- ARTICLES & ANNOUNCEMENTS (NATIONAL FOCUS)
- (1) “Pi Day” Resources
- (2) Final Meeting of the National Math Panel
- (3) As Schools Spend More Time on Reading and Math, Magnitude of Curriculum- Narrowing Effect is Revealed
- (4) “Mathematics for the President and Congress” by Keith Devlin
- (5) “Numbers Guy [Stanislas Dehaene]: Are our brains wired for math?”by Jim Holt
ARTICLES & ANNOUNCEMENTS (NATIONAL FOCUS)
Mathematics Educators of Greater St. Louis (MEGSL) sponsors a Pi Day (March 14) Web page with numerous informative and interesting links related to pi and to Sierpinski and Einstein, whose birthdays fall on March 14. Visit http://www.mobot.org/education/megsl/pi.html to peruse the extensive list of links.
Download an extensive collection of pi facts, activities, and resources (the collection has been divided into two sections to speed download time and is in Microsoft Word format):
Also visit http://TeachPi.org/, a “one-stop Pi Day Shop for teachers and number lovers.”
The Pi Searcher at http://www.angio.net/pi/piquery lets you search for any string of digits (up to 120) in the first 200 million digits of Pi. (Try entering your phone number.) You can also show any substring of Pi.
The final meeting of the National Mathematics Advisory Panel will be held on 13 March 2008 at Longfellow Middle School in Fairfax, VA. At this meeting, the Final Report will be discussed, adopted, and released during an open session from 9:00-10:45 a.m.
If you would like to attend this meeting, please send your name, title, organization you represent, address, phone number, and email address to Jennifer Graban at Jennifer.email@example.com no later than 2 p.m. (PT) on March 5. Walk-in registration will also be available. (Note: There will be no public comment session at this meeting.)
Please email any final comments to the National Math Panel at NationalMathPanel@ed.gov by March 5.
(3) As Schools Spend More Time on Reading and Math, Magnitude of Curriculum- Narrowing Effect is Revealed
Source: Center on Education Policy – 20 February 2008
Last summer, a groundbreaking report verified what many in the education and policy communities had long suspected: that a majority of the nation’s school districts were increasing time spent on reading and math in elementary schools since the No Child Left Behind Act became law in 2002, while most of these districts cut back on time spent on other subjects. Last week, a follow-up report issued by the Washington, D.C.-based Center on Education Policy provides an unprecedented look at the magnitude of those changes.
In its earlier report, CEP found that a majority of school districts–62%–had increased time for English language arts (ELA) and/or math in elementary schools since school year 2001-02. Meanwhile, 44% had increased time for ELA and/or math at the elementary level, while simultaneously cutting time from one or more areas including science, social studies, art and music, physical education, recess, and lunch.
CEP’s new report, Instructional Time in Elementary Schools: A Closer Look at Changes for Specific Subjects, examines the size of the shifts in those districts, in order to determine just how extensive the changes were.
According to the report, districts increasing time for ELA and math had done so by an average of 43%, or about three hours each week. To make room for the added time for ELA and math, districts reducing time in other areas averaged cuts of about 32% across those subjects, nearly 2.5 hours each week. Some of the districts reduced their time in one subject, while other districts decreased instructional time in several areas. “We knew that many school districts had made shifts in the time spent teaching different subjects since the No Child Left Behind was enacted, but we had little evidence of the magnitude of these changes within those districts,” said Jack Jennings, president and CEO of CEP. “Digging deeper into the data, we now know that the amount of time spent teaching reading, math and other subjects has changed substantially. In other words, changes in curriculum are not only widespread but also deep.”
According to the report, eight out of 10 of the districts that increased time for ELA did so by at least 75 minutes per week, and more than half (54%) increased by 150 minutes or more per week, or at least 30 minutes per day. Of the districts adding time for math, 63% increased by at least 75 minutes per week, with 19% adding 150 minutes or more per week.
Of the districts that both increased time for ELA or math and reduced time in other subjects, a large majority (72%) cut time by at least 75 minutes per week for one or more of the other subjects. For example, more than half (53%) of these districts cut instructional time by at least 75 minutes per week in social studies, and the same percentage cut time by at least 75 minutes per week in science.
