September 2001
Math Anxiety and Math Competence, how are they related? A study by Mark Ashcraft and Elizabeth Kirk, of Cleveland State University, picked up in a piece by B. Bower in the June 30, 2001 Science News, describes experiments giving evidence that math anxiety can be a cause of low math competence and not just a consequence. The most striking experiments show that mathanxious students who can perform as well as their peers on pencilandpaper tests fall behind on mental arithmetic tasks, especially those involving the ``carrying'' operation. The authors explain the lower performance in terms of ``working memory:'' ``We propose that there is an online reduction in the available workingmemory capacity of highmathanxiety individuals when their anxiety is aroused.'' Full text of the AshcraftKirk article is available online. What is Math about? The question comes up in a ``Words'' piece by John Casti (Technical University of Vienna) in the May 31, 2001 Nature. Casti's piece is titled ``Formally speaking'' and deals mostly with the mathematical process of formalizing the informal (``...`rational' decisionmaking and game theory, `closeness' and topology, `infinity' and calculus''). But it includes his interesting take on Gödel's incompleteness theorem: ``... Gödel's incompleteness theorem shows that there is an irreducible semantic component to mathematics. It is not just a game of shuffling meaningless strings of symbols about in accordance with a set of transformation rules.'' Or, as he puts it elsewhere, ``... one must always remember that number theory is about numbers.'' Tying and untying molecular knots. A report in the June 14, 2001 Nature describes a synthetic oligomer which can be chemically directed to knot itself, to unknot and to reknot. The work, by a sixperson team (Henry Adams, Eleanor Ashworth, Gloria Breault, Jun Guo, Christopher Hunter, Paul Mayers) based in Britain, involves creating a string of three rigid ``linkers'' joined by two ``ligands.'' When zinc perchlorate is added to the solution, the linkers assemble themselves in an octahedronlike cluster about a zinc ion, and as a result the ligandlinker chain becomes knotted.
Wiles at the Olympiad. Andrew Wiles was ``greeted like a rock star by the current generation of young math stars at the close of their global competition.'' This from the lead paragraph of ``Young Math Competitors Pay Tribute to Their Hero'' in the July 14, 2001 New York Times. Wiles addressed the International Mathematical Olympiad, hosted in Washington D.C. this year. ``You can tell by the response here that he is a hero among the math community,'' said silver medalist Oaz Nir, 17. Mr. Nir, David Shin (silver) and Reid Barton, Gabriel Carroll, Ian Le and Tiankai Liu (all gold medalists) made up the U.S. team, which tied with Russia for second place. First place went to China. Barton, who is 18, won a gold medal for the fourth time. He and Carroll had perfect scores on the examination. Is pi normal? Which means, do all digit sequences of the same length appear with the same frequency in its decimal expansion? Statistical evidence favors normality. For example, in the first 200 of the 206 billion digits recently computed by Yasumasa Kanada et al. at the University of Tokyo, 7 occurred 19,999,967,594 times. This information is from a piece by Ivars Peterson in the September 1, 2001 Science News. It seems sort of obvious that there should be no incestuous relationship between pi and 10, but establishing a proof is another matter. Recent progress has been made, however. It builds on a 1995 discovery by David Bailey (Lawrence Berkeley National Lab), Peter Borwein (Simon Fraser) and Simon Plouffe (University of Quebec at Montreal), who ``unexpectedly found a simple formula that enables one to calculate isolated digits of pi say, the trillionth digit without computing and keeping track of all the preceding digits.'' This formula only works for the base 2 and base 16 expansions, not the decimal, but it seems like a step towards determining the normality of pi in those bases. Now Bailey and Richard Crandall (Reed) have proved the equivalence between the base2 normality of pi (and ln 2) and the equidistribution property for the orbit of certain selfmaps of the interval. Peterson tells us which map works for ln 2: x_{n} = 2x_{n1} + 1/n (mod 1), and relates the pessimistic opinion of Jeff Lagarias (AT&T labs), that the new problem may be as intractable as the old. As usual, pi brings out the puns: Peterson called his piece ``Pi à la mode,'' while the Nature comment was titled ``Pi shared fairly.''
Tony Phillips

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