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"Wagering with Zero," by Brian Hayes. American Scientist, May/June 2008, pages 194-199.
In this column, Brian Hayes explores a game he refers to as a "Zeno gambling process." The game was first introduced by Hayes in a previous column called "Follow the Money" (American Scientist, September/October 2002) and is named after the Greek philosopher Zeno, who thought of problems such as a runner who runs half the distance to his destination, then half the remaining half, and so on, without ever reaching it. The wagering process involves two players who initially have the same amount of money. Every turn they flip a fair coin and wager half of the holdings of the poorer player. So for example, if each player starts out with 1/2 a unit of some imaginary currency, the first wager would be 1/4. After flipping a coin, the winning player has 3/4 and the losing player has 1/4, so the next wager would be 1/8, and so on. This imaginary currency has the property that it's always possible to split it in half, so one can imagine playing this game indefinitely. Hayes remarks that it's clear that there is no way for one person to get all the money or equivalently lose all the money in a finite number of plays, since every time the most you lose is half of what you have. One not-so-obvious observation is the fact that after the beginning, there is also no possibility for a tie. Hayes attempts in this article to understand the long-term behavior of this game. By using random walks, he realizes that it is very possible that once a player has the lead, he or she is likely to hold on to it. All the numbers that could appear as the amount of money the players have at a given time are dyadic rationals (rationals with denominator a power of 2). But Hayes observes that some numbers, which he calls Zeno's favorite numbers, appear with more frequency than others (and some numbers don't appear at all). He explores why this could be true by using binary trees representing all possible outcomes and finds that the probabilities of Zeno's numbers appearing are much higher. Hayes believes there are still many questions he'd like to answer and wants to study this further, perhaps by using other models in probability theory.
--- Adriana Salerno
"US scientist gives Israeli prize to Palestinians," by Amy Teibel. Associated Press, 26 May 2008.
Brown University mathematician and Fields Medalist David Mumford was one of three recipients of the 2008 Wolf Prize in Mathematics (the other two were Pierre Deligne and Phillip Griffiths, both of the Institute for Advanced Study in Princeton). The prize, presented by the Israel-based Wolf Foundation, carries a cash award of US$100,000. This article reports that Mumford decided to donate his share of the prize to Bir Zeit University, the West Bank's flagship university, and Gisha, an Israeli organization that works to protect the rights of Palestinian students in the Gaza Strip. Noting that he is not a political person, Mumford said he was motivated by his conviction that "higher education, access to mathematical knowledge, is something that should be shared and should be accessible to everyone." The Associated Press story was picked up by newspapers all over the world.
--- Allyn Jackson
"The Science of Fun," by Alex Bellos. The Guardian (U.K.), 31 May 2008.
Most mathematicians and math fans are familiar with Martin Gardner's work as the ultimate popularizer of puzzles and math games, mainly through his column "Mathematical Games" in Scientific American, and also through the publication of dozens of books. In 1993, one of Gardner's biggest fans, Atlanta businessman Tom Rodgers, decided to hold a conference in Gardner's honor, called the Gathering for Gardner, or G4G, meant to bring together experts in math, magic, and puzzles. It was so successful that it was held again in 1996 and every two years since. In this article, Bellos describes his experiences at the 2008 G4G. John Conway, from Princeton, counting pine cone spirals in the woods, looking for Fibonacci numbers. Erik Demaine, a 27-year old professor from MIT, receiving a standing ovation after revealing his breakthrough in extending the famous Haberdasher's Puzzle. Carolyn Yackel, lecturing on how to knit "hyperbolic" pants. An "exhibition" of mathematical objects such as origami and Daniel Erdelyi's "Spidrons". Raymond Smullyan, performing magic tricks on unsuspecting passersby. These are some of the images that Bello describes witnessing during the conference. Gardner is now too frail to attend the conference, but Bellos visited him at his home and describes his conversation with him. Gardner, citing magic as his gateway to math and philosophy as his first love, thinks of himself more as a journalist than a mathematician, stating that "beyond calculus, I am lost," and that he himself is actually "not very good" at solving puzzles. But Bellos points out that despite his lack of ego, which makes him so endearing, his work has inspired generations to take up math, and even influenced the direction of research. Gardner is still writing and publishing, showing no signs of slowing down. Read "Interview with Martin Gardner", by Allyn Jackson in the Notices of the AMS, June/July 2005, page 602.
--- Adriana Salerno
"Math Skills for Nonmajors Don't Add Up," by Jeffrey Brainard. Chronicle of Higher Education, 30 May 2008, page A11.
This sidebar is part of an article about science courses for undergraduate non-science majors. The sidebar briefly discusses teaching "numeracy" to non-math majors. The courses try to teach students "some grasp of percentages, statistics, and quantitative analysis," presenting subjects less abstract than calculus, but more advanced than balancing a checkbook. The article mentions that many undergraduates do need instruction in the latter skill.
--- Mike Breen
"Vital statistics." The Economist, 29 May 2008.
Disparity between the genders in mathematical academic performance may be due more to cultural influence than biological factors, according to a recent study from researchers at the European University Institute in Florence. A comparison of the difference in mathematical achievement and the level of cultural gender inequality—measured using indexes of education levels, attitudes towards men, and political and economic activity of women—revealed that countries with greater gender inequality had larger test score gaps. Two other interesting findings, however, were that gender inequality appeared to have no correlation to comparatively lower female skill levels in geometry or comparatively higher female reading scores. These findings may indicate that women often end up in non-mathematical fields because of a comparative advantage in that area over males, not because of a biological disadvantage in mathematical processing abilities.
--- Lisa Dekeukelaere
"Number keys promise safer data." BBC News, 21 May 2008.
"Wie Babys Statistik anwenden können (How babies can use statistics)", by George Szpiro. Neue Zürcher Zeitung, 18 May 2008.
This installment of Szpiro's monthly column about mathematics deals with the surprising ability of babies to draw statistical inferences about a group of objects, based on only a small sample, and with the fact that mathematical abilities can apparently be taught better by using abstract symbols and equations than by having recourse to practical examples.
--- Allyn Jackson
"Tones from Ancient Greece." Random Samples, Science, 16 May 2008, page 855.
Andrew Barker (University of Birmingham, UK) has constructed a helikon, a device invented by Ptolemy to understand musical scales. Barker used Ptolemy's descriptions of the mathematics of music to create the first modern-day helikon. The device has a sliding bridge and adjustable strings, which allow the user to create scales on the basis of mathematical principles.
--- Mike Breen
"Variationen zu einer Vermutung Eulers (Variation on a proof of Euler)", by George Szpiro. Neue Zürcher Zeitung, 14 May 2008.
Recently physicist Lee Jacobi and mathematician Daniel Madden proved that there are inifinitely many integer solutions to the equation a4 + b4 + c4 d4 = (a + b + c + d)4. The article traces the history of this equation back to Pierre de Fermat and to Leonhard Euler. The proof by Jacobi and Madden appeared in 2008 in the American Mathematical Monthly.
--- Allyn Jackson
"Measuring The China Earthquake's Magnitude," by Carl Bialik. Numbers Guy Blog, Wall Street Journal, 12 May 2008.
"Some swans are grey," by Robert Matthews. New Scientist, 10 May 2008, pages 44-47.
This article begins with a discussion of the ideas of the philosopher of science Karl Popper, who identified the defining characteristic of science as falsifiability: An idea or theory can be called scientific only if there is a way to test whether it is false. This is not just an academic exercise. According to the article, there is a good deal of debate among physicists today about whether some ideas now arising in theoretical physics, such as the notion of the "multiverse", are falsifiable---that is, whether they are truly scientific. But Popper's definition of science might not reflect how science is really carried out. Scientists typically do not propose theories and then try to falsify them. Rather, they try to amass evidence for theories. Even when evidence seems to contradict a theory, scientists might hold onto the theory anyway; Matthews gives an example where Einstein insisted his own ideas were "more plausible" than the alternatives even when some experiments seemed to indicate that he was wrong. Matthews presents an alternative view to Popper's black-and-white definition of science that allows for more shades of gray. This view depends on Bayesian analysis, a branch of probability theory that allows one to weigh evidence for various theories in order to assess which theory seems the most plausible. Bayesian analysis "shows that seemingly implausible theories require a hefty weight of evidence before they can be taken seriously," Matthews writes. "The Bayesian view also gives vague or contrived theories that fit pretty much any data set a tough time in the quest for crediblity." Bayesian analysis is now getting more attention from philosophers of science, as well as from "scientists in fields from archaeology to zoology."
--- Allyn Jackson
"Victorville math students learn how trigonometry is used in crime scene investigations," by Melanie C. Johnson. Press-Enterprise, 9 May 2008.
Early in May, students at two San Bernardino, California, high schools got the chance to apply mathematics to investigating crime scenes. They learned about the geometry of blood spatter, and, given actual crime scene photos, applied this knowledge to the types of problems that actual investigators have to address, including cause of death and angle of impact. Three times a year, the San Bernardino County Sheriff Department's Scientific Investigations Division, along with the San Bernardino County Superintendent of School's Alliance for Education program, hosts field studies like this for area high school students.
Kim Terry, the Alliance for Education's math curriculum specialist, emphasized the importance of increasing the graduation rate, in part by engaging students with relevant material that can answer the question: "When am I ever going to use this?" Math teacher Sharon Gollmyer sees how much students enjoy applying mathematics outside of the classroom: "It's in their faces. They are so intense... You can just see the future." And the students? Of those interviewed, one was considering forensics as a career, another wants to become a teacher, and all enjoyed the experience. High school junior Carmelina Figueroa commented that "I think it's pretty cool because we get to see what [these investigators] actually do and it's not a TV show."
--- Claudia Clark
"Math Group Tries to Help Young Teachers Stay the Course," by Sean Cavanagh. Education Week, 7 May 2008.
While math teachers may not be quitting any more quickly than those in English, there are not nearly as many waiting in line to replace the defectors. This fact concerns the National Council of Teachers of Mathematics, a group whose most recent conference in Salt Lake City included seminars dedicated to retaining new teachers. These seminars were created in response to surveys completed by incoming teachers who are often pressed for time to prepare their lessons and whose pay is not competitive with that of jobs requiring a similar skill level. The National Science Teachers Association has responded to a lack of new science teachers by launching new educational opportunities for incoming teachers and setting up student chapters at universities to help prepare and encourage prospective teachers. An encouraging quote from a new teacher cites her satisfaction with her job as being worth the pay cut, but how long will she remain this idealistic?
--- Brie Finegold
"Wishing for an African Einstein," by Daniel Clery, and "An African Showcase for Math Studies," by Robert Koenig. Science, 2 May 2008, pages 604-605.
In 2001, mathematical physicist Neil Turok visited his childhood home in South Africa after a 25-year absence. Turok was dismayed by the idea that, of the thousands of students who graduate from African universities with degrees in mathematics, "most are not able to find work and are frustrated because they can't do the interesting stuff." At the urging of his father, an anti-apartheid activist and member of the South African Parliament, Turok began planning and gathering support for what would become the African Institute for Mathematical Studies (AIMS) in Muizenberg, South Africa.
Currently, reports Koenig, 59 mathematics graduates from all over Africa are participating in the programs at AIMS doing this "interesting stuff": taking three-week intensive courses in pure math or physics as well as the "problem-solving realm of what [Institute director Fritz] Hahne calls 'relevant' mathematics---for example, related to bioinformatics, finance, or astronomy." In the process, Turok notes that they gain the confidence to pursue advanced degrees in South Africa, Europe, and North America. And of the 12 students interviewed by Science for this article, all plan to return to, and address problems in, their native lands.
--- Claudia Clark
In his essay, Caltech mathematician Jonathan Farley applies "thinking outside of the box" to come up with some unusual ideas for improving math and science teaching in public schools in his hometown of Rochester, New York. "We need to stop worrying so much about the `at risk' students and start worrying about the student who might become the next Benjamin Banneker, African-American mathematician and astronomer; or Ernest Everett Just, pioneering African-American biologist," Farley writes. "When these role models emerge, the rest will rise." Eric Gaze, a mathematician at Alfred University in New York state, takes a different viewpoint on the woeful state of math education in the U.S., commenting on the need for mathematical literacy among all students. "I have come to realize how illiterate we are when it comes to communicating with ratios, rates and percentages," he writes. "These are all middle-school math topics, so how do students get through high-school math without mastering them?" Gaze describes a new master's degree program at Alfred University, which shows future teachers of all subjects how to infuse mathematical literacy into their teaching.
--- Allyn Jackson
"Sin cities," by Mark Buchanan. New Scientist, 3 May 2008, pages 36-39.
This article discusses recent efforts to use mathematical modeling and computation to understand crime patterns in cities. One idea basic to this work is "routine activity theory", which puts the emphasis not on the deviant mind of the criminal but on the human habits and environments out of which crimes emerge. Routine activity theory suggests that "crime is a normal, if undesirable, outcome of ordinary social interactions." Often crimes like burglary spread like a communicable disease. This makes sense in the context of routine activity theory, which says burglars spend their time on routine, non-criminal activities and then commit crimes in areas they are familiar with. "Finding solutions to crime is much easier once a trend has been identified," the article says, "and now mathematically minded criminologists say that computer models based on routine activity theory have the potential to make sense of a far more complex mix of social and physical factors that may influence crime." Researchers have found that remarkably similar crime patterns show up in cities spread around the globe. Another key notion is "space syntax", which provides a way to describe how cities are built out of smaller elements of space, such as parks, roads, buildings, etc. Detailed study of the use of space and its influence on people "can explain a lot of human interaction, including crime."
--- Allyn Jackson
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