of ideal- course there is (or will be) X, Y and Z ism running through this letter. That’s who started out in the service courses but Briefly, the idea is that the curriculum be how we’re going to start. We’ll temper it because they were so interesting and regarded as a set of integral works of art. as we go. lively (!) decided to switch to a math Because such works have a wholeness in major and find themselves now in third themselves, it is not so crucial whether All the best, year. You seem to suggest that these stu- the student has mastered this technical Peter dents, from C to Z, would have great dif- skill or that, rather what counts is the ficulty filling in the gaps on their own, nature and level of sophistication of the Peter Taylor is Professor of Mathematics, that if we do not teach them a “proper” technical skills he has mastered, and it is Biology and Education at Queen’s Univer- systematic course they will not manage this that should increase from year to sity. In July, 2004 he will assume duties as to carry the subject effectively forward. year. As each new work is brought for- Head of the Department of Mathematics ward, crucial technical skills can be, and and Statistics at Queen’s University. He is Well, in my experience, they do not now indeed ought to be, reinvented in the new determined that this will not cut into his carry the subject effectively forward. context, and the experience that is squash. Many of them take my MATH 382 and I needed for this to succeed is of a general can attest that they are not very good at rather than a specific nature. using simple (basic!) ideas of calculus and linear algebra to solve new problems. The artist and the poet do not to tell the Have You Moved? [Actually I don’t mean to suggest that entire tale as they know it. Rather they The MAA makes it easy to change your they should be able to do that in third display it on the canvas, or on the page, address. Please inform the MAA Service year; the point I’m making is that they in a way that is true both to the story and Center about your change of address by don’t come away (from the teaching and to the constraints imposed by the me- using the electronic combined mem- learning) with what you might think, dium. The response of the artist is there- bership list at MAA Online that while an alternate approach might fore highly conditioned, bowing to form, (www.maa.org) or call (800) 331-1622, be no better than the traditional one, it and to technique, but it is the more pow- fax (301) 206-9789, email: might equally well be no worse. In fact I erful for that; this is the wonderful para- [email protected], or mail to the have reasons for thinking that a lighter, dox of artistic expression. Good stories MAA, PO Box 90973, Washington, DC livelier approach might actually work become art when they are told in simple 20090. better and gain us more students. As an ways which capture their richness. It is a

9 FOCUS May/June 2004 What I Learned by Teaching Proof and Logic By Peter Ruane

Most of my years in mathematics edu- around questions such as ‘How can we of symmetry) and then ran out of ideas. cation have been spent in primary tell if a number is even?’ etc and then to Calling the class together for discussion, teacher training in the UK. Naturally, this the matter of generalising results about one of the boys (Glen), who had been has involved much time in primary the addition of three or more odd num- working quietly in a corner of the class- schools, working with teachers and do- bers etc. room, told us that there were ‘lots’ of ing maths with children between the ages ways of doing this. I was amazed by of 5 and 11 years. By the phrase ‘doing Most of the attempts to justify the claim Glen’s justification, which was based maths’ I mean that I always get the chil- that O+O =E were based upon state- upon rotational symmetry as follows: dren actively exploring a problem or pro- ments of the sort ‘It always works’. Prob- vide them with heuristic exercises that ing more deeply, I challenged the chil- • Find the centre, C, of the generate discussion and the exchange of dren to provide a counter-example, square by pencilling the diagonals ideas. It involves observation of pattern, which, of course, they could not. What then erase the pencil lines. forming hypotheses, explaining and jus- did emerge were two particularly strik- • Join C, to another point, P, on one tifying (however informally). ing contributions, one from Brendan of the sides by drawing any line another from Clare. (straight, curve or squiggly). On the question of ‘logic’ and ‘proof’, I • Using tracing paper, rotate the rarely embark upon explicit teaching of Brendan’s explanation was based upon a segment PC through 180o (about C) such themes but they often arise as side last digit argument. He pointed out that to meet the opposite side at a point issues, and very valuably so. Looking since the sum of any pair from 1, 3, 5, 7, Q. back, many instances spring to mind, but 9 would be even, then the sum of any two • Line PCQ divides the square into I recall some particular teaching episodes large odd numbers would also be even. two congruent parts. that illustrate children’s capacity for logi- For example, since 7 + 5 is even, then 127 • Check this by cutting out the two cal thought and proof. What I have + 105 is even, and so on. On the other pieces and superimposing. learned from this has consequently af- hand, Clare’s explanation was intuitively fected the way I have taught mathemat- based upon that used in a formal proof. Of course, Glen didn’t word his response ics to trainee teachers. She held up both fists, each with just one like this, but he explained this method finger sticking out. She said ‘If this is odd, in his own vernacular, showing us his Below, I recount some past teaching epi- there’s one left over, and if that’s odd, diagrams. Equipped with scissors, trac- sodes as a means of illustrating this phi- there’s one left over. The left overs make ing paper, coloured pencils etc, the rest losophy. Some of the conversations with two, so everything’s even’. of the class then embarked upon Glen’s children have been reconstructed from approach and started to manufacture memory, but the gist of the dialogue is Formally, if x, y are odd then x = 2m+1 their own sets of examples. authentic. (Clare’s left fist) and y = 2n+1 (Clare’s right fist), where m and n are non-nega- Classifying by Symmetry Odds and Evens tive integers. So x + y = (2m+1) + (2n+1) = 2m+2n+2 and ‘everything’s even’. Symmetry is one of those topics that have This was a lesson with a class of children long been included in school syllabuses aged between 8 and 9 years, the aim be- Halving a Square in the UK, but most teachers don’t know ing to extend their understanding of ba- why that is, so they end up treating the sic addition. The activity was centered For this activity, the children were about topic by providing recreational colouring upon exercises that involved adding pairs ten years of age and I provided each of and cutting activities with no ostensible of even numbers, or adding pairs of odd them with a sheet with lots of squares geometrical content. What I intended numbers, or adding a mixed pair (one drawn on it. After an opening discussion, was to use symmetry to extend children’s odd, the other even). Initially, I didn’t I asked them to find different ways of awareness of properties of shape by get- explain what was going on, but I got the dividing a square into two equal pieces ting them to investigate questions such children doing the additions and, after a by use of a single line. I actually meant as: while, asking them what they’d observed. ‘congruent’ pieces, but some of the children took this to mean ‘equal by area’ but • How many lines of symmetry Nearly all the children had noticed the not necessarily congruent (so what?). can a triangle have? simple patterns: E + E = E, O + O = E, • Find, if you can, quadrilaterals E + O = O and O + E = O and this was Many drew in the obvious dividing lines having 0, 1, 2, 3 or 4 lines of followed by quite a bit of discussion (vertical, horizontal and diagonal lines symmetry.

