In This Issue...
Source: The New York Times - 16 October 2010
Benoît B. Mandelbrot, a maverick mathematician who developed the field of fractal geometry and applied it to physics, biology, finance and many other fields, died on Thursday [October 14] in Cambridge, Mass. He was 85...
Dr. Mandelbrot coined the term "fractal" to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature.
"Applied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found," said David Mumford, a professor of mathematics at Brown University. [See article below about Mumford's latest honor.] "He was one of the primary people who realized these were legitimate objects of study."
In a seminal book, The Fractal Geometry of Nature, published in 1982, Dr. Mandelbrot defended mathematical objects that he said others had dismissed as "monstrous" and "pathological." Using fractal geometry, he argued, the complex outlines of clouds and coastlines, once considered unmeasurable, could now "be approached in rigorous and vigorous quantitative fashion."
For most of his career, Dr. Mandelbrot had a reputation as an outsider to the mathematical establishment. From his perch as a researcher for I.B.M. in New York, where he worked for decades before accepting a position at Yale University, he noticed patterns that other researchers may have overlooked in their own data, then often swooped in to collaborate.
"He knew everybody, with interests going off in every possible direction," Professor Mumford said. "Every time he gave a talk, it was about something different"...
In the 1950s, Dr. Mandelbrot proposed a simple but radical way to quantify the crookedness of such an object by assigning it a "fractal dimension," an insight that has proved useful well beyond the field of cartography.
Over nearly seven decades, working with dozens of scientists, Dr. Mandelbrot contributed to the fields of geology, medicine, cosmology and engineering. He used the geometry of fractals to explain how galaxies cluster, how wheat prices change over time and how mammalian brains fold as they grow, among other phenomena.
His influence has also been felt within the field of geometry, where he was one of the first to use computer graphics to study mathematical objects like the Mandelbrot set, which was named in his honor.
"I decided to go into fields where mathematicians would never go because the problems were badly stated," Dr. Mandelbrot said. "I have played a strange role that none of my students dare to take"...
Instead of rigorously proving his insights in each field, he said he preferred to "stimulate the field by making bold and crazy conjectures" - and then move on before his claims had been verified. This habit earned him some skepticism in mathematical circles.
"He doesn't spend months or years proving what he has observed," said Heinz-Otto Peitgen, a professor of mathematics and biomedical sciences at the University of Bremen. And for that, he said, Dr. Mandelbrot "has received quite a bit of criticism."
"But if we talk about impact inside mathematics, and applications in the sciences," Professor Peitgen said, "he is one of the most important figures of the last 50 years"...
[For more details, visit the NYT Web site above.]
Source: Guardian - 17 October 2010
...Mandelbrot, born into a Lithuanian-Jewish family living in Warsaw, showed an early love for geometry and excelled at chess: he later admitted that he did not think the game through logically, but geometrically. Maps were another inspiration. His father was crazy about them, and the house was full of them...
At the start of his groundbreaking work, The Fractal Geometry of Nature, [Mandelbrot] asks: "Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree." The fractal geometry that he developed helps us to describe nature as we actually see it, and so expand our way of thinking.
The world we live in is not naturally smooth-edged and regularly shaped like the familiar cones, circles, spheres and straight lines of Euclid's geometry: it is rough-edged, wrinkled, crinkled and irregular. "Fractals" was the name he applied to irregular mathematical shapes similar to those in nature, with structures that are self-similar over many scales, the same pattern being repeated over and over. Fractal geometry offers a systematic way of approaching phenomena that look more elaborate the more they are magnified, and the images it generates are themselves a source of great fascination...
While the ideas behind fractals, iteration and self-similarity are ancient, it took the coining of the term "fractal geometry" in 1975 and the publication of The Fractal Geometry of Nature in French in the same year to give the quest an identity. As Mandelbrot put it, "to have a name is to be" -- and the field exploded...
