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# This month's topics:

## Math in the American Educator

A detail of one of the series of images by William Duke illustrating Wu's article in American Educator, used with permission. The full series, including the cover, is available here.

American Educator, the quarterly journal of the American Federation of Teachers, has circulation around 900,000. The cover story for the Fall, 2011 issue is "Phoenix Rising: Bringing the Common Core State Mathematics Standards to Life", by Hung-Hsi Wu, recently retired from the Berkeley Mathematics Department and a long-time student of American mathematics education. Wu blocks out what needs to be done if the Common Core State Mathematics Standards (CCSMS), which have been adopted almost nationwide and are scheduled to be phased in in 2014, are to achieve their goal of standardizing, updating and strengthening K-12 mathematics education in the United States. Concentrating on the middle and high school grades, Wu surveys math as presented in the CCSMS and contrasts it with the "de facto national mathematics curriculum" set out and perpetuated by the "several widely used textbooks;" he calls that curriculum Textbook School Mathematics (TSM). "TSM is much too vague and has far too many errors to be aligned with the CCSMS.... We must start from scratch." And, significantly, "Since teacher education in mathematics has long been based on TSM, both pre-service and in-service training must also be created anew." Wu gives two detailed examples, one on adding fractions, one on multiplying negative numbers, to explain how "the leap from TSM to the mathematical demands of CCSMS is a gigantic one."

How will teachers be able to make this leap? Professional development needs radical reform. U.S. higher mathematics education, through ignorance and indolence, bears much of the responsibility for the current "catastrophic education mess." Wu dismisses the "Intellectual Trickle-Down Theory" that many mathematics teacher-education programs seem to have espoused: that by building up teachers' general mathematical competence, we make them better mathematics teachers. Rather, "What colleges and universities should do is erase the damage done by TSM and revamp future high school teachers' knowledge of the algebra, geometry, trigonometry, etc. that they will be teaching;" so it will no longer happen that "TSM is recycled in K-12 from generation to generation." This means refocusing teacher-preparation courses on content knowledge relevant to K-12 classrooms." Wu advocates: "Get to know what they need, and teach it."

The end of the article treats the obstacles that treaten to block a genuine implementation of the CCSMS, with some suggestions as to how they may be overcome.

## Nobel Prize for Quasicrystals

Electron diffraction pattern of an icosahedral Zn-Mg-Ho quasicrystal. Image from Wikimedia Commons.

When Daniel Schechtman, on April 8, 1982, fired a beam of electrons at a sample of aluminum and manganese alloy (as recounted in Science, October 14, 2011, in a News & Analysis piece by Daniel Clery) he "saw a diffraction pattern unlike any he'd seen before: concentric circles of 10 bright dots," a picture like the one above. He wrote in his notebook: "10 fold ???" . The problem is that 5- or 10-fold rotational symmetry is incompatible with translational symmetry (*) and so should never occur in a crystal. He repeated and varied the experiment, always recording the same anomalous phenomenon. "When he finally told colleagues about his discovery, he was met with dismissal and ridicule." But he persevered, published, convinced his peers and now has been awarded the 2011 Nobel Prize for Chemistry. The award is for the discovery of quasicrystals, as configurations like this one came to ne named. As Clery tells the story, it was realized soon after Schechtman's discovery that a mathematical model already existed for how the atoms in the alloy could be arranged so as to produce such a pattern: at the vertices of a Penrose tiling. (See, for example, David Austin's Feature Column 1, Feature Column 2, on this website). "Much remains mysterious about quasicrystals," but they are here to stay: the International Union of Crystallography moved in 1992 to change the definition of crystal from a regularly repeating array of atoms to "any solid having an essentially discrete diffraction pattern."
(*) An elegant geometric proof of this statement is contained in the Scientific Background posted by the Nobel Committee. It runs as follows.

 "Two 4-fold ... or 6-fold ... axes of rotation generate new rotational axes at the same distance of separation as in the original pair. Repeating the procedure yields periodicity. For the pair of 5-fold axes (the second image), the procedure instead generates a new, shorter distance. An iterative procedure will thus fill the plane densely with 5-fold axes, and no periodicity will result." Illustrations by Johan Jarnestad © Royal Swedish Academy of Sciences, used with permission.