10 FOCUS

• Which of these can have a right- Obviously, such work commences by ‘Charlie’s in that bit ‘cos he’s got a dog, angle? How many right angles sorting objects according to one attribute but no cat’ at most? only (e.g., using Logiblocs, sort accord- (~ cd∧ ) • Which of your quadrilaterals ing to thick/thin or square/non-square have rotational symmetry? etc) and leads to sorting by two or more ‘Ruth’s got a cat and not a dog’ attributes. However, by using Venn or • Which quadrilaterals only have Carroll diagrams to record the results of cd~ lines of symmetry that are di- ( ∧ ) sorting, opportunity arises for the use of agonals? etc. logical connectives and the formation of ‘Hannah’s got nothing: she’s outside’ compound statements from simple state- The work was practical (templates, scis- ments as a means of describing the re- ~~cd sors, sticky paper, glue etc) and, for some ( ∧ ) gions of the diagrams (truth sets). of these questions, the children recorded This goes on for a while and we return the results by cutting out and sticking A typical lesson on this was based upon to this type of activity in various guises, their shapes into appropriate boxes of a sorting a group of children according to always developing children’s ability to two-attribute Carroll diagram. what sort of pets they may own. So, we describe and logically classify math- put a large closed rope loop on the class- ematical or everyday objects. Really, this When it came to summarising the results, room floor, with the fifteen five-year olds is a prelude to symbolic logic, but, of and reaching agreed conclusions, we excitedly gathered round. ‘WHO has a course, that stage is never reached and were really employing proof by exhaus- DOG?’ I ask in an exaggeratedly dramatic teachers confine the results to describ- tion (deriving all possible cases). It was way. Eight eager responses are given, so ing subsets of truth sets by use of every- agreed, for example, that a triangle could into the hoop those eight children go. day language. have 0 lines of symmetry, 1 line of symmetry or three lines of symmetry (be- Qn: Why isn’t Alice in the hoop?’ I have often applied the same idea to cause the children had found such ex- symmetry activities for the classification amples). However, as to providing a rea- Ans: ‘Because she hasn’t got a dog!’ of triangles, quadrilaterals etc. For ex- son for not being able to find a triangle shouts Ruth (she isn’t in the ample, with two attributes ‘has rotational with exactly two lines of symmetry most hoop either) symmetry’ and ‘has mirror symmetry’, children rather lost interest — except two Qn: ‘What pets do you have Ruth? older children will be asked to insert at of the girls (both 11 yrs old). Ans: ‘A cat and a tortoise’ She replied. least one type of quadrilateral into each of the four regions of a Venn (or Carroll) Later, the girls came up with the argu- Qn: ‘Who else owns a cat?’ diagram and give their reasons for plac- ment that, for a triangle with two lines ing a particular shape in a particular re- of symmetry, all three sides must be equal There are about five positive replies to gion. Descriptions will involve com- and so it is equilateral and therefore has this, including two from children in the pound statements of the sort ~ mr∧ a third line of symmetry. In other words, ‘dog hoop’. We put down another hoop (parallelogram), mr∧ ~ (trapezium), from an activity based upon exhausting for the cat owners, non-overlapping with mr∧ (rhombus or rectangle) etc. With all possible cases, there emerged a nice the dog set. some children, it’s possible to establish piece of deductive reasoning that was equivalences like ~~mr ~ mr, accepted as a convincing proof by most Problem! Children with cats go into the ∨⇔( ∧) but only in the form of ordinary English of the class (and, of course, myself). cat hoop, but some of those are also in the dog hoop. How do we get those chil- and in a specific practical context. Sorting and Classifying dren into both sets? After some discussion, it is decided to make the hoops Summary Working with children as young as five overlap to produce a four-region Venn In the past five years or more, primary years old, there is much scope for activi- diagram (including the outside). mathematics in the UK has been based ties based upon sorting and classifying upon the principle that teaching has to everyday objects into sets. This facilitates Each of the four regions contains at least be interactive. This means that children’s language development and particularly one child and we discuss why any par- ideas (right or wrong) must be incorpo- the refinement of vocabulary connected ticular child is, or is not, in a particular rated into lessons and teachers should with length, weight etc. By extending region. We have two logical connectives encourage children to explain, question such activities to work with 2d and 3d ‘and’, ‘or’ and two simple statements and and demonstrate. In this way, the capac- geometric objects, shape names are in- compound statements are elicited from ity for ‘logical’ thought will gradually troduced and various intuitive geomet- the diminutive participants. Replies to develop. ric properties of shape begin to evolve. my questions were typically like this:

11 FOCUS May/June 2004 Moreover, the notion of ‘proof’ has wid- Mathematics Magazine Editor Search ened from that used in formal mathematics and we now acknowledge differ- he Mathematical Association of interest containing his or her ideas about ent levels of proof, beginning with T America seeks to identify candidates to the journal. These can be emailed with children’s reasonable explanations as to succeed Frank Farris as Editor of Math- attachments as Word or plain-text TeX why they believe something to be true ematics Magazine when his term expires documents to the chair of the Search or false. in December, 2005. The Search Commit- Committee, Deanna Haunsperger tee plans to make a recommendation ([email protected]), or mailed to: In the examples that I have provided, I during the summer of 2004 so that the am suggesting that many children are new editor can be approved by the Board Deanna Haunsperger quite capable of achieving quite sophis- of Governors and begin handling all new Department of Mathematics ticated proofs and that, even for young manuscript submissions in January, and Computer Science children, deductive proof, proof by ex- 2005. The new editor would be Editor Carleton College haustion, and proof by counterexample Elect during 2005 and would serve as One North College are not beyond the bounds of possibil- Editor for the five years 2006-2010. Northfield, MN 55057 ity. Teaching activities such as those above have also taught me that Boolean Questions about the nature of the posi- Nominations are also welcome if you algebra has its pedagogical roots in sort- tion and its workload can be addressed know someone who would be an out- ing activities with very young children. to Frank Farris ([email protected]); ques- standing editor. Applications and nomi- tions about MAA support for the editor’s nations will be accepted until the posi- What did I learn by teaching proof and work can be addressed to the MAA’s Di- tion is filled, although preference will be logic to undergraduate maths majors? rector of Publications, Don Albers given to applications received by early That’s a different story! ([email protected]). May. Each applicant should submit a resume, Peter Ruane has taught mathematics in names of references, and a statement of schools to children across the age-range (5 yrs to 18 yrs) and he has been involved in the mathematical education of teachers in England for over thirty years.

There is no basis for your independence

Sign next to a bridge in Beijing, China August, 2002. Photograph by Colm Mulcahy.

12 FOCUS 2004 Summer Short Course——Teaching and Doing Knot Theory

The Ohio Section will sponsor a sum- • Do research in knot theory supplements to calculus. Having mer short course on Teaching and Do- authored a variety of research articles on • Direct student research in knot ing Knot Theory, which will be held June knot theory and hyperbolic 3-manifolds, theory 2–4 at Ohio Northern University in Ada, he is also known for giving mathemati- Ohio. Colin Adams of Williams College cal lectures in the guise of Mel Slugbate, Participants will have the opportunity to will be the instructor. a sleazy real estate agent. A recipient of conjecture wildly, throw around ideas, the Deborah and Franklin Tepper Haimo and work on original research. Knot theory is a great topic for exciting Distinguished Teaching Award from the students about mathematics. It is visual Mathematical Association of America There will be morning and afternoon and hands on. Students can begin work- (MAA) in 1998, he was a Pólya Lecturer sessions on Wednesday and Thursday ing on problems the first day with their for the MAA for 1998-2000 and was a and a morning session on Friday. shoelaces. Knot theory is also an incred- Sigma Xi Distinguished Lecturer for ibly active field. There is a tremendous 2000-2002. He is also the author of math- Colin Adams is the Francis Christopher amount of work going on currently, and ematical humor column called “Math- Oakley Third Century Professor of one can easily state open problems. It also ematically Bent” which appears in the Mathematics at Williams College. He re- has important applications to chemistry, Mathematical Intelligencer. ceived his Ph.D. from the University of biochemistry, and physics. Wisconsin-Madison in 1983. He is par- Registration will begin in April and con- ticularly interested in the mathematical This workshop is aimed at anyone who tinue until all seats are filled. The regis- theory of knots, their applications and is interested in knowing more about knot tration fee is $150.00. their connections with hyperbolic geom- theory. There is no assumption of previ- etry. He is the author of The Knot Book. ous background in the field. For more information contact Don Hunt an elementary introduction to the math- at [email protected] or call (419) 772- ematical theory of knots and co-author Participants completing the workshop 2351. with Joel Hass and Abigail Thompson of will learn how they can: How to Ace Calculus: The Streetwise You may also visit: http://www.onu.edu/ Guide, and How to Ace the Rest of Calcu- • Teach an undergraduate course in a+s/math/NewFiles/maa/shortcourse.htm. lus: the Streetwise Guide, humorous knot theory

MAA Member Frank Leighton Elected to the National Academy of Engineering

Frank Thompson Leighton, professor gineering is among the highest profes- 2003 Annual Survery of applied mathematics at the Massachu- sional distinctions accorded to an engi- of the Mathematical setts Institute of Technology and an neer. Academy membership honors those MAA member, has been elected to the who have made “important contribu- Sciences National Academy of Engineering “for tions to engineering theory and practice, contributions to the design of networks including significant contributions to the The first report of the 2003 Annual and circuits and for technology for Web literature of engineering theory and Survey of the Mathematical Sciences has content delivery.” practice,” and those who have demon- just been released and is available strated accomplishment in “the pioneer- online at http://www.maa.org/news/ The National Academy of Engineering ing of new fields of engineering, making 2003survey.html. It focuses on new (NAE) elected 76 new members and 11 major advancements in traditional fields doctoral recipients and the faculty sal- foreign associates this year, which brings of engineering, or developing/imple- ary survey. The second report will be the total U.S. membership to 2,174 and menting innovative approaches to engi- published in August, 2004. the number of foreign associates to 172. neering education.” Election to the National Academy of En-

13 FOCUS May/June 2004 Remembering Bill Chinn By Jean Bee Chan

Long time MAA member and sup- Bill also taught in junior high and senior casions we met for dinner, the bill was porter William (Bill) G. Chinn passed high schools in San Francisco, where stu- always magically paid. It turned out Bill away quietly on March 23, 2004, in San dents enjoyed his humorous and witty always paid for dinner while the rest of Francisco. Bill was an active member of approach to classroom teaching. He us were enjoying ourselves. the MAA. He served on national and sec- maintained a mathematics “laboratory” tional MAA committees, was the North- for gifted students, and served as assis- MAA members Leonard Klosinski, Jerry ern California Section Chair, and in tant to the Curriculum Coordinator of Alexanderson, David Sklar, and I were at 1981–82 served as Second Vice President the San Francisco Unified School Dis- the memorial service for Bill Chinn in of the MAA. He was a generous donor trict. He authored and co-authored nu- Oakland, California, at the Chapel of the to both the MAA and the American merous books, including First Concepts Chimes on March 30, 2004. Bill was al- Mathematical Society. in Topology, with Norman Steenrod, and ways cheerful, always generous. He was 3.1416 And All That, with Philip Davis. a great friend and a selfless contributor An alumnus of Lowell High School and to the mathematical community. He was University of California at Berkeley, Bill I first met Bill when he was Newsletter a good role model for all of us, and he served in Europe and Africa during Editor for the Northern California Sec- will live forever in our hearts. World War II, first as a pharmaceutical tion of the MAA. Later he and I served technician and then as a meteorologist. together on the MAA Development After his discharge he resumed graduate Committee. Bill loved to take pictures of study in mathematics. He was a Profes- people at meetings. Afterward, he would sor of Mathematics at the City College circulate pictures with colleagues and of San Francisco for many years. students properly identified. On the oc-

