Maybe there is something to this new math afterall

In Uncategorized on September 17, 2008 at 5:48 pm

NATALIE ANGIER Basics SEP. 16, 2008

Gut Instinct’s Surprising Role in Math

You are shopping in a busy supermarket and you’re ready to pay up and go home. You perform a quick visual sweep of the checkout options and immediately start ramming your cart through traffic toward an appealingly unpeopled line halfway across the store. As you wait in line and start reading nutrition labels, you can’t help but calculate that the 529 calories contained in a single slice of your Key lime cheesecake amounts to one-fourth of your recommended daily caloric allowance and will take you 90 minutes on the elliptical to burn off and you’d better just stick the thing behind this stack of Soap Opera Digests and hope a clerk finds it before it melts.

One shopping spree, two distinct number systems in play. Whenever we choose a shorter grocery line over a longer one, or a bustling restaurant over an unpopular one, we rally our approximate number system, an ancient and intuitive sense that we are born with and that we share with many other animals. Rats, pigeons, monkeys, babies — all can tell more from fewer, abundant from stingy. An approximate number sense is essential to brute survival: how else can a bird find the best patch of berries, or two baboons know better than to pick a fight with a gang of six?

When it comes to genuine computation, however, to seeing a self-important number like 529 and panicking when you divide it into 2,200, or realizing that, hey, it’s the square of 23! well, that calls for a very different number system, one that is specific, symbolic and highly abstract. By all evidence, scientists say, the capacity to do mathematics, to manipulate representations of numbers and explore the quantitative texture of our world is a uniquely human and very recent skill. People have been at it only for the last few millennia, it’s not universal to all cultures, and it takes years of education to master. Math-making seems the opposite of automatic, which is why scientists long thought it had nothing to do with our ancient, pre-verbal size-em-up ways.

Yet a host of new studies suggests that the two number systems, the bestial and celestial, may be profoundly related, an insight with potentially broad implications for math education.

One research team has found that how readily people rally their approximate number sense is linked over time to success in even the most advanced and abstruse mathematics courses. Other scientists have shown that preschool children are remarkably good at approximating the impact of adding to or subtracting from large groups of items but are poor at translating the approximate into the specific. Taken together, the new research suggests that math teachers might do well to emphasize the power of the ballpark figure, to focus less on arithmetic precision and more on general reckoning.

“When mathematicians and physicists are left alone in a room, one of the games they’ll play is called a Fermi problem, in which they try to figure out the approximate answer to an arbitrary problem,” said Rebecca Saxe, a cognitive neuroscientist at the Massachusetts Institute of Technology who is married to a physicist. “They’ll ask, how many piano tuners are there in Chicago, or what contribution to the ocean’s temperature do fish make, and they’ll try to come up with a plausible answer.”

“What this suggests to me,” she added, “is that the people whom we think of as being the most involved in the symbolic part of math intuitively know that they have to practice those other, nonsymbolic, approximating skills.”

This month in the journal Nature, Justin Halberda and Lisa Feigenson of Johns Hopkins University and Michele Mazzocco of the Kennedy Krieger Institute in Baltimore described their study of 64 14-year-olds who were tested at length on the discriminating power of their approximate number sense. The teenagers sat at a computer as a series of slides with varying numbers of yellow and blue dots flashed on a screen for 200 milliseconds each — barely as long as an eye blink. After each slide, the students pressed a button indicating whether they thought there had been more yellow dots or blue. (Take a version of the test.)

Given the antiquity and ubiquity of the nonverbal number sense, the researchers were impressed by how widely it varied in acuity. There were kids with fine powers of discrimination, able to distinguish ratios on the order of 9 blue dots for every 10 yellows, Dr. Feigenson said. “Others performed at a level comparable to a 9-month-old,” barely able to tell if five yellows outgunned three blues. Comparing the acuity scores with other test results that Dr. Mazzocco had collected from the students over the past 10 years, the researchers found a robust correlation between dot-spotting prowess at age 14 and strong performance on a raft of standardized math tests from kindergarten onward. “We can’t draw causal arrows one way or another,” Dr. Feigenson said, “but your evolutionarily endowed sense of approximation is related to how good you are at formal math.”