Both of CEP’s reports on curriculum, including Instructional Time in Elementary Schools: A Closer Look at Changes for Specific Subjects, and Choices, Changes, and Challenges: Curriculum and Instruction in the NCLB Era (July 2007) are based on CEP’s nationally representative survey of 349 school districts conducted between November 2006 and February 2007. Both reports are part of CEP’s multiyear effort to track the impact of the No Child Left Behind Act since it became law in 2002, and are available online at www.cep-dc.org
Source: “MAA [Mathematical Association of America] Online” – February 2008
In last month’s column, “American Mathematics in a Flat World” (http://www.maa.org/devlin/devlin_01_08.html), I examined the implications of globalization for mathematics–how it is used in business, commerce, and society. As promised then, this month I take a look at the implications for how we teach mathematics–what gets taught and how.
As you will gather if you read my last piece, I believe that the entire mathematics education system needs to be rethought, particularly at the K-12 level. But that is a national issue, on which an individual teacher or college mathematics instructor has little influence. What college instructors and mathematics departments can do is design and give courses appropriate for the changing needs of society. In particular, courses that will prepare “non-quants” (graduates other than in mathematics, science, engineering, economics and finance, which includes the vast majority in Congress and the White House) for lives in the global economy.
In the globalized knowledge economy–the world of today, and increasingly of tomorrow–a good, but appropriate, knowledge of mathematics will be particularly crucial for those who run major businesses and the country. A CEO, a Member of Congress, or the President of the USA, does not need to be able to do mathematics. Others can do that for them. But in order to make informed decisions, they do need to have a sound overall sense of what mathematics can and cannot do, where it can be used and where not, and when to believe the figures and when to be skeptical…
Some of that need can be met by what are generally called “quantitative literacy” requirements across the curriculum, though few colleges have fully implemented such a requirement. Broadly speaking, QL encompasses a general sense of number and size, estimation skills, the ability to understand graphs, pie-charts and tables and to read a spreadsheet, the ability to reason logically and numerically, and a reasonable understanding of basic probability and statistics. Since the importance of such skills lies in their applications, QL should not be the subject of a course, which would surely fail to meet its goal, rather should be viewed as a requirement to be met across the entire curriculum.
QL is important in the “flat world” described by Thomas Friedman in his book The World is Flat. But the flat world creates a need for another kind of mathematical knowledge as well, one that I think is at least as important as quantitative literacy….[The complete article is available at the above Web site. The article also briefly describes Devlin’s “Overview of Mathematics” course that he teaches at Stanford. For the course outline and philosophy, visit http://www.stanford.edu/~kdevlin/courses08.html]
Mathematician Keith Devlin (email: firstname.lastname@example.org) is the Executive Director of the Center for the Study of Language and Information at Stanford University and is The Math Guy on NPR’s Weekend Edition (http://www.npr.org/templates/rundowns/rundown.php?prgId=7).
* Visit http://www.stanford.edu/~kdevlin/MathGuy.html for the complete sound archive from National Public Radio’s Weekend Edition Saturday. Topics include “Did Ben Franklin Add Up? A discussion of Franklin’s mathematical abilities,” “A Math Great Gets His Due. The 300th anniversary of the birth of Leonhard Euler,” and “Danica McKellar’s Mathematical Theorem. The TV star has a mathematical theorem to her name.”)
Source: New Yorker Magazine – 3 March 2008
According to Stanislas Dehaene, humans have an inbuilt “number sense” capable of some basic calculations and estimates. The problems start when we learn mathematics and have to perform procedures that are anything but instinctive.
…In cognitive science, incidents of brain damage are nature’s experiments. If a lesion knocks out one ability but leaves another intact, it is evidence that they are wired into different neural circuits… Stanislas Dehaene theorized that our ability to learn sophisticated mathematical procedures resided in an entirely different part of the brain from a rougher quantitative sense. Over the decades, evidence concerning cognitive deficits in brain-damaged patients has accumulated, and researchers have concluded that we have a sense of number that is independent of language, memory, and reasoning in general. Within neuroscience, numerical cognition has emerged as a vibrant field, and Dehaene, now in his early forties, has become one of its foremost researchers. His work is “completely pioneering,” Susan Carey, a psychology professor at Harvard who has studied numerical cognition, told me [author of this piece, Jim Holt]. “If you want to make sure the math that children are learning is meaningful, you have to know something about how the brain represents number at the kind of level that Stan is trying to understand.”
