Standards in School Mathematics

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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

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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. Treves, as ematicians in writing its scheduled U.S. Department of Education, but to well as virtually all other researchers new edition (i.e., PSSM), and it did. a chorus of public protests, some of in the field, had not been aware As Joan Ferrini-Mundy, its principal it from mathematicians. Because many that this celebrated CR extension editor, explained in her September of us with children—or grandchil- theorem was proved twenty years Notices article, NCTM this time com- dren—in today’s schools have now before H. Lewy’s paper in a remark- missioned the commentary of many seen these programs in action, the able 1936 paper by Hellmuth Kneser, mathematicians, including commit- public protests are still mounting. come back to light only recently (Die tees of AMS, MAA, and SIAM, upon an PSSM may prove a marginally better Randwerte einer analytischen Funk- earlier draft prepared for us. I myself theoretical guide to further such tion zweier Veränderlichen, Monatsh. served on the AMS committee and projects than the 1989 version, but we Math. Phys. 43 (1936), 364–380). (by commission) as an individual too. deserve better than this. If the world This important paper, together with NCTM solicited public advice at large, of mathematics, sadly divorced from earlier contributions by W. Wirtinger and I know several who also attempted the world of school mathematics (1926) and F. Severi (1931), documents to link the mathematical world with education, pays no more effective that CR functions have a history much the new document, but the effort attention to the schools in the next ten older than commonly recognized.

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Letters to the Editor

Related to this matter is the wide- models. At that time only such relative for his Paris address. It was concerned spread confusion regarding the consistency proofs were known, and with finding criteria for finding global CR extension theorem, first this may account for Padoa’s and simplest proofs of theorems in math- proved by G. Fichera in 1957 and Peano’s confusion. ematics in general—once again, more often attributed mistakenly to Next, Grattan-Guinness added: “Un- a programme than a problem in the S. Bochner. For the record, there is fortunately Hilbert did not make normal sense. He left it out of his final no evidence whatsoever in Bochner’s amends in the Archiv version (pre- version, I suspect after realising that, 1943 paper to suggest that Bochner sumably lack of Italian again), but in ironically, simplicity is an extremely had even remotely been thinking L’Enseignement Mathématique Padoa complicated notion to capture in a about CR functions and the corre- explicitly discussed this problem…”. general way, so that nothing useful sponding version of Hartogs’s famous The fact that Hilbert did not modify could be said here. extension theorem. More details his article when it was later reprinted Concerning the complaints about may be found in my forthcoming in the Archiv der Mathematik und the presentation of the lectures at historical article in The Mathematical Physik to reflect Padoa’s Congress the Paris Congress which I reported, Intelligencer. article was not due to Hilbert’s not Professor Wilfrid Hodges (University knowing Italian, since Padoa’s article of London) tells me that recently he —R. Michael Range was in French. Rather, it was because himself lectured in the same room State University of New York at Padoa’s Congress article contributed that Hilbert had used. Apparently the Albany nothing to the solution of Hilbert’s acoustics there are terrible! Second Problem, despite Padoa’s claim (Received October 31, 2000) in L’Enseignement Mathématique to —Ivor Grattan-Guinness have solved it. Middlesex University Grattan-Guinness leaves the reader Correction to the History of with the impression that Peano and (Received November 20, 2000) Hilbert’s Problems Padoa were right in claiming that In the August issue of the Notices ap- the Second Problem was solved and peared Grattan-Guinness’s intriguing Hilbert wrong. But, in fact, the oppo- article “A Sideways Look at Hilbert’s site is true. As is well known, the Twenty-three Problems of 1900”. One Second Problem was not solved until claim made in that article calls for 1931 by Kurt Gödel in the profound correction. Grattan-Guinness, dis- result now known as his Second cussing the comments after Hilbert’s Incompleteness Theorem. 1900 lecture at the International Congress of Mathematicians, stated —Gregory H. Moore that “Peano…remarked that [Hilbert’s] McMaster University Second Problem on [proving] the consistency of arithmetic was already (Received November 7, 2000) essentially solved by colleagues working on his project of mathematical logic and that the forthcoming Response to Moore’s Letter Congress lecture by Alessandro On consistency, it is clear that Hilbert Padoa…was pertinent to it” (p. 756). required an absolute version for arith- Actually, Peano made a much more metic, and I should have distinguished direct and unqualified assertion: it from relative ones. It is a pity, “Monsieur Padoa’s later communica- though, that Padoa’s foundational tion will answer Hilbert’s Second work has been overshadowed by that Problem” (p. 21 of the Congress proof others. His Paris lecture did not ceedings). Despite Peano’s claim, appear till 1901, after the Archiv Padoa’s article did not solve Hilbert’s version of Hilbert’s paper anyway. Second Problem by proving the Two pieces of information from consistency of arithmetic, but only readers of my article are worth pass- stated that “to prove the consistency ing on. of a postulate system, one must find The first has some kinship with an interpretation of the undefined the above point. Professor Rüdiger symbols which satisfies all the postu- Thiele (Halle University) has found lates simultaneously” (p. 249 of in a notebook in file 600 of Hilbert’s the Congress proceedings). What mountainous Nachlass at Göttingen Hilbert wanted to do was, in fact, to University Library an apparently find an absolute consistency proof, undated passage in which Hilbert not a relative consistency proof using recalled including a 24th problem

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