(b) Mandelbrot's Web Page: http://www.math.yale.edu/mandelbrot/
(c) NOVA Interview with Mandelbrot: http://www.pbs.org/wgbh/nova/physics/mandelbrot-fractal.html
Source (Para. 1): The White House
On Friday, October 15, President Obama named ten eminent researchers as recipients of the National Medal of Science, and three individuals and one team as recipients of the National Medal of Technology and Innovation, the highest honors bestowed by the United States government on scientists, engineers, and inventors. The National Medal of Science was established by Congress in August 1959 and is to be conferred directly by the President. The recipients will receive their awards from President Obama at a White House ceremony later this year.
One of the recipients, David Mumford, is a professor emeritus of applied mathematics at Brown University. "As collaborator and catalyst, David Mumford was an early contributor to fields of inquiry that have blossomed at Brown--brain science, computer vision, neurobiology, cognitive science, the biology and psychology of perception--and to his own areas of pure and applied mathematics," said Brown President Ruth J. Simmons. "He continues to inspire collaborators in many fields, former students now in productive careers, and his professional colleagues in the United States and abroad."
Mumford’s contributions to mathematics fundamentally changed algebraic geometry and brought him a variety of honors including the Fields Medal, the highest award in mathematics (1974), and a MacArthur Foundation fellowship (1987-92). He is perhaps best known for inventing geometric invariant theory, a key tool in moduli theory, the study of how the geometric structures in algebraic geometry vary. His subsequent studies on the moduli space of curves have been an important tool in string theory.
"For the first half of my career--about 20 years--I worked in pure math, although I always had lots of interests outside of that,” Mumford said. A conversation with a collaborator in Italy led to his decision to turn toward applied mathematics, which he did while at Harvard. His interest in applied math and his dedication to a collaborative approach helped develop "a really terrific group jointly at Harvard, MIT, and Brown." He joined the Brown faculty in 1996 as a University professor in the Division of Applied Mathematics.
Mumford’s work in computer vision and pattern theory introduced new mathematical tools and models from analysis and differential geometry. His work in neurobiology in collaboration with Tai Sing Lee led to new insights about the nature of computation in the human brain, and he helped start Brown’s vigorous interdisciplinary Brain Science Program. He is now turning his attention once again to pure mathematics and to the history of mathematics.
He also maintains collaborations and communication with professional colleagues, whose work he values and understands. While honored and grateful for his most recent honor, Mumford keeps those colleagues in his thoughts. "I did some nice things, but so did a lot of other people," he said. I’m pleased that this medal will bring attention to the important role of science and mathematics in our society."
Source: University of Wisconsin, Madison - 11 October 2010
The mathematical skills of boys and girls, as well as of men and women, are substantially equal, according to a new examination of existing studies published in the current issue of Psychological Bulletin.
One portion of the new study looked systematically at 242 articles that assessed the mathematics skills of 1,286,350 people, says chief author Janet Hyde, a professor of psychology and women's studies at the University of Wisconsin-Madison.
These studies, all published between 1990 and 2007, looked at people from grade school to college and beyond. A second portion of the new study examined the results of several large, long-term scientific studies, including the National Assessment of Educational Progress.
In both cases, Hyde says, the difference between the two sexes was so close as to be meaningless.
Sara Lindberg, now a postdoctoral fellow in women's health at the UW-Madison School of Medicine and Public Health, was the primary author of the meta-analysis.
The idea that both genders have equal math abilities is widely accepted among social scientists, Hyde adds, but word has been slow to reach teachers and parents, who can play a negative role by guiding girls away from math-heavy sciences and engineering. "One reason I am still spending time on this is because parents and teachers continue to hold stereotypes that boys are better in math, and that can have a tremendous impact on individual girls who are told to stay away from engineering or the physical sciences because 'Girls can't do the math.'"
Scientists now know that stereotypes affect performance, Hyde adds. "There is lots of evidence that what we call 'stereotype threat' can hold women back in math. If, before a test, you imply that the women should expect to do a little worse than the men, that hurts performance. It's a self-fulfilling prophecy.
"Parents and teachers give little implicit messages about how good they expect kids to be at different subjects," Hyde adds, "and that powerfully affects their self-concept of their ability. When you are deciding about a major in physics, this can become a huge factor."