Dehaene has spent most of his career plotting the contours of our number sense and puzzling over which aspects of our mathematical ability are innate and which are learned, and how the two systems overlap and affect each other. He has approached the problem from every imaginable angle. Working with colleagues both in France and in the United States, he has carried out experiments that probe the way numbers are coded in our minds. He has studied the numerical abilities of animals, of Amazon tribespeople, of top French mathematics students. He has used brain-scanning technology to investigate precisely where in the folds and crevices of the cerebral cortex our numerical faculties are nestled. And he has weighed the extent to which some languages make numbers more difficult than others. His work raises crucial issues about the way mathematics is taught. In Dehaene’s view, we are all born with an evolutionarily ancient mathematical instinct. To become numerate, children must capitalize on this instinct, but they must also unlearn certain tendencies that were helpful to our primate ancestors but that clash with skills needed today. And some societies are evidently better than others at getting kids to do this. In both France and the United States, mathematics education is often felt to be in a state of crisis. The math skills of American children fare poorly in comparison with those of their peers in countries like Singapore, South Korea, and Japan. Fixing this state of affairs means grappling with the question that has taken up much of Dehaene’s career: What is it about the brain that makes numbers sometimes so easy and sometimes so hard?…
Dehaene conjectured that, when we see numerals or hear number words, our brains automatically map them onto a number line that grows increasingly fuzzy above 3 or 4. He found that no amount of training can change this. “It is a basic structural property of how our brains represent number, not just a lack of facility,” he told me…
Dehaene has become a scanning virtuoso. On returning to France after his time with [Michael] Posner, he pressed on with the use of imaging technologies to study how the mind processes numbers. The existence of an evolved number ability had long been hypothesized, based on research with animals and infants, and evidence from brain-damaged patients gave clues to where in the brain it might be found. Dehaene set about localizing this facility more precisely and describing its architecture. “In one experiment I particularly liked,” he recalled, “we tried to map the whole parietal lobe in a half hour, by having the subject perform functions like moving the eyes and hands, pointing with fingers, grasping an object, engaging in various language tasks, and, of course, making small calculations, like thirteen minus four. We found there was a beautiful geometrical organization to the areas that were activated. The eye movements were at the back, the hand movements were in the middle, grasping was in the front, and so on. And right in the middle, we were able to confirm, was an area that cared about number.”
The number area lies deep within a fold in the parietal lobe called the intraparietal sulcus (just behind the crown of the head). But it isn’t easy to tell what the neurons there are actually doing. Brain imaging, for all the sophistication of its technology, yields a fairly crude picture of what’s going on inside the skull, and the same spot in the brain might light up for two tasks even though different neurons are involved. “Some people believe that psychology is just being replaced by brain imaging, but I don’t think that’s the case at all,” Dehaene said. “We need psychology to refine our idea of what the imagery is going to show us. That’s why we do behavioral experiments, see patients. It’s the confrontation of all these different methods that creates knowledge”…
Last winter, I saw Dehaene in the ornate setting of the Institut de France, across the Seine from the Louvre. There he accepted a prize of a quarter of a million euros from Liliane Bettencourt, whose father created the cosmetics group L’Oréal. In a salon hung with tapestries, Dehaene described his research to a small audience that included a former Prime Minister of France. New techniques of neuroimaging, he explained, promise to reveal how a thought process like calculation unfolds in the brain. This isn’t just a matter of pure knowledge, he added. Since the brain’s architecture determines the sort of abilities that come naturally to us, a detailed understanding of that architecture should lead to better ways of teaching children mathematics and may help close the educational gap that separates children in the West from those in several Asian countries. The fundamental problem with learning mathematics is that while the number sense may be genetic, exact calculation requires cultural tools–symbols and algorithms–that have been around for only a few thousand years and must therefore be absorbed by areas of the brain that evolved for other purposes. The process is made easier when what we are learning harmonizes with built-in circuitry. If we can’t change the architecture of our brains, we can at least adapt our teaching methods to the constraints it imposes…