The researchers caution that they have no idea yet how the two number systems interact. Brain imaging studies have traced the approximate number sense to a specific neural structure called the intraparietal sulcus, which also helps assess features like an object’s magnitude and distance. Symbolic math, by contrast, operates along a more widely distributed circuitry, activating many of the prefrontal regions of the brain that we associate with being human. Somewhere, local and global must be hooked up to a party line.

Other open questions include how malleable our inborn number sense may be, whether it can be improved with training, and whether those improvements would pay off in a greater appetite and aptitude for math. If children start training with the flashing dot game at age 4, will they be supernumerate by middle school?

Dr. Halberda, who happens to be Dr. Feigenson’s spouse, relishes the work’s philosophical implications. “What’s interesting and surprising in our results is that the same system we spend years trying to acquire in school, and that we use to send a man to the moon, and that has inspired the likes of Plato, Einstein and Stephen Hawking, has something in common with what a rat is doing when it’s out hunting for food,” he said. “I find that deeply moving.”

Behind every great leap of our computational mind lies the pitter-patter of rats’ feet, the little squeak of rodent kind.■

  1. Some Background on Ms. Angier.

    Q&A with Natalie Angier
    By Harvey Blume, Boston Globe | May 13, 2007

    When I called New York Times science writer Natalie Angier to discuss her new book, "The Canon: A Whirligig Tour of the Beautiful Basics of Science," I started by asking why, in the new work, is there little of the impatience with religion she has expressed in some of her essays? In "Confessions of a Lonely Atheist," for example, she complained that for nonbelievers like herself America's "current climate of religiosity can be stifling." In "My God Problem" she challenged scientists who felt similarly to step up: "Why is it," she demanded, "that most scientists avoid criticizing religion even as they decry the supernatural mind-set?"

    For a worrisome interval after I put my question to her the phone line went silent. Then Angier slowly and deliberately replied: "I don't want to be known as a professional atheist. I don't want to have that be my full-time job. I've been a science writer my whole career, trying to elucidate the ideas of science. That's what I think makes life worth living." And in any case, she added, excess religiosity hardly explains why Americans are badly educated, not just about science, but, "go down the list — about history, geography, literature, and philosophy."

    In "The Canon," Angier agitates energetically for scientific literacy by highlighting key elements of scientific thinking, and by devoting chapters to, as she puts it, the "sciences generally awarded the preamble 'hard.'" The chapter on astronomy, for example, centers on the ineffable instant in which our universe blossomed out of the Big Bang. The section on molecular biology features a reprise of the high-speed commotion that prevails within a human cell even before it's time to split the DNA and divide.

    And one finds that Angier's polemical edge, when she cares to display it, is as keen as ever: She writes, for example, that proponents of creationism and/or intelligent design strike her as subscribing to sadly "data-deprived ideologies."

    IDEAS: What was your goal with "The Canon"?

    ANGIER: In order to follow science, even in the newspapers, you have to have some confidence that you get the basic lay of the land, the geography of the scientific continent. I was trying to convey the basic ideas behind scientific thinking in a way people would understand.

    IDEAS: Is there any special reason why Americans are poorly educated in science?

    ANGIER: Our obsession with money plays into it. I think there is some truth to David Baltimore's observation that people used to making a lot of money don't get that interested in science, science being a sort of blue-collar profession that requires a lot of hands-on work and that is probably not going to make you rich.

    IDEAS: Is writing easy for you?

    ANGIER: No. Mostly it's a question of trying to quiet the dybbuks — all the voices that tell you you're no good, you can't do it, every kind of criticism you can come up with. You're just trying to shut them up and let yourself go. I'd say I spend 50 percent of my time trying to get them out of the way. There are times when I do enjoy writing, but they are definitely in a minority.