Call for 2005 PREP Proposals

projects who plan to of- he Mathematical Association of T fer a workshop to apply America PRofessional Enhance- for inclusion under the ment Program (PREP) seeks pro- PREP umbrella. posals for 2005 PREP workshops. Through this process, a PREP is a comprehensive, profes- workshop will gain ac- sional career enhancement project cess to the wide audi- of the MAA funded by NSF DUE ence, logistical support grant # 0341481. and evaluation services developed for the PREP Programs for all aspects of career program. Programs enhancement are encouraged. considered for inclusion PREP workshops that present in this way must meet modern mathematical ideas or current ricula and pedagogical strategies are also the same guidelines as core PREP work- interest in research and interdisciplinary welcome. All programs must appeal to a shops. applications are especially welcome. Pro- general audience of mathematical programs that present new developments in fessionals. More information regarding submission teaching and learning, innovative ap- of workshop proposals is available proaches to leadership and career en- In addition to PREP-funded workshops, through the program website, http:// hancement, or focus on innovative cur- we invite directors of other funded www.maa.org/PREP.

14 FOCUS Letters to the Editor

Paul Monsky’s Real School

In FOCUS, March 2004, bottom of page 28, it is mentioned that “The only per- fect score on the 1950 Mathematical Contest was Paul Monsky, from Stuyvesant High School in New York City”. This is incorrect. Paul was at Brooklyn Tech when I was there.

Peter Kahn SUNY Stony Brook

What is an Acronym?

I was amused at the argument about the “acronym” SPSS in the Letters section of the March FOCUS. The point is, SPSS can never be an “acronym” for anything. A sequence of initial letters of a phrase or definition is an acronym, if it forms a pronounceable word. The operative term is “pronounceable.” RADAR, for example, is pronounceable, hence is an acronym. It would have been permissible to include the “A” from “Package” (the RA from RADAR stands for “radio”), or the “A” from “and” from the 2nd example giving SPASS, an acronym.

Oren N. Dalton El Paso, TX

Well, the OED seems to agree with you, since it defines an acronym as “A word formed from the initial letters of other words.” The computer community, however, seems to call any sequence of letters an acronym. See, for example, the definition of “TLA” in the Jargon File: http:// www.catb.org/~esr/jargon/html/T/ TLA.html.

Recently Reviewed Online

The FOCUS Online section of the The Art of the Infinite Geometry: Euclid and Beyond MAA web site includes a very active book by Robert and Ellen Kaplan by Robin Hartshorne review column called Read This!. You can A Smoother Pebble Dr. Math Gets You Ready for Algebra find the main reviews site at http:// by Donald C. Benson by The Math Forum www.maa.org/reviews/reviews.html. The archive of past reviews is huge: more than When Least is Best A Topological Aperitif 400 books have been reviewed. Here is a by Paul Nahin by Stephen Huggett and David Jordan list of some recently reviewed books:

15 FOCUS May/June 2004 MAA Contributed Paper Sessions Atlanta Joint Mathematics Meeting, January 5-8, 2005 PRELIMINARY ANNOUNCEMENT

The organizers listed below solicit contributed papers perti- ability, and selected post-calculus topics. This session invites nent to their sessions. Sessions generally limit presentations to 1) demos that introduce a topic, 2) demos that illustrate how ten minutes, but selected participants may extend their contri- concepts are applicable, 3) demos that tell a story or describe butions up to twenty minutes. Please note that the dates and the development of a procedure, and 4) demos that lead to an times scheduled for these sessions remain tentative. See the end activity that involves the class. Presenters of demos are encour- of this announcement for specific submission procedures and aged to give the demonstration, if time and equipment allow, other details. Each session room contains an overhead projec- and to discuss how to use it in a classroom setting. Proposals tor and screen; black boards will not be available. Persons need- should describe how the demo fits into a course, the use of ing additional equipment should contact, as soon as possible, technology or technology requirements, if any, and the effect but prior to September 14, 2004, the session organizer whose of the demo on student attitudes toward mathematics. name is followed by an asterisk (*). MAA CP C1 Courses Below Calculus: A New Focus MAA CP A1 Getting Students To Discuss Wednesday morning and Friday afternoon and To Write About Mathematics Mary Robinson*, University of New Mexico-Valencia Campus Wednesday morning and Thursday afternoon [email protected] Sarah L. Mabrouk*, Framingham State College Florence S. Gordon, New York Institute of Technology Laurette Foster, Prairie View A&M University This session invites papers about assignments and projects that Arlene Kleinstein, Farmingdale State University of New York require students to communicate mathematics through in-class Norma Agras, Miami Dade Community College oral presentations, in-class discussions that they must lead and Linda Martin, Albuquerque T-VI motivate, and written assignments and/or papers. These assignments/projects can include analysis and applications of An unprecedented collaborative effort has been developed mathematics, presentations of and analysis of proofs, presen- among members of MAA, AMATYC, and NCTM to launch a tations about famous mathematicians and the mathematics that national initiative to refocus courses below calculus. The goal they studied, and assignments/projects that utilize creative of the initiative is to encourage development and implementa- writing. Presenters are encouraged to discuss how the use of tion of courses that place greater emphasis on conceptual un- the assignment/project helps the student to gain greater understanding and realistic applications of the mathematics. derstanding of mathematics as well as to improve his/her un- Courses that better motivate students for and better prepare derstanding of mathematics language and his/her ability to them to take subsequent mathematics courses, including cal- communicate mathematics. Of particular interest is the effect culus, statistics and quantitative methods, are needed to better of such projects/assignments/presentations throughout the serve the needs of the quantitative disciplines and better pre- course on the student’s understanding of mathematics, his/her pare students to function effectively in today’s workplace, as communication of mathematics, and his/her attitude toward well as function effectively as citizens in today’s increasingly mathematics. quantitative society. Accordingly, for this session, we specifi- cally seek to address all of the college level courses below cal- MAA CP B1 My Favorite Demo—Innovative Strategies for culus, with particular emphasis on offerings in college algebra Mathematics Instructors and precalculus. We seek proposals for presentations that offer Wednesday morning and Thursday afternoon new visions for such courses, discuss implementation issues David R. Hill*, Temple University, [email protected] (such as faculty training, placement tests, introduction of al- Lila F. Roberts, Georgia College and State University ternative tracks for different groups of students, transferabil- ity problems, etc) related to offering such courses, present re- Mathematics instructors use a myriad of innovative techniques sults of studies on student performance and tracking data in for teaching mathematical concepts. Technology readily avail- both traditional and new versions of these courses and in fol- able in colleges and universities has provided a means to boost low-up courses, discuss the needs of other disciplines and the creativity and flexibility in lesson design. Tools an instructor workplace from courses at this level, discuss connections to utilizes may include specialized computer applications, anima- the changing school curricula and implications for teacher edu- tions and other multimedia tools, java applets, physical devices, cation. This session is co-sponsored by CRAFTY, the Com- games, etc. This contributed paper session will focus on novel mittee on Two Year Colleges, and the Committee on Service demos that mathematics instructors have successfully used in Courses. their classrooms. Rather than focus on projects or student group activities, this contributed paper session will focus on the MAA CP D1 Mathematics and Sports instructor’s activities to facilitate learning. Mathematical con- Wednesday morning and Friday afternoon tent areas will include pre-calculus, calculus, elementary prob- Doug Drinen*, University of the South, [email protected]