Piaget’s view had become standard by the nineteen-fifties, but psychologists have since come to believe that he underrated the arithmetic competence of small children. Six-month-old babies, exposed simultaneously to images of common objects and sequences of drumbeats, consistently gaze longer at the collection of objects that matches the number of drumbeats. By now, it is generally agreed that infants come equipped with a rudimentary ability to perceive and represent number. (The same appears to be true for many kinds of animals, including salamanders, pigeons, raccoons, dolphins, parrots, and monkeys.) And if evolution has equipped us with one way of representing number, embodied in the primitive number sense, culture furnishes two more: numerals and number words. These three modes of thinking about number, Dehaene believes, correspond to distinct areas of the brain. The number sense is lodged in the parietal lobe, the part of the brain that relates to space and location; numerals are dealt with by the visual areas; and number words are processed by the language areas…
Our inbuilt ineptness when it comes to more complex mathematical processes has led Dehaene to question why we insist on drilling procedures like long division into our children at all. There is, after all, an alternative: the electronic calculator. “Give a calculator to a five-year-old, and you will teach him how to make friends with numbers instead of despising them,” he has written. By removing the need to spend hundreds of hours memorizing boring procedures, he says, calculators can free children to concentrate on the meaning of these procedures, which is neglected under the educational status quo.
This attitude might make Dehaene sound like a natural ally of educators who advocate reform math, and a natural foe of parents who want their children’s math teachers to go “back to basics.” But when I asked him about reform math he wasn’t especially sympathetic. “The idea that all children are different, and that they need to discover things their own way–I don’t buy it at all,” he said. “I believe there is one brain organization. We see it in babies, we see it in adults. Basically, with a few variations, we’re all traveling on the same road.” He admires the mathematics curricula of Asian countries like China and Japan, which provide children with a highly structured experience, anticipating the kind of responses they make at each stage and presenting them with challenges designed to minimize the number of errors. “That’s what we’re trying to get back to in France,” he said…
Despite our shared brain organization, cultural differences in how we handle numbers persist, and they are not confined to the classroom. Evolution may have endowed us with an approximate number line, but it takes a system of symbols to make numbers precise—to “crystallize” them, in Dehaene’s metaphor…
Today, Arabic numerals are in use pretty much around the world, while the words with which we name numbers naturally differ from language to language. And, as Dehaene and others have noted, these differences are far from trivial. English is cumbersome. There are special words for the numbers from 11 to 19, and for the decades from 20 to 90. This makes counting a challenge for English-speaking children, who are prone to such errors as “twenty-eight, twenty-nine, twenty-ten, twenty-eleven.” French is just as bad, with vestigial base-twenty monstrosities, like quatre-vingt-dix-neuf (“four twenty ten nine”) for 99. Chinese, by contrast, is simplicity itself; its number syntax perfectly mirrors the base-ten form of Arabic numerals, with a minimum of terms. Consequently, the average Chinese four-year-old can count up to forty, whereas American children of the same age struggle to get to fifteen. And the advantages extend to adults. Because Chinese number words are so brief–they take less than a quarter of a second to say, on average, compared with a third of a second for English–the average Chinese speaker has a memory span of nine digits, versus seven digits for English speakers. (Speakers of the marvellously efficient Cantonese dialect, common in Hong Kong, can juggle ten digits in active memory.)
In 2005, Dehaene was elected to the chair in experimental cognitive psychology at the Collège de France, a highly prestigious institution founded by Francis I in 1530. The faculty consists of just fifty-two scholars, and Dehaene is the youngest member. In his inaugural lecture, Dehaene marvelled at the fact that mathematics is simultaneously a product of the human mind and a powerful instrument for discovering the laws by which the human mind operates. He spoke of the confrontation between new technologies like brain imaging and ancient philosophical questions concerning number, space, and time. And he pronounced himself lucky to be living in an era when advances in psychology and neuroimaging are combining to “render visible” the hitherto invisible realm of thought…[Visit the above Web site to read the entire article on Dehaene and his research.]