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Standards in School Mathematics commentary.qxp 1/3/01 11:09 AM Page 165 Opinion discussing French history is not a powerful image. Andrew Geek Chic Wiles in the attic for seven years is, for them, ideal. Most disconcerting to us are images that join mathematics and It’s a thing that nonmathematicians don’t realize. Mathe- madness, a theme in almost all the popular works. matics is actually an aesthetic subject almost entirely. The key to the success of “Proof: A Symposium” and —John H. Conway similar ventures is to bring together people from inside and outside the mathematical community. “Proof: A Sym- We mathematicians know well the aesthetic pleasures of posium” was cosponsored by the Sloan Foundation, which our subject. G. H. Hardy wrote in A Mathematician’s is taking a leading role in the popularization of science and Apology that a “mathematician’s patterns, like the painter’s mathematics; the Manhattan Theatre Club, which produced or the poet’s, must be beautiful; the ideas, like the colors Proof ; and the Courant Institute. (Full disclosure: While or the words, must fit together in a harmonious way.” How Courant is my institutional home, I had no part in the frustrating it is that these pleasures cannot be transmitted organization of the symposium.) The Mathematical Sciences to the general public. How frustrating that our intelligent Research Institute in Berkeley has had a series of successful and otherwise highly literate friends have so little appre- public events of this kind, beginning with the “Fermat Fest” ciation for our labors. in 1993. More recent events have included a conversation I have some good news to report. There is right now in the between mathematician Robert Osserman of Stanford popular culture a wave of interest in mathematics. Even University and playwright Tom Stoppard concerning better, the image that has captured the public imagination Stoppard’s work Arcadia, and a mix of conversations and is that of a mathematician working intensely and usually theatre about the life of Galileo. alone on the most difficult and abstract problems. What was On the screen Good Will Hunting and will soon be once considered hopelessly geeky is suddenly au courant. joined by a film based on A Beautiful Mind, Sylvia Nasar’s A recent manifestation of this phenomenon is the play biography of John Nash. Proof joins Copenhagen on Proof, in which three of the four characters are mathemati- Broadway. Bookstores are stocked with biographies of cians. The plot centers on a notebook of uncertain authorship Paul Erdo˝s; the latest book by the renowned mathematics that may or may not contain a great proof. Ben Shenkman is expositor Keith Devlin; and novels with mathematical excellent as Hal, a graduate student that many will recog- themes, such as Uncle Petros and Goldbach’s Conjecture nize. Yes, he is geeky and callow (the repartee about his and The Wild Numbers. All this attention is most enjoyable. rock band is hilarious) but simultaneously very human and Let’s take it as an opportunity to communicate to the humane. The star, Mary Louise Parker playing Catherine, is public the beauty and centrality of our subject. amazing. I’ve often been disappointed by theatrical attempts to portray genius. Parker’s Catherine has an authentic —Joel Spencer genius. She struggles with family and with mental illness. Courant Institute, New York University She has enormous intensity and vivid sexuality. The disparate pieces come together (this is magic to me) in an utterly believable and compelling portrayal. Reviews of the Proof: A Symposium play, including one that appeared in the Notices (October October 16, 2000 2000, pages 1082–1084), have been excellent. Buoyed by this New York University success, Proof moved to Broadway in October 2000. Panel I: What’s a Proof and What’s It Worth?: Peter How can we mathematicians ride this wave? “Proof: A Sarnak (moderator), Princeton University; Kit Fine, NYU; Symposium”, held last October at New York University, Arthur Jaffe, Harvard University and Clay Mathematics provides an outstanding model. The symposium was a Institute; Dusa McDuff, SUNY Stony Brook; Thomas forum for discussion by mathematicians and nonmathe- Nagel, NYU; Michael Rabin, Harvard University; Jack maticians of some of the mathematical themes and ideas Schwartz, Courant Institute, NYU. raised in Proof. The symposium’s first panel discussion, on Panel II: Women and Proof: Margaret H. Wright the nature of proof, was the most mathematical. The star (moderator), Lucent Technologies; Dusa McDuff, SUNY was Thomas Nagel, a distinguished philosopher whose Stony Brook; Cathleen Morawetz, Courant Institute, incisive views on objective truth were of interest to math- NYU; Mary Pugh, University of Pennsylvania; Jean E. ematician and nonmathematician alike. The second panel, Taylor, Rutgers University; Karen Uhlenbeck, Univer- “Women and Proof”, had an all-star cast. The stories sity of Texas at Austin. these women told were at turns funny and