16 FOCUS Sean Forman, St. Joseph’s University ticians to Educate Teachers, MAA, 2003–2006) call for math- Howard Penn, US Naval Academy ematics faculty to re-examine what they teach and how they teach in mathematics courses for prospective teachers. A key When applied to the sporting arena, mathematics can provide component of this re-examination is careful consideration of both compelling classroom examples and interesting research the mathematical understanding and thinking of the prospec- problems. Baseball has long been mined for interesting statis- tive teachers in our courses. Doing so informs faculty decisions tics examples ranging from regression and probability to the about curriculum and pedagogy, and directs their instructional game theoretic aspects of in-game strategy. Recent books on effort toward the individuals in the course. This session invites jai alai, football, and a few other sports have studied those sports papers on the mathematical preparation of teachers in which through a mathematical lens. The economics of sports is now what is taught and how it is taught is being informed by the covered by its own journal and the statistics publication Chance understandings and thinking of the prospective teachers. This routinely discusses statistical examples from sporting events. session is sponsored by COMET, the MAA Committee on the This session invites papers describing interesting classroom Mathematical Education of Teachers. examples utilizing examples from sports and papers discuss- ing the application of mathematics to sporting events. MAA CP H1 History of Undergraduate Mathematics in America, 1900–2000 MAA CP E1 Mathematics in the Islamic World Thursday morning Wednesday afternoon Jack Winn*, SUNY Farmingdale, [email protected] Glen Van Brummelen* Walter Meyer, Adelphi University Bennington College, [email protected] Joseph Malkevitch, York College of CUNY Victor Katz, University of the District of Columbia Amy Shell-Gellasch, Grafenwoehr, Germany

This contributed paper session solicits presentations on all fac- This session will sketch how the last hundred years or so have ets of the history of the mathematical sciences in the Islamic led us to today’s state of undergraduate mathematics. Ques- world, including the relationship of Islamic mathematics to tions that are appropriate to discuss include: what curricular Western mathematics and to Indian or Chinese mathematics. changes occurred; how the changes depended on changes in We hope to elaborate both the unity and diversity of Muslim mathematical knowledge; other reasons why changes in teach- contributions to both pure and applied mathematical disci- ing occurred; what effects flowed from the changes; and how plines. changes affected student learning. Papers may focus on: important individuals; important movements involving curricu- MAA CP F1 Mathlets for Teaching lum or styles of instruction; the evolution or disappearance of and Learning Mathematics particular courses; case studies of particular institutions; the Wednesday afternoon history and role of important organizations such as NSF and David Strong* MAA; key events or circumstances external to the mathemati- Pepperdine University, [email protected] cal community; etc. The speaker’s personal views about the best Thomas Leathrum, Jacksonville State University way to teach certain topics are discouraged, unless those views Joe Yanik, Emporia State University are part of, or help explain historical issues.

This session seeks to provide a forum in which presenters may MAA CP I1 Initializing and Sustaining Undergraduate demonstrate mathlets and related materials that they have cre- Research Projects and Programs ated or further developed. Mathlets are small computer-based Thursday morning (but ideally platform-independent) interactive tools for teach- Margaret M. Robinson*, Mount Holyoke College ing math, frequently developed as World Wide Web materials [email protected] such as scripts or Java applets, but there may be many other Suzanne Lenhart, University of Tennessee innovative variations. Mathlets allow students to experiment with and visualize a variety of mathematical concepts, and they Papers are requested describing undergraduate research can be easily shared by mathematics instructors around the projects, courses, and programs. Of particular interest will be world. The session is sponsored by the MAA Committee on descriptions of innovative ways to get administrative support Computers in Mathematics Education (CCIME). or other support that creates a sustainable program. Also of interest will be papers indicating where to find appropriate MAA CP G1 Drawing on Our Students’ Thinking to Improve problems and how to gauge the right level. Also descriptions the Mathematical Education of Teachers of courses with undergraduate research as the main goal would Wednesday afternoon be included. This session is sponsored by the CUPM Subcom- Dale R. Oliver* mittee on Undergraduate Research. Humboldt State University, [email protected] Mary Kay Abbey, Montgomery College MAA CP J1 Projects and Demonstrations that Enhance a Differential Equations Course The MET document (The Mathematical Education of Teach- Thursday morning ers, CBMS, 2001) and the PMET project (Preparing Mathema-