    IDEAS: Your writing has a lot of imagery and wordplay. Why?

    ANGIER: When I write I go into an almost stream-of-consciousness way of looking at things. Do I think that way when I'm not writing? Sometimes. I try to understand things metabolically, by really digesting it, having it on a gut level, feeling it's inside you. I always try to get that for myself in grappling with various topics. When I write I try to get someone to go through that process with me, investigating the material from the inside out.

    IDEAS: It feels as though the imagery allows you to assert your femininity as a science writer. Is that so?

    ANGIER: No, it's not about a concern with femininity. It's about trying to feel some kind of passion. Don't you want to have a more heightened experience? Isn't that what you're always reaching for? It's what I'm always reaching for, in the way I look at the world and in the way I write. It's the same with my attempts at humor. The goal is to expand and rejoice as opposed to being an unhappy, angry person, which I am by nature. And when you can play around with language, it takes care of the fear, somehow.

    IDEAS: When it comes to the situation of women in the sciences, do you see progress?

    ANGIER: I've looked at the roster of the National Academy of Sciences to see what percentage of new members are women. In one piece I did for the Times, I saw signs of progress. But women have to continue to fight because you do have people like Larry Summers who comes along and casually says, Oh, here's a provocative question we shouldn't be afraid to ask. Why are there so few women at the genius level in science? Is it because they're inferior in science? Or is it maybe because they're not driven as much as men are? Anyway, let's talk about it.

    This gets thrown out there, and it's one more thing we have to deal with. Do we really need Larry Summers shooting from the lip?

    IDEAS: In writing about the Big Bang, you convey how amazing it was. But isn't it a cold kind of amazement? How are we supposed to feel about the origin of the universe? How is it supposed to matter in our lives?

    ANGIER: Well, the fact that the Big Bang leads to intelligent life says something fundamental about the nature of matter and energy, I think, and its tendency to form complex patterns. I have this debate with my colleague Dennis Overbye, who argues that the universe is cold, the universe doesn't care.

    IDEAS: You don't agree?

    ANGIER: No. I think he's setting himself apart from the universe. I say to Dennis, do you believe your life is meaningless? He says, no. Do you believe you're part of the universe? He says, yes. So how can you say the universe has no meaning? You are meaning, you are part of it. I think it's legitimate to see the universe as wanting to know itself.

    IDEAS: Are you saying that we were intended from the beginning?

    ANGIER: Was the universe stewing on us for the last 13.4 billion years? No. But it's the outcome. We're here. It's a cold fact that we're here, and we are incorrigible meaning-generators.

    Of course, I also believe, with no evidence, that there are many other civilizations like ours out there, so you could say the universe is filled with meaning. But did the universe intend that at the beginning? [laughs] As a meaning-generating character I can confidently say I don't think so.

    But let's just say that we decide — and it would be a great thing to decide — that our purpose in life is to understand the universe. We've done a spectacular job so far, and have a lot more work to do. I really wish we were doing that instead of spending a trillion dollars on the Iraq war. I really wish we could figure out how to get to the point where most of us wish that. Will we get to that point? I don't know.

  2. A ‘thumbs down’ review of Ms. Angier’s “The Canon: A Whirligig Tour of the Beautiful Basics of Science”

    It’s the smugness what got me, November 25, 2007
    By L. D. Gasman (Charlottesville, VA United States) – See all my reviews

    The wordiness of The Canon is just as the other one star reviewers say. At times it is embarrassing. But it was the smugness that got me. This is really to the fore in the audio version which is read by a woman with a voice shrill enough to make your bones ache. Yet, she the fits Angier’s writing perfectly; one gets the impression throughout that Angier is just dying to write, “I’m clever and know stuff, and you don’t.”