moving. Skillful Panel III: Proof in Performance and Prose: Michael organization kept the strong-minded panelists on com- Janeway (moderator), Columbia University; David mon themes. The final panel, “Images of Proof”, consisted Auburn, author of Proof ; Rebecca Goldstein, novelist; of writers and actors. Hearing these panelists discuss Sylvia Nasar, Columbia University; Ben Shenkman, actor my world was an eye-opener. They are always searching who plays Hal in Proof. for the best image for their works. Andrew Wiles at lunch FEBRUARY 2001 NOTICES OF THE AMS 165 commentary.qxp 1/3/01 11:09 AM Page 166 Letters to the Editor Letters to the Editor was to little avail; the message—the years than it has in the past thirty, “vision” of PSSM—remains, in my the country is in for a meager intel- vision, much the same as that of lectual future. Standards in School the original 1989 Standards. True, it is not the primary busi- Mathematics PSSM continues to abhor direct ness of mathematicians to study The Notices for September and instruction in, among other things, the problems of school mathematics October 2000 featured some discus- standard algorithms, Euclidean geom- programs, let alone engage in the sion of the Principles and Standards etry, and the uses of memory. Though political struggle needed to make a for School Mathematics (“PSSM”) of the like its predecessor it has the word difference in the public schools them- National Council of Teachers of “standards” in its title, it is not a set selves, but I appeal to all who read this Mathematics (NCTM), this manifesto of standards in the usual meaning of letter to obtain a copy of PSSM being for the most part a revision of the term, for it refuses to say what ex- (http://www.nctm.org/) and reflect NCTM’s 1989 Standards. The earlier actly a child should learn in thirteen on what such a document means to document, mainly unnoticed by the years of schooling. Long division? the future of the children in today’s mathematical profession at the time, Quadratic formula? How to compute schools. I warn you that these “prin- offered as its principal vision that the quotient of two fractions? (See ciples and standards” cannot be school mathematics need not be p. 218 of PSSM for an enlightening appreciated by reading only a few difficult or dull and that the cure discussion.) Proof of a theorem on in- pages. In the small the document was to remove the mathematical scribed angles? Trigonometric identi- sometimes sounds good. But if PSSM content from it, leaving behind the ties? PSSM will neither affirm or deny, in the large informs our vision, then mathematical concepts as a sort of lest it seem to dictate content. self-esteem is better than knowledge, Cheshire Cat grin. There is no place Joan Ferrini-Mundy has publicly dictionaries can replace a ready (mem- here for detail, for which see the averred that both PSSM and its pre- orized?) vocabulary, and higher-order “Mathematically Correct” Web page decessor have been misunderstood thinking skills will boil stones into (http://mathematicallycorrect. and that NCTM does indeed advocate soup. com/) or find a copy in a library and learning the multiplication tables. This see for yourself. is almost true for the multiplication —Ralph A. Raimi Needless to say, not everyone table, though only as a last resort University of Rochester agrees with the above assessment of (PSSM, p. 152). Other such concessions the import of the 1989 Standards, but are harder to find. Almost anything in (Received October 20, 2000) by the end of the 1990s enough math- the way of content to be remembered ematicians—notable among them can be omitted from a school mathe- Richard Askey, the late Han Sah, and matics program without running afoul Hellmuth Kneser’s Forgotten CR Hung-Hsi Wu—had developed a of PSSM, providing the pedagogy is Extension Theorem loathing for NCTM doctrine that man- right and the process suitably “ex- F. Treves’ interesting and informative aged to attract the attention of NCTM ploratory”. “Explore”, “develop”, and article in the November 2000 issue itself. Other opposition has also “understand”, and their variants, are discusses several major themes in emerged, mainly from parents’ groups much more prominent in the text than multidimensional complex analysis enraged at the NCTM-blessed mathe- “know”, “prove”, and “remember”. in the concrete context of the hyper- matics programs beginning to spread Under the color of NCTM’s vision of quadric. in their schools. (“Mathematically Cor- mathematics as expressed in the 1989 In particular, the author recalls the rect”, which speaks for some scien- Standards have been written a num- local extension theorem for Cauchy- tists and mathematicians as well, was ber of school mathematics programs Riemann functions (Theorem 1, page a pioneer among these.) Clearly NCTM recently officially recognized as “ex- 1248), with reference to Hans Lewy’s would have to take account of math- emplary” or at least “promising” by the well-known 1956 paper.
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