17 FOCUS May/June 2004

Rich Marchand*, Slippery Rock University Bill Stone, New Mexico Institute of Mining & Technology [email protected] Shawnee McMurran, California State University We seek presentations that deal with all aspects of the peda- San Bernardino gogy and the modeling of environmental problems suitable for general education, calculus, and above. Readers are invited to Differential equations is a diverse mathematical field that af- take up the challenge of searching the natural sciences, as well fords educators a great deal of flexibility in terms of content. as economics, environmental science, and environmental edu- The course can be highly theoretical, applied, or a combina- cation for problems that can be clarified, extended or solved tion of each. This session invites novel projects or demonstra- by undergraduate mathematics. We encourage contributions tions that enhance a differential equations course either through that emphasize computational, visual or qualitative approaches. the facilitation of mathematical theory or exposure to interdisciplinary fields. New and interesting case studies are encour- MAA CP N1 Teaching Visualization Skills aged, especially those that require computational or qualita- Friday morning tive techniques. Demonstrations may be virtual, physical or Mary L. Platt*, Salem State College, [email protected] mathematical. Examples include, but are not limited to, novel Catherine A. Gorini, Maharishi University of Management proofs, mathlets, or physical demonstrations. Sarah J. Greenwald, Appalachian State University

MAA CP K1 Countering “I Can’t Do Math”: Strategies for The ability to understand, use, and create diagrams, graphs, Teaching Under-Prepared, Math-Anxious Students and illustrations is essential for students in every area of math- Thursday morning ematics. Computers have made graphics of every form widely Suzanne Doree*, Augsburg College, [email protected] available, so there is an increasing need to help students de- Bonnie Gold, Monmouth University velop their ability to handle visual information. This session Richard Jardine, Keene State College invites papers on all aspects of visualization in the college classroom: which skills are needed for success in mathematics, how How can we create a comfortable learning environment for to train students to use visual information, examples of class- under-prepared or math-anxious students and, in particular, room activities that help develop visualization skills, and ways how can we constructively assess student learning? What class- to assess a student’s visualization skills. room practices are especially effective with such students and how does research on student learning inform those practices? MAA CP 01 Teaching and Assessing Problem Solving How might the recommendations of the 2004 CUPM Curricu- Friday morning lum Guide influence our approach in teaching developmental Alex Heidenberg*, US Military Academy or introductory courses to better reach these students? This [email protected] session invites papers on all aspects of “what works” in teach- Michael Huber, US Military Academy ing under-prepared, math-anxious students. Developing problem-solving skills in the modeling sense is a MAA CP L1 Using Real-World Data to Illustrate central component in refocusing courses to emphasize process, Statistical Concepts conceptual understanding, and student growth. Universities Thursday afternoon and Friday morning and colleges are now writing institutional goals that address Thomas L. Moore*, Grinnell College, [email protected] the capabilities of their graduates. How do we measure success John D. McKenzie, Jr., Babson College in teaching our students to be effective problem-solvers? This session invites presentations about courses that focus on the Guidelines in statistical education emphasize the use of real process of problem-solving as a vehicle to learning mathemat- data instead of the small, contrived data sets that appear in ics at the pre-calculus / introductory calculus levels, with spe- some textbooks. Faculty who have used real-world data to il- cial emphasis on modeling. These presentations can include lustrate statistical concepts are invited to submit proposals that course composition, philosophy, teaching ideas, and/or past describe the data set, its location on the web, and their use of projects, examinations, or other successful methods of assess- the data set to teach ideas related to an introductory course in ment where students have become competent and confident statistics: (1) data collection (sampling, design of experiments, problem-solvers. Each presentation should address the spe- potential biases); (2) data description (numerical summaries, cific goals in developing problem-solvers as well as the assess- graphical displays); (3) sampling distributions; (4) elementary ment techniques used to measure attainment of those goals. In inference (interval estimation and hypothesis testing); (5) other addition, presenters should address how technology (calcula- applications, such as ANOVA, regression, and chi-square tests. tors, computer algebra systems, etc.) is incorporated into the teaching plan. MAA CP M1 Environmental Mathematics and the Interdisciplinary MAA CP P1 Philosophy of Mathematics Friday morning Friday afternoon Dr. Karen Bolinger* Charles R. Hampton* Clarion University, [email protected] The College of Wooster, [email protected] Ben Fusaro, Florida State University Bonnie Gold, Monmouth University 18 FOCUS

This session, sponsored by the SIGMAA on the Philosophy of [email protected] Mathematics, invites papers on any topic in the philosophy of G. Elton Graves, Rose Hulman Institute of Technology mathematics except logic and set theory. Possible topics include David A. Smith, Duke University the nature of mathematics, the nature of mathematical objects, the nature of mathematical knowledge, the relation between “Biology as Information,” the title of the 2004 Gibbs lecture by mathematics and the physical world, the role of esthetics in Eric S. Lander, Professor of Biology at MIT, emphasizes the the development of mathematics. fundamental changes that the science of biology is undergo- ing, especially since the connections between biology and math- MAA CP Q1 Using Handheld Technology To Facilitate Stu- ematics are becoming broader and deeper. This contributed dent-Centered Teaching/Learning Activities At The Develop- paper session will provide a forum for mathematicians with mental Algebra Level experience working with the interface of mathematics and bi- Friday afternoon ology to present papers which discuss the mathematics needed Ed Laughbaum*, The Ohio State University by contemporary biologists, the opportunities for mathemati- [email protected] cians and biologists to collaborate in teaching, curriculum de- Maria DeLucia, Middlesex County College velopment, student research projects, or professional research. Talks especially valued are those that make practical sugges- Lecture is the predominate method of choice for teaching re- tions concerning how to establish fruitful communication be- medial level algebra, but handheld graphing devices are often tween mathematicians and biologists, and, how to stimulate integrated by faculty. However, in many cases, the teaching/ mathematics and biology students to prepare themselves to learning is still instructor centered. Handheld devices offer the participate in this swiftly changing field. This session is spon- flexibility of enhancing teaching and learning through student- sored by the Subcommittee on Mathematics Across the Disci- centered activities, which can be used outside of class, or dur- plines. ing class through group work. Anecdotal evidence shows developmental algebra teachers often supplement textbook ma- MAA CP T1 Mathematics Experiences in Business, Industry terials with “graphing calculator” activities because even “re- and Government form” textbooks do not offer appropriate ancillary packages. Saturday morning Therefore, we invite developmental algebra faculty to submit Phil Gustafson*, Mesa State College, [email protected] proposals on creative teaching/learning activities that are stu- Michael Monticino, University of North Texas dent centered, provide a diverse learning environment, offer options for learning and teaching, and use handheld devices. This contributed paper session will provide a forum for mathematicians with experience in Business, Industry and Govern- MAA CP R1 My Three Favorite Original Calculus Problems ment (BIG) to present papers or discuss projects involving the Saturday morning application of mathematics to BIG problems. BIG mathemati- J.D. Phillips*, Wabash College, [email protected] cians as well as faculty and students in academia who are inter- Tim Pennings, Hope College ested in learning more about BIG practitioners, projects, and issues, will find this session of interest. This session is spon- This session is for those who, while teaching single and multi- sored by the MAA Business, Industry and Government Special variable calculus over the years, have thought of a few clever or Interest Group (BIG SIGMAA). novel problems with solid pedagogical value, which they would like to share with others. In particular, we are looking for origi- MAA CP U1 Mathematical Experiences for Students Outside nal problems suitable for homework assignments or challeng- the Classroom ing test questions. (We are not looking for extended modeling Saturday afternoon projects and open-ended problems since good collections of Kay Somers*, Moravian College, [email protected] these already exist.) We hope to organize these into a booklet Jody Sorensen, Grand Valley State University for publication, which could be used as a resource for calculus courses. Thus, we ask that each submission adhere to the fol- Mathematics “happens” both inside and outside the classroom lowing template: i) Statement of the problem, ii) Brief expla- and, in fact, many mathematics majors are drawn to the sub- nation of why it is interesting and pedagogically valuable, iii) ject through a special event sponsored by a Student Chapter or Complete solution leading to an answer in closed form. Sub- Math Club. This session seeks presentations by academic, in- missions may include from two to four problems. Participants dustrial, business, and/or student mathematicians so that the should bring copies of their problems to the session for distri- audience will be encouraged to organize and run bution. Each problem should begin on a new page. To include special events for their students. Descriptions of non-class- as many as possible, each participant will be given 10 minutes room activities could include, but are not limited to, special for presentation of the problems. lectures, workshops for students, Math Days, Math Fairs, research projects for students, Math Career Days, student con- MAA CP S1 MEETING the CHALLENGE: Relationship ferences, recreational mathematics activities, problem solving st Between Mathematics and Biology in the 21 Century activities and contests, general community-building activities, Saturday morning and student consulting projects. Information on how such Catherine M. Murphy*, Purdue University Calumet 19 FOCUS May/June 2004 activities are organized and carried out, what activities especially grab students’ interests, how students are contacted and Submission Procedures for MAA encouraged to participate, and how the events are funded will Contributed Papers be especially helpful. This session is organized by the MAA Committee on Undergraduate Student Activities and Chapters. Send your abstract directly to the AMS. At the same time, MAA CP V1 Research on the Teaching and Learning of send a detailed one-page summary of your paper, preferably Undergraduate Mathematics by e-mail, directly to the organizer–indicated with an (*). In Saturday afternoon order to enable the organizer to evaluate the appropriateness Bill Martin*, North Dakota State University of your paper, include as much detailed information as pos- [email protected] sible within the one-page limitation. The summary need not Barbara Edwards, Oregon State University duplicate the information in the abstract. A proposal should Draga Vidakovic, Georgia State University not be sent to more than one organizer. Participants may speak in at most two MAA contributed paper sessions. If your Research papers that address issues concerning the teaching paper cannot be accommodated in the session it was sub- and learning of undergraduate mathematics are invited. Ap- mitted, it will be automatically considered for the general propriate for this session are theoretical or empirical investi- session. Speakers in the general session will be limited to one gations conducted within clearly defined theoretical frame- talk because of time constraints. The summary must reach works, using either qualitative or quantitative methodologies. the organizer by Tuesday, September 14, 2004. Abstract must Of highest priority are proposals that report on completed stud- reach the AMS by Tuesday, October 5, 2004. ies that further existing work in the field. The AMS will publish abstracts for the MAA talks. Abstracts MAA CP W1 In-Service Training Programs For must be submitted electronically to the AMS. No knowledge K-12 Mathematics Teachers of LaTeX is necessary, however, LaTeX and AMSLaTeX are Saturday afternoon the only typesetting systems that can be used if mathematics Zsuzsanna Szaniszlo*, Valparaiso University is included. The abstracts’ submissions page is at http:// [email protected] www.ams.org/abstracts/instructions.html. Simply fill in each Judith Covington, Louisiana State University, Shreveport field as instructed. Submitters will be able to view their ab- Tamas Szabo, Weber State University stracts before final submission. All questions concerning the submission of abstracts should be addressed to: All over the country many small and large scale projects exist aimed at providing in-service training for K-12 mathematics [email protected]. teachers. The directors of these projects will share their experiences developing and implementing the projects, including Here are the codes you will need: MEETING NUMBER: 1003 both mathematical and organizational issues. Mathematicians contemplating starting similar projects will be able to learn The EVENT CODE is the seven characters appearing before about successful strategies and potential pitfalls for these out- the title of the sessions shown below, e.g., MAA CP A1 reach activities. The session invites talks that showcase successful in-service training programs for K-12 mathematics teach- The SUBJECT CODE is the last two-character letter/num- ers. The talks should reflect on every aspect of the program ber combination from the event code list, i.e., A1. including funding sources, organizational details, information on co-operations with the school districts, mathematical content and methodology, follow-up, evaluation and dissemina- tion. Programs that are easily replicable will be given priority. New 2004 MAA Section Governors MAA CP X1 General Contributed Paper Session Eastern PA/Delaware Doug Ensley Wednesday, Thursday, Friday, Saturday mornings and afternoons Florida David Kerr Daniel E. Otero, Xavier University Illinois Richard Wilders [email protected] Intermountain David G. Wright Thomas A. Hern, Bowling Green State University Iowa James Freeman James K. Strayer, Lock Haven University of Pennsylvania Louisiana-Mississippi Roger A. Waggoner Michael A. Jones, Montclair State University Maryland/DC/Virginia David Carothers Michigan Ruth G. Favro Papers may be presented on any mathematical topic. Papers North Central Dan Kemp that fit into one of the other sessions should be sent to that Southern California-Nevada Arthur Benjamin organizer, not to this session. Texas Elizabeth M. Bator