## Archimedes in the New York Times

Edward Rothstein's exhibition review "Finding Archimedes in the Shadows," appeared in the New York Times on October 16, 2011. The exhibition in question ("Lost and Found: The Secrets of Archimedes") opened that day at the Walters Art Museum in Baltimore and runs through January 1, 2012; it tells the story of the "Archimedes Palimpsest" and documents the efforts, by a team based at the Walters, to extract maximum information from the original Archimedean text. As Rothstein explains, that text, copied onto parchment around 1000 AD, had been erased; the parchment was reused as a Greek Orthodox prayer book which was treasured for centuries in a monastery but somehow lost in the early 20th century. There the story turns murky, and the manuscript undergoes further degradation; until in 2008 it appears on the market, is bought by an anonymous benefactor, and entrusted to the Walters for decipherment. Rothstein takes us through with him: "At the exhibition's start you come face to face with two leaves from the Palimpsest; all you see is a fragment of a ruined manuscript, charred, stained and inscribed with prayers. And you can also make out the ghost of a diagram, a spiral. Above these leaves a series of slides shows the same pages under colored lights, revealing various details." His only reservation, about the final gallery, is that there is not enough space allotted to the documents' substance: "Instead of ... detailing other restoration projects, it would have been far more illuminating to extend this mathematical section further."

The Times showed several images from the exhibition, including a nice, big reproduction of this one:

"A processed image showing the Archimedes text of 'Floating Bodies.'" Significantly, the diagram does not correspond to any of those published in Heath's 1897 edition of "On Floating Bodies." See a higher-resolution image. Images © The Owner of the Archimedes Palimpsest, reproduced courtesy of the Walters Art Museum.

## String theory - the new calculus?

Fans of Eugene Wigner must been elated recently by the apparently completely unreasonable mathematical overlap between string theory and condensed matter physics. The story so far is told by Zeeya Merali in a Feature/News piece in Nature (October 20, 2011). To set the stage, we have string theory: " ...mathematically rich, ... undeniable aesthetic appeal. But it is all about what physics might be at scales of $10^{-35}$ metres ... about 20 orders of magnitude smaller than a proton, putting the theory hopelessly beyond the reach of any direct experimental test." Then there is the research area of Dam Thanh Son, who first stumbled upon the overlap: "he was trying to understand the properties of quark-gluon plasmas, the short-lived, super-hot fireballs that form when heavy nuclei ... are smashed together in accelerators." Merali recounts that when Son visited his 1980s college dorm-mate Andrei Starinets, back in 1999 (both were then working in New York City), he "saw the string-theory calculations that Starinets had been working on with fellow PhD student Giuseppe Policastro [and] recognized the equations as the same ones he had been using to analyse the plasma."

How could 3-dimensional quark-gluon equations turn up in a string-theory calculation? As Starinets explained (here in Merali's words), the answer stems from a duality proposed in 1997 by Juan Maldacena: "string theory predicts a mathematical equivalence between two hypothetical universes, one of which would be similar to our own. It would have the same three dimensions of space and one dimension of time, for example, and be filled with much the same types of elementary particle.... But it would not contain strings - or gravity. The other universe would be the opposite: it would contain both strings and gravity... but no elementary particles. It would also have an additional dimension of space. ... That was why Son was seeing quark-gluon equations in a string-theory calculation... : they were the three-dimensional equivalent of the gravitational fields that he [Starinets] and Policastro had been studying in the four-dimensional universe."

Son, Starinets and Policastro derived a strategy to solve difficult quantum-field calculations: "map [them] into the four-dimensional world, in which the equations tend to be much easier to solve. Then they could map the results back to the three-dimensional world and read off the answer." By this method they were able to predict "the value of the shear viscosity of a plasma,... a key parameter of the quark-gluon fireball." Their prediction was experimentally confirmed seven years later.

For a final take on the phenomenon, Merali called on Andrew Green (St Andrews, UK): "Maybe string theory is not a unique theory of reality, but something deeper--a set of mathematical principles that can be used to relate all physical theories. Maybe string theory is the new calculus."

## "Biology is too complex to be unified by mathematics"

Marc Feldman (Professor of mathematical and evolutionary biology at Stanford) dissects Ian Stewart's new book "The Mathematics of Life" in a Nature review (August 25, 2011). His final paragraph: "The extreme diversity between the different disciplines of biological research [militates] against a unification in terms of mathematical modelling. It is not sufficient to claim, as Stewart does, that such a unification could be built around a theory of complexity. Although Stewart's many examples of mathematical models of biological phenomena are interesting to read, each requires a different part of mathematics. The fragmentation of biology and its maths is likely to continue for a long time."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu