Math Digest

This article discusses new research, reported in the 25 April issue of Science magazine, ("The Advantage of Abstract Examples in Learning Math," Kaminski, Sloutsky, and Heckler) that seems to suggest that teaching mathematics through abstract principles works better than teaching with concrete problems. So-called "word problems", involving concrete situations like trains heading in opposite directions or the mixing of liquids in a cup, contain details that might simply distract students from the core mathematical principles. In a randomized, controlled experiment, Ohio State University researchers taught some students a mathematical concept using abstract mathematical ideas, and taught a control group of other students the same concept using word problems. Afterward, students were tested by seeing how well they figured out the rules of a mathematically based game. "The students who learned the math abstractly did well with figuring out the rules of the game," Chang writes. "Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing." The article quotes David Bressoud of Macalester College, currently the president of the Mathematical Association of America, who called the study "fascinating".

--- Allyn Jackson

Edward Lorenz was "the first to realize what is now called chaotic behavior in the mathematical modeling of weather systems." His obituary issued by the Massachusetts Institute of Technology (where he served many years in the Department of Meteorology) also states that "in the early 1960s, he realized that small differences in a dynamic system such as the atmosphere---or a model of the atmosphere---could trigger vast and often unsuspected results." He wrote about these conclusions in a 1972 paper, "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?," which lead to what became known as "the butterfly effect." The Los Angeles Times obituary notes that his work influenced a wide range of basic sciences, and "although the chaos theory was initially applied to weather forecasting, it subsequently found its way into a wide variety of scientific and nonscientific applications, including the geometry of snowflakes and the predictability of which movies will become blockbusters." Lorenz was a member of the National Academy of Sciences and won numerous awards, honors, and honorary degrees including the Crafoord Prize (a prize established by the Royal Swedish Academy of Sciences to recognize fields not eligible for Nobel Prizes) and the Kyoto Prize "for establishing the theoretical basis of weather and climate predictability, as well as the basis for computer-aided atmospheric physics and meteorology." The MIT obituary has more information.

--- Annette Emerson

The theme for Mathematics Awareness Month for 2008 is the mathematics of voting. In recognition of this theme, Paulos, a professor of mathematics at Philadelphia's Temple University, writes about the varied methods of voting and how they might change the outcome of an election. He points out that only one method is currently being used in the state primaries, the plurality vote, in which the candidate with the most votes "wins". But winning Republican nominees take all the votes from that state whereas the democratic primaries split the votes proportionally. Unfortunately, no detailed description is given for other voting systems, and a celebrated but perhaps disheartening result of Kenneth Arrow is discussed. Arrow showed that there is no method to derive group preference while satisfying a set of five rules for fairness, including that individuals' rankings of candidates be taken into account and that the winning ranking not mimic any one person's vote. Although Paulos's presentation of Arrow's Theorem might lead us to believe that "No voting system can be fair", this does not necessarily mean that different rules for fairness might not yield a different outcome or that all voting methods are equal in their reflection of the public's will. Paulos closes his article by endorsing Barack Obama.

Some of the same topics are covered in "Vote of no confidence", by Phil McKenna, New Scientist, 12 April 2008, pages 30-33.

--- Brie Finegold

Ce Bian and Andrew Booker, two mathematicians from the University of Bristol, England, recently published a finding that could help solve the Riemann Hypothesis---one of the most challenging unsolved problems in mathematics, now that Fermat's Last Theorem and the Poincaré Conjecture have both been cracked. The Riemann Hypothesis involves a function, the Riemann zeta function, that holds information on how the prime numbers are distributed. In some places many primes are close by, but in other places there are large gaps. Decoding the distribution of the primes is an especially important problem given the nature of primes as building blocks for all other numbers. The Riemann zeta function falls into a class called L-functions, and the two mathematicians in Bristol expended over 10,000 computer hours to discover another L-function. Insights learned from their recently derived function could help them understand the secrets of Riemann's. (Image: Visualization of the Riemann zeta function, with colors indicating different function values, by Jean-Francois Colonna, CMAP/École Polytechnique.)

--- Lisa DeKeukelaere

In this article, Diana Dabby, an assistant professor of electrical engineering and music at the Olin College of Engineering, writes about musical variation, "the technique of altering musical material to create something related, yet new." Adding additional notes around a theme, known as ornamentation, is one way to create variation. Joseph Haydn's F Minor Variations provide many examples of this. Composers may also create variation by using permutations and combinations of notes. Igor Stravinsky used this technique to compose his Variations: Aldous Huxley in Memoriam.

Composers have not limited themselves to rearranging groups of notes. For example, Pierre Henri mixed the recordings of three sounds---"a breathed sigh, the sung sigh of a musical saw, and a squeaking door"---to create his 1963 composition Variations pour une porte et un soupir. The composer John Cage takes the idea of variation even further: His "score" for Variations IV "consists of handwritten instructions providing a schematic that enables chance not only to decide the musical material [consisting of any sounds] but also to determine its order" so that each performance is vastly different. Another way to introduce variation is to use "chaotic mapping" with an existing work. This technique results in variations that can be "close to the original, diverge from it substantially, or achieve degrees of variability between these two extremes."

The Science website has some samples of the Haydn F Minor Variations, a chaotic mapping of the Bach Prelude in C (Bach is pictured at left), and a composition based on this chaotic mapping.

--- Claudia Clark

It is 7 feet high and 11 feet long, weighs about 3 tons, contains a multitude of gears, is operated by turning a crank, and can calculate solutions involving large polynomials without multiplying. It was designed in the 1840s by Charles Babbage but not built until the late 1980s by London's Science Museum. A second unit has been built and has been unveiled in the United States at the Computer History Museum in Mountain View, CA. "It" is Difference Engine No. 2, one of Babbage's attempts to eliminate human error by automating the creation of tables of values. These include logarithmic and trigonometric tables, widely used in his time for navigation, banking, and engineering.

The Difference Engine performs multiplication by using the method of differences---a means of finding successive values by adding previously calculated values. With the input of some initial values and muscle power, this machine can calculate and print the solutions to equations with 7th degree polynomials to 31 digits of accuracy, notes Andrew Carol, designer of a simpler LEGO Difference Engine. Carol also contends that, among all the machines designed and built to automate the process of creating tables of values, Babbage's design was "the most advanced and the most famous." (Photo of Difference Engine No. 2, courtesy of the Science Museum, London.)

--- Claudia Clark

It seems like the Aztecs had as much trouble as present-day Americans do when it came to figuring out their taxes. Two ancient codices (from A.D. 1540 to 1544) from the city-state of Tepetlaoztoc, recently deciphered by scientists Maria del Carmen Jorge y Jorge (National Autonomous University in Mexico City, Mexico) and B.J. Williams (University of Wisconsin-Rock County), give records of each household and its number of members, amount of land owned and soil types. Since landowners often had to pay tribute according to the value of their holdings, Jorge y Jorge explains, these texts were very detailed. The report, "Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic," published in the 4 April 2008 issue of Science, shows how the size of each parcel was calculated using a series of five algorithms, one of which coincidentally was also used by ancient Sumerians. One of the most puzzling features of the arithmetic in these records involved fractional symbols like hearts, hands, and arrows. But as Jorge y Jorge explains, these were actually very natural measurements related to the human body. The hand, for example, likely symbolizes the distance from the tips of the fingers of one outstreched arm to those of the other. (Image: Map (circa 1540) depicting parcels of land, courtesy of Library of Congress, Geography and Map Division.)

--- Adriana Salerno

On March 28, 2008, the legendary mathematician Alexander Grothendieck turned 80 years old. He had a huge impact on mathematics in the 20th century by building modern foundations for algebraic geometry. Since around 1990, he has cut himself off from society and has lived in an undisclosed village in the French Pyrenées. These two articles describe the extraordinary life of this singular genius. One of the authors, Winfried Scharlau, has written (in German) the first volume in a projected three-volume biography of Grothendieck called Wer ist Alexander Grothendieck: Anarchie, Mathematik, Spiritualität (Who is Alexander Grothendieck?: Anarchy, Mathematics, Spirituality). The book is self-published, and information about it is available on the Grothendieck Circle web site. A two-part article describing the life of Grothendieck appeared in the October 2004 and November 2004 issues of the Notices of the AMS.

--- Allyn Jackson

On 27 March 2008, the Norwegian Academy of Science and Letters held a news conference with live video feed to announce that the 2008 Abel Prize is awarded to John Griggs Thompson, Graduate Research Professor, University of Florida, and Jacques Tits, Professor Emeritus, Collège de France "for their profound achievements in algebra and in particular for shaping modern group theory." Thompson "revolutionized the theory of finite groups by proving extraordinarily deep theorems that laid the foundation for the complete classification of finite simple groups, one of the greatest achievements of twentieth century mathematics," and "Tits's geometric approach was essential in the study and realization of the sporadic groups, including the Monster." Newswires and newspapers from around the world that picked up the Academy's news release included the Associated Press, The Hindu (India), The Sun Herald (US), Cape Times (South Africa), FOXBusiness (US), The Guardian (UK), Earthtimes (UK), Xinhua (China), and Inquirer.net (Philippines). The prize amount is 6,000,000 Norwegian kroner (over US$1,000,000). Thompson and Tits will receive their prize in a ceremony in Oslo on 20 May 2008. The Abel Prize website has more information on the winners, their work, and the prize.

--- Annette Emerson

Click here for a list of links to web pages of publications covered in the Digest.