    But we are warned. The book is called “The Canon” for a reason. I get the impression that Angier truly believes that what she has put down in her tedious book is what all right thinking people should believe. And just to ram the point home she makes snide comments about the Bush Administration and Creationists and — by extension — all of us who don’t buy into the whole “science is the way, the truth and the light” thing.

    As a result the opening sections on scientific methodology and probability theory are critical to what Ms. Angier seems to be trying to achieve. And boy are those sections l-o-n-g. They also have a sort of frantic air about them as Ms. Angier dances back and forwards trying to explain why scientists are so often wrong, but why we should believe everything that the latest scientific consensus says anyway.

    Although the title is accurate in the sense outlined above, a better title would surely be “The Politically Correct Guide to Science.” For example,if this was the only book that one had ever read on science, one might be forgiven for believing that almost all American scientists were women, since that is mostly who Angier interviews.

    In the past few years there seems to have been an extreme outburst of insecurity among parts of the scientific community. We are told that if we do not do/think as we commanded by the scientific elites on matters of public policy, philosophy and religion, the world will come crashing down on us. None of the advocates of this point of view have ever said what makes them expert on anything outside their field of science. This should make us all very comfortable in rejecting what we hear from scientists, whether it is their adolescent atheism or the perverse ludditism that many of them see as the only reasonable response to global warming. (Or is it Global Warming these days.)

    But if scientists are really worried about the state of the world, The Canon is there to give them some comfort. No doubt people of a certain kind will lap it all up as if. . . well as if it was really canon law.

  3. Another review…

    A good idea, mostly spoiled, July 3, 2007
    By Porter (Atlanta) – See all my reviews

    Natalie Angier, a writer for the New York Times, wrote this book for adult nonscientists who want to understand the basics of science. As a 43 year old accountant whose education includes a graduate degree but only the minimum of science, I’m certainly in the right demographic. There is something to learn from this book, but the author’s relentless desire to be clever creates some awful, exasperating prose.

    I would guess a journalist feels a sense of liberation when getting to write outside a newspaper’s staid style guide. And an occasional flourish with language can make a book more interesting, just as a sprinkling of pepper makes a pork chop taste better. But Angier raids the literary spice rack with complete abandon. Almost every paragraph includes a pun, an alliteration, or some goofy metaphor. I wanted a quick read about the basics of science, but I had to pause repeatedly to translate into plain English phrases like, “that solar toady of a planet named after the Roman god with feathers on his shoes.” Dammit, just say “Mercury.”

    I read with special interest the chapter on evolution, since that’s the one area where there is outright disbelief of the scientific consensus. I think Angier badly misunderstands the opposition to evolution, at least the opposition I’m aware of as a member of a Methodist church. (Angier erroneously refers to the Book of Revelation as a “gospel,” which suggests that she’s not very knowledgeable about Christianity.) She names her chapter “Evolutionary Biology” and focuses narrowly on evolution as an account of the history of life and an explanation for the physical features of life forms. She doesn’t mention Social Darwinism or the phrase “survival of the fittest.” She makes only a brief and neutral reference to evolutionary psychology, which combines evolution-made-me-do-it excuses for reprehensible behavior, like a man leaving his wife for a younger woman, with cynical explanations for altruistic actions.

    Most Christians I know would make peace with evolution focused only on biology. But they bitterly, and properly, resist evolution as the authority for defining normal and expected human behavior. Angier would have been better off dealing forthrightly with the noxious philosophical movements that have coupled their cars to the evolutionary train, instead of thinking that one more recitation of the fossil record would make evolutionists of us all.

  4. More…

    The author’s blatantly dismissive and often subtly derisive attitude towards religious belief may be a bit off-putting for many readers. For an author trying so very hard to engage her general public, and publishing in a market wherein a substantial majority claim belief in a Higher Power, this snidely secularist perspective seems a bit ill-advised.

  5. Comments from another blog on Ms. Angier’s Science Section article…

    Wednesday, September 17, 2008
    Please visit an actual classroom before you make recommendations, III
    This time the recommendations come from Natalie Angier, a science reporter with the New York Times.

