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This month's topics:
"Israel Gelfand, Math Giant, Dies at 96." That's the headline for Kenneth Chang's obituary (in the October 7, 2009 New York Times) for the great Russian mathematician. Chang's article is remarkably specific about Gelfand's stature in the mathematical community and the special nature of his mathematical personality. He quotes Vladimir Retakh (Gelfand's colleague at Rutgers), who puts him near Euler, Hilbert and Poincaré, and tells how Vladimir Arnold (Gelfand's neighbor in the Pantheon) characterizes his type of mathematics by contrasting him with Andrei Kolmogorov (another neighbor): "Suppose they both arrived in a country with a lot of mountains. Kolmogorov would immediately try to climb the highest mountain. Gelfand would immediately start to build roads." Or, in Chang's paraphrase: "Dr. Gelfand did not achieve fame from attacking and solving famous, intractable problems. Instead, he was a pioneer in untrodden mathematical fields, laying the foundation and creating tools for others to use." The long and detailed obituary ends with a quote from an interview Gelfand gave the Times in 2003. "It is important not to separate mathematics from life. You can explain fractions even to heavy drinkers. If you ask them, 'Which is larger, 2/3 or 3/5?' it is likely they will not know. But if you ask, 'Which is better, two bottles of vodka for three people, or three bottles of vodka for five people?' they will answer you immediately. They will say two for three, of course."
Thomas Maugh's Gelfand obituary in the Los Angeles Times (October 11; "His research laid the mathematical foundation for the imaging abilities of MRI and CT scanners.") covers much of the same material (including Arnold's comparison and the vodka story) but adds some new details. "At 15, he contracted appendicitis, which then required a 12-day stay in the hospital. On the way there he asked his parents to buy him a calculus text, and he mastered it in his bed." Maugh spoke with Edward Frenkel (Berkeley, worked with Gelfand in Moscow) who sketches out for us the scene at Gelfand's "legendary mathematics seminar, held every Monday night for nearly 50 years on the 14th floor of the Moscow university building," and where "The meetings, which often lasted well into the night, were more like a social event than a traditional seminar." Frenkel: "[Gelfand] would walk the aisles, stop and chat with people, interrupt and ask questions, pull a member of the audience to the blackboard and ask them to repeat what had just been said or to find a mistake in it. His interest was always in the development of the next generation of mathematicians."
Le Monde interviewed Eric Charbonnier, an analyst at the Organization for Economic Cooperation and Development Directorate for Education, in an on-line chat posted October 13, 2009. The headline: "In France, math has become too important in the educational system." The problem is that in France success in mathematics is the key to accessing higher education, as opposed to the other OECD member states "where literary or economic concentrations [in High School] can lead to the best universities," according to Charbonnier (my translations for this item). "Today, in France, if you want to study economics or journalism in the university, you need to concentrate in science or mathematics in high school. And that's the problem."
Charbonnier also remarks how France spends 23% more per student than the average OECD country and has only average results. But things could be worse: "The United States spend much more than France, and their students' performance is significantly lower."
The Fall 2009 issue of American Educator (a quarterly publication of the American Federation of Teachers) features a substantial article by H. H. Wu (Berkeley) with this title; the piece is available online in PDF format. Wu: "I am an advocate for having math instruction delivered by math teachers as early as possible, starting no later than fourth grade." He starts the argument here by leading us through the addition of whole numbers. "This discussion ... may not convince you that math teachers are a necessity in the first through third grades, but it will give you an idea of the important mathematical foundation that is being laid in the early grades." Wu then goes on to fractions, "a main source of math phobia." He emphasizes "the inherently abstract nature of the concept of a fraction," remarking that the conventional explanation in terms of slices of pie does not help students when it comes to arithmetic operations: "how do you multiply two pieces of pie, or use a pie to solve speed or ratio problems?" As he explains, the concept must be built up from a robust understanding of division in terms of multiplication ("the meaning of 24/6 = 4 is 24 = 4 X 6."). Wu ends with what he calls "the bigger picture." Mathematics in elementary school must incorporate the fundamental properties of mathematics: Coherence, precision, reasoning. "The fact that many elementary teachers lack the knowledge to teach mathematics with coherence, precision, and reasoning is a systemic problem with grave consequences." But "Given that there are over 2 million elementary teachers, the problem of raising the mathematical proficiency of all elementary teachers is so enormous as to be beyond comprehension. A viable alternative is to produce a much smaller corps of mathematics teachers with strong content knowledge who would be solely in charge of teaching mathematics at least beginning with grade 4."
John Tierney's "Findings" column for October 19, 2009 was dedicated to Martin Gardner, who turned 95 that week. "With more than 70 books to his name, he is the world's best-known recreational mathematician, and has probably introduced more people to the joys of math than anyone in history." Or, as Ronald Graham explains, "Martin has turned thousands of children into mathematicians, and thousands of mathematicians into children." Besides telling the story of Gardner's Recreational Mathematics column, which ran in the Scientific American from 1956 to 1981, Tierney has Gardner speculate on why people enjoy mathematical puzzles: "Evolution has developed the brain's ability to solve puzzles, and at the same time has produced in our brain a pleasure of solving problems." We also learn that Gardner is, mathematically, an "unashamed Platonist." Tierney quotes from his writing: "If all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime even if no one had proved them prime."
This is the title of an "Opinion" piece in the October 15 2009 Nature. Timothy Gowers (Cambridge; co-author Michael Nielsen) explains the Polymath Project: how he "used blogs and a wiki to mediate a fully open collaboration" aimed at an unsolved problem in mathematics. The problem was "to find an elementary proof of a special case of the density Hales-Jewett theorem (DHJ), which is a central result of combinatorics ... ." (A box illustrates the theorem in terms of hypercubic tic-tac-toe, k cubes on a side in n dimensions, each cube to be filled in with an integer from 1 to k. "The aim of the polymath project was to tackle the first truly difficult case of DHJ, which is when k = 3.") Gowers announced the project on January 27, 2009; collaborative discussion began on February 1. "Over the next 37 days, 27 people contributed approximately 800 substantive comments, containing 170,000 words," with Gowers acting as moderator. "Progress came far faster than anyone expected. On 10 March, Gowers announced that he was confident that the Polymath participants had found an elementary proof of the special case of DHJ, but also that ... the argument could be straightforwardly generalized to prove the full theorem."
The article goes on to discuss problems with authorship (how the credit should be shared), preservation (the working record is currently spread "across two blogs and a wiki, leaving it vulnerable should any of those sites disappear") and scale-up (for a larger collaboration, the linear nature of the discussion would become unwieldy). Finally: "The working record of the Polymath Project is a remarkable resource for students of mathematics and for historians and philosophers of science. For the first time one can see on full display a complete account of how a serious mathematical result was discovered. It shows vividly how ideas grow, change, improve and are discarded, and how advances in understanding may come not in a single giant leap, but through the aggregation and refinement of many smaller insights. It shows the persistence required to solve a difficult problem, often in the face of considerable uncertainty, and how even the best mathematicians can make basic mistakes and pursue many failed ideas. There are ups, downs and real tension as the participants close in on a solution. Who would have guessed that the working record of a mathematical project would read like a thriller?"
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