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MAA President Ronald L. Graham Testifies Before House Appropriations Subcommittee on Funding for Mathematics Education By Harry Waldman

On March 25, 2004, believe that the nation Ronald L. Graham, President would be well served of the Mathematical Asso- by continuing to in- ciation of America, testified crease mathematical before the House Subcom- sciences research mittee on Veterans Affairs, funding. However, we Housing and Urban Devel- also believe that re- opment, on NSF funding for search funding alone mathematics education. will not solve the nation’s difficulties in Thanking the Subcommittee attracting and educat- for its steadfast support of ing students and the National Science Foun- teachers in the math- dation in recent years—dur- ematical sciences.” ing which it provided in- MAA President Ronald L. Graham testifying before the Appropriations Sub- creases well above the committee. The Division of Un- Administration’s requests dergraduate Educa- considered essential to ensure that the for scientific education and research— tion at NSF aims to improve mathemat- number of mathematically trained U.S. Graham urged continued support for the ics and science education through the citizens does not drop below the level NSF’s investments that “have been and reform of courses, curricula, and instruc- needed to maintain U.S. technological continue to be vitally important for this tional materials, and to increase the qual- leadership. Nation’s long-term economic and na- ity and quantity of the mathematics and tional security.” science workforce. DUE programs fund DUE and DMS, for instance, collabo- innovative efforts at colleges and univer- rated in funding the Calculus Reform Right from the start President Graham sities in every state and at every level, all effort of the first half of the 1990’s. Not focused in on the one area of the NSF’s in an effort to find the best ways to in- only did this program change and mod- portfolio which, he said, “affects not only crease the scientific and technological lit- ernize the teaching of calculus in high every field of science and engineering but eracy of the nation’s youth. schools and colleges, the program was also, the nation’s long-term economic largely responsible for spurring renewal development, national security, and gen- DUE administers the Advanced Techno- across science, technology, engineering eral welfare. This area is mathematics logical Education program, which is a and mathematics undergraduate educa- education, particularly at the under- program that was created by a member tion throughout the country. In graduate level.” of the Subcommittee—Congressman Graham’s view, he told the Subcommit- David Price. The ATE program enables tee, no single educational program “has He called undergraduate education “the two-year colleges to collaborate with in- ever had such widespread impact.” crucial link in the educational system dustry in the development of programs preparing mathematicians, scientists, and courses for the education of the Further, a National Academy of Sciences health care professionals, engineers, tech- nation’s technological workforce. DUE is report in 1998 concluded that, “based on nological workforce, and the teachers of responsible for programs in teacher present trends, it is unlikely that the U.S. the next generation of students.” The training. will be able to maintain world leadership National Security Agency, he pointed out, in the mathematical sciences.” is the largest employer in the world of “Not getting the education of teachers mathematicians at all levels of educa- right the first time,” said Graham, “results In response to this report and others, tional background. in large expenditures for teacher train- NSF and Congress have increased funding later.” ing for mathematical sciences research The NSF funds undergraduate math- significantly in the past 5 years, doubling ematics education through two divisions: While the NSF budget has grown over- the research budget from about $100 the Division of Undergraduate Educa- all, Graham noted, “undergraduate edu- million in FY 1998 to $200 million in FY tion (DUE), located in the Education and cation has actually declined.” In FY 2004 2005. Human Resources Directorate, and the alone, he said, DUE was cut by nearly $20 Division of Mathematical Sciences million from the FY 2003 level. In FY “We at MAA,” Graham said, “strongly (DMS), located in the Mathematics and 2005, DUE would receive $159 million support this increased investment, and Physical Sciences Directorate. Both are

21 FOCUS May/June 2004 under the Administration’s request – an graduate students, and undergraduate improve the pipeline from the under- increase of 2%. students on research projects and edu- graduate years through positions in the workforce and academe. “Report after report has stated that the key to continued economic vitality is a “Not getting the educa- Graham also briefly touched upon the better educated workforce, particularly Mathematics and Science Partnership in mathematics and science,” stressed tion of teachers right program, which is slated to be trans- Graham. Thus, the MAA “urges Congress ferred to the Department of Education to provide $200 million for NSF’s DUE the first time results in under the FY 2005 budget request. MSP program in FY 2005. This amount would large expenditures for funds projects aimed at addressing the reverse the nearly $20 million reduction lackluster performance of U.S. children these programs suffered in FY 2004 and teacher training later.” in K-12 science and mathematics by cre- put it on a path that will enable the ating partnerships among teachers and nation’s institutions of higher educa- researchers. tion—including our community col- --Ron Graham leges—to more fully participate in NSF’s “The MAA believes,” Graham said, “that undergraduate programs.” if transferred to the Department, MSP cation. Universities with VIGRE grants funds will likely be distributed via block Another program that Graham urged the have reported a dramatic increase in the grants, which could spread the money Subcommittee to support, which is of- numbers of highly qualified U.S. citizens too thinly to do any real good and which fered by DMS, is VIGRE (Vertical Inte- pursuing graduate degrees in the math- will, in all likelihood, result in much of gration Grants for Research and Educa- ematical sciences. The program was re- the funding being redirected at the state tion in the Mathematics Sciences), which cently broadened to support faculty and level to programs outside the scope of brings together university professors, students from all types of institutions to MSP’s original intent. We at MAA urge you to keep MSP at the National Science Foundation.”