    In this week’s Science Section, she reports on two studies showing connections between two cognitive number mechanisms:

    1. The approximate number system: in Angier’s words, “an ancient and intuitive sense that we are born with and that we share with many other animals.”

    2. The abstract, symbolic system that allows us to “manipulate representations of numbers” and make precise calculations.

    One study shows that a person’s facility with the approximate system is connected to his/her facility with the symbolic system. The other shows that, in Angier’s words:

    “[P]reschool children are remarkably good at approximating the impact of adding to or subtracting from large groups of items but are poor at translating the approximate into the specific.”

    All this, Angier reports, has “potentially broad implications for math education.”

    “Taken together, the new research suggests that math teachers might do well to emphasize the power of the ballpark figure, to focus less on arithmetic precision and more on general reckoning.”

    Huh? Less on symbolic and more on approximate? This is a non-sequitor, unless we know that the causality flows from approximate to symbolic.

    But as Angier quotes one of the researchers (Lisa Feigenson of Johns Hopkins) as saying, “We can’t draw causal arrows one way or another” between “your evolutionarily endowed sense of approximation” and “how good you are at formal math.”

    And, as Angier herself writes: “The researchers caution that they have no idea yet how the two number systems interact,” and that it’s currently an “open question[] …how malleable our inborn number sense may be, whether it can be improved with training, and whether those improvements would pay off in a greater appetite and aptitude for math.”

    So how does Angier leap to the conclusion that schools should be stressing approximate number sense over symbolic numerical reasoning?

    It seems that our science reporter has managed to:

    1. avoid visiting actual classrooms, where she would see how much symbolic math has been jettisoned the sake of “number sense,” and by how much overall levels of math achievement have therefore declined.

    2. fall under the influence of the reigning ed school orthodoxy, which is as enamored of intuition as it is contemptuous of abstract reasoning.

    3. take, on faith, the bizarre claims by one of the researchers about the parlor games played by mathematicians:

    “When mathematicians and physicists are left alone in a room, one of the games they’ll play is called a Fermi problem, in which they try to figure out the approximate answer to an arbitrary problem,” said Rebecca Saxe, a cognitive neuroscientist at the Massachusetts Institute of Technology who is married to a physicist. “They’ll ask, how many piano tuners are there in Chicago, or what contribution to the ocean’s temperature do fish make, and they’ll try to come up with a plausible answer.”

    Not the mathematicians I know!

    Why doesn’t anyone ask them about what they think of what’s going on in today’s actual classrooms?

  6. Another comment from the same website (oilf.blogspot.com)…

    Niels Henrik Abel said:

    Focus more on general reckoning and less on precision? The students I see can’t even handle estimation. They wouldn’t know a reasonable answer if it came up and bit them in the behind.

    Substituting general reckoning for precision is capitulating to innumeracy, not providing an antidote. This chick is off her rocker.

    September 17, 2008 9:38 AM

  7. Don’t fall for this malarkey, folks.

    Here’s a reminder of the undeniable content deficit as between Everyday Math and a real mathematics curriculum…

    Math problem of the week: Everyday Math vs. Singapore Math

    1. From unit 2 of the Everyday Math grade 2 Student Math Journal, volume 1, p. 44 (volume 1, of 2, 164 pages total):

    There are 10 houses on Jerry’s block. There are 15 houses on Nancy’s block. How many more houses are on Nancy’s block?


    Write a number model

    A pack of gum costs 25¢. Sean bought 3 packs. How much money did he spend?

    2. From unit 2 of the Singapore Math grade 2 Primary Mathematics Workbook, volume 1 (2a), p. 54 (volume 1, of 2, 192 pages total):

    Mrs. Bates bought an oven for $393.
    She also bought a refrigerator for $438. How much did she spend altogether?

    She spend $____ altogether

    There are 468 desks in a hall. There are 156 more chairs than desks. How many chairs are there in the hall?

    There are ___ chairs in the hall.


    Place value and borrowing/carrying (regrouping): the bigger picture:

    Grade 2 Singapore Math teaches place value in Unit 1 and regrouping in Unit 2.

    Grade 2 Everyday Math teaches place value in Unit 3. Not until Unit 12, the final unit of the the grade 2 curriculum, do students do problems comparable to the Singapore problems above. I.e., problems like 356 + 275 (cf. p. 302), which are specially classified as “harder” problems, and for which you are invited to use “your favorite addition strategy.”

  8. More trenchant commentary on the shoddy work of biased NYTimes authors…


    Tuesday, March 11, 2008
    The assault on reason is bipartisan

    In her NY Times review of Susan Jacoby’s “The Age of American Unreason,” Michiko Kakutani singles out conservatives as the main political force against science and reason. They include the Bush Administration, religious fundamentalists, and those who insist on local control and funding of public schools. The other culprits, Jacoby and Kakutani agree, are psycho-cultural: the rise of video, the internet, and a “culture of distraction”; the decline of print and attention spans; the triumph over self-education of self-improvement and self-esteem.

    But within education, religious home schoolers and Creationist school boards aside, the biggest assault on science and reason comes from the political left–from the public education establishment that is one of the largest blocs of reliably Democratic voters in this country.

    It is, in part, the power-brokers within this monolith–curriculum consultants, education professors, and the National Council of Teachers of Mathematics–whom we can thank for the facts that, quoting Kakutani, “two-thirds of Americans cannot name a single Supreme Court justice,” and “American 15-year-olds rank 24th out of those of 29 countries in mathematics literacy.”

    With near unanimity, our education experts speak out against the “rote learning of facts” that underlies knowledge, and the “mindless algorithms” and “mere calculation” that underly math. What Kakutani says about conservatives–“conservatives have turned ‘intellectual’ into a dirty word in politics”–applies as well to the arbiters of our curricula.

    Jacoby’s book bemoans the absence of national education standards. In fact we have them, especially for math: the National Standards of the National Council of Teachers of Mathematics; the state tests, mandated by No Child Left Behind, that measure students against these standards; and the curriculum packages that enact them, designed and funded by the Directorate for Education and Human Resources of the National Science Foundation.

    Jacoby laments America’s insistence on local control. But most of our public schools lie within large municipalities that have consolidated their control over education. No longer can such schools choose their textbooks and curricula; instead they must defer to superintendents and boards of education, which, in turn, defer to education experts who preach their party line.

    Compared to other countries, we in America enjoy the worst of both worlds. Localized funding, with all the self-reinforcing economic disparities it brings, and centralized control by unelected non-experts in math, science, and knowledge, accountable neither to voters nor to reason.

  9. More trenchant commentary on the shoddy work of biased NYTimes authors…

    Yeah, if we only read the outstanding work down by Newsmax & the Washington Times authors, we'd be so much better off.

  10. It seems that the NY Times is in love with secular constructionists as well as secular social engineering by the left.

    How nice.

    Maybe this is why their readership is at an all time low.

  11. “Maybe there is something to this new math afterall”

    Don’t fall for this rubbish: Skill with reckoning is very weak without first building skill in precise mathematical computation.

    Perhaps it was ‘general reckoning’ instead of rigorous, mathematical risk analysis that caused today’s mortgage mess.

  12. I wonder why people still read The New York Times. Don’t they know it’s a little Pravda mixed in with The Communist Manifesto?

    Since we live in a capitalist system, aren’t they concerned that such pretentions as displayed in the NYT merely destroys any legitimate understanding of our world?

  13. no, they’re not concerned 10:26. the times is so far left, it’s left the country. the paper now writes for people in germany, france, italy, etc. or for those who would go to such places and give apologetic appeasement speeches to the great adulation of new (and old) socialists.

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