Letters to the Editor: The Lady Doth Protest Too MuchSeptember 13, 2001
To the Editor:
I read with interest Philip Davis's review of the new book by Diane Ravitch. It seems the lady doth protest too much! Math teaching (like teaching in general) is subject to fashions.
I was taught in India and did my PhD in the early '70s in the applied math department at Caltech. I hated the "rote learning" and the failure to clarify concepts inherent in the school and college curricula at that time in India, and the pressure to chase grades (which Ravitch would presumably call "external evaluations of an objective kind") as opposed to genuine understanding. However, I had very committed teachers, and their collective influence enabled me to survive (and even benefit from) the hot-house that was Caltech. While we can all deplore the wildest excesses of Bourbakiism and the New Math, the fact that people learn at different rates and are motivated by different stimuli surely cannot be denied. Ravitch's call for "orderly" curricula sounds like some nostalgic hankering for what G.H. Hardy and J.E. Littlewood described as the "senior wrangler mentality" of 19th-century Cambridge.
In England (where I live) and perhaps also in the U.S., there is, however, a serious failing in "modern" math (and physics) teaching: total omission of the cultural history of the subject, side-by-side with the actual mathematics (pure or applied). The usual excuse (there really is no time; it is up to the students to read about the history) is simply an admission of failure. No self-respecting course in math, physics, or astronomy should ignore the immense struggles over the centuries to formulate and understand the key concepts; the steady evolution of the notions of rigour and proof; the terrible controversies, jealousies, and professional misconduct that have punctuated the growth of science; and the remarkable simultaneous contributions in many different places by very different people.
It was by reading E.T. Bell's Men of Mathematics as a 16-year-old that I had the joy of finding out how nontrivial math really is and how delightful (and, of course, how beautiful). The aesthetic and historical perspectives provide vital enrichment of purely technical skills. Only then is a truly "liberal" education possible. Many extremely competent young people these days have virtually no idea of the tortuous way in which, for example, the theory of functions of a real variable was created by the pioneers and the many false starts and dead ends en route. Instead, students are taken through a dry series of theorems and lemmas, rather akin to studying Mozart without ever listening to the music.
An interesting principle of biology states that "ontogeny recapitulates phylogeny." This could be usefully applied to math and physics teaching, where exposure to the failed approaches of the past (say phlogiston theory or misinterpretations of the theory of divergent/asymptotic series) could actually serve to reassure students that they are in excellent company in their apparent inability to understand subtle and hard-won concepts and to help them grasp better the context and content of modern, refined theories. In my view, only history can demonstrate that mathematical concepts are no more "discovered" than are all the myriad chess games, but are in fact invented under the stress of the need to solve real problems within an accepted framework of logic and structure.
Anantanarayanan Thyagaraja, Theoretical Physics Group, UKAEA Fusion, Culham Science Centre, Abingdon, UK.