
New York State Regents Math Flap. The first stories broke on June 19, 2003. Newsday reported that "The state Department of Education is ... speeding an analysis of Tuesday's standardized math tests because of initial reports of unusually low grades." In fact the failure rates on the June 17 Regents Math A exam turned out to be close to double those typical of the Regents Math I exam that it supercedes. The Math A exam, covering the first three terms of high school mathematics, is mostly taken by Sophomores. But many of those who failed were Seniors who needed a pass to graduate. Tears ensued ("Kim could not speak. She was crying too hard."  Michael Winerip, New York Times June 25; "the Math A test that sent students sobbing from their desks"  Newsday editorial, June 26), and anguished calls from parents to legislators. By June 21, Newsday's John Hildebrand, under the headine "Testy Reactions," was reporting that the Regents were under political pressure to "revoke scores and expunge them from students' records." An accompanying box ("Not Your Father's 'Rithmetic  The New Regents Math A exam contains some real tough questions") presented these two "stumpers:"
Question: The diagram shows a square with side y inside a square with side x. Which expression represents the area of the shaded region? 1. x^{2} 3. y^{2}  x^{2} 2. y^{2} 4. x^{2}  y^{2}  Question: A straw is placed in a rectangular box that is 3 inches by 4 inches by 8 inches. If the straw fits exactly in the box diagonally from the bottom left front corner to the top right back corner, how long is the straw, to the nearest tenth of an inch? 
On June 25, the crisis broke: as the New York Times reported, "Citing Flaw, New York State Voids Math Scores"
The controversy gave many New Yorkers a chance to express their views on mathematics and math education. Letters to the Editor ran the gamut from "How many collegeeducated adults can solve that problem [the straw] accurately? Are such skills really necessary to be a productive member of our society?" (New York Times, June 30) to "I dropped out of school in 1948. Yet I can pass that test  barely. What does that say about the quality of the teachers and the methods used today?" (New York Post, June 23).
Try the test yourself on the Regents' website.
"Pure Math  Pure Joy" was a piece by Dennis Overbye in the New York Times' Week in Review, June 29, 2003. Overbye recounts a visit to the Mathematical Sciences Research Institute in Berkeley. The tone is ecstatic. "Consider it an embassy of another world, a Platonic realm of clarity and beauty," etc. Overbye remarks usefully on the relevance of higher mathematics ("ever since Archimedes leaped out of his bath shouting 'Eureka!' more than 2,000 years ago") and gives us a nice quote about string theory from Brian Greene: "Since our theories are so far ahead of experimental capabilities, we are forced to use mathematics as our eyes. That's why we follow it where it takes us even if we can't see where we're going." The article is illustrated by some samples from Ed Alcock's photo essay on MSRI; the entire collection is available on the MSRI website.
Gravitational caustics. In the May 2 2003 Science a 7member team led by Chris Carilli (National Radio Astronomy Observatory) published "A Molecular Einstein Ring: Imaging a Starburst Disk Surrounding a QuasiStellar Object." The QSO in question is PSS J2322+1944; images both in the Infrared (CO emission) and at 1.4 GHz show the "Einstein ring" diagnostic of "strong gravitational lensing by an intervening galaxy." In the absence of information about that particular lens, the team worked from a better known one and experimented with "various source configurations" until they could get a close match to the observed pattern. The model they derive is illustrated here.
"A gravitational lens model for the CO emission in PSS J2322+1944. ... The left panel presents the source plane distribution, corresponding to the true (i.e., undistorted by lensing) morphology of the system. The image plane distribution is presented in the right panel, corresponding to the observed morphology after being distorted by the gravitational lens. The pointlike QSO is represented by a black asterisk in the left panel and by two black asterisks in the right panel. The green solid lines are the caustics and critical lines in the source and image planes, respectively ... . The CO emission is modeled as an inclined disk (i ~ 60°) around the QSO, and the north and south parts of the disk are colorcoded red and blue, respectively, corresponding to different velocity regions on opposite sides of the QSO." (Image ©2003 Science, used with permission). 
Where do we do math? The June 2003 Nature Reviews Neuroscience ran the survey "Neural foundations of logical and mathematical cognition" by Olivier Houdé and Nathalie TzourioMazoyer (CNRS, Caen/Paris). The authors review what is known about where logical reasoning occurs in the brain, with special attention to the question of to what extent visual and/or linguistic circuits intervene. Among the studies cited was one of their own, which showed that "although there is a general arithmetic ability for small numbers that is shared by preverbal infants and monkeys, the ontogeny of this initial knowledge in humans follows different performance patterns, depending on what language the children speak." In fact they claim that for Frenchspeaking 2yearolds verbal arithmetic facility is hindered by the use of un both as a counting number and as the masculine singular indefinite article, in that language. In another study, they used Positron Emission Tomography to study the brain of a calculating prodigy (Rüdiger Gamm, "remarkable in that he has the ability to calculate, for example, the quotient of two primes to 60 decimal places with incredible accuracy.") They found that while Gamm also used the parts of the brain relied on by nonexpert calculators, his "calculation processes recruited a system of brain areas that are implicated in episodic memory." Whereas "the rest of us rely on the limited span of working memory." They speculate "that experts might develop a way of exploiting the unlimited storage capacity of longterm memory to retain taskrelevant information, such as the sequence of steps and intermediate results that are needed for complex arithmetic operations."
Beyond Buckyballs. The chemist Achim Müller contributed "The Beauty of Symmetry" to the May 2 2003 Science. His piece was occasioned by the appearance, in that issue, of two papers treating "fullerenelike clusters." In one Bai et al reported the synthesis of a completely inorganic spherical molecule, based on a PhosphorusCopper skeleton, with the same geometry as a Buckyball (but 3 times the size); in the other, Moses et al develop a molecule in which an icosahedron with 12 Nickelatom vertices interpenetrates a dodecahedron with 20 Arsenicatom vertices. Müller mentions and illustrates (see at left) the molecule nicknamed "Mo_{132}" which exhibits "an unprecedented series of interpenetrating distorted Platonic and Archimedean solids (all of which are shown in the figure), with all atoms located in the surface." Image courtesy Hartmut Bögge, Bielefeld. Click for large image of full series. 
Tony Phillips
Stony Brook
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