Hyde hopes the new results will slow the trend toward single-sex schools, which are sometimes justified on the basis of differential math skills. It may also affect standardized tests, which gained clout with the passage of No Child Left Behind, and tend to emphasize lower-level math skills such as multiplication, Hyde says. "High-stakes testing really needs to include higher-level problem-solving, which tends to be more important in jobs that require math skills. But because many teachers teach to the test, they will not teach higher reasoning unless the tests start to include it."
The new findings reinforce a recent study that ranked gender dead last among nine factors, including parental education, family income, and school effectiveness, in influencing the math performance of 10-year-olds.
Hyde acknowledges that women have made significant advances in technical fields. Half of medical school students are female, as are 48 percent of undergraduate math majors. "If women can't do math, how are they getting these majors?" she asks.
Because progress in physics and engineering is much slower, "we have lots of work to do," Hyde says. "This persistent stereotyping disadvantages girls. My message to parents is that they should have confidence in their daughters' math performance. They need to realize that women can do math just as well as men. These changes will encourage women to pursue occupations that require lots of math."
Source: National Council of Teachers of Mathematics (NCTM)
The National Council of Teachers of Mathematics (NCTM) has posted a PowerPoint file on its Web site "to inform teachers and to support them in implementation of the Common Core State Standards [(CCSS)]. Other presentations for grade bands are under development and will be made available soon." Download the file from the Web page above.
Also on this Web page is a link to the joint statement in support of the CCSS issued by NCTM, the Association of Mathematics Teacher Educators (AMTE), the Association of State Supervisors of Mathematics (ASSM), and the National Council of Supervisors of Mathematics.
For more information about NCTM's initiatives related to the CCSS, read NCTM President J. Michael Shaughnessy's report located at http://www.nctm.org/about/content.aspx?id=26483
Source: The New York Times - 30 September 2010
By the time they get to kindergarten, children in this well-to-do suburb [(Franklin Lakes, N.J.)] already know their numbers, so their teachers worried that a new math program was too easy when it covered just 1 and 2--for a whole week.
"Talk about the number 1 for 45 minutes?" said Chris Covello, who teaches 16 students ages 5 and 6. "I was like, I don't know. But then I found you really could. Before, we had a lot of ground to cover, and now it's more open-ended and gets kids thinking."
The slower pace is a cornerstone of the district's new approach to teaching math, which is based on the national math system of Singapore and aims to emulate that country's success by promoting a deeper understanding of numbers and math concepts. Students in Singapore have repeatedly ranked at or near the top on international math exams since the mid-1990s.
Franklin Lakes, about 30 miles northwest of Manhattan, is one of dozens of districts, from Scarsdale, N.Y., to Lexington, Ky., that in recent years have adopted Singapore math, as it is called, amid growing concerns that too many American students lack the higher-order math skills called for in a global economy.
For decades, efforts to improve math skills have driven schools to embrace one math program after another, abandoning a program when it does not work and moving on to something purportedly better. In the 1960s there was the "new math," whose focus on abstract theories spurred a back-to-basics movement, emphasizing rote learning and drills. After that came "reform math," whose focus on problem solving and conceptual understanding has been derided by critics as the "new new math."
Singapore math may well be a fad, too, but supporters say it seems to address one of the difficulties in teaching math: all children learn differently. In contrast to the most common math programs in the United States, Singapore math devotes more time to fewer topics, to ensure that children master the material through detailed instruction, questions, problem solving, and visual and hands-on aids like blocks, cards and bar charts. Ideally, they do not move on until they have thoroughly learned a topic.
Principals and teachers say that slowing down the learning process gives students a solid math foundation upon which to build increasingly complex skills, and makes it less likely that they will forget and have to be retaught the same thing in later years.
And with Singapore math, the pace can accelerate by fourth and fifth grades, putting children as much as a year ahead of students in other math programs as they grasp complex problems more quickly...
[Visit http://www.nytimes.com/2010/10/01/education/01math.html to read the rest of this informative article.]
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